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GROWTH AND FORM




 CAMBRIDGE UNIVERSITY PRESS

 C. F. CLAY, MANAGER

 London: FETTER LANE, E.C.

 Edinburgh: 100 PRINCES STREET

 [Illustration]

 New York: G. P. PUTNAM’S SONS

 Bombay, Calcutta and Madras: MACMILLAN AND Co., LTD.

 Toronto: J. M. DENT AND SONS, LTD.

 Tokyo: THE MARUZEN-KABUSHIKI-KAISHA


 _All rights reserved_




 ON GROWTH AND FORM

 BY

 D’ARCY WENTWORTH THOMPSON


 Cambridge:
 at the University Press
 1917




“The reasonings about the wonderful and intricate operations of nature
are so full of uncertainty, that, as the Wise-man truly observes,
_hardly do we guess aright at the things that are upon earth, and with
labour do we find the things that are before us_.” Stephen Hales,
_Vegetable Staticks_ (1727), p. 318, 1738.




PREFATORY NOTE


This book of mine has little need of preface, for indeed it is
“all preface” from beginning to end. I have written it as an easy
introduction to the study of organic Form, by methods which are the
common-places of physical science, which are by no means novel in their
application to natural history, but which nevertheless naturalists are
little accustomed to employ.

It is not the biologist with an inkling of mathematics, but the
skilled and learned mathematician who must ultimately deal with such
problems as are merely sketched and adumbrated here. I pretend to no
mathematical skill, but I have made what use I could of what tools I
had; I have dealt with simple cases, and the mathematical methods which
I have introduced are of the easiest and simplest kind. Elementary
as they are, my book has not been written without the help—the
indispensable help—of many friends. Like Mr Pope translating Homer,
when I felt myself deficient I sought assistance! And the experience
which Johnson attributed to Pope has been mine also, that men of
learning did not refuse to help me.

My debts are many, and I will not try to proclaim them all: but I beg
to record my particular obligations to Professor Claxton Fidler, Sir
George Greenhill, Sir Joseph Larmor, and Professor A. McKenzie; to a
much younger but very helpful friend, Mr John Marshall, Scholar of
Trinity; lastly, and (if I may say so) most of all, to my colleague
Professor William Peddie, whose advice has made many useful additions
to my book and whose criticism has spared me many a fault and blunder.

I am under obligations also to the authors and publishers of many books
from which illustrations have been borrowed, and especially to the
following:―

To the Controller of H.M. Stationery Office, for leave to reproduce a
number of figures, chiefly of Foraminifera and of Radiolaria, from the
Reports of the Challenger Expedition. {vi}

To the Council of the Royal Society of Edinburgh, and to that of the
Zoological Society of London:—the former for letting me reprint from
their _Transactions_ the greater part of the text and illustrations of
my concluding chapter, the latter for the use of a number of figures
for my chapter on Horns.

To Professor E. B. Wilson, for his well-known and all but indispensable
figures of the cell (figs. 42–51, 53); to M. A. Prenant, for other
figures (41, 48) in the same chapter; to Sir Donald MacAlister and Mr
Edwin Arnold for certain figures (335–7), and to Sir Edward Schäfer
and Messrs Longmans for another (334), illustrating the minute
trabecular structure of bone. To Mr Gerhard Heilmann, of Copenhagen,
for his beautiful diagrams (figs. 388–93, 401, 402) included in my
last chapter. To Professor Claxton Fidler and to Messrs Griffin, for
letting me use, with more or less modification or simplification,
a number of illustrations (figs. 339–346) from Professor Fidler’s
_Textbook of Bridge Construction_. To Messrs Blackwood and Sons, for
several cuts (figs. 127–9, 131, 173) from Professor Alleyne Nicholson’s
_Palaeontology_; to Mr Heinemann, for certain figures (57, 122,
123, 205) from Dr Stéphane Leduc’s _Mechanism of Life_; to Mr A. M.
Worthington and to Messrs Longmans, for figures (71, 75) from _A Study
of Splashes_, and to Mr C. R. Darling and to Messrs E. and S. Spon
for those (fig. 85) from Mr Darling’s _Liquid Drops and Globules_.
To Messrs Macmillan and Co. for two figures (304, 305) from Zittel’s
_Palaeontology_, to the Oxford University Press for a diagram (fig.
28) from Mr J. W. Jenkinson’s _Experimental Embryology_; and to the
Cambridge University Press for a number of figures from Professor
Henry Woods’s _Invertebrate Palaeontology_, for one (fig. 210) from Dr
Willey’s _Zoological Results_, and for another (fig. 321) from “Thomson
and Tait.”

Many more, and by much the greater part of my diagrams, I owe to the
untiring help of Dr Doris L. Mackinnon, D.Sc., and of Miss Helen
Ogilvie, M.A., B.Sc., of this College.

 D’ARCY WENTWORTH THOMPSON.

 UNIVERSITY COLLEGE, DUNDEE.

 _December, 1916._




CONTENTS


 CHAP.                                                     PAGE
    I. INTRODUCTORY                                           1

   II. ON MAGNITUDE                                          16

  III. THE RATE OF GROWTH                                    50

   IV. ON THE INTERNAL FORM AND STRUCTURE OF THE CELL       156

    V. THE FORMS OF CELLS                                   201

   VI. A NOTE ON ADSORPTION                                 277

  VII. THE FORMS OF TISSUES, OR CELL-AGGREGATES             293

 VIII. THE SAME (_continued_)                               346

   IX. ON CONCRETIONS, SPICULES, AND SPICULAR SKELETONS     411

    X. A PARENTHETIC NOTE ON GEODETICS                      488

   XI. THE LOGARITHMIC SPIRAL                               493

  XII. THE SPIRAL SHELLS OF THE FORAMINIFERA                587

 XIII. THE SHAPES OF HORNS, AND OF TEETH OR TUSKS: WITH
       A NOTE ON TORSION                                    612

  XIV. ON LEAF-ARRANGEMENT, OR PHYLLOTAXIS                  635

   XV. ON THE SHAPES OF EGGS, AND OF CERTAIN OTHER HOLLOW
       STRUCTURES                                           652

  XVI. ON FORM AND MECHANICAL EFFICIENCY                    670

 XVII. ON THE THEORY OF TRANSFORMATIONS, OR THE COMPARISON
       OF RELATED FORMS                                     719

       EPILOGUE                                             778

       INDEX                                                780




LIST OF ILLUSTRATIONS


 1. Nerve-cells, from larger and smaller animals (Minot, after Irving
 Hardesty) . . . 37

 2. Relative magnitudes of some minute organisms (Zsigmondy) . . . 39

 3. Curves of growth in man (Quetelet and Bowditch) . . . 61

 4, 5. Mean annual increments of stature and weight in man (_do._)
 . . . 66, 69

 6. The ratio, throughout life, of female weight to male (_do._)
 . . . 71

 7–9. Curves of growth of child, before and after birth (His and
 Rüssow) . . . 74–6

 10. Curve of growth of bamboo (Ostwald, after Kraus) . . . 77

 11. Coefficients of variability in human stature (Boas and Wissler)
 . . . 80

 12. Growth in weight of mouse (Wolfgang Ostwald) . . . 83

 13. _Do._ of silkworm (Luciani and Lo Monaco) . . . 84

 14. _Do._ of tadpole (Ostwald, after Schaper) . . . 85

 15. Larval eels, or _Leptocephali_, and young elver (Joh. Schmidt)
 . . . 86

 16. Growth in length of _Spirogyra_ (Hofmeister) . . . 87

 17. Pulsations of growth in _Crocus_ (Bose) . . . 88

 18. Relative growth of brain, heart and body of man (Quetelet) . . . 90

 19. Ratio of stature to span of arms (_do._) . . . 94

 20. Rates of growth near the tip of a bean-root (Sachs) . . . 96

 21, 22. The weight-length ratio of the plaice, and its annual periodic
 changes . . . 99, 100

 23. Variability of tail-forceps in earwigs (Bateson) . . . 104

 24. Variability of body-length in plaice . . . 105

 25. Rate of growth in plants in relation to temperature (Sachs)
 . . . 109

 26. _Do._ in maize, observed (Köppen), and calculated curves . . . 112

 27. _Do._ in roots of peas (Miss I. Leitch) . . . 113

 28, 29. Rate of growth of frog in relation to temperature (Jenkinson,
 after O. Hertwig), and calculated curves of _do._ . . . 115, 6

 30. Seasonal fluctuation of rate of growth in man (Daffner) . . . 119

 31. _Do._ in the rate of growth of trees (C. E. Hall) . . . 120

 32. Long-period fluctuation in the rate of growth of Arizona trees (A.
 E. Douglass) . . . 122

 33, 34. The varying form of brine-shrimps (_Artemia_), in relation to
 salinity (Abonyi) . . . 128, 9

 35–39. Curves of regenerative growth in tadpoles’ tails (M. L. Durbin)
 . . . 140–145

 40. Relation between amount of tail removed, amount restored, and time
 required for restoration (M. M. Ellis) . . . 148

 41. Caryokinesis in trout’s egg (Prenant, after Prof. P. Bouin)
 . . . 169

 42–51. Diagrams of mitotic cell-division (Prof. E. B. Wilson)
 . . . 171–5

 52. Chromosomes in course of splitting and separation (Hatschek and
 Flemming) . . . 180

 53. Annular chromosomes of mole-cricket (Wilson, after vom Rath)
 . . . 181

 54–56. Diagrams illustrating a hypothetic field of force in
 caryokinesis (Prof. W. Peddie) . . . 182–4

 57. An artificial figure of caryokinesis (Leduc) . . . 186

 58. A segmented egg of _Cerebratulus_ (Prenant, after Coe) . . . 189

 59. Diagram of a field of force with two like poles . . . 189

 60. A budding yeast-cell . . . 213

 61. The roulettes of the conic sections . . . 218

 62. Mode of development of an unduloid from a cylindrical tube
 . . . 220

 63–65. Cylindrical, unduloid, nodoid and catenoid oil-globules
 (Plateau) . . . 222, 3

 66. Diagram of the nodoid, or elastic curve . . . 224

 67. Diagram of a cylinder capped by the corresponding portion of a
 sphere . . . 226

 68. A liquid cylinder breaking up into spheres . . . 227

 69. The same phenomenon in a protoplasmic cell of _Trianea_ . . . 234

 70. Some phases of a splash (A. M. Worthington) . . . 235

 71. A breaking wave (_do._) . . . 236

 72. The calycles of some campanularian zoophytes . . . 237

 73. A flagellate monad, _Distigma proteus_ (Saville Kent) . . . 246

 74. _Noctiluca miliaris_, diagrammatic . . . 246

 75. Various species of _Vorticella_ (Saville Kent and others) . . . 247

 76. Various species of _Salpingoeca_ (_do._) . . . 248

 77. Species of _Tintinnus_, _Dinobryon_ and _Codonella_ (_do._)
 . . . 248

 78. The tube or cup of _Vaginicola_ . . . 248

 79. The same of _Folliculina_ . . . 249

 80. _Trachelophyllum_ (Wreszniowski) . . . 249

 81. _Trichodina pediculus_ . . . 252

 82. _Dinenymplia gracilis_ (Leidy) . . . 253

 83. A “collar-cell” of _Codosiga_ . . . 254

 84. Various species of _Lagena_ (Brady) . . . 256

 85. Hanging drops, to illustrate the unduloid form (C. R. Darling)
 . . . 257

 86. Diagram of a fluted cylinder . . . 260

 87. _Nodosaria scalaris_ (Brady) . . . 262

 88. Fluted and pleated gonangia of certain Campanularians (Allman)
 . . . 262

 89. Various species of _Nodosaria_, _Sagrina_ and _Rheophax_ (Brady)
 . . . 263

 90. _Trypanosoma tineae_ and _Spirochaeta anodontae_, to shew
 undulating membranes (Minchin and Fantham) . . . 266

 91. Some species of _Trichomastix_ and _Trichomonas_ (Kofoid) . . . 267

 92. _Herpetomonas_ assuming the undulatory membrane of a Trypanosome
 (D. L. Mackinnon) . . . 268

 93. Diagram of a human blood-corpuscle . . . 271

 94. Sperm-cells of decapod crustacea, _Inachus_ and _Galathea_
 (Koltzoff) . . . 273

 95. The same, in saline solutions of varying density (_do._) . . . 274

 96. A sperm-cell of _Dromia_ (_do._) . . . 275

 97. Chondriosomes in cells of kidney and pancreas (Barratt and
 Mathews) . . . 285

 98. Adsorptive concentration of potassium salts in various plant-cells
 (Macallum) . . . 290

 99–101. Equilibrium of surface-tension in a floating drop . . . 294, 5

 102. Plateau’s “bourrelet” in plant-cells; diagrammatic (Berthold)
 . . . 298

 103. Parenchyma of maize, shewing the same phenomenon . . . 298

 104, 5. Diagrams of the partition-wall between two soap-bubbles
 . . . 299, 300

 106. Diagram of a partition in a conical cell . . . 300

 107. Chains of cells in _Nostoc_, _Anabaena_ and other low algae
 . . . 300

 108. Diagram of a symmetrically divided soap-bubble . . . 301

 109. Arrangement of partitions in dividing spores of _Pellia_
 (Campbell) . . . 302

 110. Cells of _Dictyota_ (Reinke) . . . 303

 111, 2. Terminal and other cells of _Chara_, and young antheridium of
 _do._ . . . 303

 113. Diagram of cell-walls and partitions under various conditions of
 tension . . . 304

 114, 5. The partition-surfaces of three interconnected bubbles
 . . . 307, 8

 116. Diagram of four interconnected cells or bubbles . . . 309

 117. Various configurations of four cells in a frog’s egg (Rauber)
 . . . 311

 118. Another diagram of two conjoined soap-bubbles . . . 313

 119. A froth of bubbles, shewing its outer or “epidermal” layer
 . . . 314

 120. A tetrahedron, or tetrahedral system, shewing its centre of
 symmetry . . . 317

 121. A group of hexagonal cells (Bonanni) . . . 319

 122, 3. Artificial cellular tissues (Leduc) . . . 320

 124. Epidermis of _Girardia_ (Goebel) . . . 321

 125. Soap-froth, and the same under compression (Rhumbler) . . . 322

 126. Epidermal cells of _Elodea canadensis_ (Berthold) . . . 322

 127. _Lithostrotion Martini_ (Nicholson) . . . 325

 128. _Cyathophyllum hexagonum_ (Nicholson, after Zittel) . . . 325

 129. _Arachnophyllum pentagonum_ (Nicholson) . . . 326

 130. _Heliolites_ (Woods) . . . 326

 131. Confluent septa in _Thamnastraea_ and _Comoseris_ (Nicholson,
 after Zittel) . . . 327

 132. Geometrical construction of a bee’s cell . . . 330

 133. Stellate cells in the pith of a rush; diagrammatic . . . 335

 134. Diagram of soap-films formed in a cubical wire skeleton (Plateau)
 . . . 337

 135. Polar furrows in systems of four soap-bubbles (Robert) . . . 341

 136–8. Diagrams illustrating the division of a cube by partitions of
 minimal area . . . 347–50

 139. Cells from hairs of _Sphacelaria_ (Berthold) . . . 351

 140. The bisection of an isosceles triangle by minimal partitions
 . . . 353

 141. The similar partitioning of spheroidal and conical cells . . . 353

 142. S-shaped partitions from cells of algae and mosses (Reinke and
 others) . . . 355

 143. Diagrammatic explanation of the S-shaped partitions . . . 356

 144. Development of _Erythrotrichia_ (Berthold) . . . 359

 145. Periclinal, anticlinal and radial partitioning of a quadrant
 . . . 359

 146. Construction for the minimal partitioning of a quadrant . . . 361

 147. Another diagram of anticlinal and periclinal partitions . . . 362

 148. Mode of segmentation of an artificially flattened frog’s egg
 (Roux) . . . 363

 149. The bisection, by minimal partitions, of a prism of small angle
 . . . 364

 150. Comparative diagram of the various modes of bisection of a
 prismatic sector . . . 365

 151. Diagram of the further growth of the two halves of a quadrantal
 cell . . . 367

 152. Diagram of the origin of an epidermic layer of cells . . . 370

 153. A discoidal cell dividing into octants . . . 371

 154. A germinating spore of _Riccia_ (after Campbell), to shew the
 manner of space-partitioning in the cellular tissue . . . 372

 155, 6. Theoretical arrangement of successive partitions in a
 discoidal cell . . . 373

 157. Sections of a moss-embryo (Kienitz-Gerloff) . . . 374

 158. Various possible arrangements of partitions in groups of four to
 eight cells . . . 375

 159. Three modes of partitioning in a system of six cells . . . 376

 160, 1. Segmenting eggs of _Trochus_ (Robert), and of _Cynthia_
 (Conklin) . . . 377

 162. Section of the apical cone of _Salvinia_ (Pringsheim) . . . 377

 163, 4. Segmenting eggs of _Pyrosoma_ (Korotneff), and of _Echinus_
 (Driesch) . . . 377

 165. Segmenting egg of a cephalopod (Watase) . . . 378

 166, 7. Eggs segmenting under pressure: of _Echinus_ and _Nereis_
 (Driesch), and of a frog (Roux) . . . 378

 168. Various arrangements of a group of eight cells on the surface of
 a frog’s egg (Rauber) . . . 381

 169. Diagram of the partitions and interfacial contacts in a system of
 eight cells . . . 383

 170. Various modes of aggregation of eight oil-drops (Roux) . . . 384

 171. Forms, or species, of _Asterolampra_ (Greville) . . . 386

 172. Diagrammatic section of an alcyonarian polype . . . 387

 173, 4. Sections of _Heterophyllia_ (Nicholson and Martin Duncan)
 . . . 388, 9

 175. Diagrammatic section of a ctenophore (_Eucharis_) . . . 391

 176, 7. Diagrams of the construction of a Pluteus larva . . . 392, 3

 178, 9. Diagrams of the development of stomata, in _Sedum_ and in the
 hyacinth . . . 394

 180. Various spores and pollen-grains (Berthold and others) . . . 396

 181. Spore of _Anthoceros_ (Campbell) . . . 397

 182, 4, 9. Diagrammatic modes of division of a cell under certain
 conditions of asymmetry . . . 400–5

 183. Development of the embryo of _Sphagnum_ (Campbell) . . . 402

 185. The gemma of a moss (_do._) . . . 403

 186. The antheridium of _Riccia_ (_do._) . . . 404

 187. Section of growing shoot of _Selaginella_, diagrammatic . . . 404

 188. An embryo of _Jungermannia_ (Kienitz-Gerloff) . . . 404

 190. Development of the sporangium of _Osmunda_ (Bower) . . . 406

 191. Embryos of _Phascum_ and of _Adiantum_ (Kienitz-Gerloff) . . . 408

 192. A section of _Girardia_ (Goebel) . . . 408

 193. An antheridium of _Pteris_ (Strasburger) . . . 409

 194. Spicules of _Siphonogorgia_ and _Anthogorgia_ (Studer) . . . 413

 195–7. Calcospherites, deposited in white of egg (Harting) . . . 421, 2

 198. Sections of the shell of _Mya_ (Carpenter) . . . 422

 199. Concretions, or spicules, artificially deposited in cartilage
 (Harting) . . . 423

 200. Further illustrations of alcyonarian spicules: _Eunicea_ (Studer)
 . . . 424

 201–3. Associated, aggregated and composite calcospherites (Harting)
 . . . 425, 6

 204. Harting’s “conostats” . . . 427

 205. Liesegang’s rings (Leduc) . . . 428

 206. Relay-crystals of common salt (Bowman) . . . 429

 207. Wheel-like crystals in a colloid medium (_do._) . . . 429

  208. A concentrically striated calcospherite or spherocrystal
 (Harting) . . . 432

 209. Otoliths of plaice, shewing “age-rings” (Wallace) . . . 432

 210. Spicules, or calcospherites, of _Astrosclera_ (Lister) . . . 436

 211. 2. C- and S-shaped spicules of sponges and holothurians (Sollas
 and Théel) . . . 442

 213. An amphidisc of _Hyalonema_ . . . 442

 214–7. Spicules of calcareous, tetractinellid and hexactinellid
 sponges, and of various holothurians (Haeckel, Schultze, Sollas and
 Théel) . . . 445–452

 218. Diagram of a solid body confined by surface-energy to a liquid
 boundary-film . . . 460

 219. _Astrorhiza limicola_ and _arenaria_ (Brady) . . . 464

 220. A nuclear “_reticulum plasmatique_” (Carnoy) . . . 468

 221. A spherical radiolarian, _Aulonia hexagona_ (Haeckel) . . . 469

 222. _Actinomma arcadophorum_ (_do._) . . . 469

 223. _Ethmosphaera conosiphonia_ (_do._) . . . 470

 224. Portions of shells of _Cenosphaera favosa_ and _vesparia_ (_do._)
 . . . 470

 225. _Aulastrum triceros_ (_do._) . . . 471

 226. Part of the skeleton of _Cannorhaphis_ (_do._) . . . 472

 227. A Nassellarian skeleton, _Callimitra carolotae_ (_do._) . . . 472

 228, 9. Portions of _Dictyocha stapedia_ (_do._) . . . 474

 230. Diagram to illustrate the conformation of _Callimitra_ . . . 476

 231. Skeletons of various radiolarians (Haeckel) . . . 479

 232. Diagrammatic structure of the skeleton of _Dorataspis_ (_do._)
 . . . 481

 233, 4. _Phatnaspis cristata_ (Haeckel), and a diagram of the same
 . . . 483

 235. _Phractaspis prototypus_ (Haeckel) . . . 484

 236. Annular and spiral thickenings in the walls of plant-cells
 . . . 488

 237. A radiograph of the shell of _Nautilus_ (Green and Gardiner)
 . . . 494

 238. A spiral foraminifer, _Globigerina_ (Brady) . . . 495

 239–42. Diagrams to illustrate the development or growth of a
 logarithmic spiral . . . 407–501

 243. A helicoid and a scorpioid cyme . . . 502

 244. An Archimedean spiral . . . 503

 245–7. More diagrams of the development of a logarithmic spiral
 . . . 505, 6

 248–57. Various diagrams illustrating the mathematical theory of
 gnomons . . . 508–13

 258. A shell of _Haliotis_, to shew how each increment of the shell
 constitutes a gnomon to the preexisting structure . . . 514

 259, 60. Spiral foraminifera, _Pulvinulina_ and _Cristellaria_, to
 illustrate the same principle . . . 514, 5

 261. Another diagram of a logarithmic spiral . . . 517

 262. A diagram of the logarithmic spiral of _Nautilus_ (Moseley)
 . . . 519

 263, 4. Opercula of _Turbo_ and of _Nerita_ (Moseley) . . . 521, 2

 265. A section of the shell of _Melo ethiopicus_ . . . 525

 266. Shells of _Harpa_ and _Dolium_, to illustrate generating curves
 and gene . . . 526

 267. D’Orbigny’s Helicometer . . . 529

 268. Section of a nautiloid shell, to shew the “protoconch” . . . 531

 269–73. Diagrams of logarithmic spirals, of various angles . . . 532–5

 274, 6, 7. Constructions for determining the angle of a logarithmic
 spiral . . . 537, 8

 275. An ammonite, to shew its corrugated surface pattern . . . 537

  278–80. Illustrations of the “angle of retardation” . . . 542–4

 281. A shell of _Macroscaphites_, to shew change of curvature . . . 550

 282. Construction for determining the length of the coiled spire
 . . . 551

 283. Section of the shell of _Triton corrugatus_ (Woodward) . . . 554

 284. _Lamellaria perspicua_ and _Sigaretus haliotoides_ (_do._)
 . . . 555

 285, 6. Sections of the shells of _Terebra maculata_ and _Trochus
 niloticus_ . . . 559, 60

 287–9. Diagrams illustrating the lines of growth on a lamellibranch
 shell . . . 563–5

 290. _Caprinella adversa_ (Woodward) . . . 567

 291. Section of the shell of _Productus_ (Woods) . . . 567

 292. The “skeletal loop” of _Terebratula_ (_do._) . . . 568

 293, 4. The spiral arms of _Spirifer_ and of _Atrypa_ (_do._) . . . 569

 295–7. Shells of _Cleodora_, _Hyalaea_ and other pteropods (Boas)
 . . . 570, 1

 298, 9. Coordinate diagrams of the shell-outline in certain pteropods
 . . . 572, 3

 300. Development of the shell of _Hyalaea tridentata_ (Tesch) . . . 573

 301. Pteropod shells, of _Cleodora_ and _Hyalaea_, viewed from the
 side (Boas) . . . 575

 302, 3. Diagrams of septa in a conical shell . . . 579

 304. A section of _Nautilus_, shewing the logarithmic spirals of the
 septa to which the shell-spiral is the evolute . . . 581

 305. Cast of the interior of the shell of _Nautilus_, to shew the
 contours of the septa at their junction with the shell-wall . . . 582

 306. _Ammonites Sowerbyi_, to shew septal outlines (Zittel, after
 Steinmann and Döderlein) . . . 584

 307. Suture-line of _Pinacoceras_ (Zittel, after Hauer) . . . 584

 308. Shells of _Hastigerina_, to shew the “mouth” (Brady) . . . 588

 309. _Nummulina antiquior_ (V. von Möller) . . . 591

 310. _Cornuspira foliacea_ and _Operculina complanata_ (Brady)
 . . . 594

 311. _Miliolina pulchella_ and _linnaeana_ (Brady) . . . 596

 312, 3. _Cyclammina cancellata_ (_do._), and diagrammatic figure of
 the same . . . 596, 7

 314. _Orbulina universa_ (Brady) . . . 598

 315. _Cristellaria reniformis_ (_do._) . . . 600

 316. _Discorbina bertheloti_ (_do._) . . . 603

 317. _Textularia trochus_ and _concava_ (_do._) . . . 604

 318. Diagrammatic figure of a ram’s horns (Sir V. Brooke) . . . 615

 319. Head of an Arabian wild goat (Sclater) . . . 616

 320. Head of _Ovis Ammon_, shewing St Venant’s curves . . . 621

 321. St Venant’s diagram of a triangular prism under torsion (Thomson
 and Tait) . . . 623

 322. Diagram of the same phenomenon in a ram’s horn . . . 623

 323. Antlers of a Swedish elk (Lönnberg) . . . 629

 324. Head and antlers of _Cervus duvauceli_ (Lydekker) . . . 630

 325, 6. Diagrams of spiral phyllotaxis (P. G. Tait) . . . 644, 5

 327. Further diagrams of phyllotaxis, to shew how various spiral
 appearances may arise out of one and the same angular leaf-divergence
 . . . 648

 328. Diagrammatic outlines of various sea-urchins . . . 664

 329, 30. Diagrams of the angle of branching in blood-vessels (Hess)
 . . . 667, 8

 331, 2. Diagrams illustrating the flexure of a beam . . . 674, 8

 333. An example of the mode of arrangement of bast-fibres in a
 plant-stem (Schwendener) . . . 680

 334. Section of the head of a femur, to shew its trabecular structure
 (Schäfer, after Robinson) . . . 681

 335. Comparative diagrams of a crane-head and the head of a femur
 (Culmann and H. Meyer) . . . 682

 336. Diagram of stress-lines in the human foot (Sir D. MacAlister,
 after H. Meyer) . . . 684

 337. Trabecular structure of the _os calcis_ (_do._) . . . 685

 338. Diagram of shearing-stress in a loaded pillar . . . 686

 339. Diagrams of tied arch, and bowstring girder (Fidler) . . . 693

 340, 1. Diagrams of a bridge: shewing proposed span, the corresponding
 stress-diagram and reciprocal plan of construction (_do._) . . . 696

 342. A loaded bracket and its reciprocal construction-diagram
 (Culmann) . . . 697

 343, 4. A cantilever bridge, with its reciprocal diagrams (Fidler)
 . . . 698

 345. A two-armed cantilever of the Forth Bridge (_do._) . . . 700

 346. A two-armed cantilever with load distributed over two pier-heads,
 as in the quadrupedal skeleton . . . 700

 347–9. Stress-diagrams. or diagrams of bending moments, in the
 backbones of the horse, of a Dinosaur, and of _Titanotherium_
 . . . 701–4

 350. The skeleton of _Stegosaurus_ . . . 707

 351. Bending-moments in a beam with fixed ends, to illustrate the
 mechanics of chevron-bones . . . 709

 352, 3. Coordinate diagrams of a circle, and its deformation into an
 ellipse . . . 729

 354. Comparison, by means of Cartesian coordinates, of the
 cannon-bones of various ruminant animals . . . 729

 355, 6. Logarithmic coordinates, and the circle of Fig. 352 inscribed
 therein . . . 729, 31

 357, 8. Diagrams of oblique and radial coordinates . . . 731

 359. Lanceolate, ovate and cordate leaves, compared by the help of
 radial coordinates . . . 732

 360. A leaf of _Begonia daedalea_ . . . 733

 361. A network of logarithmic spiral coordinates . . . 735

 362, 3. Feet of ox, sheep and giraffe, compared by means of Cartesian
 coordinates . . . 738, 40

 364, 6. “Proportional diagrams” of human physiognomy (Albert Dürer)
 . . . 740, 2

 365. Median and lateral toes of a tapir, compared by means of
 rectangular and oblique coordinates . . . 741

 367, 8. A comparison of the copepods _Oithona_ and _Sapphirina_
 . . . 742

 369. The carapaces of certain crabs, _Geryon_, _Corystes_ and others,
 compared by means of rectilinear and curvilinear coordinates . . . 744

 370. A comparison of certain amphipods, _Harpinia_, _Stegocephalus_
 and _Hyperia_ . . . 746

 371. The calycles of certain campanularian zoophytes, inscribed in
 corresponding Cartesian networks . . . 747

 372. The calycles of certain species of _Aglaophenia_, similarly
 compared by means of curvilinear coordinates . . . 748

 373, 4. The fishes _Argyropelecus_ and _Sternoptyx_, compared by means
 of rectangular and oblique coordinate systems . . . 748

 375, 6. _Scarus_ and _Pomacanthus_, similarly compared by means of
 rectangular and coaxial systems . . . 749

 377–80. A comparison of the fishes _Polyprion_, _Pseudopriacanthus_,
 _Scorpaena_ and _Antigonia_ . . . 750

 381, 2. A similar comparison of _Diodon_ and _Orthagoriscus_ . . . 751

 383. The same of various crocodiles: _C. porosus_, _C. americanus_ and
 _Notosuchus terrestris_ . . . 753

 384. The pelvic girdles of _Stegosaurus_ and _Camptosaurus_ . . . 754

 385, 6. The shoulder-girdles of _Cryptocleidus_ and of _Ichthyosaurus_
 . . . 755

 387. The skulls of _Dimorphodon_ and of _Pteranodon_ . . . 756

 388–92. The pelves of _Archaeopteryx_ and of _Apatornis_ compared, and
 a method illustrated whereby intermediate configurations may be found
 by interpolation (G. Heilmann) . . . 757–9

 393. The same pelves, together with three of the intermediate or
 interpolated forms . . . 760

 394, 5. Comparison of the skulls of two extinct rhinoceroses,
 _Hyrachyus_ and _Aceratherium_ (Osborn) . . . 761

 396. Occipital views of various extinct rhinoceroses (_do._) . . . 762

 397–400. Comparison with each other, and with the skull of
 _Hyrachyus_, of the skulls of _Titanotherium_, tapir, horse and rabbit
 . . . 763, 4

 401, 2. Coordinate diagrams of the skulls of _Eohippus_ and of
 _Equus_, with various actual and hypothetical intermediate types
 (Heilmann) . . . 765–7

 403. A comparison of various human scapulae (Dwight) . . . 769

 404. A human skull, inscribed in Cartesian coordinates . . . 770

 405. The same coordinates on a new projection, adapted to the skull of
 the chimpanzee . . . 770

 406. Chimpanzee’s skull, inscribed in the network of Fig. 405 . . . 771

 407, 8. Corresponding diagrams of a baboon’s skull, and of a dog’s
 . . . 771, 3




“Cum formarum naturalium et corporalium esse non consistat nisi in
unione ad materiam, ejusdem agentis esse videtur eas producere cujus
est materiam transmutare. Secundo, quia cum hujusmodi formae non
excedant virtutem et ordinem et facultatem principiorum agentium in
natura, nulla videtur necessitas eorum originem in principia reducere
altiora.” Aquinas, _De Pot. Q._ iii, a, 11. (Quoted in _Brit. Assoc.
Address_, _Section D_, 1911.)

“...I would that all other natural phenomena might similarly be
deduced from mechanical principles. For many things move me to suspect
that everything depends upon certain forces, in virtue of which the
particles of bodies, through forces not yet understood, are either
impelled together so as to cohere in regular figures, or are repelled
and recede from one another.” Newton, in Preface to the _Principia_.
(Quoted by Mr W. Spottiswoode, _Brit. Assoc. Presidential Address_,
1878.)

“When Science shall have subjected all natural phenomena to the laws
of Theoretical Mechanics, when she shall be able to predict the result
of every combination as unerringly as Hamilton predicted conical
refraction, or Adams revealed to us the existence of Neptune,—that we
cannot say. That day may never come, and it is certainly far in the dim
future. We may not anticipate it, we may not even call it possible. But
none the less are we bound to look to that day, and to labour for it
as the crowning triumph of Science:—when Theoretical Mechanics shall
be recognised as the key to every physical enigma, the chart for every
traveller through the dark Infinite of Nature.” J. H. Jellett, in
_Brit. Assoc. Address_, _Section A_, 1874.

{1}




CHAPTER I

INTRODUCTORY


Of the chemistry of his day and generation, Kant declared that it
was “a science, but not science,”—“eine Wissenschaft, aber nicht
Wissenschaft”; for that the criterion of physical science lay in its
relation to mathematics. And a hundred years later Du Bois Reymond,
profound student of the many sciences on which physiology is based,
recalled and reiterated the old saying, declaring that chemistry would
only reach the rank of science, in the high and strict sense, when it
should be found possible to explain chemical reactions in the light of
their causal relation to the velocities, tensions and conditions of
equilibrium of the component molecules; that, in short, the chemistry
of the future must deal with molecular mechanics, by the methods and
in the strict language of mathematics, as the astronomy of Newton
and Laplace dealt with the stars in their courses. We know how great
a step has been made towards this distant and once hopeless goal,
as Kant defined it, since van’t Hoff laid the firm foundations of a
mathematical chemistry, and earned his proud epitaph, _Physicam chemiae
adiunxit_[1].

We need not wait for the full realisation of Kant’s desire, in order
to apply to the natural sciences the principle which he urged. Though
chemistry fall short of its ultimate goal in mathematical mechanics,
nevertheless physiology is vastly strengthened and enlarged by making
use of the chemistry, as of the physics, of the age. Little by little
it draws nearer to our conception of a true science, with each branch
of physical science which it {2} brings into relation with itself:
with every physical law and every mathematical theorem which it learns
to take into its employ. Between the physiology of Haller, fine as it
was, and that of Helmholtz, Ludwig, Claude Bernard, there was all the
difference in the world.

As soon as we adventure on the paths of the physicist, we learn to
_weigh_ and to _measure_, to deal with time and space and mass and
their related concepts, and to find more and more our knowledge
expressed and our needs satisfied through the concept of _number_, as
in the dreams and visions of Plato and Pythagoras; for modern chemistry
would have gladdened the hearts of those great philosophic dreamers.

But the zoologist or morphologist has been slow, where the physiologist
has long been eager, to invoke the aid of the physical or mathematical
sciences; and the reasons for this difference lie deep, and in part
are rooted in old traditions. The zoologist has scarce begun to dream
of defining, in mathematical language, even the simpler organic
forms. When he finds a simple geometrical construction, for instance
in the honey-comb, he would fain refer it to psychical instinct or
design rather than to the operation of physical forces; when he sees
in snail, or nautilus, or tiny foraminiferal or radiolarian shell, a
close approach to the perfect sphere or spiral, he is prone, of old
habit, to believe that it is after all something more than a spiral or
a sphere, and that in this “something more” there lies what neither
physics nor mathematics can explain. In short he is deeply reluctant
to compare the living with the dead, or to explain by geometry or by
dynamics the things which have their part in the mystery of life.
Moreover he is little inclined to feel the need of such explanations
or of such extension of his field of thought. He is not without some
justification if he feels that in admiration of nature’s handiwork he
has an horizon open before his eyes as wide as any man requires. He
has the help of many fascinating theories within the bounds of his own
science, which, though a little lacking in precision, serve the purpose
of ordering his thoughts and of suggesting new objects of enquiry.
His art of classification becomes a ceaseless and an endless search
after the blood-relationships of things living, and the pedigrees of
things {3} dead and gone. The facts of embryology become for him, as
Wolff, von Baer and Fritz Müller proclaimed, a record not only of the
life-history of the individual but of the annals of its race. The facts
of geographical distribution or even of the migration of birds lead
on and on to speculations regarding lost continents, sunken islands,
or bridges across ancient seas. Every nesting bird, every ant-hill
or spider’s web displays its psychological problems of instinct or
intelligence. Above all, in things both great and small, the naturalist
is rightfully impressed, and finally engrossed, by the peculiar beauty
which is manifested in apparent fitness or “adaptation,”—the flower for
the bee, the berry for the bird.

Time out of mind, it has been by way of the “final cause,” by the
teleological concept of “end,” of “purpose,” or of “design,” in one or
another of its many forms (for its moods are many), that men have been
chiefly wont to explain the phenomena of the living world; and it will
be so while men have eyes to see and ears to hear withal. With Galen,
as with Aristotle, it was the physician’s way; with John Ray, as with
Aristotle, it was the naturalist’s way; with Kant, as with Aristotle,
it was the philosopher’s way. It was the old Hebrew way, and has its
splendid setting in the story that God made “every plant of the field
before it was in the earth, and every herb of the field before it
grew.” It is a common way, and a great way; for it brings with it a
glimpse of a great vision, and it lies deep as the love of nature in
the hearts of men.

Half overshadowing the “efficient” or physical cause, the argument of
the final cause appears in eighteenth century physics, in the hands of
such men as Euler[2] and Maupertuis, to whom Leibniz[3] had passed it
on. Half overshadowed by the mechanical concept, it runs through Claude
Bernard’s _Leçons sur les {4} phénomènes de la Vie_[4], and abides in
much of modern physiology[5]. Inherited from Hegel, it dominated Oken’s
_Naturphilosophie_ and lingered among his later disciples, who were
wont to liken the course of organic evolution not to the straggling
branches of a tree, but to the building of a temple, divinely planned,
and the crowning of it with its polished minarets[6].

It is retained, somewhat crudely, in modern embryology, by those
who see in the early processes of growth a significance “rather
prospective than retrospective,” such that the embryonic phenomena
must be “referred directly to their usefulness in building the body
of the future animal[7]”:—which is no more, and no less, than to say,
with Aristotle, that the organism is the τέλος, or final cause, of
its own processes of generation and development. It is writ large in
that Entelechy[8] which Driesch rediscovered, and which he made known
to many who had neither learned of it from Aristotle, nor studied it
with Leibniz, nor laughed at it with Voltaire. And, though it is in a
very curious way, we are told that teleology was “refounded, reformed
or rehabilitated[9]” by Darwin’s theory of natural selection, whereby
“every variety of form and colour was urgently and absolutely called
upon to produce its title to existence either as an active useful
agent, or as a survival” of such active usefulness in the past. But
in this last, and very important case, we have reached a “teleology”
without a τέλος, {5} as men like Butler and Janet have been prompt to
shew: a teleology in which the final cause becomes little more, if
anything, than the mere expression or resultant of a process of sifting
out of the good from the bad, or of the better from the worse, in short
of a process of mechanism[10]. The apparent manifestations of “purpose”
or adaptation become part of a mechanical philosophy, according to
which “chaque chose finit toujours par s’accommoder à son milieu[11].”
In short, by a road which resembles but is not the same as Maupertuis’s
road, we find our way to the very world in which we are living, and
find that if it be not, it is ever tending to become, “the best of all
possible worlds[12].”

But the use of the teleological principle is but one way, not the
whole or the only way, by which we may seek to learn how things came to
be, and to take their places in the harmonious complexity of the world.
To seek not for ends but for “antecedents” is the way of the physicist,
who finds “causes” in what he has learned to recognise as fundamental
properties, or inseparable concomitants, or unchanging laws, of matter
and of energy. In Aristotle’s parable, the house is there that men
may live in it; but it is also there because the builders have laid
one stone upon another: and it is as a _mechanism_, or a mechanical
construction, that the physicist looks upon the world. Like warp and
woof, mechanism and teleology are interwoven together, and we must not
cleave to the one and despise the other; for their union is “rooted in
the very nature of totality[13].”

Nevertheless, when philosophy bids us hearken and obey the lessons both
of mechanical and of teleological interpretation, the precept is hard
to follow: so that oftentimes it has come to pass, just as in Bacon’s
day, that a leaning to the side of the final cause “hath intercepted
the severe and diligent inquiry of all {6} real and physical causes,”
and has brought it about that “the search of the physical cause
hath been neglected and passed in silence.” So long and so far as
“fortuitous variation[14]” and the “survival of the fittest” remain
engrained as fundamental and satisfactory hypotheses in the philosophy
of biology, so long will these “satisfactory and specious causes”
tend to stay “severe and diligent inquiry,” “to the great arrest and
prejudice of future discovery.”

The difficulties which surround the concept of active or “real”
causation, in Bacon’s sense of the word, difficulties of which Hume and
Locke and Aristotle were little aware, need scarcely hinder us in our
physical enquiry. As students of mathematical and of empirical physics,
we are content to deal with those antecedents, or concomitants, of our
phenomena, without which the phenomenon does not occur,—with causes,
in short, which, _aliae ex aliis aptae et necessitate nexae_, are no
more, and no less, than conditions _sine quâ non_. Our purpose is still
adequately fulfilled: inasmuch as we are still enabled to correlate,
and to equate, our particular phenomena with more and ever more of the
physical phenomena around, and so to weave a web of connection and
interdependence which shall serve our turn, though the metaphysician
withhold from that interdependence the title of causality. We come in
touch with what the schoolmen called a _ratio cognoscendi_, though
the true _ratio efficiendi_ is still enwrapped in many mysteries. And
so handled, the quest of physical causes merges with another great
Aristotelian theme,—the search for relations between things apparently
disconnected, and for “similitude in things to common view unlike.”
Newton did not shew the cause of the apple falling, but he shewed a
similitude between the apple and the stars.

Moreover, the naturalist and the physicist will continue to speak
of “causes,” just as of old, though it may be with some mental
reservations: for, as a French philosopher said, in a kindred
difficulty: “ce sont là des manières de s’exprimer, {7} et si elles
sont interdites il faut renoncer à parler de ces choses.”

The search for differences or essential contrasts between the phenomena
of organic and inorganic, of animate and inanimate things has occupied
many mens’ minds, while the search for community of principles, or
essential similitudes, has been followed by few; and the contrasts are
apt to loom too large, great as they may be. M. Dunan, discussing the
“Problème de la Vie[15]” in an essay which M. Bergson greatly commends,
declares: “Les lois physico-chimiques sont aveugles et brutales; là
où elles règnent seules, au lieu d’un ordre et d’un concert, il ne
peut y avoir qu’incohérence et chaos.” But the physicist proclaims
aloud that the physical phenomena which meet us by the way have their
manifestations of form, not less beautiful and scarce less varied than
those which move us to admiration among living things. The waves of the
sea, the little ripples on the shore, the sweeping curve of the sandy
bay between its headlands, the outline of the hills, the shape of the
clouds, all these are so many riddles of form, so many problems of
morphology, and all of them the physicist can more or less easily read
and adequately solve: solving them by reference to their antecedent
phenomena, in the material system of mechanical forces to which they
belong, and to which we interpret them as being due. They have also,
doubtless, their _immanent_ teleological significance; but it is on
another plane of thought from the physicist’s that we contemplate their
intrinsic harmony and perfection, and “see that they are good.”

Nor is it otherwise with the material forms of living things. Cell
and tissue, shell and bone, leaf and flower, are so many portions
of matter, and it is in obedience to the laws of physics that their
particles have been moved, moulded and conformed[16]. {8} They are no
exception to the rule that Θεὸς ἀεὶ γεωμετρεῖ. Their problems of form
are in the first instance mathematical problems, and their problems
of growth are essentially physical problems; and the morphologist is,
_ipso facto_, a student of physical science.

Apart from the physico-chemical problems of modern physiology, the road
of physico-mathematical or dynamical investigation in morphology has
had few to follow it; but the pathway is old. The way of the old Ionian
physicians, of Anaxagoras[17], of Empedocles and his disciples in the
days before Aristotle, lay just by that highwayside. It was Galileo’s
and Borelli’s way. It was little trodden for long afterwards, but once
in a while Swammerdam and Réaumur looked that way. And of later years,
Moseley and Meyer, Berthold, Errera and Roux have been among the little
band of travellers. We need not wonder if the way be hard to follow,
and if these wayfarers have yet gathered little. A harvest has been
reaped by others, and the gleaning of the grapes is slow.

It behoves us always to remember that in physics it has taken great
men to discover simple things. They are very great names indeed that
we couple with the explanation of the path of a stone, the droop of
a chain, the tints of a bubble, the shadows in a cup. It is but the
slightest adumbration of a dynamical morphology that we can hope to
have, until the physicist and the mathematician shall have made these
problems of ours their own, or till a new Boscovich shall have written
for the naturalist the new _Theoria Philosophiae Naturalis_.

How far, even then, mathematics will _suffice_ to describe, and physics
to explain, the fabric of the body no man can foresee. It may be that
all the laws of energy, and all the properties of matter, and all the
chemistry of all the colloids are as powerless to explain the body as
they are impotent to comprehend the soul. For my part, I think it is
not so. Of how it is that the soul informs the body, physical science
teaches me nothing: consciousness is not explained to my comprehension
by all the nerve-paths and “neurones” of the physiologist; nor do I
ask of physics how goodness shines in one man’s face, and evil betrays
itself in another. But of the construction and growth and working {9}
of the body, as of all that is of the earth earthy, physical science
is, in my humble opinion, our only teacher and guide[18].

Often and often it happens that our physical knowledge is inadequate
to explain the mechanical working of the organism; the phenomena are
superlatively complex, the procedure is involved and entangled, and the
investigation has occupied but a few short lives of men. When physical
science falls short of explaining the order which reigns throughout
these manifold phenomena,—an order more characteristic in its totality
than any of its phenomena in themselves,—men hasten to invoke a
guiding principle, an entelechy, or call it what you will. But all the
while, so far as I am aware, no physical law, any more than that of
gravity itself, not even among the puzzles of chemical “stereometry,”
or of physiological “surface-action” or “osmosis,” is known to be
_transgressed_ by the bodily mechanism.

Some physicists declare, as Maxwell did, that atoms or molecules more
complicated by far than the chemist’s hypotheses demand are requisite
to explain the phenomena of life. If what is implied be an explanation
of psychical phenomena, let the point be granted at once; we may go
yet further, and decline, with Maxwell, to believe that anything of
the nature of _physical_ complexity, however exalted, could ever
suffice. Other physicists, like Auerbach[19], or Larmor[20], or
Joly[21], assure us that our laws of thermodynamics do not suffice, or
are “inappropriate,” to explain the maintenance or (in Joly’s phrase)
the “accelerative absorption” {10} of the bodily energies, and the
long battle against the cold and darkness which is death. With these
weighty problems I am not for the moment concerned. My sole purpose is
to correlate with mathematical statement and physical law certain of
the simpler outward phenomena of organic growth and structure or form:
while all the while regarding, _ex hypothesi_, for the purposes of this
correlation, the fabric of the organism as a material and mechanical
configuration.

Physical science and philosophy stand side by side, and one upholds the
other. Without something of the strength of physics, philosophy would
be weak; and without something of philosophy’s wealth, physical science
would be poor. “Rien ne retirera du tissu de la science les fils d’or
que la main du philosophe y a introduits[22].” But there are fields
where each, for a while at least, must work alone; and where physical
science reaches its limitations, physical science itself must help us
to discover. Meanwhile the appropriate and legitimate postulate of the
physicist, in approaching the physical problems of the body, is that
with these physical phenomena no alien influence interferes. But the
postulate, though it is certainly legitimate, and though it is the
proper and necessary prelude to scientific enquiry, may some day be
proven to be untrue; and its disproof will not be to the physicist’s
confusion, but will come as his reward. In dealing with forms which are
so concomitant with life that they are seemingly controlled by life, it
is in no spirit of arrogant assertiveness that the physicist begins his
argument, after the fashion of a most illustrious exemplar, with the
old formulary of scholastic challenge,—_An Vita sit? Dico quod non._

――――――――――

The terms Form and Growth, which make up the title of this little
book, are to be understood, as I need hardly say, in their relation
to the science of organisms. We want to see how, in some cases at
least, the forms of living things, and of the parts of living things,
can be explained by physical considerations, and to realise that, in
general, no organic forms exist save such as are in conformity with
ordinary physical laws. And while growth is a somewhat vague word for a
complex matter, which may {11} depend on various things, from simple
imbibition of water to the complicated results of the chemistry of
nutrition, it deserves to be studied in relation to form, whether it
proceed by simple increase of size without obvious alteration of form,
or whether it so proceed as to bring about a gradual change of form and
the slow development of a more or less complicated structure.

In the Newtonian language of elementary physics, force is recognised by
its action in producing or in changing motion, or in preventing change
of motion or in maintaining rest. When we deal with matter in the
concrete, force does not, strictly speaking, enter into the question,
for force, unlike matter, has no independent objective existence. It is
energy in its various forms, known or unknown, that acts upon matter.
But when we abstract our thoughts from the material to its form, or
from the thing moved to its motions, when we deal with the subjective
conceptions of form, or movement, or the movements that change of form
implies, then force is the appropriate term for our conception of the
causes by which these forms and changes of form are brought about. When
we use the term force, we use it, as the physicist always does, for
the sake of brevity, using a symbol for the magnitude and direction of
an action in reference to the symbol or diagram of a material thing.
It is a term as subjective and symbolic as form itself, and so is
appropriately to be used in connection therewith.

The form, then, of any portion of matter, whether it be living or dead,
and the changes of form that are apparent in its movements and in its
growth, may in all cases alike be described as due to the action of
force. In short, the form of an object is a “diagram of forces,” in
this sense, at least, that from it we can judge of or deduce the forces
that are acting or have acted upon it: in this strict and particular
sense, it is a diagram,—in the case of a solid, of the forces that
_have_ been impressed upon it when its conformation was produced,
together with those that enable it to retain its conformation; in the
case of a liquid (or of a gas) of the forces that are for the moment
acting on it to restrain or balance its own inherent mobility. In an
organism, great or small, it is not merely the nature of the _motions_
of the living substance that we must interpret in terms of force
(according to kinetics), but also {12} the _conformation_ of the
organism itself, whose permanence or equilibrium is explained by the
interaction or balance of forces, as described in statics.

If we look at the living cell of an Amoeba or a Spirogyra, we see
a something which exhibits certain active movements, and a certain
fluctuating, or more or less lasting, form; and its form at a given
moment, just like its motions, is to be investigated by the help of
physical methods, and explained by the invocation of the mathematical
conception of force.

Now the state, including the shape or form, of a portion of matter,
is the resultant of a number of forces, which represent or symbolise
the manifestations of various kinds of energy; and it is obvious,
accordingly, that a great part of physical science must be understood
or taken for granted as the necessary preliminary to the discussion
on which we are engaged. But we may at least try to indicate, very
briefly, the nature of the principal forces and the principal
properties of matter with which our subject obliges us to deal. Let
us imagine, for instance, the case of a so-called “simple” organism,
such as _Amoeba_; and if our short list of its physical properties and
conditions be helpful to our further discussion, we need not consider
how far it be complete or adequate from the wider physical point of
view[23].

This portion of matter, then, is kept together by the intermolecular
force of cohesion; in the movements of its particles relatively to
one another, and in its own movements relative to adjacent matter, it
meets with the opposing force of friction. It is acted on by gravity,
and this force tends (though slightly, owing to the Amoeba’s small
mass, and to the small difference between its density and that of the
surrounding fluid), to flatten it down upon the solid substance on
which it may be creeping. Our Amoeba tends, in the next place, to be
deformed by any pressure from outside, even though slight, which may be
applied to it, and this circumstance shews it to consist of matter in a
fluid, or at least semi-fluid, state: which state is further indicated
when we observe streaming or current motions in its interior. {13}
Like other fluid bodies, its surface, whatsoever other substance, gas,
liquid or solid, it be in contact with, and in varying degree according
to the nature of that adjacent substance, is the seat of molecular
force exhibiting itself as a surface-tension, from the action of which
many important consequences follow, which greatly affect the form of
the fluid surface.

While the protoplasm of the Amoeba reacts to the slightest pressure,
and tends to “flow,” and while we therefore speak of it as a fluid,
it is evidently far less mobile than such a fluid, for instance, as
water, but is rather like treacle in its slow creeping movements as
it changes its shape in response to force. Such fluids are said to
have a high viscosity, and this viscosity obviously acts in the way of
retarding change of form, or in other words of retarding the effects
of any disturbing action of force. When the viscous fluid is capable
of being drawn out into fine threads, a property in which we know that
the material of some Amoebae differs greatly from that of others, we
say that the fluid is also _viscid_, or exhibits viscidity. Again, not
by virtue of our Amoeba being liquid, but at the same time in vastly
greater measure than if it were a solid (though far less rapidly than
if it were a gas), a process of molecular diffusion is constantly going
on within its substance, by which its particles interchange their
places within the mass, while surrounding fluids, gases and solids in
solution diffuse into and out of it. In so far as the outer wall of
the cell is different in character from the interior, whether it be
a mere pellicle as in Amoeba or a firm cell-wall as in Protococcus,
the diffusion which takes place _through_ this wall is sometimes
distinguished under the term _osmosis_.

Within the cell, chemical forces are at work, and so also in all
probability (to judge by analogy) are electrical forces; and the
organism reacts also to forces from without, that have their origin
in chemical, electrical and thermal influences. The processes of
diffusion and of chemical activity within the cell result, by the
drawing in of water, salts, and food-material with or without chemical
transformation into protoplasm, in growth, and this complex phenomenon
we shall usually, without discussing its nature and origin, describe
and picture as a _force_. Indeed we shall manifestly be inclined to use
the term growth in two senses, {14} just indeed as we do in the case
of attraction or gravitation, on the one hand as a _process_, and on
the other hand as a _force_.

In the phenomena of cell-division, in the attractions or repulsions of
the parts of the dividing nucleus and in the “caryokinetic” figures
that appear in connection with it, we seem to see in operation forces
and the effects of forces, that have, to say the least of it, a close
analogy with known physical phenomena; and to this matter we shall
afterwards recur. But though they resemble known physical phenomena,
their nature is still the subject of much discussion, and neither the
forms produced nor the forces at work can yet be satisfactorily and
simply explained. We may readily admit, then, that besides phenomena
which are obviously physical in their nature, there are actions
visible as well as invisible taking place within living cells which
our knowledge does not permit us to ascribe with certainty to any
known physical force; and it may or may not be that these phenomena
will yield in time to the methods of physical investigation. Whether
or no, it is plain that we have no clear rule or guide as to what
is “vital” and what is not; the whole assemblage of so-called vital
phenomena, or properties of the organism, cannot be clearly classified
into those that are physical in origin and those that are _sui generis_
and peculiar to living things. All we can do meanwhile is to analyse,
bit by bit, those parts of the whole to which the ordinary laws of the
physical forces more or less obviously and clearly and indubitably
apply.

Morphology then is not only a study of material things and of the
forms of material things, but has its dynamical aspect, under which
we deal with the interpretation, in terms of force, of the operations
of Energy. And here it is well worth while to remark that, in dealing
with the facts of embryology or the phenomena of inheritance, the
common language of the books seems to deal too much with the _material_
elements concerned, as the causes of development, of variation or of
hereditary transmission. Matter as such produces nothing, changes
nothing, does nothing; and however convenient it may afterwards be
to abbreviate our nomenclature and our descriptions, we must most
carefully realise in the outset that the spermatozoon, the nucleus,
{15} the chromosomes or the germ-plasm can never _act_ as matter alone,
but only as seats of energy and as centres of force. And this is but an
adaptation (in the light, or rather in the conventional symbolism, of
modern physical science) of the old saying of the philosopher: ἀρχὴ
γὰρ ἡ φύσις μᾶλλον τῆς ὕλης.

{16}




CHAPTER II

ON MAGNITUDE


To terms of magnitude, and of direction, must we refer all our
conceptions of form. For the form of an object is defined when we know
its magnitude, actual or relative, in various directions; and growth
involves the same conceptions of magnitude and direction, with this
addition, that they are supposed to alter in time. Before we proceed
to the consideration of specific form, it will be worth our while to
consider, for a little while, certain phenomena of spatial magnitude,
or of the extension of a body in the several dimensions of space[24].

We are taught by elementary mathematics that, in similar solid figures,
the surface increases as the square, and the volume as the cube, of the
linear dimensions. If we take the simple case of a sphere, with radius
_r_, the area of its surface is equal to 4π_r_^2, and its volume to
(4/3)π_r_^3; from which it follows that the ratio of volume to surface,
or _V_/_S_, is (1/3)_r_. In other words, the greater the radius (or
the larger the sphere) the greater will be its volume, or its mass (if
it be uniformly dense throughout), in comparison with its superficial
area. And, taking _L_ to represent any linear dimension, we may write
the general equations in the form

 _S_ ∝ _L_^2, _V_ ∝ _L_^3,
 or _S_ = _k_ ⋅ _L_^2, and _V_ = _k′_ ⋅ _L_^3;
 and _V_/_S_ ∝ _L_.

From these elementary principles a great number of consequences follow,
all more or less interesting, and some of them of great importance.
In the first place, though growth in length (let {17} us say) and
growth in volume (which is usually tantamount to mass or weight) are
parts of one and the same process or phenomenon, the one attracts
our _attention_ by its increase, very much more than the other. For
instance a fish, in doubling its length, multiplies its weight by no
less than eight times; and it all but doubles its weight in growing
from four inches long to five.

In the second place we see that a knowledge of the correlation between
length and weight in any particular species of animal, in other words
a determination of _k_ in the formula _W_ = _k_ ⋅ _L_^3, enables us
at any time to translate the one magnitude into the other, and (so to
speak) to weigh the animal with a measuring-rod; this however being
always subject to the condition that the animal shall in no way have
altered its form, nor its specific gravity. That its specific gravity
or density should materially or rapidly alter is not very likely; but
as long as growth lasts, changes of form, even though inappreciable
to the eye, are likely to go on. Now weighing is a far easier and far
more accurate operation than measuring; and the measurements which
would reveal slight and otherwise imperceptible changes in the form of
a fish—slight relative differences between length, breadth and depth,
for instance,—would need to be very delicate indeed. But if we can make
fairly accurate determinations of the length, which is very much the
easiest dimension to measure, and then correlate it with the weight,
then the value of _k_, according to whether it varies or remains
constant, will tell us at once whether there has or has not been a
tendency to gradual alteration in the general form. To this subject we
shall return, when we come to consider more particularly the rate of
growth.

But a much deeper interest arises out of this changing ratio of
dimensions when we come to consider the inevitable changes of physical
relations with which it is bound up. We are apt, and even accustomed,
to think that magnitude is so purely relative that differences of
magnitude make no other or more essential difference; that Lilliput and
Brobdingnag are all alike, according as we look at them through one
end of the glass or the other. But this is by no means so; for _scale_
has a very marked effect upon physical phenomena, and the effect of
scale constitutes what is known as the principle of similitude, or of
dynamical similarity. {18}

This effect of scale is simply due to the fact that, of the physical
forces, some act either directly at the surface of a body, or otherwise
in _proportion_ to the area of surface; and others, such as gravity,
act on all particles, internal and external alike, and exert a force
which is proportional to the mass, and so usually to the volume, of the
body.

The strength of an iron girder obviously varies with the cross-section
of its members, and each cross-section varies as the square of a linear
dimension; but the weight of the whole structure varies as the cube of
its linear dimensions. And it follows at once that, if we build two
bridges geometrically similar, the larger is the weaker of the two[25].
It was elementary engineering experience such as this that led Herbert
Spencer[26] to apply the principle of similitude to biology.

The same principle had been admirably applied, in a few clear
instances, by Lesage[27], a celebrated eighteenth century physician
of Geneva, in an unfinished and unpublished work[28]. Lesage argued,
for instance, that the larger ratio of surface to mass would lead in a
small animal to excessive transpiration, were the skin as “porous” as
our own; and that we may hence account for the hardened or thickened
skins of insects and other small terrestrial animals. Again, since
the weight of a fruit increases as the cube of its dimensions, while
the strength of the stalk increases as the square, it follows that
the stalk should grow out of apparent due proportion to the fruit; or
alternatively, that tall trees should not bear large fruit on slender
branches, and that melons and pumpkins must lie upon the ground. And
again, that in quadrupeds a large head must be supported on a neck
which is either {19} excessively thick and strong, like a bull’s, or
very short like the neck of an elephant.

But it was Galileo who, wellnigh 300 years ago, had first laid down
this general principle which we now know by the name of the principle
of similitude; and he did so with the utmost possible clearness, and
with a great wealth of illustration, drawn from structures living and
dead[29]. He showed that neither can man build a house nor can nature
construct an animal beyond a certain size, while retaining the same
proportions and employing the same materials as sufficed in the case
of a smaller structure[30]. The thing will fall to pieces of its own
weight unless we either change its relative proportions, which will at
length cause it to become clumsy, monstrous and inefficient, or else
we must find a new material, harder and stronger than was used before.
Both processes are familiar to us in nature and in art, and practical
applications, undreamed of by Galileo, meet us at every turn in this
modern age of steel.

Again, as Galileo was also careful to explain, besides the questions
of pure stress and strain, of the strength of muscles to lift an
increasing weight or of bones to resist its crushing stress, we have
the very important question of _bending moments_. This question enters,
more or less, into our whole range of problems; it affects, as we shall
afterwards see, or even determines the whole form of the skeleton, and
is very important in such a case as that of a tall tree[31].

Here we have to determine the point at which the tree will curve
under its own weight, if it be ever so little displaced from the
perpendicular[32]. In such an investigation we have to make {20} some
assumptions,—for instance, with regard to the trunk, that it tapers
uniformly, and with regard to the branches that their sectional area
varies according to some definite law, or (as Ruskin assumed[33]) tends
to be constant in any horizontal plane; and the mathematical treatment
is apt to be somewhat difficult. But Greenhill has shewn that (on such
assumptions as the above), a certain British Columbian pine-tree, which
yielded the Kew flagstaff measuring 221 ft. in height with a diameter
at the base of 21 inches, could not possibly, by theory, have grown
to more than about 300 ft. It is very curious that Galileo suggested
precisely the same height (_dugento braccia alta_) as the utmost limit
of the growth of a tree. In general, as Greenhill shews, the diameter
of a homogeneous body must increase as the power 3/2 of the height,
which accounts for the slender proportions of young trees, compared
with the stunted appearance of old and large ones[34]. In short, as
Goethe says in _Wahrheit und Dichtung_, “Es ist dafür gesorgt dass
die Bäume nicht in den Himmel wachsen.” But Eiffel’s great tree of
steel (1000 feet high) is built to a very different plan; for here
the profile of the tower follows the logarithmic curve, giving _equal
strength_ throughout, according to a principle which we shall have
occasion to discuss when we come to treat of “form and mechanical
efficiency” in connection with the skeletons of animals.

Among animals, we may see in a general way, without the help of
mathematics or of physics, that exaggerated bulk brings with it a
certain clumsiness, a certain inefficiency, a new element of risk
and hazard, a vague preponderance of disadvantage. The case was
well put by Owen, in a passage which has an interest of its own as
a premonition (somewhat like De Candolle’s) of the “struggle for
existence.” Owen wrote as follows[35]: “In proportion to the bulk of a
species is the difficulty of the contest which, as a living organised
whole, the individual of such species {21} has to maintain against
the surrounding agencies that are ever tending to dissolve the vital
bond, and subjugate the living matter to the ordinary chemical and
physical forces. Any changes, therefore, in such external conditions
as a species may have been originally adapted to exist in, will
militate against that existence in a degree proportionate, perhaps in
a geometrical ratio, to the bulk of the species. If a dry season be
greatly prolonged, the large mammal will suffer from the drought sooner
than the small one; if any alteration of climate affect the quantity
of vegetable food, the bulky Herbivore will first feel the effects of
stinted nourishment.”

But the principle of Galileo carries us much further and along more
certain lines.

The tensile strength of a muscle, like that of a rope or of our girder,
varies with its cross-section; and the resistance of a bone to a
crushing stress varies, again like our girder, with its cross-section.
But in a terrestrial animal the weight which tends to crush its limbs
or which its muscles have to move, varies as the cube of its linear
dimensions; and so, to the possible magnitude of an animal, living
under the direct action of gravity, there is a definite limit set.
The elephant, in the dimensions of its limb-bones, is already shewing
signs of a tendency to disproportionate thickness as compared with
the smaller mammals; its movements are in many ways hampered and its
agility diminished: it is already tending towards the maximal limit of
size which the physical forces permit. But, as Galileo also saw, if
the animal be wholly immersed in water, like the whale, (or if it be
partly so, as was in all probability the case with the giant reptiles
of our secondary rocks), then the weight is counterpoised to the extent
of an equivalent volume of water, and is completely counterpoised if
the density of the animal’s body, with the included air, be identical
(as in a whale it very nearly is) with the water around. Under these
circumstances there is no longer a physical barrier to the indefinite
growth in magnitude of the animal[36]. Indeed, {22} in the case of the
aquatic animal there is, as Spencer pointed out, a distinct advantage,
in that the larger it grows the greater is its velocity. For its
available energy depends on the mass of its muscles; while its motion
through the water is opposed, not by gravity, but by “skin-friction,”
which increases only as the square of its dimensions; all other things
being equal, the bigger the ship, or the bigger the fish, the faster it
tends to go, but only in the ratio of the square root of the increasing
length. For the mechanical work (_W_) of which the fish is capable
being proportional to the mass of its muscles, or the cube of its
linear dimensions: and again this work being wholly done in producing a
velocity (_V_) against a resistance (_R_) which increases as the square
of the said linear dimensions; we have at once

 _W_ = _l_^3,

 and also _W_ = _R_ _V_^2 = _l_^2 _V_^2.

 Therefore _l_^3 = _l_^2 _V_^2, and _V_ = √_l_.

This is what is known as Froude’s Law of the _correspondence of
speeds_.

But there is often another side to these questions, which makes them
too complicated to answer in a word. For instance, the work (per
stroke) of which two similar engines are capable should obviously vary
as the cubes of their linear dimensions, for it varies on the one hand
with the _surface_ of the piston, and on the other, with the _length_
of the stroke; so is it likewise in the animal, where the corresponding
variation depends on the cross-section of the muscle, and on the space
through which it contracts. But in two precisely similar engines,
the actual available horse-power varies as the square of the linear
dimensions, and not as the cube; and this for the obvious reason that
the actual energy developed depends upon the _heating-surface_ of
the boiler[37]. So likewise must there be a similar tendency, among
animals, for the rate of supply of kinetic energy to vary with the
surface of the {23} lung, that is to say (other things being equal)
with the _square_ of the linear dimensions of the animal. We may of
course (departing from the condition of similarity) increase the
heating-surface of the boiler, by means of an internal system of tubes,
without increasing its outward dimensions, and in this very way nature
increases the respiratory surface of a lung by a complex system of
branching tubes and minute air-cells; but nevertheless in two similar
and closely related animals, as also in two steam-engines of precisely
the same make, the law is bound to hold that the rate of working must
tend to vary with the square of the linear dimensions, according to
Froude’s law of _steamship comparison_. In the case of a very large
ship, built for speed, the difficulty is got over by increasing the
size and number of the boilers, till the ratio between boiler-room
and engine-room is far beyond what is required in an ordinary small
vessel[38]; but though we find lung-space increased among animals where
greater rate of working is required, as in general among birds, I do
not know that it can be shewn to increase, as in the “over-boilered”
ship, with the size of the animal, and in a ratio which outstrips
that of the other bodily dimensions. If it be the case then, that
the working mechanism of the muscles should be able to exert a force
proportionate to the cube of the linear bodily dimensions, while the
respiratory mechanism can only supply a store of energy at a rate
proportional to the square of the said dimensions, the singular result
ought to follow that, in swimming for instance, the larger fish ought
to be able to put on a spurt of speed far in excess of the smaller
one; but the distance travelled by the year’s end should be very much
alike for both of them. And it should also follow that the curve of
fatigue {24} should be a steeper one, and the staying power should be
less, in the smaller than in the larger individual. This is the case of
long-distance racing, where the big winner puts on his big spurt at the
end. And for an analogous reason, wise men know that in the ’Varsity
boat-race it is judicious and prudent to bet on the heavier crew.

Leaving aside the question of the supply of energy, and keeping to
that of the mechanical efficiency of the machine, we may find endless
biological illustrations of the principle of similitude.

In the case of the flying bird (apart from the initial difficulty of
raising itself into the air, which involves another problem) it may be
shewn that the bigger it gets (all its proportions remaining the same)
the more difficult it is for it to maintain itself aloft in flight. The
argument is as follows:

In order to keep aloft, the bird must communicate to the air a downward
momentum equivalent to its own weight, and therefore proportional
to _the cube of its own linear dimensions_. But the momentum so
communicated is proportional to the mass of air driven downwards, and
to the rate at which it is driven: the mass being proportional to the
bird’s wing-area, and also (with any given slope of wing) to the speed
of the bird, and the rate being again proportional to the bird’s speed;
accordingly the whole momentum varies as the wing-area, i.e. as _the
square of the linear dimensions, and also as the square of the speed_.
Therefore, in order that the bird may maintain level flight, its speed
must be proportional to _the square root of its linear dimensions_.

Now the rate at which the bird, in steady flight, has to work in order
to drive itself forward, is the rate at which it communicates energy to
the air; and this is proportional to _m_ _V_^2, i.e. to the mass and
to the square of the velocity of the air displaced. But the mass of
air displaced per second is proportional to the wing-area and to the
speed of the bird’s motion, and therefore to the power 2½ of the linear
dimensions; and the speed at which it is displaced is proportional
to the bird’s speed, and therefore to the square root of the linear
dimensions. Therefore the energy communicated per second (being
proportional to the mass and to the square of the speed) is jointly
proportional to the power 2½ of the linear dimensions, as above, and
to the first power thereof: {25} that is to say, it increases in
proportion _to the power_ 3½ _of the linear dimensions_, and therefore
faster than the weight of the bird increases.

Put in mathematical form, the equations are as follows:

(_m_ = the mass of air thrust downwards; _V_ its velocity, proportional
to that of the bird; _M_ its momentum; _l_ a linear dimension of the
bird; _w_ its weight; _W_ the work done in moving itself forward.)

 _M_ = _w_ = _l_^3.

 But _M_ = _m_ _V_, and _m_ = _l_^2 _V_.

 Therefore _M_ = _l_^2 _V_^2,
       and _l_^2 _V_^2 = _l_^3,
        or _V_ = √_l_.

 But, again, _W_ = _m_ _V_^2 = _l_^2 _V_ × _V_^2
                            = _l_^2 × √_l_ × _l_
                            = _l_^{3½}.

The work requiring to be done, then, varies as the power 3½ of the
bird’s linear dimensions, while the work of which the bird is capable
depends on the mass of its muscles, and therefore varies as the cube
of its linear dimensions[39]. The disproportion does not seem at first
sight very great, but it is quite enough to tell. It is as much as
to say that, every time we double the linear dimensions of the bird,
the difficulty of flight is increased in the ratio of 2^3 : 2^{3½},
or 8 : 11·3, or, say, 1 : 1·4. If we take the ostrich to exceed the
sparrow in linear dimensions as 25 : 1, which seems well within the
mark, we have the ratio between 25^{3½} and 25^3, or between 5^7 : 5^6;
in other words, flight is just five times more difficult for the larger
than for the smaller bird[40].

The above investigation includes, besides the final result, a number
of others, explicit or implied, which are of not less importance. Of
these the simplest and also the most important is {26} contained in
the equation _V_ = √_l_, a result which happens to be identical with
one we had also arrived at in the case of the fish. In the bird’s case
it has a deeper significance than in the other; because it implies here
not merely that the velocity will tend to increase in a certain ratio
with the length, but that it _must_ do so as an essential and primary
condition of the bird’s remaining aloft. It is accordingly of great
practical importance in aeronautics, for it shews how a provision of
increasing speed must accompany every enlargement of our aeroplanes. If
a given machine weighing, say, 500 lbs. be stable at 40 miles an hour,
then one geometrically similar which weighs, say, a couple of tons must
have its speed determined as follows:

 _W_ : _w_ :: _L_^3 : _l_^3 :: 8 : 1.

 Therefore       _L_ : _l_ :: 2 : 1.

 But       _V_^2 : _v_^2 :: _L_ : _l_.

 Therefore       _V_ : _v_ :: √2 : 1 = 1·414 : 1.

That is to say, the larger machine must be capable of a speed equal
to 1·414 × 40, or about 56½ miles per hour.

It is highly probable, as Lanchester[41] remarks, that Lilienthal
met his untimely death not so much from any intrinsic fault in
the design or construction of his machine, but simply because his
engine fell somewhat short of the power required to give the speed
which was necessary for stability. An arrow is a very imperfectly
designed aeroplane, but nevertheless it is evidently capable, to a
certain extent and at a high velocity, of acquiring “stability” and
hence of actual “flight”: the duration and consequent range of its
trajectory, as compared with a bullet of similar initial velocity,
being correspondingly benefited. When we return to our birds, and
again compare the ostrich with the sparrow, we know little or nothing
about the speed in flight of the latter, but that of the swift is
estimated[42] to vary from a minimum of 20 to 50 feet or more per
second,—say from 14 to 35 miles per hour. Let us take the same lower
limit as not far from the minimal velocity of the sparrow’s flight
also; and it {27} would follow that the ostrich, of 25 times the
sparrow’s linear dimensions, would be compelled to fly (if it flew at
all) with a _minimum_ velocity of 5 × 14, or 70 miles an hour.

The same principle of _necessary speed_, or the indispensable relation
between the dimensions of a flying object and the minimum velocity at
which it is stable, accounts for a great number of observed phenomena.
It tells us why the larger birds have a marked difficulty in rising
from the ground, that is to say, in acquiring to begin with the
horizontal velocity necessary for their support; and why accordingly,
as Mouillard[43] and others have observed, the heavier birds, even
those weighing no more than a pound or two, can be effectively “caged”
in a small enclosure open to the sky. It tells us why very small birds,
especially those as small as humming-birds, and _à fortiori_ the still
smaller insects, are capable of “stationary flight,” a very slight and
scarcely perceptible velocity _relatively to the air_ being sufficient
for their support and stability. And again, since it is in all cases
velocity relative to the air that we are speaking of, we comprehend the
reason why one may always tell which way the wind blows by watching the
direction in which a bird _starts_ to fly.

It is not improbable that the ostrich has already reached a magnitude,
and we may take it for certain that the moa did so, at which flight by
muscular action, according to the normal anatomy of a bird, has become
physiologically impossible. The same reasoning applies to the case of
man. It would be very difficult, and probably absolutely impossible,
for a bird to fly were it the bigness of a man. But Borelli, in
discussing this question, laid even greater stress on the obvious fact
that a man’s pectoral muscles are so immensely less in proportion than
those of a bird, that however we may fit ourselves with wings we can
never expect to move them by any power of our own relatively weaker
muscles; so it is that artificial flight only became possible when
an engine was devised whose efficiency was extraordinarily great in
comparison with its weight and size.

Had Leonardo da Vinci known what Galileo knew, he would not have spent
a great part of his life on vain efforts to make to himself wings.
Borelli had learned the lesson thoroughly, and {28} in one of his
chapters he deals with the proposition, “Est impossible, ut homines
propriis viribus artificiose volare possint[44].”

But just as it is easier to swim than to fly, so is it obvious that,
in a denser atmosphere, the conditions of flight would be altered, and
flight facilitated. We know that in the carboniferous epoch there lived
giant dragon-flies, with wings of a span far greater than nowadays they
ever attain; and the small bodies and huge extended wings of the fossil
pterodactyles would seem in like manner to be quite abnormal according
to our present standards, and to be beyond the limits of mechanical
efficiency under present conditions. But as Harlé suggests[45],
following upon a suggestion of Arrhenius, we have only to suppose that
in carboniferous and jurassic days the terrestrial atmosphere was
notably denser than it is at present, by reason, for instance, of its
containing a much larger proportion of carbonic acid, and we have at
once a means of reconciling the apparent mechanical discrepancy.

Very similar problems, involving in various ways the principle of
dynamical similitude, occur all through the physiology of locomotion:
as, for instance, when we see that a cockchafer can carry a plate,
many times his own weight, upon his back, or that a flea can jump many
inches high.

Problems of this latter class have been admirably treated both by
Galileo and by Borelli, but many later writers have remained ignorant
of their work. Linnaeus, for instance, remarked that, if an elephant
were as strong in proportion as a stag-beetle, it would be able to pull
up rocks by the root, and to level mountains. And Kirby and Spence have
a well-known passage directed to shew that such powers as have been
conferred upon the insect have been withheld from the higher animals,
for the reason that had these latter been endued therewith they would
have “caused the early desolation of the world[46].” {29}

Such problems as that which is presented by the flea’s jumping powers,
though essentially physiological in their nature, have their interest
for us here: because a steady, progressive diminution of activity with
increasing size would tend to set limits to the possible growth in
magnitude of an animal just as surely as those factors which tend to
break and crush the living fabric under its own weight. In the case
of a leap, we have to do rather with a sudden impulse than with a
continued strain, and this impulse should be measured in terms of the
velocity imparted. The velocity is proportional to the impulse (_x_),
and inversely proportional to the mass (_M_) moved: _V_ = _x_/_M_. But,
according to what we still speak of as “Borelli’s law,” the impulse
(i.e. the work of the impulse) is proportional to the volume of the
muscle by which it is produced[47], that is to say (in similarly
constructed animals) to the mass of the whole body; for the impulse is
proportional on the one hand to the cross-section of the muscle, and
on the other to the distance through which it contracts. It follows at
once from this that the velocity is constant, whatever be the size of
the animals: in other words, that all animals, provided always that
they are similarly fashioned, with their various levers etc., in like
proportion, ought to jump, not to the same relative, but to the same
actual height[48]. According to this, then, the flea is not a better,
but rather a worse jumper than a horse or a man. As a matter of fact,
Borelli is careful to point out that in the act of leaping the impulse
is not actually instantaneous, as in the blow of a hammer, but takes
some little time, during which the levers are being extended by which
the centre of gravity of the animal is being propelled forwards; and
this interval of time will be longer in the case of the longer levers
of the larger animal. To some extent, then, this principle acts as a
corrective to the more general one, {30} and tends to leave a certain
balance of advantage, in regard to leaping power, on the side of the
larger animal[49].

But on the other hand, the question of strength of materials comes in
once more, and the factors of stress and strain and bending moment make
it, so to speak, more and more difficult for nature to endow the larger
animal with the length of lever with which she has provided the flea or
the grasshopper.

To Kirby and Spence it seemed that “This wonderful strength of insects
is doubtless the result of something peculiar in the structure and
arrangement of their muscles, and principally their extraordinary power
of contraction.” This hypothesis, which is so easily seen, on physical
grounds, to be unnecessary, has been amply disproved in a series of
excellent papers by F. Plateau[50].

A somewhat simple problem is presented to us by the act of walking.
It is obvious that there will be a great economy of work, if the leg
swing at its normal _pendulum-rate_; and, though this rate is hard to
calculate, owing to the shape and the jointing of the limb, we may
easily convince ourselves, by counting our steps, that the leg does
actually swing, or tend to swing, just as a pendulum does, at a certain
definite rate[51]. When we walk quicker, we cause the leg-pendulum to
describe a greater arc, but we do not appreciably cause it to swing, or
vibrate, quicker, until we shorten the pendulum and begin to run. Now
let two individuals, _A_ and _B_, walk in a similar fashion, that is
to say, with a similar _angle_ of swing. The _arc_ through which the
leg swings, or the _amplitude_ of each step, will therefore vary as the
length of leg, or say as _a_/_b_; but the time of swing will vary as
the square {31} root of the pendulum-length, or √_a_/√_b_. Therefore
the velocity, which is measured by amplitude/time, will also vary as
the square-roots of the length of leg: that is to say, the average
velocities of _A_ and _B_ are in the ratio of √_a_ : √_b_.

The smaller man, or smaller animal, is so far at a disadvantage
compared with the larger in speed, but only to the extent of the ratio
between the square roots of their linear dimensions: whereas, if the
rate of movement of the limb were identical, irrespective of the size
of the animal,—if the limbs of the mouse for instance swung at the same
rate as those of the horse,—then, as F. Plateau said, the mouse would
be as slow or slower in its gait than the tortoise. M. Delisle[52]
observed a “minute fly” walk three inches in half-a-second. This was
good steady walking. When we walk five miles an hour we go about 88
inches in a second, or 88/6 = 14·7 times the pace of M. Delisle’s
fly. We should walk at just about the fly’s pace if our stature were
1/(14·7)^2, or 1/216 of our present height,—say 72/216 inches, or
one-third of an inch high.

But the leg comprises a complicated system of levers, by whose various
exercise we shall obtain very different results. For instance, by
being careful to rise upon our instep, we considerably increase the
length or amplitude of our stride, and very considerably increase
our speed accordingly. On the other hand, in running, we bend and
so shorten the leg, in order to accommodate it to a quicker rate of
pendulum-swing[53]. In short, the jointed structure of the leg permits
us to use it as the shortest possible pendulum when it is swinging, and
as the longest possible lever when it is exerting its propulsive force.

Apart from such modifications as that described in the last
paragraph,—apart, that is to say, from differences in mechanical
construction or in the manner in which the mechanism is used,—we have
now arrived at a curiously simple and uniform result. For in all the
three forms of locomotion which we have attempted {32} to study, alike
in swimming, in flight and in walking, the general result, attained
under very different conditions and arrived at by very different modes
of reasoning, is in every case that the velocity tends to vary as the
square root of the linear dimensions of the organism.

From all the foregoing discussion we learn that, as Crookes once upon
a time remarked[54], the form as well as the actions of our bodies
are entirely conditioned (save for certain exceptions in the case of
aquatic animals, nicely balanced with the density of the surrounding
medium) by the strength of gravity upon this globe. Were the force of
gravity to be doubled, our bipedal form would be a failure, and the
majority of terrestrial animals would resemble short-legged saurians,
or else serpents. Birds and insects would also suffer, though there
would be some compensation for them in the increased density of the
air. While on the other hand if gravity were halved, we should get a
lighter, more graceful, more active type, requiring less energy and
less heat, less heart, less lungs, less blood.

Throughout the whole field of morphology we may find examples of a
tendency (referable doubtless in each case to some definite physical
cause) for surface to keep pace with volume, through some alteration
of its form. The development of “villi” on the inner surface of the
stomach and intestine (which enlarge its surface much as we enlarge
the effective surface of a bath-towel), the various valvular folds
of the intestinal lining, including the remarkable “spiral fold” of
the shark’s gut, the convolutions of the brain, whose complexity is
evidently correlated (in part at least) with the magnitude of the
animal,—all these and many more are cases in which a more or less
constant ratio tends to be maintained between mass and surface, which
ratio would have been more and more departed from had it not been for
the alterations of surface-form[55]. {33}

In the case of very small animals, and of individual cells, the
principle becomes especially important, in consequence of the molecular
forces whose action is strictly limited to the superficial layer.
In the cases just mentioned, action is _facilitated_ by increase of
surface: diffusion, for instance, of nutrient liquids or respiratory
gases is rendered more rapid by the greater area of surface; but
there are other cases in which the ratio of surface to mass may
make an essential change in the whole condition of the system. We
know, for instance, that iron rusts when exposed to moist air, but
that it rusts ever so much faster, and is soon eaten away, if the
iron be first reduced to a heap of small filings; this is a mere
difference of degree. But the spherical surface of the raindrop and
the spherical surface of the ocean (though both happen to be alike in
mathematical form) are two totally different phenomena, the one due
to surface-energy, and the other to that form of mass-energy which
we ascribe to gravity. The contrast is still more clearly seen in
the case of waves: for the little ripple, whose form and manner of
propagation are governed by surface-tension, is found to travel with
a velocity which is inversely as the square root of its length; while
the ordinary big waves, controlled by gravitation, have a velocity
directly proportional to the square root of their wave-length. In like
manner we shall find that the form of all small organisms is largely
independent of gravity, and largely if not mainly due to the force of
surface-tension: either as the direct result of the continued action of
surface tension on the semi-fluid body, or else as the result of its
action at a prior stage of development, in bringing about a form which
subsequent chemical changes have rendered rigid and lasting. In either
case, we shall find a very great tendency in small organisms to assume
either the spherical form or other simple forms related to ordinary
inanimate surface-tension phenomena; which forms do not recur in the
external morphology of large animals, or if they in part recur it is
for other reasons. {34}

Now this is a very important matter, and is a notable illustration of
that principle of similitude which we have already discussed in regard
to several of its manifestations. We are coming easily to a conclusion
which will affect the whole course of our argument throughout this
book, namely that there is an essential difference in kind between
the phenomena of form in the larger and the smaller organisms. I have
called this book a study of _Growth and Form_, because in the most
familiar illustrations of organic form, as in our own bodies for
example, these two factors are inseparably associated, and because we
are here justified in thinking of form as the direct resultant and
consequence of growth: of growth, whose varying rate in one direction
or another has produced, by its gradual and unequal increments, the
successive stages of development and the final configuration of
the whole material structure. But it is by no means true that form
and growth are in this direct and simple fashion correlative or
complementary in the case of minute portions of living matter. For in
the smaller organisms, and in the individual cells of the larger, we
have reached an order of magnitude in which the intermolecular forces
strive under favourable conditions with, and at length altogether
outweigh, the force of gravity, and also those other forces leading to
movements of convection which are the prevailing factors in the larger
material aggregate.

However we shall require to deal more fully with this matter in our
discussion of the rate of growth, and we may leave it meanwhile, in
order to deal with other matters more or less directly concerned with
the magnitude of the cell.

The living cell is a very complex field of energy, and of energy of
many kinds, surface-energy included. Now the whole surface-energy of
the cell is by no means restricted to its _outer_ surface; for the cell
is a very heterogeneous structure, and all its protoplasmic alveoli and
other visible (as well as invisible) heterogeneities make up a great
system of internal surfaces, at every part of which one “phase” comes
in contact with another “phase,” and surface-energy is accordingly
manifested. But still, the external surface is a definite portion of
the system, with a definite “phase” of its own, and however little we
may know of the distribution of the total energy of the system, it
is at least plain that {35} the conditions which favour equilibrium
will be greatly altered by the changed ratio of external surface to
mass which a change of magnitude, unaccompanied by change of form,
produces in the cell. In short, however it may be brought about, the
phenomenon of division of the cell will be precisely what is required
to keep approximately constant the ratio between surface and mass,
and to restore the balance between the surface-energy and the other
energies of the system. When a germ-cell, for instance, divides or
“segments” into two, it does not increase in mass; at least if there be
some slight alleged tendency for the egg to increase in mass or volume
during segmentation, it is very slight indeed, generally imperceptible,
and wholly denied by some[56]. The development or growth of the egg
from a one-celled stage to stages of two or many cells, is thus a
somewhat peculiar kind of growth; it is growth which is limited to
increase of surface, unaccompanied by growth in volume or in mass.

In the case of a soap-bubble, by the way, if it divide into two
bubbles, the volume is actually diminished[57] while the surface-area
is greatly increased. This is due to a cause which we shall have to
study later, namely to the increased pressure due to the greater
curvature of the smaller bubbles.

An immediate and remarkable result of the principles just described is
a tendency on the part of all cells, according to their kind, to vary
but little about a certain mean size, and to have, in fact, certain
absolute limitations of magnitude.

Sachs[58] pointed out, in 1895, that there is a tendency for each
nucleus to be only able to gather around itself a certain definite
amount of protoplasm. Driesch[59], a little later, found that, by
artificial subdivision of the egg, it was possible to rear dwarf
sea-urchin larvae, one-half, one-quarter, or even one-eighth of their
{36} normal size; and that these dwarf bodies were composed of
only a half, a quarter or an eighth of the normal number of cells.
Similar observations have been often repeated and amply confirmed.
For instance, in the development of _Crepidula_ (a little American
“slipper-limpet,” now much at home on our own oyster-beds), Conklin[60]
has succeeded in rearing dwarf and giant individuals, of which the
latter may be as much as twenty-five times as big as the former. But
nevertheless, the individual cells, of skin, gut, liver, muscle, and
of all the other tissues, are just the same size in one as in the
other,—in dwarf and in giant[61]. Driesch has laid particular stress
upon this principle of a “fixed cell-size.”

We get an excellent, and more familiar illustration of the same
principle in comparing the large brain-cells or ganglion-cells, both of
the lower and of the higher animals[62].

[Illustration: Fig. 1. Motor ganglion-cells, from the cervical spinal
cord. (From Minot, after Irving Hardesty.)]

In Fig. 1 we have certain identical nerve-cells taken from various
mammals, from the mouse to the elephant, all represented on the same
scale of magnification; and we see at once that they are all of much
the same _order_ of magnitude. The nerve-cell of the elephant is about
twice that of the mouse in linear dimensions, and therefore about
eight times greater in volume, or mass. But making some allowance for
difference of shape, the linear dimensions of the elephant are to
those of the mouse in a ratio certainly not less than one to fifty;
from which it would follow that the bulk of the larger animal is
something like 125,000 times that of the less. And it also follows,
the size of the nerve-cells being {37} about as eight to one, that, in
corresponding parts of the nervous system of the two animals, there
are more than 15,000 times as many individual cells in one as in
the other. In short we may (with Enriques) lay it down as a general
law that among animals, whether large or small, the ganglion-cells
vary in size within narrow limits; and that, amidst all the great
variety of structural type of ganglion observed in different classes
of animals, it is always found that the smaller species have simpler
ganglia than the larger, that is to say ganglia containing a smaller
number of cellular elements[63]. The bearing of such simple facts as
this upon the cell-theory in general is not to be disregarded; and the
warning is especially clear against exaggerated attempts to correlate
physiological processes with the visible mechanism of associated cells,
rather than with the system of energies, or the field of force, which
is associated with them. For the life of {38} the body is more than
the _sum_ of the properties of the cells of which it is composed: as
Goethe said, “Das Lebendige ist zwar in Elemente zerlegt, aber man kann
es aus diesen nicht wieder zusammenstellen und beleben.”

Among certain lower and microscopic organisms, such for instance as
the Rotifera, we are still more palpably struck by the small number of
cells which go to constitute a usually complex organ, such as kidney,
stomach, ovary, etc. We can sometimes number them in a few units, in
place of the thousands that make up such an organ in larger, if not
always higher, animals. These facts constitute one among many arguments
which combine to teach us that, however important and advantageous the
subdivision of organisms into cells may be from the constructional, or
from the dynamical point of view, the phenomenon has less essential
importance in theoretical biology than was once, and is often still,
assigned to it.

Again, just as Sachs shewed that there was a limit to the amount of
cytoplasm which could gather round a single nucleus, so Boveri has
demonstrated that the nucleus itself has definite limitations of size,
and that, in cell-division after fertilisation, each new nucleus has
the same size as its parent-nucleus[64].

In all these cases, then, there are reasons, partly no doubt
physiological, but in very large part purely physical, which set limits
to the normal magnitude of the organism or of the cell. But as we have
already discussed the existence of absolute and definite limitations,
of a physical kind, to the _possible_ increase in magnitude of an
organism, let us now enquire whether there be not also a lower limit,
below which the very existence of an organism is impossible, or
at least where, under changed conditions, its very nature must be
profoundly modified.

Among the smallest of known organisms we have, for instance,
_Micromonas mesnili_, Bonel, a flagellate infusorian, which measures
about ·34 _µ_, or ·00034 mm., by ·00025 mm.; smaller even than this
we have a pathogenic micrococcus of the rabbit, _M. progrediens_,
Schröter, the diameter of which is said to be only ·00015 mm. or
·15 _µ_, or 1·5 × 10^{−5} cm.,—about equal to the thickness of {39}
the thinnest gold-leaf; and as small if not smaller still are a few
bacteria and their spores. But here we have reached, or all but reached
the utmost limits of ordinary microscopic vision; and there remain
still smaller organisms, the so-called “filter-passers,” which the
ultra-microscope reveals, but which are mainly brought within our ken
only by the maladies, such as hydrophobia, foot-and-mouth disease, or
the “mosaic” disease of the tobacco-plant, to which these invisible
micro-organisms give rise[65]. Accordingly, since it is only by the
diseases which they occasion that these tiny bodies are made known to
us, we might be tempted to suppose that innumerable other invisible
organisms, smaller and yet smaller, exist unseen and unrecognised by
man.

[Illustration: Fig. 2. Relative magnitudes of: A, human blood-corpuscle
(7·5 µ in diameter); B, _Bacillus anthracis_ (4 – 15 µ × 1 µ); C,
various Micrococci (diam. 0·5 – 1 µ, rarely 2 µ); D, _Micromonas
progrediens_, Schröter (diam. 0·15 µ).]

To illustrate some of these small magnitudes I have adapted the
preceding diagram from one given by Zsigmondy[66]. Upon the {40} same
scale the minute ultramicroscopic particles of colloid gold would be
represented by the finest dots which we could make visible to the naked
eye upon the paper.

A bacillus of ordinary, typical size is, say, 1 µ in length. The length
(or height) of a man is about a million and three-quarter times as
great, i.e. 1·75 metres, or 1·75 × 10^6 µ; and the mass of the man is
in the neighbourhood of five million, million, million (5 × 10^{18})
times greater than that of the bacillus. If we ask whether there may
not exist organisms as much less than the bacillus as the bacillus is
less than the dimensions of a man, it is very easy to see that this
is quite impossible, for we are rapidly approaching a point where the
question of molecular dimensions, and of the ultimate divisibility of
matter, begins to call for our attention, and to obtrude itself as a
crucial factor in the case.

Clerk Maxwell dealt with this matter in his article “Atom[67],” and, in
somewhat greater detail, Errera discusses the question on the following
lines[68]. The weight of a hydrogen molecule is, according to the
physical chemists, somewhere about 8·6 × 2 × 10^{−22} milligrammes; and
that of any other element, whose molecular weight is _M_, is given by
the equation

 (_M_) = 8·6 × _M_ × 10^{−22}.

Accordingly, the weight of the atom of sulphur may be taken as

 8·6 × 32 × 10^{−22} mgm. = 275 × 10^{−22} mgm.

The analysis of ordinary bacteria shews them to consist[69] of about
85% of water, and 15% of solids; while the solid residue of vegetable
protoplasm contains about one part in a thousand of sulphur. We may
assume, therefore, that the living protoplasm contains about

 1/1000 × 15/100 = 15 × 10^{−5}

parts of sulphur, taking the total weight as = 1.

But our little micrococcus, of 0·15 µ in diameter, would, if it were
spherical, have a volume of

 π/6 × 0·15^3 µ = 18 × 10^{−4} cubic microns; {41}

and therefore (taking its density as equal to that of water), a weight
of

 18 × 10^{−4} × 10^{−9} = 18 × 10^{−13} mgm.

But of this total weight, the sulphur represents only

 18 × 10^{−13} × 15 × 10^{−5} = 27 × 10^{−17} mgm.

And if we divide this by the weight of an atom of sulphur, we have

 (27 × 10^{−17}) ÷ (275 × 10^{−22}) = 10,000, or thereby.

According to this estimate, then, our little _Micrococcus
progrediens_ should contain only about 10,000 atoms of sulphur,
an element indispensable to its protoplasmic constitution; and it
follows that an organism of one-tenth the diameter of our micrococcus
would only contain 10 sulphur-atoms, and therefore only ten chemical
“molecules” or units of protoplasm!

It may be open to doubt whether the presence of sulphur be really
essential to the constitution of the proteid or “protoplasmic”
molecule; but Errera gives us yet another illustration of a similar
kind, which is free from this objection or dubiety. The molecule of
albumin, as is generally agreed, can scarcely be less than a thousand
times the size of that of such an element as sulphur: according to
one particular determination[70], serum albumin has a constitution
corresponding to a molecular weight of 10,166, and even this may be
far short of the true complexity of a typical albuminoid molecule. The
weight of such a molecule is

 8·6 × 10166 × 10^{−22} = 8·7 × 10^{−18} mgm.

Now the bacteria contain about 14% of albuminoids, these constituting
by far the greater part of the dry residue; and therefore (from
equation (5)), the weight of albumin in our micrococcus is about

 14/100 × 18 × 10^{−13} = 2·5 × 10^{−13} mgm.

If we divide this weight by that which we have arrived at as the
weight of an albumin molecule, we have

 (2·5 × 10^{−13}) ÷ (8·7 × 10^{−18}) = 2·9 × 10^{−4},

in other words, our micrococcus apparently contains something less
than 30,000 molecules of albumin. {42}

According to the most recent estimates, the weight of the hydrogen
molecule is somewhat less than that on which Errera based his
calculations, namely about 16 × 10^{−22} mgms. and according to
this value, our micrococcus would contain just about 27,000 albumin
molecules. In other words, whichever determination we accept, we see
that an organism one-tenth as large as our micrococcus, in linear
dimensions, would only contain some thirty molecules of albumin; or, in
other words, our micrococcus is only about thirty times as large, in
linear dimensions, as a single albumin molecule[71].

We must doubtless make large allowances for uncertainty in the
assumptions and estimates upon which these calculations are based; and
we must also remember that the data with which the physicist provides
us in regard to molecular magnitudes are, to a very great extent,
_maximal_ values, above which the molecular magnitude (or rather
the sphere of the molecule’s range of motion) is not likely to lie:
but below which there is a greater element of uncertainty as to its
possibly greater minuteness. But nevertheless, when we shall have made
all reasonable allowances for uncertainty upon the physical side, it
will still be clear that the smallest known bodies which are described
as organisms draw nigh towards molecular magnitudes, and we must
recognise that the subdivision of the organism cannot proceed to an
indefinite extent, and in all probability cannot go very much further
than it appears to have done in these already discovered forms. For,
even, after giving all due regard to the complexity of our unit (that
is to say the albumin-molecule), with all the increased possibilities
of interrelation with its neighbours which this complexity implies, we
cannot but see that physiologically, and comparatively speaking, we
have come down to a very simple thing.

While such considerations as these, based on the chemical composition
of the organism, teach us that there must be a definite lower limit
to its magnitude, other considerations of a purely physical kind
lead us to the same conclusion. For our discussion of the principle
of similitude has already taught us that, long before we reach these
almost infinitesimal magnitudes, the {43} diminishing organism will
have greatly changed in all its physical relations, and must at length
arrive under conditions which must surely be incompatible with anything
such as we understand by life, at least in its full and ordinary
development and manifestation.

We are told, for instance, that the powerful force of surface-tension,
or capillarity, begins to act within a range of about 1/500,000 of an
inch, or say 0·05 µ. A soap-film, or a film of oil upon water, may be
attenuated to far less magnitudes than this; the black spots upon a
soap-bubble are known, by various concordant methods of measurement, to
be only about 6 × 10^{−7} cm., or about ·006 µ thick, and Lord Rayleigh
and M. Devaux[72] have obtained films of oil of ·002 µ, or even ·001 µ
in thickness.

But while it is possible for a fluid film to exist in these almost
molecular dimensions, it is certain that, long before we reach them,
there must arise new conditions of which we have little knowledge and
which it is not easy even to imagine.

It would seem that, in an organism of ·1 µ in diameter, or even rather
more, there can be no essential distinction between the interior
and the surface layers. No hollow vesicle, I take it, can exist of
these dimensions, or at least, if it be possible for it to do so,
the contained gas or fluid must be under pressures of a formidable
kind[73], and of which we have no knowledge or experience. Nor, I
imagine, can there be any real complexity, or heterogeneity, of its
fluid or semi-fluid contents; there can be no vacuoles within such a
cell, nor any layers defined within its fluid substance, for something
of the nature of a boundary-film is the necessary condition of the
existence of such layers. Moreover, the whole organism, provided that
it be fluid or semi-fluid, can only be spherical in form. What, then,
can we attribute, in the way of properties, to an organism of a size as
small as, or smaller than, say ·05 µ? It must, in all probability, be
a homogeneous, structureless sphere, composed of a very small number
of albuminoid or other molecules. Its vital properties and functions
must be extraordinarily limited; its specific outward characters, even
if we could see it, must be _nil_; and its specific properties must
be little more than those of an ion-laden corpuscle, enabling it to
perform {44} this or that chemical reaction, or to produce this or
that pathogenic effect. Even among inorganic, non-living bodies, there
must be a certain grade of minuteness at which the ordinary properties
become modified. For instance, while under ordinary circumstances
crystallisation starts in a solution about a minute solid fragment or
crystal of the salt, Ostwald has shewn that we may have particles so
minute that they fail to serve as a nucleus for crystallisation,—which
is as much as to say that they are too minute to have the form and
properties of a “crystal”; and again, in his thin oil-films, Lord
Rayleigh has noted the striking change of physical properties which
ensues when the film becomes attenuated to something less than one
close-packed layer of molecules[74].

Thus, as Clerk Maxwell put it, “molecular science sets us face to face
with physiological theories. It forbids the physiologist from imagining
that structural details of infinitely small dimensions [such as Leibniz
assumed, one within another, _ad infinitum_] can furnish an explanation
of the infinite variety which exists in the properties and functions
of the most minute organisms.” And for this reason he reprobates, with
not undue severity, those advocates of pangenesis and similar theories
of heredity, who would place “a whole world of wonders within a body
so small and so devoid of visible structure as a germ.” But indeed it
scarcely needed Maxwell’s criticism to shew forth the immense physical
difficulties of Darwin’s theory of Pangenesis: which, after all, is as
old as Democritus, and is no other than that Promethean _particulam
undique desectam_ of which we have read, and at which we have smiled,
in our Horace.

There are many other ways in which, when we “make a long excursion
into space,” we find our ordinary rules of physical behaviour entirely
upset. A very familiar case, analysed by Stokes, is that the viscosity
of the surrounding medium has a relatively powerful effect upon bodies
below a certain size. A droplet of water, a thousandth of an inch (25
µ) in diameter, cannot fall in still air quicker than about an inch and
a half per second; and as its size decreases, its resistance varies
as the diameter, and not (as with larger bodies) as the surface of
the {45} drop. Thus a drop one-tenth of that size (2·5 µ), the size,
apparently, of the drops of water in a light cloud, will fall a hundred
times slower, or say an inch a minute; and one again a tenth of this
diameter (say ·25 µ, or about twice as big, in linear dimensions, as
our micrococcus), will scarcely fall an inch in two hours. By reason
of this principle, not only do the smaller bacteria fall very slowly
through the air, but all minute bodies meet with great proportionate
resistance to their movements in a fluid. Even such comparatively large
organisms as the diatoms and the foraminifera, laden though they are
with a heavy shell of flint or lime, seem to be poised in the water of
the ocean, and fall in it with exceeding slowness.

The Brownian movement has also to be reckoned with,—that remarkable
phenomenon studied nearly a century ago (1827) by Robert Brown,
_facile princeps botanicorum_. It is one more of those fundamental
physical phenomena which the biologists have contributed, or helped to
contribute, to the science of physics.

The quivering motion, accompanied by rotation, and even by
translation, manifested by the fine granular particles issuing from
a crushed pollen-grain, and which Robert Brown proved to have no
vital significance but to be manifested also by all minute particles
whatsoever, organic and inorganic, was for many years unexplained.
Nearly fifty years after Brown wrote, it was said to be “due, either
directly to some calorical changes continually taking place in the
fluid, or to some obscure chemical action between the solid particles
and the fluid which is indirectly promoted by heat[75].” Very shortly
after these last words were written, it was ascribed by Wiener to
molecular action, and we now know that it is indeed due to the impact
or bombardment of molecules upon a body so small that these impacts do
not for the moment, as it were, “average out” to approximate equality
on all sides. The movement becomes manifest with particles of somewhere
about 20 µ in diameter, it is admirably displayed by particles of about
12 µ in diameter, and becomes more marked the smaller the particles
are. The bombardment causes our particles to behave just like molecules
of uncommon size, and this {46} behaviour is manifested in several
ways[76]. Firstly, we have the quivering movement of the particles;
secondly, their movement backwards and forwards, in short, straight,
disjointed paths; thirdly, the particles rotate, and do so the more
rapidly the smaller they are, and by theory, confirmed by observation,
it is found that particles of 1 µ in diameter rotate on an average
through 100° per second, while particles of 13 µ in diameter turn
through only 14° per minute. Lastly, the very curious result appears,
that in a layer of fluid the particles are not equally distributed, nor
do they all ever fall, under the influence of gravity, to the bottom.
But just as the molecules of the atmosphere are so distributed, under
the influence of gravity, that the density (and therefore the number
of molecules per unit volume) falls off in geometrical progression as
we ascend to higher and higher layers, so is it with our particles,
even within the narrow limits of the little portion of fluid under
our microscope. It is only in regard to particles of the simplest
form that these phenomena have been theoretically investigated[77],
and we may take it as certain that more complex particles, such as
the twisted body of a Spirillum, would show other and still more
complicated manifestations. It is at least clear that, just as the
early microscopists in the days before Robert Brown never doubted but
that these phenomena were purely vital, so we also may still be apt
to confuse, in certain cases, the one phenomenon with the other. We
cannot, indeed, without the most careful scrutiny, decide whether the
movements of our minutest organisms are intrinsically “vital” (in the
sense of being beyond a physical mechanism, or working model) or not.
For example, Schaudinn has suggested that the undulating movements of
_Spirochaete pallida_ must be due to the presence of a minute, unseen,
“undulating membrane”; and Doflein says of the same species that “sie
verharrt oft mit eigenthümlich zitternden Bewegungen zu einem Orte.”
Both movements, the trembling or quivering {47} movement described
by Doflein, and the undulating or rotating movement described by
Schaudinn, are just such as may be easily and naturally interpreted as
part and parcel of the Brownian phenomenon.

While the Brownian movement may thus simulate in a deceptive way
the active movements of an organism, the reverse statement also to
a certain extent holds good. One sometimes lies awake of a summer’s
morning watching the flies as they dance under the ceiling. It is a
very remarkable dance. The dancers do not whirl or gyrate, either in
company or alone; but they advance and retire; they seem to jostle and
rebound; between the rebounds they dart hither or thither in short
straight snatches of hurried flight; and turn again sharply in a new
rebound at the end of each little rush. Their motions are wholly
“erratic,” independent of one another, and devoid of common purpose.
This is nothing else than a vastly magnified picture, or simulacrum, of
the Brownian movement; the parallel between the two cases lies in their
complete irregularity, but this in itself implies a close resemblance.
One might see the same thing in a crowded market-place, always provided
that the bustling crowd had no _business_ whatsoever. In like manner
Lucretius, and Epicurus before him, watched the dust-motes quivering
in the beam, and saw in them a mimic representation, _rei simulacrum
et imago_, of the eternal motions of the atoms. Again the same
phenomenon may be witnessed under the microscope, in a drop of water
swarming with Paramoecia or suchlike Infusoria; and here the analogy
has been put to a numerical test. Following with a pencil the track
of each little swimmer, and dotting its place every few seconds (to
the beat of a metronome), Karl Przibram found that the mean successive
distances from a common base-line obeyed with great exactitude the
“Einstein formula,” that is to say the particular form of the “law of
chance” which is applicable to the case of the Brownian movement[78].
The phenomenon is (of course) merely analogous, and by no means
identical with the Brownian movement; for the range of motion of the
little active organisms, whether they be gnats or infusoria, is vastly
greater than that of the minute particles which are {48} passive under
bombardment; but nevertheless Przibram is inclined to think that even
his comparatively large infusoria are small enough for the molecular
bombardment to be a stimulus, though not the actual cause, of their
irregular and interrupted movements.

There is yet another very remarkable phenomenon which may come into
play in the case of the minutest of organisms; and this is their
relation to the rays of light, as Arrhenius has told us. On the waves
of a beam of light, a very minute particle (_in vacuo_) should be
actually caught up, and carried along with an immense velocity; and
this “radiant pressure” exercises its most powerful influence on bodies
which (if they be of spherical form) are just about ·00016 mm., or
·16 µ in diameter. This is just about the size, as we have seen, of
some of our smallest known protozoa and bacteria, while we have some
reason to believe that others yet unseen, and perhaps the spores of
many, are smaller still. Now we have seen that such minute particles
fall with extreme slowness in air, even at ordinary atmospheric
pressures: our organism measuring ·16 µ would fall but 83 metres in a
year, which is as much as to say that its weight offers practically no
impediment to its transference, by the slightest current, to the very
highest regions of the atmosphere. Beyond the atmosphere, however, it
cannot go, until some new force enable it to resist the attraction of
terrestrial gravity, which the viscosity of an atmosphere is no longer
at hand to oppose. But it is conceivable that our particle _may_ go yet
farther, and actually break loose from the bonds of earth. For in the
upper regions of the atmosphere, say fifty miles high, it will come in
contact with the rays and flashes of the Northern Lights, which consist
(as Arrhenius maintains) of a fine dust, or cloud of vapour-drops,
laden with a charge of negative electricity, and projected outwards
from the sun. As soon as our particle acquires a charge of negative
electricity it will begin to be repelled by the similarly laden auroral
particles, and the amount of charge necessary to enable a particle
of given size (such as our little monad of ·16 µ) to resist the
attraction of gravity may be calculated, and is found to be such as
the actual conditions can easily supply. Finally, when once set free
from the entanglement of the earth’s {49} atmosphere, the particle
may be propelled by the “radiant pressure” of light, with a velocity
which will carry it.—like Uriel gliding on a sunbeam,—as far as the
orbit of Mars in twenty days, of Jupiter in eighty days, and as far as
the nearest fixed star in three thousand years! This, and much more,
is Arrhenius’s contribution towards the acceptance of Lord Kelvin’s
hypothesis that life may be, and may have been, disseminated across the
bounds of space, throughout the solar system and the whole universe!

It may well be that we need attach no great practical importance to
this bold conception; for even though stellar space be shewn to be
_mare liberum_ to minute material travellers, we may be sure that those
which reach a stellar or even a planetary bourne are infinitely, or all
but infinitely, few. But whether or no, the remote possibilities of the
case serve to illustrate in a very vivid way the profound differences
of physical property and potentiality which are associated in the scale
of magnitude with simple differences of degree.

{50}




CHAPTER III

THE RATE OF GROWTH


When we study magnitude by itself, apart, that is to say, from the
gradual changes to which it may be subject, we are dealing with a
something which may be adequately represented by a number, or by means
of a line of definite length; it is what mathematicians call a _scalar_
phenomenon. When we introduce the conception of change of magnitude,
of magnitude which varies as we pass from one direction to another in
space, or from one instant to another in time, our phenomenon becomes
capable of representation by means of a line of which we define both
the length and the direction; it is (in this particular aspect) what is
called a _vector_ phenomenon.

When we deal with magnitude in relation to the dimensions of space, the
vector diagram which we draw plots magnitude in one direction against
magnitude in another,—length against height, for instance, or against
breadth; and the result is simply what we call a picture or drawing of
an object, or (more correctly) a “plane projection” of the object. In
other words, what we call Form is a _ratio of magnitudes_, referred to
direction in space.

When in dealing with magnitude we refer its variations to successive
intervals of time (or when, as it is said, we _equate_ it with time),
we are then dealing with the phenomenon of _growth_; and it is evident,
therefore, that this term growth has wide meanings. For growth may
obviously be positive or negative; that is to say, a thing may grow
larger or smaller, greater or less; and by extension of the primitive
concrete signification of the word, we easily and legitimately apply
it to non-material things, such as temperature, and say, for instance,
that a body “grows” hot or cold. When in a two-dimensional diagram, we
represent a magnitude (for instance length) in relation to time (or
“plot” {51} length against time, as the phrase is), we get that kind
of vector diagram which is commonly known as a “curve of growth.” We
perceive, accordingly, that the phenomenon which we are now studying is
a _velocity_ (whose “dimensions” are Space/Time or _L_/_T_); and this
phenomenon we shall speak of, simply, as a rate of growth.

In various conventional ways we can convert a two-dimensional into
a three-dimensional diagram. We do so, for example, by means of
the geometrical method of “perspective” when we represent upon
a sheet of paper the length, breadth and depth of an object in
three-dimensional space; but we do it more simply, as a rule, by means
of “contour-lines,” and always when time is one of the dimensions
to be represented. If we superimpose upon one another (or even set
side by side) pictures, or plane projections, of an organism, drawn
at successive intervals of time, we have such a three-dimensional
diagram, which is a partial representation (limited to two dimensions
of _space_) of the organism’s gradual change of form, or course of
development; and in such a case our contour-lines may, for the purposes
of the embryologist, be separated by intervals representing a few hours
or days, or, for the purposes of the palaeontologist, by interspaces of
unnumbered and innumerable years[79].

Such a diagram represents in two of its three dimensions form, and in
two, or three, of its dimensions growth; and so we see how intimately
the two conceptions are correlated or inter-related to one another.
In short, it is obvious that the form of an animal is determined by
its specific rate of growth in various directions; accordingly, the
phenomenon of rate of growth deserves to be studied as a necessary
preliminary to the theoretical study of form, and, mathematically
speaking, organic form itself appears to us as a _function of
time_[80]. {52}

At the same time, we need only consider this part of our subject
somewhat briefly. Though it has an essential bearing on the problems
of morphology, it is in greater degree involved with physiological
problems; and furthermore, the statistical or numerical aspect of the
question is peculiarly adapted for the mathematical study of variation
and correlation. On these important subjects we shall scarcely touch;
for our main purpose will be sufficiently served if we consider the
characteristics of a rate of growth in a few illustrative cases,
and recognise that this rate of growth is a very important specific
property, with its own characteristic value in this organism or that,
in this or that part of each organism, and in this or that phase of its
existence.

The statement which we have just made that “the form of an organism is
determined by its rate of growth in various directions,” is one which
calls (as we have partly seen in the foregoing chapter) for further
explanation and for some measure of qualification. Among organic forms
we shall have frequent occasion to see that form is in many cases due
to the immediate or direct action of certain molecular forces, of
which surface-tension is that which plays the greatest part. Now when
surface-tension (for instance) causes a minute semi-fluid organism to
assume a spherical form, or gives the form of a catenary or an elastic
curve to a film of protoplasm in contact with some solid skeletal rod,
or when it acts in various other ways which are productive of definite
contours, this is a process of conformation that, both in appearance
and reality, is very different from the process by which an ordinary
plant or animal _grows_ into its specific form. In both cases, change
of form is brought about by the movement of portions of matter, and in
both cases it is _ultimately_ due to the action of molecular forces;
but in the one case the movements of the particles of matter lie for
the most part _within molecular range_, while in the other we have
to deal chiefly with the transference of portions of matter into the
system from without, and from one widely distant part of the organism
to another. It is to this latter class of phenomena that we usually
restrict the term growth; and it is in regard to them that we are in
a position to study the _rate of action_ in different directions,
and to see that it is merely on a difference of velocities that the
modification of form essentially depends. {53} The difference between
the two classes of phenomena is somewhat akin to the difference between
the forces which determine the form of a rain-drop and those which, by
the flowing of the waters and the sculpturing of the solid earth, have
brought about the complex configuration of a river; _molecular_ forces
are paramount in the conformation of the one, and _molar_ forces are
dominant in the other.

At the same time it is perfectly true that _all_ changes of form,
inasmuch as they necessarily involve changes of actual and relative
magnitude, may, in a sense, be properly looked upon as phenomena of
growth; and it is also true, since the movement of matter must always
involve an element of time[81], that in all cases the rate of growth
is a phenomenon to be considered. Even though the molecular forces
which play their part in modifying the form of an organism exert an
action which is, theoretically, all but instantaneous, that action is
apt to be dragged out to an appreciable interval of time by reason of
viscosity or some other form of resistance in the material. From the
physical or physiological point of view the rate of action even in such
cases may be well worth studying; for example, a study of the rate of
cell-division in a segmenting egg may teach us something about the work
done, and about the various energies concerned. But in such cases the
action is, as a rule, so homogeneous, and the form finally attained is
so definite and so little dependent on the time taken to effect it,
that the specific rate of change, or rate of growth, does not enter
into the _morphological_ problem.

To sum up, we may lay down the following general statements. The form
of organisms is a phenomenon to be referred in part to the direct
action of molecular forces, in part to a more complex and slower
process, indirectly resulting from chemical, osmotic and other forces,
by which material is introduced into the organism and transferred from
one part of it to another. It is this latter complex phenomenon which
we usually speak of as “growth.” {54}

Every growing organism, and every part of such a growing organism, has
its own specific rate of growth, referred to a particular direction.
It is the ratio between the rates of growth in various directions by
which we must account for the external forms of all, save certain
very minute, organisms. This ratio between rates of growth in various
directions may sometimes be of a _simple_ kind, as when it results in
the mathematically definable outline of a shell, or in the smooth curve
of the margin of a leaf. It may sometimes be a very _constant_ one, in
which case the organism, while growing in bulk, suffers little or no
perceptible change in form; but such equilibrium seldom endures for
more than a season, and when the _ratio_ tends to alter, then we have
the phenomenon of morphological “development,” or steady and persistent
change of form.

This elementary concept of Form, as determined by varying rates of
Growth, was clearly apprehended by the mathematical mind of Haller,—who
had learned his mathematics of the great John Bernoulli, as the latter
in turn had learned his physiology from the writings of Borelli. Indeed
it was this very point, the apparently unlimited extent to which, in
the development of the chick, inequalities of growth could and did
produce changes of form and changes of anatomical “structure,” that
led Haller to surmise that the process was actually without limits,
and that all development was but an unfolding, or “_evolutio_,” in
which no part came into being which had not essentially existed
before[82]. In short the celebrated doctrine of “preformation” implied
on the one hand a clear recognition of what, throughout the later
stages of development, growth can do, by hastening the increase in
size of one part, hindering that of another, changing their relative
magnitudes and positions, and altering their forms; while on the other
hand it betrayed a failure (inevitable in those days) to recognise
the essential difference between these movements of masses and the
molecular processes which precede and accompany {55} them, and which
are characteristic of another order of magnitude.

By other writers besides Haller the very general, though not strictly
universal connection between form and rate of growth has been clearly
recognised. Such a connection is implicit in those “proportional
diagrams” by which Dürer and some of his brother artists were wont to
illustrate the successive changes of form, or of relative dimensions,
which attend the growth of the child, to boyhood and to manhood. The
same connection was recognised, more explicitly, by some of the older
embryologists, for instance by Pander[83], and appears, as a survival
of the doctrine of preformation, in his study of the development of
the chick. And long afterwards, the embryological aspect of the case
was emphasised by His, who pointed out, for instance, that the various
foldings of the blastoderm, by which the neural and amniotic folds
were brought into being, were essentially and obviously the resultant
of unequal rates of growth,—of local accelerations or retardations
of growth,—in what to begin with was an even and uniform layer of
embryonic tissue. If we imagine a flat sheet of paper, parts of which
are caused (as by moisture or evaporation) to expand or to contract,
the plane surface is at once dimpled, or “buckled,” or folded, by
the resultant forces of expansion or contraction: and the various
distortions to which the plane surface of the “germinal disc” is
subject, as His shewed once and for all, are precisely analogous.
An experimental demonstration still more closely comparable to the
actual case of the blastoderm, is obtained by making an “artificial
blastoderm,” of little pills or pellets of dough, which are caused to
grow, with varying velocities, by the addition of varying quantities of
yeast. Here, as Roux is careful to point out[84], we observe that it
is not only the _growth_ of the individual cells, but the _traction_
exercised through their mutual interconnections, which brings about the
foldings and other distortions of the entire structure. {56}

But this again was clearly present to Haller’s mind, and formed an
essential part of his embryological doctrine. For he has no sooner
treated of _incrementum_, or _celeritas incrementi_, than he proceeds
to deal with the contributory and complementary phenomena of expansion,
traction (_adtractio_)[85], and pressure, and the more subtle
influences which he denominates _vis derivationis et revulsionis_[86]:
these latter being the secondary and correlated effects on growth in
one part, brought about, through such changes as are produced (for
instance) in the circulation, by the growth of another.

Let us admit that, on the physiological side, Haller’s or His’s methods
of explanation carry us back but a little way; yet even this little
way is something gained. Nevertheless, I can well remember the harsh
criticism, and even contempt, which His’s doctrine met with, not merely
on the ground that it was inadequate, but because such an explanation
was deemed wholly inappropriate, and was utterly disavowed[87].
Hertwig, for instance, asserted that, in embryology, when we found one
embryonic stage preceding another, the existence of the former was,
for the embryologist, an all-sufficient “causal explanation” of the
latter. “We consider (he says), that we are studying and explaining a
causal relation when we have demonstrated that the gastrula arises by
invagination of a blastosphere, or the neural canal by the infolding
of a cell plate so as to constitute a tube[88].” For Hertwig,
therefore, as {57} Roux remarks, the task of investigating a physical
mechanism in embryology,—“der Ziel das Wirken zu erforschen,”—has no
existence at all. For Balfour also, as for Hertwig, the mechanical or
physical aspect of organic development had little or no attraction.
In one notable instance, Balfour himself adduced a physical, or
quasi-physical, explanation of an organic process, when he referred the
various modes of segmentation of an ovum, complete or partial, equal or
unequal and so forth, to the varying amount or the varying distribution
of food yolk in association with the germinal protoplasm of the
egg[89]. But in the main, Balfour, like all the other embryologists of
his day, was engrossed by the problems of phylogeny, and he expressly
defined the aims of comparative embryology (as exemplified in his own
textbook) as being “twofold: (1) to form a basis for Phylogeny. and
(2) to form a basis for Organogeny or the origin and evolution of
organs[90].”

It has been the great service of Roux and his fellow-workers of the
school of “Entwickelungsmechanik,” and of many other students to
whose work we shall refer, to try, as His tried[91] to import into
embryology, wherever possible, the simpler concepts of physics, to
introduce along with them the method of experiment, and to refuse to be
bound by the narrow limitations which such teaching as that of Hertwig
would of necessity impose on the work and the thought and on the whole
philosophy of the biologist.

――――――――――

Before we pass from this general discussion to study some of the
particular phenomena of growth, let me give a single illustration, from
Darwin, of a point of view which is in marked contrast to Haller’s
simple but essentially mathematical conception of Form.

There is a curious passage in the _Origin of Species_[92], where Darwin
is discussing the leading facts of embryology, and in particular Von
Baer’s “law of embryonic resemblance.” Here Darwin says “We are so
much accustomed to see a difference in {58} structure between the
embryo and the adult, that we are tempted to look at this difference
as in some necessary manner contingent on growth. _But there is no
reason why, for instance, the wing of a bat, or the fin of a porpoise,
should not have been sketched out with all their parts in proper
proportion, as soon as any part became visible._” After pointing out
with his habitual care various exceptions, Darwin proceeds to lay down
two general principles, viz. “that slight variations generally appear
at a not very early period of life,” and secondly, that “at whatever
age a variation first appears in the parent, it tends to reappear
at a corresponding age in the offspring.” He then argues that it is
with nature as with the fancier, who does not care what his pigeons
look like in the embryo, so long as the full-grown bird possesses
the desired qualities; and that the process of selection takes place
when the birds or other animals are nearly grown up,—at least on the
part of the breeder, and presumably in nature as a general rule. The
illustration of these principles is set forth as follows; “Let us take
a group of birds, descended from some ancient form and modified through
natural selection for different habits. Then, from the many successive
variations having supervened in the several species at a not very early
age, and having been inherited at a corresponding age, the young will
still resemble each other much more closely than do the adults,—just
as we have seen with the breeds of the pigeon .... Whatever influence
long-continued use or disuse may have had in modifying the limbs or
other parts of any species, this will chiefly or solely have affected
it when nearly mature, when it was compelled to use its full powers
to gain its own living; and the effects thus produced will have been
transmitted to the offspring at a corresponding nearly mature age.
Thus the young will not be modified, or will be modified only in a
slight degree, through the effects of the increased use or disuse of
parts.” This whole argument is remarkable, in more ways than we need
try to deal with here; but it is especially remarkable that Darwin
should begin by casting doubt upon the broad fact that a “difference
in structure between the embryo and the adult” is “in some necessary
manner contingent on growth”; and that he should see no reason why
complicated structures of the adult “should not have been sketched out
{59} with all their parts in proper proportion, as soon as any part
became visible.” It would seem to me that even the most elementary
attention to form in its relation to growth would have removed most of
Darwin’s difficulties in regard to the particular phenomena which he
is here considering. For these phenomena are phenomena of form, and
therefore of relative magnitude; and the magnitudes in question are
attained by growth, proceeding with certain specific velocities, and
lasting for certain long periods of time. And it is accordingly obvious
that in any two related individuals (whether specifically identical or
not) the differences between them must manifest themselves gradually,
and be but little apparent in the young. It is for the same simple
reason that animals which are of very different sizes when adult,
differ less and less in size (as well as in form) as we trace them
backwards through the foetal stages.

――――――――――

Though we study the visible effects of varying rates of growth
throughout wellnigh all the problems of morphology, it is not very
often that we can directly measure the velocities concerned. But owing
to the obvious underlying importance which the phenomenon has to the
morphologist we must make shift to study it where we can, even though
our illustrative cases may seem to have little immediate bearing on the
morphological problem[93].

In a very simple organism, of spherical symmetry, such as the single
spherical cell of Protococcus or of Orbulina, growth is reduced to
its simplest terms, and indeed it becomes so simple in its outward
manifestations that it is no longer of special interest to the
morphologist. The rate of growth is measured by the rate of change in
length of a radius, i.e. _V_ = (_R′_ − _R_)/_T_, and from this we may
calculate, as already indicated, the rate of growth in terms of surface
and of volume. The growing body remains of constant form, owing to the
symmetry of the system; because, that is to say, on the one hand the
pressure exerted by the growing protoplasm is exerted equally in all
directions, after the manner of a hydrostatic pressure, which indeed it
actually is: while on the other hand, the “skin” or surface layer of
the cell is sufficiently {60} homogeneous to exert at every point an
approximately uniform resistance. Under these conditions then, the rate
of growth is uniform in all directions, and does not affect the form of
the organism.

But in a larger or a more complex organism the study of growth, and of
the rate of growth, presents us with a variety of problems, and the
whole phenomenon becomes a factor of great morphological importance. We
no longer find that it tends to be uniform in all directions, nor have
we any right to expect that it should. The resistances which it meets
with will no longer be uniform. In one direction but not in others it
will be opposed by the important resistance of gravity; and within the
growing system itself all manner of structural differences will come
into play, setting up unequal resistances to growth by the varying
rigidity or viscosity of the material substance in one direction or
another. At the same time, the actual sources of growth, the chemical
and osmotic forces which lead to the intussusception of new matter, are
not uniformly distributed; one tissue or one organ may well manifest a
tendency to increase while another does not; a series of bones, their
intervening cartilages, and their surrounding muscles, may all be
capable of very different rates of increment. The differences of form
which are the resultants of these differences in rate of growth are
especially manifested during that part of life when growth itself is
rapid: when the organism, as we say, is undergoing its _development_.
When growth in general has become slow, the relative differences in
rate between different parts of the organism may still exist, and
may be made manifest by careful observation, but in many, or perhaps
in most cases, the resultant change of form does not strike the eye.
Great as are the differences between the rates of growth in different
parts of an organism, the marvel is that the ratios between them are
so nicely balanced as they actually are, and so capable, accordingly,
of keeping for long periods of time the form of the growing organism
all but unchanged. There is the nicest possible balance of forces and
resistances in every part of the complex body; and when this normal
equilibrium is disturbed, then we get abnormal growth, in the shape of
tumours, exostoses, and malformations of every kind. {61}


_The rate of growth in Man._

Man will serve us as well as another organism for our first
illustrations of rate of growth; and we cannot do better than go for
our first data concerning him to Quetelet’s _Anthropométrie_[94], an
epoch-making book for the biologist. For not only is it packed with
information, some of it still unsurpassed, in regard to human growth
and form, but it also merits our highest admiration as the first great
essay in scientific statistics, and the first work in which organic
variation was discussed from the point of view of the mathematical
theory of probabilities.

[Illustration: Fig. 3. Curve of Growth in Man, from birth to 20 yrs
(♂); from Quetelet’s Belgian data. The upper curve of stature from
Bowditch’s Boston data.]

If the child be some 20 inches, or say 50 cm. tall at birth, and
the man some six feet high, or say 180 cm., at twenty, we may say
that his _average_ rate of growth has been (180 − 50)/20 cm., or 6·5
centimetres per annum. But we know very well that this is {62} but
a very rough preliminary statement, and that the boy grew quickly
during some, and slowly during other, of his twenty years. It becomes
necessary therefore to study the phenomenon of growth in successive
small portions; to study, that is to say, the successive lengths, or
the successive small differences, or increments, of length (or of
weight, etc.), attained in successive short increments of time. This
we do in the first instance in the usual way, by the “graphic method”
of plotting length against time, and so constructing our “curve of
growth.” Our curve of growth, whether of weight or length (Fig. 3), has
always a certain characteristic form, or characteristic _curvature_.
This is our immediate proof of the fact that the _rate of growth_
changes as time goes on; for had it not been so, had an equal increment
of length been added in each equal interval of time, our “curve” would
have appeared as a straight line. Such as it is, it tells us not
only that the rate of growth tends to alter, but that it alters in a
definite and orderly way; for, subject to various minor interruptions,
due to secondary causes, our curves of growth are, on the whole,
“smooth” curves.

The curve of growth for length or stature in man indicates a rapid
increase at the outset, that is to say during the quick growth of
babyhood; a long period of slower, but still rapid and almost steady
growth in early boyhood; as a rule a marked quickening soon after the
boy is in his teens, when he comes to “the growing age”; and finally
a gradual arrest of growth as the boy “comes to his full height,” and
reaches manhood.

If we carried the curve further, we should see a very curious thing.
We should see that a man’s full stature endures but for a spell;
long before fifty[95] it has begun to abate, by sixty it is notably
lessened, in extreme old age the old man’s frame is shrunken and
it is but a memory that “he once was tall.” We have already seen,
and here we see again, that growth may have a “negative value.” The
phenomenon of negative growth in old age extends to weight also, and is
evidently largely chemical in origin: the organism can no longer add
new material to its fabric fast enough to keep pace with the wastage of
time. Our curve {63} of growth is in fact a diagram of activity, or
“time-energy” diagram[96]. As the organism grows it is absorbing energy
beyond its daily needs, and accumulating it at a rate depicted in our

 _Stature, weight, and span of outstretched arms._

 (_After Quetelet_, _pp._ 193, 346.)

     Stature in metres         Weight in kgm.        Span of     % ratio
                                                       arms,   of stature
 Age  Male     Female  % F/M   Male   Female  % F/M   male      to span
  0   0·500    0·494    98·8    3·2     2·9    90·7   0·496      100·8
  1   0·698    0·690    98·8    9·4     8·8    93·6   0·695      100·4
  2   0·791    0·781    98·7   11·3    10·7    94·7   0·789      100·3
  3   0·864    0·854    98·8   12·4    11·8    95·2   0·863      100·1
  4   0·927    0·915    98·7   14·2    13·0    91·5   0·927      100·0
  5   0·987    0·974    98·7   15·8    14·4    91·1   0·988       99·9
  6   1·046    1·031    98·5   17·2    16·0    93·0   1·048       99·8
  7   1·104    1·087    98·4   19·1    17·5    91·6   1·107       99·7
  8   1·162    1·142    98·2   20·8    19·1    91·8   1·166       99·6
  9   1·218    1·196    98·2   22·6    21·4    94·7   1·224       99·5
 10   1·273    1·249    98·1   24·5    23·5    95·9   1·281       99·4
 11   1·325    1·301    98·2   27·1    25·6    94·5   1·335       99·2
 12   1·375    1·352    98·3   29·8    29·8   100·0   1·388       99·1
 13   1·423    1·400    98·4   34·4    32·9    95·6   1·438       98·9
 14   1·469    1·446    98·4   38·8    36·7    94·6   1·489       98·7
 15   1·513    1·488    98·3   43·6    40·4    92·7   1·538       99·4
 16   1·554    1·521    97·8   49·7    43·6    87·7   1·584       98·1
 17   1·594    1·546    97·0   52·8    47·3    89·6   1·630       97·9
 18   1·630    1·563    95·9   57·8    49·0    84·8   1·670       97·6
 19   1·655    1·570    94·9   58·0    51·6    89·0   1·705       97·1
 20   1·669    1·574    94·3   60·1    52·3    87·0   1·728       96·6
 25   1·682    1·578    93·8   62·9    53·3    84·7   1·731       97·2
 30   1·686    1·580    93·7   63·7    54·3    85·3   1·766       95·5
 40   1·686    1·580    93·7   63·7    55·2    86·7   1·766       95·5
 50   1·686    1·580    93·7   63·5    56·2    88·4     —           —
 60   1·676    1·571    93·7   61·9    54·3    87·7     —           —
 70   1·660    1·556    93·7   59·5    51·5    86·5     —           —
 80   1·636    1·534    93·8   57·8    49·4    85·5     —           —
 90   1·610    1·510    93·8   57·8    49·3    85·3     —           —

curve; but the time comes when it accumulates no longer, and at last
it is constrained to draw upon its dwindling store. But in part, the
slow decline in stature is an expression of an unequal contest between
our bodily powers and the unchanging force of gravity, {64} which
draws us down when we would fain rise up[97]. For against gravity we
fight all our days, in every movement of our limbs, in every beat of
our hearts; it is the indomitable force that defeats us in the end,
that lays us on our deathbed, that lowers us to the grave[98].

Side by side with the curve which represents growth in length, or
stature, our diagram shows the curve of weight[99]. That this curve
is of a very different shape from the former one, is accounted for in
the main (though not wholly) by the fact which we have already dealt
with, that, whatever be the law of increment in a linear dimension,
the law of increase in volume, and therefore in weight, will be that
these latter magnitudes tend to vary as the cubes of the linear
dimensions. This however does not account for the change of direction,
or “point of inflection” which we observe in the curve of weight at
about one or two years old, nor for certain other differences between
our two curves which the scale of our diagram does not yet make clear.
These differences are due to the fact that the form of the child is
altering with growth, that other linear dimensions are varying somewhat
differently from length or stature, and that consequently the growth in
bulk or weight is following a more complicated law.

Our curve of growth, whether for weight or length, is a direct picture
of velocity, for it represents, as a connected series, the successive
epochs of time at which successive weights or lengths are attained.
But, as we have already in part seen, a great part of the interest
of our curve lies in the fact that we can see from it, not only that
length (or some other magnitude) is changing, but that the _rate of
change_ of magnitude, or rate of growth, is itself changing. We have,
in short, to study the phenomenon of _acceleration_: we have begun by
studying a velocity, or rate of {65} change of magnitude; we must
now study an acceleration, or rate of change of velocity. The rate,
or velocity, of growth is measured by the _slope_ of the curve; where
the curve is steep, it means that growth is rapid, and when growth
ceases the curve appears as a horizontal line. If we can find a means,
then, of representing at successive epochs the corresponding slope,
or steepness, of the curve, we shall have obtained a picture of the
rate of change of velocity, or the acceleration of growth. The measure
of the steepness of a curve is given by the tangent to the curve, or
we may estimate it by taking for equal intervals of time (strictly
speaking, for each infinitesimal interval of time) the actual increment
added during that interval of time: and in practice this simply amounts
to taking the successive _differences_ between the values of length (or
of weight) for the successive ages which we have begun by studying. If
we then plot these successive _differences_ against time, we obtain
a curve each point upon which represents a velocity, and the whole
curve indicates the rate of change of velocity, and we call it an
acceleration-curve. It contains, in truth, nothing whatsoever that was
not implicit in our former curve; but it makes clear to our eye, and
brings within the reach of further investigation, phenomena that were
hard to see in the other mode of representation.

The acceleration-curve of height, which we here illustrate, in Fig. 4,
is very different in form from the curve of growth which we have just
been looking at; and it happens that, in this case, there is a very
marked difference between the curve which we obtain from Quetelet’s
data of growth in height and that which we may draw from any other
series of observations known to me from British, French, American or
German writers. It begins (as will be seen from our next table) at
a very high level, such as it never afterwards attains; and still
stands too high, during the first three or four years of life, to be
represented on the scale of the accompanying diagram. From these high
velocities it falls away, on the whole, until the age when growth
itself ceases, and when the rate of growth, accordingly, has, for
some years together, the constant value of _nil_; but the rate of
fall, or rate of change of velocity, is subject to several changes or
interruptions. During the first three or four years of life the fall is
continuous and rapid, {66} but it is somewhat arrested for a while in
childhood, from about five years old to eight. According to Quetelet’s
data, there is another slight interruption in the falling rate between
the ages of about fourteen and sixteen; but in place of this almost
insignificant interruption, the English and other statistics indicate a
sudden

[Illustration: Fig. 4. Mean annual increments of stature (♂), Belgian
and American.]

and very marked acceleration of growth beginning at about twelve
years of age, and lasting for three or four years; when this period
of acceleration is over, the rate begins to fall again, and does
so with great rapidity. We do not know how far the absence of this
striking feature in the Belgian curve is due to the imperfections of
Quetelet’s data, or whether it is a real and significant feature in the
small-statured race which he investigated.

 _Annual Increment of Stature (in cm.) from Belgian and American
 Statistics._

 D: Belgian (Quetelet, p. 344)
 E: Paris* (Variot et Chaumet, p. 55)
 F: Toronto† (Boas, p. 1547)
 G: Worcester‡, Mass. (Boas, p. 1548)
 H: Ann. increment
 I: Increment
 J: Boys
 K: Girls
 V: Variability of do.

     ────[D]───  ───────[E]───────  ──────[F]──────  ────────[G]───────
     Height        Height     [I]   Height [V]       [H]       [H]
 Age (Boys) [H]  [J]   [K]  [J] [K] (Boys) (6)  [H] ([J]) [V] ([K]) [V]
  0   50·0   —     —     —    —   —    —     —    —    —    —    —    —
  1   69·8 19·8  74·2  73·6   —   —    —     —    —    —    —    —    —
  2   79·1  9·3  82·7  81·8 8·5 8·2    —     —    —    —    —    —    —
  3   86·4  7·3  89·1  88·4 6·4 6·6    —     —    —    —    —    —    —
  4   92·7  6·3  96·8  95·8 7·7 7·4    —     —    —    —    —    —    —
  5   98·7  6·0 103·3 101·9 6·5 6·1 105·90 4·40   —    —    —    —    —
  6  104·0  5·9 109·9 108·9 6·6 7·0 111·58 4·62 5·68 6·55 1·57 5·75 0·88
  7  110·4  5·8 114·4 113·8 4·5 4·9 116·83 4·93 5·25 5·70 0·68 5·90 0·98
  8  116·2  5·8 119·7 119·5 5·3 5·7 122·04 5·34 5·21 5·37 0·86 5·70 1·10
  9  121·8  5·6 125·0 124·7 5·3 4·8 126·91 5·49 4·87 4·89 0·96 5·50 0·97
 10  127·3  5·5 130·3 129·5 5·3 5·2 131·78 5·75 4·87 5·10 1·03 5·97 1·23
 11  132·5  5·2 133·6 134·4 3·3 4·9 136·20 6·19 4·42 5·02 0·88 6·17 1·85
 12  137·5  5·0 137·6 141·5 4·0 7·1 140·74 6·66 4·54 4·99 1·26 6·98 1·89
 13  142·3  4·8 145·1 148·6 7·5 7·1 146·00 7·54 5·26 5·91 1·86 6·71 2·06
 14  146·9  4·6 153·8 152·9 8·7 4·3 152·39 8·49 6·39 7·88 2·39 5·44 2·89
 15  151·3  4·4 159·6 154·2 5·8 1·3 159·72 8·78 7·33 6·23 2·91 5·34 2·71
 16  155·4  4·1   —     —    —    — 164·90 7·73 5·18 5·64 3·46   —    —
 17  159·4  4·0   —     —    —    — 168·91 7·22 4·01   —    —    —    —
 18  163·0  3·6   —     —    —    — 171·07 6·74 2·16   —    —    —    —
 19  165·5  2·5   —     —    —    —   —     —     —    —    —    —    —
 20  167·0  1·5   —     —    —    —   —     —     —    —    —    —    —

 * Ages from 1–2, 2–3, etc.

 † The epochs are, in this table, 5·5, 6·5, years, etc.

 ‡ Direct observations on actual, or individualised,
 increase of stature from year to year: between the ages of
 5–6, 6–7, etc.

Even apart from these data of Quetelet’s (which seem to constitute
an extreme case), it is evident that there are very {68} marked
differences between different races, as we shall presently see there
are between the two sexes, in regard to the epochs of acceleration of
growth, in other words, in the “phase” of the curve.

It is evident that, if we pleased, we might represent the _rate of
change of acceleration_ on yet another curve, by constructing a table
of “second differences”; this would bring out certain very interesting
phenomena, which here however we must not stay to discuss.

 _Annual Increment of Weight in Man_ (_kgm._).

 (After Quetelet, _Anthropométrie_, p. 346*.)

              Increment
  Age      Male    Female
  0–1       5·9     5·6
  1–2       2·0     2·4
  2–3       1·5     1·4
  3–4       1·5     1·5
  4–5       1·9     1·4
  5–6       1·9     1·4
  6–7       1·9     1·1
  7–8       1·9     1·2
  8–9       1·9     2·0
  9–10      1·7     2·1
 10–11      1·8     2·4
 11–12      2·0     3·5
 12–13      4·1     3·5
 13–14      4·0     3·8
 14–15      4·1     3·7
 15–16      4·2     3·5
 16–17      4·3     3·3
 17–18      4·2     3·0
 18–19      3·7     2·3
 19–20      1·9     1·1
 20–21      1·7     1·1
 21–22      1·7     0·5
 22–23      1·6     0·4
 23–24      0·9    −0·2
 24–25      0·8    −0·2

 * The values given in this table are not in precise accord
 with those of the Table on p. 63. The latter represent
 Quetelet’s results arrived at in 1835; the former are the
 means of his determinations in 1835–40.

The acceleration-curve for man’s weight (Fig. 5), whether we draw
it from Quetelet’s data, or from the British, American and other
statistics of later writers, is on the whole similar to that which
we deduce from the statistics of these latter writers in regard to
height or stature; that is to say, it is not a curve which continually
descends, but it indicates a rate of growth which is subject to
important fluctuations at certain epochs of life. We see that it begins
at a high level, and falls continuously and rapidly[100] {69} during
the first two or three years of life. After a slight recovery, it runs
nearly level during boyhood from about five to twelve years old; it
then rapidly rises, in the “growing period” of the early teens, and
slowly and steadily falls from about the age of sixteen onwards. It
does not reach the base-line till the man is about seven or eight and
twenty, for normal increase of weight continues during the years when
the man is “filling out,” long after growth in height has ceased;
but at last, somewhere about thirty, the velocity reaches zero, and
even falls below it, for then the man usually begins to lose weight a
little. The subsequent slow changes in this acceleration-curve we need
not stop to deal with.

[Illustration: Fig. 5. Mean annual increments of weight, in man and
woman; from Quetelet’s data.]

In the same diagram (Fig. 5) I have set forth the acceleration-curves
in respect of increment of weight for both man and woman, according to
Quetelet. That growth in boyhood and growth in girlhood follow a very
different course is a matter of common knowledge; but if we simply
plot the ordinary curve of growth, or velocity-curve, the difference,
on the small scale of our diagrams, {70} is not very apparent. It is
admirably brought out, however, in the acceleration-curves. Here we see
that, after infancy, say from three years old to eight, the velocity in
the girl is steady, just as in the boy, but it stands on a lower level
in her case than in his: the little maid at this age is growing slower
than the boy. But very soon, and while his acceleration-curve is still
represented by a straight line, hers has begun to ascend, and until
the girl is about thirteen or fourteen it continues to ascend rapidly.
After that age, as after sixteen or seventeen in the boy’s case, it
begins to descend. In short, throughout all this period, it is a very
_similar_ curve in the two sexes; but it has its notable differences,
in amplitude and especially in _phase_. Last of all, we may notice that
while the acceleration-curve falls to a negative value in the male
about or even a little before the age of thirty years, this does not
happen among women. They continue to grow in weight, though slowly,
till very much later in life; until there comes a final period, in both
sexes alike, during which weight, and height and strength all alike
diminish.

 From certain corrected, or “typical” values, given for American
 children by Boas and Wissler (_l.c._ p. 42), we obtain the following
 still clearer comparison of the annual increments of _stature_ in boys
 and girls: the typical stature at the commencement of the period, i.e.
 at the age of eleven, being 135·1 cm. and 136·9 cm. for the boys and
 girls respectively, and the annual increments being as follows:

 Age           12    13    14    15    16    17    18    19    20
 Boys (cm.)    4·1   6·3   8·7   7·9   5·2   3·2   1·9   0·9   0·3
 Girls (cm.)   7·5   7·0   4·6   2·1   0·9   0·4   0·1   0·0   0·0
 Difference   −3·4  −0·7   4·1   5·8   4·3   2·8   1·8   0·9   0·3

The result of these differences (which are essentially
_phase_-differences) between the two sexes in regard to the velocity
of growth and to the rate of change of that velocity, is to cause the
_ratio_ between the weights of the two sexes to fluctuate in a somewhat
complicated manner. At birth the baby-girl weighs on the average nearly
10 per cent. less than the boy. Till about two years old she tends to
gain upon him, but she then loses again until the age of about five;
from five she gains for a few years somewhat rapidly, and the girl of
ten to twelve is only some 3 per cent. less in weight than the boy. The
boy in his teens gains {71} steadily, and the young woman of twenty
is nearly 15 per cent. lighter than the man. This ratio of difference
again slowly diminishes, and between fifty and sixty stands at about
12 per cent., or not far from the mean for all ages; but once more as
old age advances, the difference tends, though very slowly, to increase
(Fig. 6).

[Illustration: Fig. 6. Percentage ratio, throughout life, of female
weight to male; from Quetelet’s data.]

While careful observations on the rate of growth in other animals are
somewhat scanty, they tend to show so far as they go that the general
features of the phenomenon are always much the same. Whether the animal
be long-lived, as man or the elephant, or short-lived, like horse or
dog, it passes through the same phases of growth[101]. In all cases
growth begins slowly; it attains a maximum velocity early in its
course, and afterwards slows down (subject to temporary accelerations)
towards a point where growth ceases altogether. But especially in the
cold-blooded animals, such as fishes, the slowing-down period is very
greatly protracted, and the size of the creature would seem never
actually to reach, but only to approach asymptotically, to a maximal
limit.

The size ultimately attained is a resultant of the rate, and of {72}
the duration, of growth. It is in the main true, as Minot has said,
that the rabbit is bigger than the guinea-pig because he grows the
faster; but that man is bigger than the rabbit because he goes on
growing for a longer time.

――――――――――

In ordinary physical investigations dealing with velocities, as for
instance with the course of a projectile, we pass at once from the
study of acceleration to that of momentum and so to that of force; for
change of momentum, which is proportional to force, is the product of
the mass of a body into its acceleration or change of velocity. But we
can take no such easy road of kinematical investigation in this case.
The “velocity” of growth is a very different thing from the “velocity”
of the projectile. The forces at work in our case are not susceptible
of direct and easy treatment; they are too varied in their nature and
too indirect in their action for us to be justified in equating them
directly with the mass of the growing structure.

 It was apparently from a feeling that the velocity of growth ought
 in some way to be equated with the mass of the growing structure
 that Minot[102] introduced a curious, and (as it seems to me) an
 unhappy method of representing growth, in the form of what he called
 “percentage-curves”; a method which has been followed by a number of
 other writers and experimenters. Minot’s method was to deal, not with
 the actual increments added in successive periods, such as years or
 days, but with these increments represented as _percentages_ of the
 amount which had been reached at the end of the former period. For
 instance, taking Quetelet’s values for the height in centimetres of a
 male infant from birth to four years old, as follows:

 Years   0     1     2     3     4
 cm.    50·0  69·8  79·1  86·4  92·7

 Minot would state the percentage growth in each of the four annual
 periods at 39·6, 13·3, 9·6 and 7·3 per cent. respectively.

 Now when we plot actual length against time, we have a perfectly
 definite thing. When we differentiate this _L_/_T_, we have
 _dL_/_dT_, which is (of course) velocity; and from this, by a second
 differentiation, we obtain _d_^2 _L_/_dT_^2, that is to say, the
 acceleration. {73}

 But when you take percentages of _y_, you are determining _dy_/_y_,
 and when you plot this against _dx_, you have

 (_dy_/_y_)/_dx_, or _dy_/(_y_ ⋅ _dx_), or (1/_y_) ⋅ (_dy_/_dx_),

 that is to say, you are multiplying the thing you wish to represent
 by another quantity which is itself continually varying; and the
 result is that you are dealing with something very much less easily
 grasped by the mind than the original factors. Professor Minot is, of
 course, dealing with a perfectly legitimate function of _x_ and _y_;
 and his method is practically tantamount to plotting log _y_ against
 _x_, that is to say, the logarithm of the increment against the time.
 This could only be defended and justified if it led to some simple
 result, for instance if it gave us a straight line, or some other
 simpler curve than our usual curves of growth. As a matter of fact, it
 is manifest that it does nothing of the kind.


_Pre-natal and post-natal growth._

In the acceleration-curves which we have shown above (Figs. 2, 3), it
will be seen that the curve starts at a considerable interval from the
actual date of birth; for the first two increments which we can as yet
compare with one another are those attained during the first and second
complete years of life. Now we can in many cases “interpolate” with
safety _between_ known points upon a curve, but it is very much less
safe, and is not very often justifiable (at least until we understand
the physical principle involved, and its mathematical expression), to
“extrapolate” beyond the limits of our observations. In short, we do
not yet know whether our curve continued to ascend as we go backwards
to the date of birth, or whether it may not have changed its direction,
and descended, perhaps, to zero-value. In regard to length, or stature,
however, we can obtain the requisite information from certain tables
of Rüssow’s[103], who gives the stature of the infant month by month
during the first year of its life, as follows:

 Age in months          0    1   2   3   4   5   6   7    8     9   10    11     12
 Length in cm.        (50)  54  58  60  62  64  65  66   67·5   68  69    70·5   72
 [Differences (in cm.)    4   4   2   2   2   1   1   1·5    ·5   1   1·5    1·5]

If we multiply these _monthly_ differences, or mean monthly velocities,
by 12, to bring them into a form comparable with the {74} _annual_
velocities already represented on our acceleration-curves, we shall see
that the one series of observations joins on very well with the other;
and in short we see at once that our acceleration-curve rises steadily
and rapidly as we pass back towards the date of birth.

[Illustration: Fig. 7. Curve of growth (in length or stature) of child,
before and after birth. (From His and Rüssow’s data.)]

But birth itself, in the case of a viviparous animal, is but an
unimportant epoch in the history of growth. It is an epoch whose
relative date varies according to the particular animal: the foal and
the lamb are born relatively later, that is to say when development
has advanced much farther, than in the case of man; the kitten and the
puppy are born earlier and therefore more helpless than we are; and the
mouse comes into the world still earlier and more inchoate, so much so
that even the little marsupial is scarcely more unformed and embryonic.
In all these cases alike, we must, in order to study the curve of
growth in its entirety, take full account of prenatal or intra-uterine
growth. {75}

According to His[104], the following are the mean lengths of the unborn
human embryo, from month to month.

 Months          0   1    2    3    4    5    6    7    8    9      10
                                                                  (Birth)
 Length in mm.   0  7·5   40   84  162  275  352  402  443  472   490–500
 Increment per
   month in mm.  —  7·5  32·5  44   78  113   77   50   41   29    18–28

[Illustration: Fig. 8. Mean monthly increments of length or stature of
child (in cms.).]

These data link on very well to those of Rüssow, which we have just
considered, and (though His’s measurements for the pre-natal months are
more detailed than are those of Rüssow for the first year of post-natal
life) we may draw a continuous curve of growth (Fig. 7) and curve of
acceleration of growth (Fig. 8) for the combined periods. It will at
once be seen that there is a “point of inflection” somewhere about
the fifth month of intra-uterine life[105]: up to that date growth
proceeds with a continually increasing {76} velocity; but after that
date, though growth is still rapid, its velocity tends to fall away.
There is a slight break between our two separate sets of statistics
at the date of birth, while this is the very epoch regarding which we
should particularly like to have precise and continuous information.
Undoubtedly there is a certain slight arrest of growth, or diminution
of the rate of growth, about the epoch of birth: the sudden change
in the {77} method of nutrition has its inevitable effect; but this
slight temporary set-back is immediately followed by a secondary, and
temporary, acceleration.

[Illustration: Fig. 9. Curve of pre-natal growth (length or stature) of
child; and corresponding curve of mean monthly increments (mm.).]

[Illustration: Fig. 10. Curve of growth of bamboo (from Ostwald, after
Kraus).]

It is worth our while to draw a separate curve to illustrate on a
larger scale His’s careful data for the ten months of pre-natal life
(Fig. 9). We see that this curve of growth is a beautifully regular
one, and is nearly symmetrical on either side of that point of
inflection of which we have already spoken; it is a curve for which
we might well hope to find a simple mathematical expression. The
acceleration-curve shown in Fig. 9 together with the pre-natal curve
of growth, is not taken directly from His’s recorded data, but is
derived from the tangents drawn to a smoothed curve, corresponding as
nearly as possible to the actual curve of growth: the rise to a maximal
velocity about the fifth month and the subsequent gradual fall are
now demonstrated even more clearly than before. In Fig. 10, which is
a curve of growth of the bamboo[106], we see (so far as it goes) the
same essential features, {78} the slow beginning, the rapid increase
of velocity, the point of inflection, and the subsequent slow negative
acceleration[107].


_Variability and Correlation of Growth._

The magnitudes and velocities which we are here dealing with are, of
course, mean values derived from a certain number, sometimes a large
number, of individual cases. But no statistical account of mean values
is complete unless we also take account of the _amount of variability_
among the individual cases from which the mean value is drawn. To do
this throughout would lead us into detailed investigations which lie
far beyond the scope of this elementary book; but we may very briefly
illustrate the nature of the process, in connection with the phenomena
of growth which we have just been studying.

It was in connection with these phenomena, in the case of man, that
Quetelet first conceived the statistical study of variation, on lines
which were afterwards expounded and developed by Galton, and which have
grown, in the hands of Karl Pearson and others, into the modern science
of Biometrics.

When Quetelet tells us, for instance, that the mean stature of the
ten-year old boy is 1·273 metres, this implies, according to the law of
error, or law of probabilities, that all the individual measurements
of ten-year-old boys group themselves _in an orderly way_, that is
to say according to a certain definite law, about this mean value of
1·273. When these individual measurements are grouped and plotted
as a curve, so as to show the number of individual cases at each
individual length, we obtain a characteristic curve of error or curve
of frequency; and the “spread” of this curve is a measure of the amount
of variability in this particular case. A certain mathematical measure
of this “spread,” as described in works upon statistics, is called the
Index of Variability, or Standard Deviation, and is usually denominated
by the letter σ. It is practically equivalent to a determination of
the point upon the frequency curve where it _changes its curvature_
on either side of the mean, and where, from being concave towards
the middle line, it spreads out to be convex thereto. When we divide
this {79} value by the mean, we get a figure which is independent
of any particular units, and which is called the Coefficient of
Variability. (It is usually multiplied by 100, to make it of a more
convenient amount; and we may then define this coefficient, _C_, as
= (σ/_M_) × 100.)

In regard to the growth of man, Pearson has determined this coefficient
of variability as follows: in male new-born infants, the coefficient
in regard to weight is 15·66, and in regard to stature, 6·50; in
male adults, for weight 10·83, and for stature, 3·66. The amount of
variability tends, therefore, to decrease with growth or age.

Similar determinations have been elaborated by Bowditch, by Boas and
Wissler, and by other writers for intermediate ages, especially from
about five years old to eighteen, so covering a great part of the whole
period of growth in man[108].

 _Coefficient of Variability (σ/_M_ × 100) in Man, at various ages._

 Age                           5      6      7     8      9
 Stature (Bowditch)           4·76   4·60   4·42  4·49   4·40
 Stature (Boas and Wissler)   4·15   4·14   4·22  4·37   4·33
 Weight  (Bowditch)          11·56  10·28  11·08  9·92  11·04

 Age                          10     11     12     13     14
 Stature (Bowditch)           4·55   4·70   4·90   5·47   5·79
 Stature (Boas and Wissler)   4·36   4·54   4·73   5·16   5·57
 Weight  (Bowditch)          11·60  11·76  13·72  13·60  16·80

 Age                          15     16     17     18
 Stature (Bowditch)           5·57   4·50   4·55   3·69
 Stature (Boas and Wissler)   5·50   4·69   4·27   3·94
 Weight  (Bowditch)          15·32  13·28  12·96  10·40

The result is very curious indeed. We see, from Fig. 11, that the
curve of variability is very similar to what we have called the
acceleration-curve (Fig. 4): that is to say, it descends when the rate
of growth diminishes, and rises very markedly again when, in late
boyhood, the rate of growth is temporarily accelerated. We {80} see,
in short, that the amount of _variability_ in stature or in weight is a
function of the _rate of growth_ in these magnitudes, though we are not
yet in a position to equate the terms precisely, one with another.

[Illustration: Fig. 11. Coefficients of variability of stature in Man
(♂). from Boas and Wissler’s data.]

 If we take not merely the variability of stature or weight at a given
 age, but the variability of the actual successive increments in each
 yearly period, we see that this latter coefficient of variability
 tends to increase steadily, and more and more rapidly, within the
 limits of age for which we have information; and this phenomenon is,
 in the main, easy of explanation. For a great part of the difference,
 in regard to rate of growth, between one individual and another is a
 difference of _phase_,—a difference in the epochs of acceleration and
 retardation, and finally in the epoch when growth comes to an end.
 And it follows that the variability of rate will be more and more
 marked, as we approach and reach the period when some individuals
 still continue, and others have already ceased, to grow. In the
 following epitomised table, {81} I have taken Boas’s determinations
 of variability (σ) (_op. cit._ p. 1548), converted them into the
 corresponding coefficients of variability ((σ/_M_) × 100), and then
 smoothed the resulting numbers.

 _Coefficients of Variability in Annual Increment of Stature._

 Age     7     8     9     10    11    12    13    14    15
 Boys   17·3  15·8  18·6  19·1  21·0  24·7  29·0  36·2  46·1
 Girls  17·1  17·8  19·2  22·7  25·9  29·3  37·0  44·8    —

 The greater variability of annual increment in the girls, as compared
 with the boys, is very marked, and is easily explained by the more
 rapid rate at which the girls run through the several phases of the
 phenomenon.

 Just as there is a marked difference in “phase” between the
 growth-curves of the two sexes, that is to say a difference in the
 periods when growth is rapid or the reverse, so also, within each sex,
 will there be room for similar, but individual phase-differences.
 Thus we may have children of accelerated development, who at a given
 epoch after birth are both rapidly growing and already “big for their
 age”; and others of retarded development who are comparatively small
 and have not reached the period of acceleration which, in greater
 or less degree, will come to them in turn. In other words, there
 must under such circumstances be a strong positive “coefficient of
 correlation” between stature and rate of growth, and also between
 the rate of growth in one year and the next. But it does not by any
 means follow that a child who is precociously big will continue to
 grow rapidly, and become a man or woman of exceptional stature. On the
 contrary, when in the case of the precocious or “accelerated” children
 growth has begun to slow down, the backward ones may still be growing
 rapidly, and so making up (more or less completely) to the others. In
 other words, the period of high positive correlation between stature
 and increment will tend to be followed by one of negative correlation.
 This interesting and important point, due to Boas and Wissler[109], is
 confirmed by the following table:―

 _Correlation of Stature and Increment in Boys and Girls._

 (_From Boas and Wissler._)

 Age                6      7      8      9      10     11     12     13     14     15
 Stature     (B)  112·7  115·5  123·2  127·4  133·2  136·8  142·7  147·3  155·9  162·2
             (G)  111·4  117·7  121·4  127·9  131·8  136·7  144·6  149·7  153·8  157·2
 Increment   (B)    5·7    5·3    4·9    5·1    5·0    4·7    5·9    7·5    6·2    5·2
             (G)    5·9    5·5    5·5    5·9    6·2    7·2    6·5    5·4    3·3    1·7
 Correlation (B)     ·25    ·11    ·08    ·25    ·18    ·18    ·48    ·29   −·42   −·44
             (G)     ·44    ·14    ·24    ·47    ·18   −·18   −·42   −·39   −·63    ·11

{82}

A minor, but very curious point brought out by the same investigators
is that, if instead of stature we deal with height in the sitting
posture (or, practically speaking, with length of trunk or back), then
the correlations between this height and its annual increment are
throughout negative. In other words, there would seem to be a general
tendency for the long trunks to grow slowly throughout the whole period
under investigation. It is a well-known anatomical fact that tallness
is in the main due not to length of body but to length of limb.

The whole phenomenon of variability in regard to magnitude and to rate
of increment is in the highest degree suggestive: inasmuch as it helps
further to remind and to impress upon us that specific rate of growth
is the real physiological factor which we want to get at, of which
specific magnitude, dimensions and form, and all the variations of
these, are merely the concrete and visible resultant. But the problems
of variability, though they are intimately related to the general
problem of growth, carry us very soon beyond our present limitations.


_Rate of growth in other organisms[110]._

Just as the human curve of growth has its slight but well-marked
interruptions, or variations in rate, coinciding with such epochs as
birth and puberty, so is it with other animals, and this phenomenon is
particularly striking in the case of animals which undergo a regular
metamorphosis.

In the accompanying curve of growth in weight of the mouse (Fig. 12),
based on W. Ostwald’s observations[111], we see a distinct slackening
of the rate when the mouse is about a fortnight old, at which period it
opens its eyes and very soon afterwards is weaned. At about six weeks
old there is another well-marked retardation of growth, following on a
very rapid period, and coinciding with the epoch of puberty. {83}

Fig. 13 shews the curve of growth of the silkworm[112], during its
whole larval life, up to the time of its entering the chrysalis stage.

The silkworm moults four times, at intervals of about a week, the first
moult being on the sixth or seventh day after hatching. A distinct
retardation of growth is exhibited on our curve in the case of the
third and fourth moults; while a similar retardation accompanies the
first and second moults also, but the scale of our diagram does not
render it visible. When the worm is about seven weeks old, a remarkable
process of “purgation” takes place, as a preliminary to entering on the
pupal, or chrysalis, stage; and the great and sudden loss of weight
which accompanies this process is the most marked feature of our curve.

[Illustration: Fig. 12. Growth in weight of Mouse. (After W. Ostwald.)]

The rate of growth in the tadpole[113] (Fig. 14) is likewise marked
by epochs of retardation, and finally by a sudden and drastic change.
There is a slight diminution in weight immediately after {84} the
little larva frees itself from the egg; there is a retardation of
growth about ten days later, when the external gills disappear; and
finally, the complete metamorphosis, with the loss of the tail, the
growth of the legs and the cessation of branchial respiration, is
accompanied by a loss of weight amounting to wellnigh half the weight
of the full-grown larva. {85}

[Illustration: Fig. 13. Growth in weight of Silkworm. (From Ostwald,
after Luciani and Lo Monaco.)]

While as a general rule, the better the animals be fed the quicker
they grow and the sooner they metamorphose, Barfürth has pointed
out the curious fact that a short spell of starvation, just before
metamorphosis is due, appears to hasten the change.

[Illustration: Fig. 14. Growth in weight of Tadpole. (From Ostwald,
after Schaper.)]

The negative growth, or actual loss of bulk and weight which often,
and perhaps always, accompanies metamorphosis, is well shewn in the
case of the eel[114]. The contrast of size is great between {87} the
flattened, lancet-shaped Leptocephalus larva and the little black
cylindrical, almost thread-like elver, whose magnitude is less than
that of the Leptocephalus in every dimension, even, at first, in length
(Fig. 15).

[Illustration: Fig. 15. Development of Eel; from Leptocephalus larvae
to young Elver. (From Ostwald after Joh. Schmidt.)]

[Illustration: Fig. 16. Growth in length of Spirogyra. (From Ostwald,
after Hofmeister.)]

From the higher study of the physiology of growth we learn that such
fluctuations as we have described are but special interruptions in
a process which is never actually continuous, but is perpetually
interrupted in a rhythmic manner[115]. Hofmeister shewed, for instance,
that the growth of Spirogyra proceeds by fits and starts, by periods
of activity and rest, which alternate with one another at intervals
of so many minutes (Fig. 16). And Bose, by very refined methods of
experiment, has shewn that plant-growth really proceeds by tiny and
perfectly rhythmical pulsations recurring at regular intervals of a few
seconds of time. Fig. 17 shews, according to Bose’s observations[116],
the growth of a crocus, under a very high magnification. The stalk
grows by little jerks, each with an amplitude of about ·002 mm., every
{88} twenty seconds or so, and after each little increment there is a
partial recoil.

[Illustration: Fig. 17. Pulsations of growth in Crocus, in
micro-millimetres. (After Bose.)]


_The rate of growth of various parts or organs[117]._

The differences in regard to rate of growth between various parts or
organs of the body, internal and external, can be amply illustrated in
the case of man, and also, but chiefly in regard to external form, in
some few other creatures[118]. It is obvious that there lies herein
an endless field for the mathematical study of correlation and of
variability, but with this aspect of the case we cannot deal.

In the accompanying table, I shew, from some of Vierordt’s data, the
_relative_ weights, at various ages, compared with the weight at birth,
of the entire body, of the brain, heart and liver; {89} and also the
percentage relation which each of these organs bears, at the several
ages, to the weight of the whole body.

 _Weight of Various Organs, compared with the Total Weight of the Human
 Body (male)._ (_After Vierordt, Anatom. Tabellen, pp. 38, 39._)

                                             Percentage weights compared
      Weight         Relative weights of       with total body-weights
     of body†     ─────────────────────────  ───────────────────────────
 Age  in kg.      Body  Brain  Heart  Liver   Body  Brain  Heart  Liver
  0    3·1        1      1      1      1      100   12·29   0·76   4·57
  1    9·0        2·90   2·48   1·75   2·35   100   10·50   0·46   3·70
  2   11·0        3·55   2·69   2·20   3·02   100    9·32   0·47   3·89
  3   12·5        4·03   2·91   2·75   3·42   100    8·86   0·52   3·88
  4   14·0        4·52   3·49   3·14   4·15   100    9·50   0·53   4·20
  5   15·9        5·13   3·32   3·43   3·80   100    7·94   0·51   3·39
  6   17·8        5·74   3·57   3·60   4·34   100    7·63   0·48   3·45
  7   19·7        6·35   3·54   3·95   4·86   100    6·84   0·47   3·49
  8   21·6        6·97   3·62   4·02   4·59   100    6·38   0·44   3·01
  9   23·5        7·58   3·74   4·59   4·95   100    6·06   0·46   2·99
 10   25·2        8·13   3·70   5·41   5·90   100    5·59   0·51   3·32
 11   27·0        8·71   3·57   5·97   6·14   100    5·04   0·52   3·22
 12   29·0        9·35   3·78  (4·13)  6·21   100    4·88  (0·34)  3·03
 13   33·1       10·68   3·90   6·95   7·31   100    4·49   0·50   3·13
 14   37·1       11·97   3·38   9·16   8·39   100    3·47   0·58   3·20
 15   41·2       13·29   3·91   8·45   9·22   100    3·62   0·48   3·17
 16   45·9       14·81   3·77   9·76   9·45   100    3·16   0·51   2·95
 17   49·7       16·03   3·70  10·63  10·46   100    2·84   0·51   2·98
 18   53·9       17·39   3·73  10·33  10·65   100    2·64   0·46   2·80
 19   57·6       18·58   3·67  11·42  11·61   100    2·43   0·51   2·86
 20   59·5       19·19   3·79  12·94  11·01   100    2·43   0·51   2·62
 21   61·2       19·74   3·71  12·59  11·48   100    2·31   0·49   2·66
 22   62·9       20·29   3·54  13·24  11·82   100    2·14   0·50   2·66
 23   64·5       20·81   3·66  12·42  10·79   100    2·16   0·46   2·37
 24     —          —     3·74  13·09  13·04   100      —      —      —
 25   66·2       21·36   3·76  12·74  12·84   100    2·16   0·46   2·75

 † From Quetelet.

From the first portion of the table, it will be seen that none of these
organs by any means keep pace with the body as a whole in regard to
growth in weight; in other words, there must be some other part of the
fabric, doubtless the muscles and the bones, which increase _more_
rapidly than the average increase of the body. Heart and liver both
grow nearly at the same rate, and by the {90} age of twenty-five they
have multiplied their weight at birth by about thirteen times, while
the weight of the entire body has been multiplied by about twenty-one;
but the weight of the brain has meanwhile been multiplied only
about three and a quarter times. In the next place, we see the very
remarkable phenomenon that the brain, growing rapidly till the child
is about four years old, then grows more much slowly till about eight
or nine years old, and after that time there is scarcely any further
perceptible increase. These phenomena are diagrammatically illustrated
in Fig. 18.

[Illustration: Fig. 18. Relative growth in weight (in Man) of Brain,
Heart, and whole Body.]

 Many statistics indicate a decrease of brain-weight during adult life.
 Boas[119] was inclined to attribute this apparent phenomenon to our
 statistical methods, and to hold that it could “hardly be explained in
 any other way than by assuming an increased death-rate among men with
 very large brains, at an age of about twenty years.” But Raymond Pearl
 has shewn that there is evidence of a steady and very gradual decline
 in the weight of the brain with advancing age, beginning at or before
 the twentieth year, and continuing throughout adult life[120]. {91}

The second part of the table shews the steadily decreasing weights of
the organs in question as compared with the body; the brain falling
from over 12 per cent. at birth to little over 2 per cent. at five and
twenty; the heart from ·75 to ·46 per cent.; and the liver from 4·57 to
2·75 per cent. of the whole bodily weight.

It is plain, then, that there is no simple and direct relation, holding
good _throughout life_, between the size of the body as a whole and
that of the organs we have just discussed; and the changing ratio
of magnitude is especially marked in the case of the brain, which,
as we have just seen, constitutes about one-eighth of the whole
bodily weight at birth, and but one-fiftieth at five and twenty.
The same change of ratio is observed in other animals, in equal or
even greater degree. For instance, Max Weber[121] tells us that in
the lion, at five weeks, four months, eleven months, and lastly when
full-grown, the brain-weight represents the following fractions of
the weight of the whole body, viz. 1/18, 1/80, 1/184, and 1/546. And
Kellicott has, in like manner, shewn that in the dogfish, while some
organs (e.g. rectal gland, pancreas, etc.) increase steadily and very
nearly proportionately to the body as a whole, the brain, and some
other organs also, grow in a diminishing ratio, which is capable of
representation, approximately, by a logarithmic curve[122].

But if we confine ourselves to the adult, then, as Raymond Pearl has
shewn in the case of man, the relation of brain-weight to age, to
stature, or to weight, becomes a comparatively simple one, and may be
sensibly expressed by a straight line, or simple equation.

 Thus, if _W_ be the brain-weight (in grammes), and _A_ be the age,
 or _S_ the stature, of the individual, then (in the case of Swedish
 males) the following simple equations suffice to give the required
 ratios:

 _W_ = 1487·8 − 1·94  _A_ = 915·06 + 2·86  _S_.

{ 92}

 These equations are applicable to ages between fifteen and eighty;
 if we take narrower limits, say between fifteen and fifty, we can
 get a closer agreement by using somewhat altered constants. In the
 two sexes, and in different races, these empirical constants will be
 greatly changed[123]. Donaldson has further shewn that the correlation
 between brain-weight and body-weight is very much closer in the rat
 than in man[124].

 The falling ratio of weight of brain to body with increase of size or
 age finds its parallel in comparative anatomy, in the general law that
 the larger the animal the less is the relative weight of the brain.

                            Weight of     Weight of
                          entire animal    brain
                               gms.         gms.       Ratio
 Marmoset                       335          12·5      1 : 26
 Spider monkey                 1845         126        1 : 15
 Felis minuta                  1234          23·6      1 : 56
 F. domestica                  3300          31        1 : 107
 Leopard                     27,700         164        1 : 168
 Lion                       119,500         219        1 : 546
 Elephant                 3,048,000        5430        1 : 560
 Whale (Globiocephalus)   1,000,000        2511        1 : 400

 For much information on this subject, see Dubois, “Abhängigkeit des
 Hirngewichtes von der Körpergrösse bei den Säugethieren,” _Arch. f.
 Anthropol._ XXV, 1897. Dubois has attempted, but I think with very
 doubtful success, to equate the weight of the brain with that of the
 animal. We may do this, in a very simple way, by representing the
 weight of the body as a _power_ of that of the brain; thus, in the
 above table of the weights of brain and body in four species of cat,
 if we call _W_ the weight of the body (in grammes), and _w_ the weight
 of the brain, then if in all four cases we express the ratio by _W_
 = _w_^{_n_}, we find that _n_ is almost constant, and differs little
 from 2·24 in all four species: the values being respectively, in the
 order of the table 2·36, 2·24, 2·18, and 2·17. But this evidently
 amounts to no more than an empirical rule; for we can easily see
 that it depends on the particular scale which we have used, and
 that if the weights had been taken, for instance, in kilogrammes
 or in milligrammes, the agreement or coincidence would not have
 occurred[125]. {93}

 _The Length of the Head in Man at various Ages._

 (_After Quetelet, p. 207._)

                        Men                       Women
          ──────────────────────────      ──────────────────────
   Age    Total height  Head   Ratio      Height  Head†    Ratio
               m.        m.                m.      m.
  Birth      0·500      0·111   4·50      0·494   0·111    4·45
  1 year     0·698      0·154   4·53      0·690   0·154    4·48
  2 years    0·791      0·173   4·57      0·781   0·172    4·54
  3 years    0·864      0·182   4·74      0·854   0·180    4·74
  5 years    0·987      0·192   5·14      0·974   0·188    5·18
 10 years    1·273      0·205   6·21      1·249   0·201    6·21
 15 years    1·513      0·215   7·04      1·488   0·213    6·99
 20 years    1·669      0·227   7·35      1·574   0·220    7·15
 30 years    1·686      0·228   7·39      1·580   0·221    7·15
 40 years    1·686      0·228   7·39      1·580   0·221    7·15

 † A smooth curve, very similar to this, for the growth in
 “auricular height” of the girl’s head, is given by Pearson,
 in _Biometrika_, III, p. 141. 1904.

As regards external form, very similar differences exist, which however
we must express in terms not of weight but of length. Thus the annexed
table shews the changing ratios of the vertical length of the head to
the entire stature; and while this ratio constantly diminishes, it will
be seen that the rate of change is greatest (or the coefficient of
acceleration highest) between the ages of about two and five years.

In one of Quetelet’s tables (_supra_, p. 63), he gives measurements
of the total span of the outstretched arms in man, from year to year,
compared with the vertical stature. The two measurements are so nearly
identical in actual magnitude that a direct comparison by means of
curves becomes unsatisfactory; but I have reduced Quetelet’s data to
percentages, and it will be seen from Fig. 19 that the percentage
proportion of span to height undergoes a remarkable and steady change
from birth to the age of twenty years; the man grows more rapidly in
stretch of arms than he does in height, and the span which was less
than {94} the stature at birth by about 1 per cent. exceeds it at the
age of twenty by about 4 per cent. After the age of twenty, Quetelet’s
data are few and irregular, but it is clear that the span goes on for
a long while increasing in proportion to the stature. How far the
phenomenon is due to actual growth of the arms and how far to the
increasing breadth of the chest is not yet ascertained.

[Illustration: Fig. 19. Ratio of stature in Man, to span of
outstretched arms.

(From Quetelet’s data.)]

The differences of rate of growth in different parts of the body
are very simply brought out by the following table, which shews the
relative growth of certain parts and organs of a young trout, at
intervals of a few days during the period of most rapid development. It
would not be difficult, from a picture of the little trout at any one
of these stages, to draw its approximate form at any other, by the help
of the numerical data here set forth[126]. {95}

 _Trout (Salmo fario): proportionate growth of various organs._

 (_From Jenkinson’s data._)

 Days  Total                  1st     Ventral   2nd              Breadth
 old   length   Eye   Head   dorsal     fin    dorsal  Tail-fin  of tail
  49   100     100    100     100      100     100      100       100
  63   129·9   129·4  148·3   148·6    148·5   108·4    173·8     155·9
  77   154·9   147·3  189·2  (203·6)  (193·6)  139·2    257·9     220·4
  92   173·4   179·4  220·0  (193·2)  (182·1)  154·5    307·6     272·2
 106   194·6   192·5  242·5   173·2    165·3   173·4    337·3     287·7

While it is inequality of growth in _different_ directions that we can
most easily comprehend as a phenomenon leading to gradual change of
outward form, we shall see in another chapter[127] that differences of
rate at different parts of a longitudinal system, though always in the
same direction, also lead to very notable and regular transformations.
Of this phenomenon, the difference in rate of longitudinal growth
between head and body is a simple case, and the difference which
accompanies and results from it in the bodily form of the child and the
man is easy to see. A like phenomenon has been studied in much greater
detail in the case of plants, by Sachs and certain other botanists,
after a method in use by Stephen Hales a hundred and fifty years
before[128].

On the growing root of a bean, ten narrow zones were marked off,
starting from the apex, each zone a millimetre in breadth. After
twenty-four hours’ growth, at a certain constant temperature, the
whole marked portion had grown from 10 mm. to 33 mm. in length; but
the individual zones had grown at very unequal rates, as shewn in the
annexed table[129].

 Zone    Increment
            mm.
 Apex       1·5
  2nd       5·8
  3rd       8·2
  4th       3·5
  5th       1·6
  6th       1·3
  7th       0·5
  8th       0·3
  9th       0·2
 10th       0·1

{96}

[Illustration: Fig. 20. Rate of growth in successive zones near the tip
of the bean-root.]

The several values in this table lie very nearly (as we see by Fig.
20) in a smooth curve; in other words a definite law, or principle of
continuity, connects the rates of growth at successive points along the
growing axis of the root. Moreover this curve, in its general features,
is singularly like those acceleration-curves which we have already
studied, in which we plotted the rate of growth against successive
intervals of time, as here we have plotted it against successive
spatial intervals of an actual growing structure. If we suppose for a
moment that the velocities of growth had been transverse to the axis,
instead of, as in this case, longitudinal and parallel with it, it is
obvious that these same velocities would have given us a leaf-shaped
structure, of which our curve in Fig. 20 (if drawn to a suitable scale)
would represent the actual outline on either side of the median axis;
or, again, if growth had been not confined to one plane but symmetrical
about the axis, we should have had a sort of turnip-shaped root, {97}
having the form of a surface of revolution generated by the same
curve. This then is a simple and not unimportant illustration of the
direct and easy passage from velocity to form.

 A kindred problem occurs when, instead of “zones” artificially marked
 out in a stem, we deal with the rates of growth in successive actual
 “internodes”; and an interesting variation of this problem occurs when
 we consider, not the actual growth of the internodes, but the varying
 number of leaves which they successively produce. Where we have whorls
 of leaves at each node, as in Equisetum and in many water-weeds, then
 the problem presents itself in a simple form, and in one such case,
 namely in Ceratophyllum, it has been carefully investigated by Mr
 Raymond Pearl[130].

 It is found that the mean number of leaves per whorl increases with
 each successive whorl; but that the rate of increment diminishes from
 whorl to whorl, as we ascend the axis. In other words, the increase
 in the number of leaves per whorl follows a logarithmic ratio; and if
 _y_ be the mean number of leaves per whorl, and _x_ the successional
 number of the whorl from the root or main stem upwards, then

 _y_ = _A_ + _C_ log(_x_ − _a_),

 where _A_, _C_, and _a_ are certain specific constants, varying with
 the part of the plant which we happen to be considering. On the main
 stem, the rate of change in the number of leaves per whorl is very
 slow; when we come to the small twigs, or “tertiary branches,” it has
 become rapid, as we see from the following abbreviated table:

 _Number of leaves per whorl on the tertiary branches of Ceratophyllum._

 Position of whorl       1     2     3     4     5      6
 Mean number of leaves  6·55  8·07  9·00  9·20  9·75  10·00
 Increment               —    1·52   ·93   ·20  (·55)  (·25)

We have seen that a slow but definite change of form is a common
accompaniment of increasing age, and is brought about as the simple
and natural result of an altered ratio between the rates of growth in
different dimensions: or rather by the progressive change necessarily
brought about by the difference in their accelerations. There are
many cases however in which the change is all but imperceptible to
ordinary measurement, and many others in which some one dimension is
easily measured, but others are hard to measure with corresponding
accuracy. {98} For instance, in any ordinary fish, such as a plaice or
a haddock, the length is not difficult to measure, but measurements of
breadth or depth are very much more uncertain. In cases such as these,
while it remains difficult to define the precise nature of the change
of form, it is easy to shew that such a change is taking place if we
make use of that ratio of length to weight which we have spoken of in
the preceding chapter. Assuming, as we may fairly do, that weight is
directly proportional to bulk or volume, we may express this relation
in the form _W_/_L_^3 = _k_, where _k_ is a constant, to be determined
for each particular case. (_W_ and _L_ are expressed in grammes and
centimetres, and it is usual to multiply the result by some figure,
such as 1000, so as to give the constant _k_ a value near to unity.)

 _Plaice caught in a certain area, March, 1907. Variation of k (the
 weight-length coefficient) with size. (Data taken from the Department
 of Agriculture and Fisheries’ Plaice-Report, vol._ I, _p._ 107, 1908.)

 Size in cm.  Weight in gm.  _W_/_L_^3 × 10,000  _W_/_L_^3 (smoothed)
     23            113              92·8                   —
     24            128              92·6                  94·3
     25            152              97·3                  96·1
     26            173              98·4                  97·9
     27            193              98·1                  99·0
     28            221             100·6                 100·4
     29            250             102·5                 101·2
     30            271             100·4                 101·2
     31            300             100·7                 100·4
     32            328             100·1                  99·8
     33            354              98·5                  98·8
     34            384              97·7                  98·0
     35            419              97·7                  97·6
     36            454              97·3                  96·7
     37            492              95·2                  96·3
     38            529              96·4                  95·6
     39            564              95·1                  95·0
     40            614              95·9                  95·0
     41            647              93·9                  93·8
     42            679              91·6                  92·5
     43            732              92·1                  92·5
     44            800              93·9                  94·0
     45            875              96·0                   —

{99}

Now while this _k_ may be spoken of as a “constant,” having a certain
mean value specific to each species of organism, and depending on the
form of the organism, any change to which it may be subject will be a
very delicate index of progressive changes of form; for we know that
our measurements of length are, on the average, very accurate, and
weighing is a still more delicate method of comparison than any linear
measurement.

[Illustration: Fig. 21. Changes in the weight-length ratio of Plaice,
with increasing size.]

Thus, in the case of plaice, when we deal with the mean values for a
large number of specimens, and when we are careful to deal only with
such as are caught in a particular locality and at a particular time,
we see that _k_ is by no means constant, but steadily increases to a
maximum, and afterwards slowly declines with the increasing size of the
fish (Fig. 21). To begin with, therefore, the weight is increasing more
rapidly than the cube of the length, and it follows that the length
itself is increasing less rapidly than some other linear dimension;
while in later life this condition is reversed. The maximum is reached
when the length of the fish is somewhere near to 30 cm., and it is
tempting to suppose that with this “point of inflection” there is
associated some well-marked epoch in the fish’s life. As a matter of
fact, the size of 30 cm. is approximately that at which sexual maturity
may be said to begin, or is at least near enough to suggest a close
connection between the two phenomena. The first step towards further
investigation of the {100} apparent coincidence would be to determine
the coefficient _k_ of the two sexes separately, and to discover
whether or not the point of inflection is reached (or sexual maturity
is reached) at a smaller size in the male than in the female plaice;
but the material for this investigation is at present scanty.

[Illustration: Fig. 22. Periodic annual change in the weight-length
ratio of Plaice.]

A still more curious and more unexpected result appears when we compare
the values of _k_ for the same fish at different seasons of the
year[131]. When for simplicity’s sake (as in the accompanying table and
Fig. 22) we restrict ourselves to fish of one particular size, it is
not necessary to determine the value of _k_, because a change in the
ratio of length to weight is obvious enough; but when we have small
numbers, and various sizes, to deal with, the determination of _k_ may
help us very much. It will be seen, then, that in the case of plaice
the ratio of weight to length exhibits a regular periodic variation
with the course of the seasons. {101}

 _Relation of Weight to Length in Plaice of 55 cm. long, from Month to
 Month. (Data taken from the Department of Agriculture and Fisheries
 Plaice-Report, vol._ II, _p._ 92, 1909.)

        Average weight
          in grammes     _W_/_L_^3 × 100   _W_/_L_^3 (smoothed)
 Jan.        2039              1·226              1·157
 Feb.        1735              1·043              1·080
 March       1616              0·971              0·989
 April       1585              0·953              0·967
 May         1624              0·976              0·985
 June        1707              1·026              1·005
 July        1686              1·013              1·037
 August      1783              1·072              1·042
 Sept.       1733              1·042              1·111
 Oct.        2029              1·220              1·160
 Nov.        2026              1·218              1·213
 Dec.        1998              1·201              1·215

With unchanging length, the weight and therefore the bulk of the fish
falls off from about November to March or April, and again between
May or June and November the bulk and weight are gradually restored.
The explanation is simple, and depends wholly on the process of
spawning, and on the subsequent building up again of the tissues and
the reproductive organs. It follows that, by this method, without ever
seeing a fish spawn, and without ever dissecting one to see the state
of its reproductive system, we can ascertain its spawning season, and
determine the beginning and end thereof, with great accuracy.

――――――――――

As a final illustration of the rate of growth, and of unequal growth in
various directions, I give the following table of data regarding the
ox, extending over the first three years, or nearly so, of the animal’s
life. The observed data are (1) the weight of the animal, month by
month, (2) the length of the back, from the occiput to the root of the
tail, and (3) the height to the withers. To these data I have added (1)
the ratio of length to height, (2) the coefficient (_k_) expressing
the ratio of weight to the cube of the length, and (3) a similar
coefficient (_k′_) for the height of the animal. It will be seen that,
while all these ratios tend to alter continuously, shewing that the
animal’s form is steadily altering as it approaches maturity, the
ratio between length and weight {102} changes comparatively little.
The simple ratio between length and height increases considerably, as
indeed we should expect; for we know that in all Ungulate animals the
legs are remarkably

 _Relations between the Weight and certain Linear Dimensions of the Ox.
 (Data from Przibram, after Cornevin†.)_

                                                             _k_ =        _k′_ =
 Age in   _W_, wt.  _L_, length     _H_,                  _W_/_L_^3    _W_/_H_^3
 months    in kg.     of back     height      _L_/_H_       × 10           × 10
   0        37          ·78         ·70        1·114        ·779          1·079
   1        55·3        ·94         ·77        1·221        ·665          1·210
   2        86·3       1·09         ·85        1·282        ·666          1·406
   3       121·3       1·207        ·94        1·284        ·690          1·460
   4       150·3       1·314        ·95        1·383        ·662          1·754
   5       179·3       1·404       1·040       1·350        ·649          1·600
   6       210·3       1·484       1·087       1·365        ·644          1·638
   7       247·3       1·524       1·122       1·358        ·699          1·751
   8       267·3       1·581       1·147       1·378        ·677          1·791
   9       282·8       1·621       1·162       1·395        ·664          1·802
  10       303·7       1·651       1·192       1·385        ·675          1·793
  11       327·7       1·694       1·215       1·394        ·674          1·794
  12       350·7       1·740       1·238       1·405        ·666          1·849
  13       374·7       1·765       1·254       1·407        ·682          1·900
  14       391·3       1·785       1·264       1·412        ·688          1·938
  15       405·9       1·804       1·270       1·420        ·692          1·982
  16       417·9       1·814       1·280       1·417        ·700          2·092
  17       423·9       1·832       1·290       1·420        ·689          1·974
  18       423·9       1·859       1·297       1·433        ·660          1·943
  19       427·9       1·875       1·307       1·435        ·649          1·916
  20       437·9       1·884       1·311       1·437        ·655          1·944
  21       447·9       1·893       1·321       1·433        ·661          1·943
  22       464·4       1·901       1·333       1·426        ·676          1·960
  23       480·9       1·909       1·345       1·419        ·691          1·977
  24       500·9       1·914       1·352       1·416        ·714          2·027
  25       520·9       1·919       1·359       1·412        ·737          2·075
  26       534·1       1·924       1·361       1·414        ·750          2·119
  27       547·3       1·929       1·363       1·415        ·762          2·162
  28       554·5       1·929       1·363       1·415        ·772          2·190
  29       561·7       1·929       1·363       1·415        ·782          2·218
  30       586·2       1·949       1·383       1·409        ·792          2·216
  31       610·7       1·969       1·403       1·403        ·800          2·211
  32       625·7       1·983       1·420       1·396        ·803          2·186
  33       640·7       1·997       1·437       1·390        ·805          2·159
  34       655·7       2·011       1·454       1·383        ·806          2·133

  † Cornevin, Ch., Études sur la croissance, _Arch. de
  Physiol. norm. et pathol._ (5), IV, p. 477, 1892.

{103}

long at birth in comparison with other dimensions of the body. It is
somewhat curious, however, that this ratio seems to fall off a little
in the third year of growth, the animal continuing to grow in height to
a marked degree after growth in length has become very slow. The ratio
between height and weight is by much the most variable of our three
ratios; the coefficient _W_/_H_^3 steadily increases, and is more than
twice as great at three years old as it was at birth. This illustrates
the important, but obvious fact, that the coefficient _k_ is most
variable in the case of that dimension which grows most uniformly, that
is to say most nearly in proportion to the general bulk of the animal.
In short, the successive values of _k_, as determined (at successive
epochs) for one dimension, are a measure of the _variability_ of the
others.

――――――――――

From the whole of the foregoing discussion we see that a certain
definite rate of growth is a characteristic or specific phenomenon,
deep-seated in the physiology of the organism; and that a very large
part of the specific morphology of the organism depends upon the
fact that there is not only an average, or aggregate, rate of growth
common to the whole, but also a variation of rate in different parts
of the organism, tending towards a specific rate characteristic of
each different part or organ. The smallest change in the relative
magnitudes of these partial or localised velocities of growth will be
soon manifested in more and more striking differences of form. This
is as much as to say that the time-element, which is implicit in the
idea of growth, can never (or very seldom) be wholly neglected in our
consideration of form[132]. It is scarcely necessary to enlarge here
upon our statement, for not only is the truth of it self-evident,
but it will find illustration again and again throughout this book.
Nevertheless, let us go out of our way for a moment to consider it in
reference to a particular case, and to enquire whether it helps to
remove any of the difficulties which that case appears to present.
{104}

[Illustration: Fig. 23. Variability of length of tail-forceps in a
sample of Earwigs. (After Bateson, _P. Z. S._ 1892, p. 588.)]

In a very well-known paper, Bateson shewed that, among a large number
of earwigs, collected in a particular locality, the males fell into two
groups, characterised by large or by small tail-forceps, with very few
instances of intermediate magnitude. This distribution into two groups,
according to magnitude, is illustrated in the accompanying diagram
(Fig. 23); and the phenomenon was described, and has been often quoted,
as one of dimorphism, or discontinuous variation. In this diagram the
time-element does not appear; but it is certain, and evident, that
it lies close behind. Suppose we take some organism which is born
not at all times of the year (as man is) but at some one particular
season (for instance a fish), then any random sample will consist of
individuals whose _ages_, and therefore whose _magnitudes_, will form
a discontinuous series; and by plotting these magnitudes on a curve in
relation to the number of individuals of each particular magnitude, we
obtain a curve such as that shewn in Fig. 24, the first practical use
of which is to enable us to analyse our sample into its constituent
“age-groups,” or in other words to determine approximately the age,
or ages of the fish. And if, instead of measuring the whole length of
our fish, we had confined ourselves to particular parts, such as head,
or {105} tail or fin, we should have obtained discontinuous curves
of distribution, precisely analogous to those for the entire animal.
Now we know that the differences with which Bateson was dealing were
entirely a question of magnitude, and we cannot help seeing that the
discontinuous distributions of magnitude represented by his earwigs’
tails are just such as are illustrated by the magnitudes of the older
and younger fish; we may indeed go so far as to say that the curves
are precisely comparable, for in both cases we see a characteristic
feature of detail, namely that the “spread” of the curve is greater in
the second wave than in the first, that is to say (in the case of the
fish) in the older as well as larger series. Over the reason for this
phenomenon, which is simple and all but obvious, we need not pause.

[Illustration: Fig. 24. Variability of length of body in a sample of
Plaice.]

It is evident, then, that in this case of “dimorphism,” the tails of
the one group of earwigs (which Bateson calls the “high males”) have
either grown _faster_, or have been growing for a longer period of
time, than those of the “low males.” If we could be certain that the
whole random sample of earwigs were of one and the same age, then we
should have to refer the phenomenon of dimorphism to a physiological
phenomenon, simple in kind (however remarkable and unexpected); viz.
that there were two alternative {106} values, very different from
one another, for the mean velocity of growth, and that the individual
earwigs varied around one or other of these mean values, in each case
according to the law of probabilities. But on the other hand, if we
could believe that the two groups of earwigs were _of different ages_,
then the phenomenon would be simplicity itself, and there would be no
more to be said about it[133].

――――――――――

Before we pass from the subject of the relative rate of growth of
different parts or organs, we may take brief note of the fact that
various experiments have been made to determine whether the normal
ratios are maintained under altered circumstances of nutrition, and
especially in the case of partial starvation. For instance, it has been
found possible to keep young rats alive for many weeks on a diet such
as is just sufficient to maintain life without permitting any increase
of weight. The rat of three weeks old weighs about 25 gms., and under a
normal diet should weigh at ten weeks old about 150 gms., in the male,
or 115 gms. in the female; but the underfed rat is still kept at ten
weeks old to the weight of 25 gms. Under normal diet the proportions
of the body change very considerably between the ages of three and ten
weeks. For instance the tail gets relatively longer; and even when the
_total_ growth of the rat is prevented by underfeeding, the _form_
continues to alter so that this increasing length of the tail is still
manifest[134]. {107}

_Full-fed Rats._

 Age in   Length of    Length of    Total
 weeks    body (mm.)   tail (mm.)   length   % of tail
   0         48·7        16·9        65·6      25·8
   1         64·5        29·4        93·9      31·3
   3         90·4        59·1       149·5      39·5
   6        128·0       110·0       238·0      46·2
  10        173·0       150·0       323·0      46·4

 _Underfed Rats._

  6         98·0        72·3      170·3      42·5
 10         99·6        83·9      183·5      45·7

Again as physiologists have long been aware, there is a marked
difference in the variation of weight of the different organs,
according to whether the animal’s total weight remain constant, or
be caused to diminish by actual starvation; and further striking
differences appear when the diet is not only scanty, but ill-balanced.
But these phenomena of abnormal growth, however interesting from
the physiological view, are of little practical importance to the
morphologist.


_The effect of temperature[135]._

The rates of growth which we have hitherto dealt with are based on
special investigations, conducted under particular local conditions.
For instance, Quetelet’s data, so far as we have used them to
illustrate the rate of growth in man, are drawn from his study of the
population of Belgium. But apart from that “fortuitous” individual
variation which we have already considered, it is obvious that the
normal rate of growth will be found to vary, in man and in other
animals, just as the average stature varies, in different localities,
and in different “races.” This phenomenon is a very complex one, and
is doubtless a resultant of many undefined contributory causes; but
we at least gain something in regard to it, when we discover that the
rate of growth is directly affected by temperature, and probably by
other physical {108} conditions. Réaumur was the first to shew, and
the observation was repeated by Bonnet[136], that the rate of growth or
development of the chick was dependent on temperature, being retarded
at temperatures below and somewhat accelerated at temperatures above
the normal temperature of incubation, that is to say the temperature
of the sitting hen. In the case of plants the fact that growth is
greatly affected by temperature is a matter of familiar knowledge; the
subject was first carefully studied by Alphonse De Candolle, and his
results and those of his followers are discussed in the textbooks of
Botany[137].

 That variation of temperature constitutes only one factor in
 determining the rate of growth is admirably illustrated in the case
 of the Bamboo. It has been stated (by Lock) that in Ceylon the rate
 of growth of the Bamboo is directly proportional to the humidity
 of the atmosphere: and again (by Shibata) that in Japan it is
 directly proportional to the temperature. The two statements have
 been ingeniously and satisfactorily reconciled by Blackman[138], who
 suggests that in Ceylon the temperature-conditions are all that can be
 desired, but moisture is apt to be deficient: while in Japan there is
 rain in abundance but the average temperature is somewhat too low. So
 that in the one country it is the one factor, and in the other country
 it is the other, which is _essentially_ variable.

The annexed diagram (Fig. 25), shewing the growth in length of the
roots of some common plants during an identical period of forty-eight
hours, at temperatures varying from about 14° to 37° C., is a
sufficient illustration of the phenomenon. We see that in all cases
there is a certain optimum temperature at which the rate of growth is
a maximum, and we can also see that on either side of this optimum
temperature the acceleration of growth, positive or negative, with
increase of temperature is rapid, while at a distance from the optimum
it is very slow. From the data given by Sachs and others, we see
further that this optimum temperature is very much the same for all the
common plants of our own climate which have as yet been studied; in
them it is {109} somewhere about 26° C. (or say 77° F.), or about the
temperature of a warm summer’s day; while it is found, very naturally,
to be considerably higher in the case of plants such as the melon or
the maize, which are at home in warmer regions that our own.

――――――――――

[Illustration: Fig. 25. Relation of rate of growth to temperature in
certain plants. (From Sachs’s data.)]

In a large number of physical phenomena, and in a very marked degree in
all chemical reactions, it is found that rate of action is affected,
and for the most part accelerated, by rise of temperature; and this
effect of temperature tends to follow a definite “exponential” law,
which holds good within a considerable range of temperature, but is
altered or departed from when we pass beyond certain normal limits. The
law, as laid down by van’t Hoff for chemical reactions, is, that for
an interval of _n_ degrees the velocity varies as _x_^{_n_}, _x_ being
called the “temperature coefficient”[139] for the reaction in question.
{110}

Van’t Hoff’s law, which has become a fundamental principle of chemical
mechanics, is likewise applicable (with certain qualifications)
to the phenomena of vital chemistry; and it follows that, on very
much the same lines, we may speak of the “temperature coefficient”
of growth. At the same time we must remember that there is a very
important difference (though we can scarcely call it a _fundamental_
one) between the purely physical and the physiological phenomenon, in
that in the former we study (or seek and profess to study) one thing
at a time, while in the latter we have always to do with various
factors which intersect and interfere; increase in the one case (or
change of any kind) tends to be continuous, in the other case it tends
to be brought to arrest. This is the simple meaning of that _Law of
Optimum_, laid down by Errera and by Sachs as a general principle of
physiology: namely that _every_ physiological process which varies
(like growth itself) with the amount or intensity of some external
influence, does so according to a law in which progressive increase
is followed by progressive decrease; in other words the function has
its _optimum_ condition, and its curve shews a definite _maximum_. In
the case of temperature, as Jost puts it, it has on the one hand its
accelerating effect which tends to follow van’t Hoff’s law. But it has
also another and a cumulative effect upon the organism: “Sie schädigt
oder sie ermüdet ihn, und je höher sie steigt, desto rascher macht sie
die Schädigung geltend und desto schneller schreitet sie voran.” It
would seem to be this double effect of temperature in the case of the
organism which gives us our “optimum” curves, which are the expression,
accordingly, not of a primary phenomenon, but of a more or less complex
resultant. Moreover, as Blackman and others have pointed out, our
“optimum” temperature is very ill-defined until we take account also
of the _duration_ of our experiment; for obviously, a high temperature
may lead to a short, but exhausting, spell of rapid growth, while
the slower rate manifested at a lower temperature may be the best in
the end. {111} The mile and the hundred yards are won by different
runners; and maximum rate of working, and maximum amount of work done,
are two very different things[140].

――――――――――

In the case of maize, a certain series of experiments shewed that
the growth in length of the roots varied with the temperature as
follows[141]:

 Temperature   Growth in 48 hours
     °C.              mm.
    18·0             1·1
    23·5             10·8
    26·6             29·6
    28·5             26·5
    30·2             64·6
    33·5             69·5
    36·5             20·7

Let us write our formula in the form

 _V__{(_t_+_n_)}/_V__{_t_} = _x_^{_n_}.

Then choosing two values out of the above experimental series (say the
second and the second-last), we have _t_ = 23·5, _n_ = 10, and _V_,
_V′_ = 10·8 and 69·5 respectively.

Accordingly 69·5/10·8 = 6·4 = _x_^{10}.

Therefore (log 6·4)/10, or ·0806 = log _x_.

And, _x_ = 1·204 (for an interval of 1° C.).

This first approximation might be considerably improved by taking
account of all the experimental values, two only of which we have as
yet made use of; but even as it is, we see by Fig. 26 that it is in
very fair accordance with the actual results of observation, _within
those particular limits_ of temperature to which the experiment is
confined. {112}

For an experiment on _Lupinus albus_, quoted by Asa Gray[142], I have
worked out the corresponding coefficient, but a little more carefully.
Its value I find to be 1·16, or very nearly identical with that we have
just found for the maize; and the correspondence between the calculated
curve and the actual observations is now a close one.

[Illustration: Fig. 26. Relation of rate of growth to temperature in
Maize. Observed values (after Köppen), and calculated curve.]

 Since the above paragraphs were written, new data have come to hand.
 Miss I. Leitch has made careful observations of the rate of growth
 of rootlets of the Pea; and I have attempted a further analysis of
 her principal results[143]. In Fig. 27 are shewn the mean rates of
 growth (based on about a hundred experiments) at some thirty-four
 different temperatures between 0·8° and 29·3°, each experiment lasting
 rather less than twenty-four hours. Working out the mean temperature
 coefficient for a great many combinations of these values, I obtain
 a value of 1·092 per C.°, or 2·41 for an interval of 10°, and a mean
 value for the whole series showing a rate of growth of just about 1
 mm. per hour at a temperature of 20°. My curve in Fig. 27 is drawn
 from these determinations; and it will be seen that, while it is by
 no means exact at the lower temperatures, and will of course fail
 us altogether at very high {113} temperatures, yet it serves as a
 very satisfactory guide to the relations between rate and temperature
 within the ordinary limits of healthy growth. Miss Leitch holds that
 the curve is _not_ a van’t Hoff curve; and this, in strict accuracy,
 we need not dispute. But the phenomenon seems to me to be one into
 which the van’t Hoff ratio enters largely, though doubtless combined
 with other factors which we cannot at present determine or eliminate.

[Illustration: Fig. 27. Relation of rate of growth to temperature in
rootlets of Pea. (From Miss I. Leitch’s data.)]

While the above results conform fairly well to the law of the
temperature coefficient, it is evident that the imbibition of water
plays so large a part in the process of elongation of the root or stem
that the phenomenon is rather a physical than a chemical one: and
on this account, as Blackman has remarked, the data commonly given
for the rate of growth in plants are apt to be {114} irregular,
and sometimes (we might even say) misleading[144]. The fact also,
which we have already learned, that the elongation of a shoot tends
to proceed by jerks, rather than smoothly, is another indication
that the phenomenon is not purely and simply a chemical one. We have
abundant illustrations, however, among animals, in which we may study
the temperature coefficient under circumstances where, though the
phenomenon is always complicated by osmotic factors, true metabolic
growth or chemical combination plays a larger role. Thus Mlle. Maltaux
and Professor Massart[145] have studied the rate of division in a
certain flagellate, _Chilomonas paramoecium_, and found the process
to take 29 minutes at 15° C., 12 at 25°, and only 5 minutes at 35°
C. These velocities are in the ratio of 1 : 2·4 : 5·76, which ratio
corresponds precisely to a temperature coefficient of 2·4 for each rise
of 10°, or about 1·092 for each degree centigrade.

By means of this principle we may throw light on the apparently
complicated results of many experiments. For instance, Fig. 28 is an
illustration, which has been often copied, of O. Hertwig’s work on the
effect of temperature on the rate of development of the tadpole[146].

From inspection of this diagram, we see that the time taken to attain
certain stages of development (denoted by the numbers III–VII) was as
follows, at 20° and at 10° C., respectively.

            At 20°    At 10°
 Stage III   2·0     6·5 days
 Stage  IV   2·7     8·1 days
 Stage   V   3·0    10·7 days
 Stage  VI   4·0    13·5 days
 Stage VII   5·0    16·8 days
 Total      16·7    55·6 days

That is to say, the time taken to produce a given result at {115} 10°
was (on the average) somewhere about 55·6/16·7, or 3·33, times as long
as was required at 20°.

[Illustration: Fig. 28. Diagram shewing time taken (in days), at
various temperatures (°C.), to reach certain stages of development
in the Frog: viz. I, gastrula; II, medullary plate; III, closure
of medullary folds; IV, tail-bud; V, tail and gills; VI, tail-fin;
VII, operculum beginning; VIII, do. closing; IX, first appearance of
hind-legs. (From Jenkinson, after O. Hertwig, 1898.)]

We may then put our equation again in the simple form, {116}

            _x_^{10} = 3·33.

            Or, 10 log _x_ = log 3·33 = ·52244.

  Therefore log _x_ = ·05224,

  and       _x_ = 1·128.

That is to say, between the intervals of 10° and 20° C., if it take
_m_ days, at a certain given temperature, for a certain stage of
development to be attained, it will take _m_ × 1·128^{_n_} days, when
the temperature is _n_ degrees less, for the same stage to be arrived
at.

[Illustration: Fig. 29. Calculated values, corresponding to preceding
figure.]

Fig. 29 is calculated throughout from this value; and it will be seen
that it is extremely concordant with the original diagram, as regards
all the stages of development and the whole range of temperatures
shewn: in spite of the fact that the coefficient on which it is based
was derived by an easy method from a very few points in the original
curves. {117}

Karl Peter[147], experimenting chiefly on echinoderm eggs, and also
making use of Hertwig’s experiments on young tadpoles, gives the normal
temperature coefficients for intervals of 10° C. (commonly written
_Q__{10}) as follows.

 Sphaerechinus      2·15,
 Echinus            2·13,
 Rana               2·86.

These values are not only concordant, but are evidently of the same
order of magnitude as the temperature-coefficient in ordinary chemical
reactions. Peter has also discovered the very interesting fact that
the temperature-coefficient alters with age, usually but not always
becoming smaller as age increases.

 Sphaerechinus;  Segmentation    _Q_^{10} = 2·29,
                 Later stages    _Q_^{10} = 2·03.
 Echinus;        Segmentation    _Q_^{10} = 2·30,
                 Later stages    _Q_^{10} = 2·08.
 Rana;           Segmentation    _Q_^{10} = 2·23,
                 Later stages    _Q_^{10} = 3·34.

Furthermore, the temperature coefficient varies with the temperature,
diminishing as the temperature rises,—a rule which van’t Hoff has shewn
to hold in ordinary chemical operations. Thus, in Rana the temperature
coefficient at low temperatures may be as high as 5·6: which is
just another way of saying that at low temperatures development is
exceptionally retarded.

――――――――――

In certain fish, such as plaice and haddock, I and others have found
clear evidence that the ascending curve of growth is subject to
seasonal interruptions, the rate during the winter months being always
slower than in the months of summer: it is as though we superimposed
a periodic, annual, sine-curve upon the continuous curve of growth.
And further, as growth itself grows less and less from year to year,
so will the difference between the winter and the summer rate also
grow less and less. The fluctuation in rate {118} will represent a
vibration which is gradually dying out; the amplitude of the sine-curve
will gradually diminish till it disappears; in short, our phenomenon is
simply expressed by what is known as a “damped sine-curve.” Exactly the
same thing occurs in man, though neither in his case nor in that of the
fish have we sufficient data for its complete illustration.

We can demonstrate the fact, however, in the case of man by the help
of certain very interesting measurements which have been recorded by
Daffner[148], of the height of German cadets, measured at half-yearly
intervals.

 _Growth in height of German military Cadets, in half-yearly periods._
 (_Daffner._)

                      Height in cent.          Increment in cm.
  Number          ───────────────────────    Winter    Summer
 observed   Age   October  April  October    ½-year   ½-year   Year
    12     11–12   139·4   141·0   143·3      1·6       2·3     3·9
    80     12–13   143·0   144·5   147·4      1·5       2·9     4·4
   146     13–14   147·5   149·5   152·5      2·0       3·0     5·0
   162     14–15   152·2   155·0   158·5      2·5       3·5     6·0
   162     15–16   158·5   160·8   163·8      2·3       3·0     5·3
   150     16–17   163·5   165·4   167·7      1·9       2·3     4·2
    82     17–18   167·7   168·9   170·4      1·2       1·5     2·7
    22     18–19   169·8   170·6   171·5      0·8       0·9     1·7
     6     19–20   170·7   171·1   171·5      0·4       0·4     0·8

In the accompanying diagram (Fig. 30) the half-yearly increments are
set forth, from the above table, and it will be seen that they form two
even and entirely separate series. The curve joining up each series of
points is an acceleration-curve; and the comparison of the two curves
gives a clear view of the relative rates of growth during winter and
summer, and the fluctuation which these velocities undergo during the
years in question. The dotted line represents, approximately, the
acceleration-curve in its continuous fluctuation of alternate seasonal
decrease and increase.

――――――――――

In the case of trees, the seasonal fluctuations of growth[149] admit
{119} of easy determination, and it is a point of considerable
interest to compare the phenomenon in evergreen and in deciduous trees.
I happen to have no measurements at hand with which to make this
comparison in the case of our native trees, but from a paper by Mr
Charles E. Hall[150] I have compiled certain mean values for growth in
the climate of Uruguay.

[Illustration: Fig. 30. Half-yearly increments of growth, in cadets of
various ages. (From Daffner’s data.)]

 _Mean monthly increase in Girth of Evergreen and Deciduous Trees,
 at San Jorge, Uruguay._ (_After C. E. Hall._) _Values expressed as
 percentages of total annual increase._

             Jan.  Feb.  Mar.  Apr.  May  June July  Aug.  Sept.  Oct.  Nov.  Dec.
 Evergreens   9·1   8·8   8·6   8·9  7·7   5·4  4·3   6·0   9·1   11·1  10·8  10·2

 Deciduous
   trees     20·3  14·6   9·0   2·3  0·8   0·3   0·7  1·3   3·5    9·9  16·7  21·0

The measurements taken were those of the girth of the tree, in mm.,
at three feet from the ground. The evergreens included species of
Pinus, Eucalyptus and Acacia; the deciduous trees included Quercus,
Populus, Robinia and Melia. I have merely taken mean values for these
two groups, and expressed the monthly values as percentages of the mean
annual increase. The result (as shewn by Fig. 31) is very much what we
might have expected. The growth of the deciduous trees is completely
arrested in winter-time, and the arrest is all but complete over {120}
a considerable period of time; moreover, during the warm season, the
monthly values are regularly graded (approximately in a sine-curve)
with a clear maximum (in the southern hemisphere) about the month of
December. In the evergreen trees, on the other hand, the amplitude
of the periodic wave is very much less; there is a notable amount of
growth all the year round, and, while there is a marked diminution in
rate during the coldest months, there is a tendency towards equality
over a considerable part of the warmer season. It is probable that some
of the species examined, and especially the pines, were definitely
retarded in growth, either by a temperature above their optimum, or by
deficiency of moisture, during the hottest period of the year; with
the result that the seasonal curve in our diagram has (as it were) its
region of maximum cut off.

[Illustration: Fig. 31. Periodic annual fluctuation in rate of growth
of trees (in the southern hemisphere).]

In the case of trees, the seasonal periodicity of growth is so well
marked that we are entitled to make use of the phenomenon in a converse
way, and to draw deductions as to variations in {121} climate during
past years from the record of varying rates of growth which the tree,
by the thickness of its annual rings, has preserved for us. Mr. A. E.
Douglass, of the University of Arizona, has made a careful study of
this question[151], and I have received (through Professor H. H. Turner
of Oxford) some measurements of the average width of the successive
annual rings in “yellow pine,” 500 years old, from Arizona, in which
trees the annual rings are very clearly distinguished. From the year
1391 to 1518, the mean of two trees was used; from 1519 to 1912, the
mean of five; and the means of these, and sometimes of larger numbers,
were found to be very concordant. A correction was applied by drawing
a long, nearly straight line through the curve for the whole period,
which line was assumed to represent the slowly diminishing mean width
of ring accompanying the increase of size, or age, of the tree; and the
actual growth as measured was equated with this diminishing mean. The
figures used give, accordingly, the ratio of the actual growth in each
year to the mean growth corresponding to the age or magnitude of the
tree at that epoch.

It was at once manifest that the rate of growth so determined shewed a
tendency to fluctuate in a long period of between 100 and 200 years. I
then smoothed in groups of 100 (according to Gauss’s method) the yearly
values, so that each number thus found represented the mean annual
increase during a century: that is to say, the value ascribed to the
year 1500 represented the _average annual growth_ during the whole
period between 1450 and 1550, and so on. These values give us a curve
of beautiful and surprising smoothness, from which we seem compelled
to draw the direct conclusion that the climate of Arizona, during the
last 500 years, has fluctuated with a regular periodicity of almost
precisely 150 years. Here again we should be left in doubt (so far
as these {123} observations go) whether the essential factor be a
fluctuation of temperature or an alternation of moisture and aridity;
but the character of the Arizona climate, and the known facts of recent
years, encourage the belief that the latter is the more direct and more
important factor.

[Illustration: Fig. 32. Long-period fluctuation in rate of growth of
Arizona trees (smoothed in 100-year periods), from A.D. 1390–1490 to
A.D. 1810–1910.]

It has been often remarked that our common European trees, such for
instance as the elm or the cherry, tend to have larger leaves the
further north we go; but in this case the phenomenon is to be ascribed
rather to the longer hours of daylight than to any difference of
temperature[152]. The point is a physiological one, and consequently of
little importance to us here[153]; the main point for the morphologist
is the very simple one that physical or climatic conditions have
greatly influenced the rate of growth. The case is analogous to the
direct influence of temperature in modifying the colouration of
organisms, such as certain butterflies. Now if temperature affects the
rate of growth in strict uniformity, alike in all directions and in all
parts or organs, its direct effect must be limited to the production
of local races or varieties differing from one another in actual
magnitude, as the Siberian goldfinch or bullfinch, for instance, differ
from our own. But if there be even ever so little of a discriminating
action in the enhancement of growth by temperature, such that it
accelerates the growth of one tissue or one organ more than another,
then it is evident that it must at once lead to an actual difference of
racial, or even “specific” form.

It is not to be doubted that the various factors of climate have
some such discriminating influence. The leaves of our northern trees
may themselves be an instance of it; and we have, {124} probably, a
still better instance of it in the case of Alpine plants[154], whose
general habit is dwarfed, though their floral organs suffer little or
no reduction. The subject, however, has been little investigated, and
great as its theoretic importance would be to us, we must meanwhile
leave it alone.


_Osmotic factors in growth._

The curves of growth which we have now been studying represent
phenomena which have at least a two-fold interest, morphological and
physiological. To the morphologist, who recognises that form is a
“function” of growth, the important facts are mainly these: (1) that
the rate of growth is an orderly phenomenon, with general features
common to very various organisms, while each particular organism has
its own characteristic phenomena, or “specific constants”; (2) that
rate of growth varies with temperature, that is to say with season
and with climate, and with various other physical factors, external
and internal; (3) that it varies in different parts of the body,
and according to various directions or axes; such variations being
definitely correlated with one another, and thus giving rise to the
characteristic proportions, or form, of the organism, and to the
changes in form which it undergoes in the course of its development.
But to the physiologist, the phenomenon suggests many other important
considerations, and throws much light on the very nature of growth
itself, as a manifestation of chemical and physical energies.

To be content to shew that a certain rate of growth occurs in a certain
organism under certain conditions, or to speak of the phenomenon as
a “reaction” of the living organism to its environment or to certain
stimuli, would be but an example of that “lack of particularity[155]”
in regard to the actual mechanism of physical cause and effect with
which we are apt in biology to be too easily satisfied. But in the case
of rate of growth we pass somewhat {125} beyond these limitations; for
the affinity with certain types of chemical reaction is plain, and has
been recognised by a great number of physiologists.

A large part of the phenomenon of growth, both in animals and still
more conspicuously in plants, is associated with “turgor,” that is to
say, is dependent on osmotic conditions; in other words, the velocity
of growth depends in great measure (as we have already seen, p. 113)
on the amount of water taken up into the living cells, as well as on
the actual amount of chemical metabolism performed by them[156]. Of the
chemical phenomena which result in the actual increase of protoplasm we
shall speak presently, but the rôle of water in growth deserves also a
passing word, even in our morphological enquiry.

It has been shewn by Loeb that in Cerianthus or Tubularia, for
instance, the cells in order to grow must be turgescent; and this
turgescence is only possible so long as the salt water in which the
cells lie does not overstep a certain limit of concentration. The
limit, in the case of Tubularia, is passed when the salt amounts to
about 5·4 per cent. Sea-water contains some 3·0 to 3·5 p.c. of salts;
but it is when the salinity falls much below this normal, to about 2·2
p.c., that Tubularia exhibits its maximal turgescence, and maximal
growth. A further dilution is said to act as a poison to the animal.
Loeb has also shewn[157] that in certain eggs (e.g. those of the
little fish _Fundulus_) an increasing concentration of the sea-water
(leading to a diminishing “water-content” of the egg) retards the rate
of segmentation and at length renders segmentation impossible; though
nuclear division, by the way, goes on for some time longer.

Among many other observations of the same kind, those of
Bialaszewicz[158], on the early growth of the frog, are notable. He
shews that the growth of the embryo while still _within the {126}
vitelline membrane_ depends wholly on the absorption of water; that
whether rate of growth be fast or slow (in accordance with temperature)
the quantity of water absorbed is constant; and that successive changes
of form correspond to definite quantities of water absorbed. The
solid residue, as Davenport has also shewn, may actually and notably
diminish, while the embryo organism is increasing rapidly in bulk and
weight.

On the other hand, in later stages and especially in the higher
animals, the percentage of water tends to diminish. This has been shewn
by Davenport in the frog, by Potts in the chick, and particularly by
Fehling in the case of man[159]. Fehling’s results are epitomised as
follows:

 Age in weeks          6    17   22   24   26   30   35   39
 Percentage of water  97·5 91·8 92·0 89·9 86·4 83·7 82·9 74·2

 And the following illustrate Davenport’s results for the frog:

 Age in weeks         1    2    5    7    9    14    41    84
 Percentage of water 56·3 58·5 76·7 89·3 93·1 95·0  90·2  87·5

To such phenomena of osmotic balance as the above, or in other words to
the dependence of growth on the uptake of water, Höber[160] and also
Loeb are inclined to refer the modifications of form which certain
phyllopod crustacea undergo, when the highly saline waters which
they inhabit are further concentrated, or are abnormally diluted.
Their growth, according to Schmankewitsch, is retarded by increase of
concentration, so that the individuals from the more saline waters
appear stunted and dwarfish; and they become altered or transformed
in other ways, which for the most part suggest “degeneration,” or
a failure to attain full and perfect development[161]. Important
physiological changes also ensue. The rate of multiplication is
increased, and parthenogenetic reproduction is encouraged. Male
individuals become plentiful in the less saline waters, and here
the females bring forth {127} their young alive; males disappear
altogether in the more concentrated brines, and then the females lay
eggs, which, however, only begin to develop when the salinity is
somewhat reduced.

The best-known case is the little “brine-shrimp,” _Artemia salina_,
found, in one form or another, all the world over, and first discovered
more than a century and a half ago in the salt-pans at Lymington. Among
many allied forms, one, _A. milhausenii_, inhabits the natron-lakes of
Egypt and Arabia, where, under the name of “loul,” or “Fezzan-worm,”
it is eaten by the Arabs[162]. This fact is interesting, because it
indicates (and investigation has apparently confirmed) that the tissues
of the creature are not impregnated with salt, as is the medium in
which it lives. The fluids of the body, the _milieu interne_ (as
Claude Bernard called them[163]), are no more salt than are those of
any ordinary crustacean or other animal, but contain only some 0·8 per
cent. of NaCl[164], while the _milieu externe_ may contain 10, 20, or
more per cent. of this and other salts; which is as much as to say
that the skin, or body-wall, of the creature acts as a “semi-permeable
membrane,” through which the dissolved salts are not permitted
to diffuse, though water passes through freely: until a statical
equilibrium (doubtless of a complex kind) is at length attained.

Among the structural changes which result from increased concentration
of the brine (partly during the life-time of the individual, but more
markedly during the short season which suffices for the development of
three or four, or perhaps more, successive generations), it is found
that the tail comes to bear fewer and fewer bristles, and the tail-fins
themselves tend at last to disappear; these changes corresponding
to what have been {128} described as the specific characters of _A.
milhausenii_, and of a still more extreme form, _A. köppeniana_;
while on the other hand, progressive dilution of the water tends to
precisely opposite conditions, resulting in forms which have also
been described as separate species, and even referred to a separate
genus, Callaonella, closely akin to Branchipus (Fig. 33). _Pari passu_
with these changes, there is a marked change in the relative lengths
of the fore and hind portions of the body, that is to say, of the
“cephalothorax” and abdomen: the latter growing relatively longer, the
salter the water. In other words, not only is the rate of growth of the
whole

[Illustration: Fig. 33. Brine-shrimps (Artemia), from more or less
saline water. Upper figures shew tail-segment and tail-fins; lower
figures, relative length of cephalothorax and abdomen. (After Abonyi.)]

animal lessened by the saline concentration, but the specific rates
of growth in the parts of its body are relatively changed. This latter
phenomenon lends itself to numerical statement, and Abonyi has lately
shewn that we may construct a very regular curve, by plotting the
proportionate length of the creature’s abdomen against the salinity,
or density, of the water; and the several species of Artemia, with all
their other correlated specific characters, are then found to occupy
successive, more or less well-defined, and more or less extended,
regions of the curve (Fig. 33). In short, the density of the water is
so clearly a “function” of the specific {129} character, that we may
briefly define the species _Artemia_ (_Callaonella_) _Jelskii_, for
instance, as the Artemia of density 1000–1010 (NaCl), or the typical
_A. salina_, or _principalis_, as the Artemia of density 1018–1025,
and so forth. It is a most interesting fact that these Artemiae, under
the protection of their semi-permeable skins, are capable of living
in waters not only of great density, but of very varied chemical
composition. The natron-lakes, for instance, contain large quantities
of magnesium

[Illustration: Fig. 34. Percentage ratio of length of abdomen to
cephalothorax in brine-shrimps, at various salinities. (After Abonyi.)]

sulphate; and the Artemiae continue to live equally well in
artificial solutions where this salt, or where calcium chloride, has
largely taken the place of sodium chloride in the more common habitat.
Furthermore, such waters as those of the natron-lakes are subject to
very great changes of chemical composition as concentration proceeds,
owing to the different solubilities of the constituent salts. It
appears that the forms which the Artemiae assume, and the changes which
they undergo, are identical or {130} indistinguishable, whichever of
the above salts happen to exist, or to predominate, in their saline
habitat. At the same time we still lack (so far as I know) the simple,
but crucial experiments which shall tell us whether, in solutions
of different chemical composition, it is _at equal densities_, or
at “_isotonic_” concentrations (that is to say, under conditions
where the osmotic pressure, and consequently the rate of diffusion,
is identical), that the same structural changes are produced, or
corresponding phases of equilibrium attained.

While Höber and others[165] have referred all these phenomena to
osmosis, Abonyi is inclined to believe that the viscosity, or
mechanical resistance, of the fluid also reacts upon the organism; and
other possible modes of operation have been suggested. But we may take
it for certain that the phenomenon as a whole is not a simple one;
and that it includes besides the passive phenomena of intermolecular
diffusion, some other form of activity which plays the part of a
regulatory mechanism[166].


_Growth and catalytic action._

In ordinary chemical reactions we have to deal (1) with a specific
velocity proper to the particular reaction, (2) with variations due
to temperature and other physical conditions, (3) according to van’t
Hoff’s “Law of Mass,” with variations due to the actual quantities
present of the reacting substances, and (4) in certain cases, with
variations due to the presence of “catalysing agents.” In the simpler
reactions, the law of mass involves a steady, gradual slowing-down of
the process, according to a logarithmic ratio, as the reaction proceeds
and as the initial amount of substance diminishes; a phenomenon,
however, which need not necessarily {131} occur in the organism, part
of whose energies are devoted to the continual bringing-up of fresh
supplies.

Catalytic action occurs when some substance, often in very minute
quantity, is present, and by its presence produces or accelerates an
action, by opening “a way round,” without the catalytic agent itself
being diminished or used up[167]. Here the velocity curve, though
quickened, is not necessarily altered in form, for gradually the law
of mass exerts its effect and the rate of the reaction gradually
diminishes. But in certain cases we have the very remarkable phenomenon
that a body acting as a catalyser is necessarily formed as a product,
or bye-product, of the main reaction, and in such a case as this the
reaction-velocity will tend to be steadily accelerated. Instead of
dwindling away, the reaction will continue with an ever-increasing
velocity: always subject to the reservation that limiting conditions
will in time make themselves felt, such as a failure of some necessary
ingredient, or a development of some substance which shall antagonise
or finally destroy the original reaction. Such an action as this we
have learned, from Ostwald, to describe as “autocatalysis.” Now we know
that certain products of protoplasmic metabolism, such as the enzymes,
are very powerful catalysers, and we are entitled to speak of an
autocatalytic action on the part of protoplasm itself. This catalytic
activity of protoplasm is a very important phenomenon. As Blackman
says, in the address already quoted, the botanists (or the zoologists)
“call it _growth_, attribute it to a specific power of protoplasm for
assimilation, and leave it alone as a fundamental phenomenon; but they
are much concerned as to the distribution of new growth in innumerable
specifically distinct forms.” While the chemist, on the other hand,
recognises it as a familiar phenomenon, and refers it to the same
category as his other known examples of _autocatalysis_. {132}

This very important, and perhaps even fundamental phenomenon of growth
would seem to have been first recognised by Professor Chodat of Geneva,
as we are told by his pupil Monnier[168]. “On peut bien, ainsi que
M. Chodat l’a proposé, considérer l’accroissement comme une réaction
chimique complexe, dans laquelle le catalysateur est la cellule
vivante, et les corps en présence sont l’eau, les sels, et l’acide
carbonique.”

Very soon afterwards a similar suggestion was made by Loeb[169], in
connection with the synthesis of _nuclein_ or nuclear protoplasm;
for he remarked that, as in an autocatalysed chemical reaction, the
velocity of the synthesis increases during the initial stage of
cell-division in proportion to the amount of nuclear matter already
synthesised. In other words, one of the products of the reaction, i.e.
one of the constituents of the nucleus, accelerates the production of
nuclear from cytoplasmic material.

The phenomenon of autocatalysis is by no means confined to living or
protoplasmic chemistry, but at the same time it is characteristically,
and apparently constantly, associated therewith. And it would seem
that to it we may ascribe a considerable part of the difference
between the growth of the organism and the simpler growth of the
crystal[170]: the fact, for instance, that the cell can grow in a
very low concentration of its nutritive solution, while the crystal
grows only in a supersaturated one; and the fundamental fact that the
nutritive solution need only contain the more or less raw materials of
the complex constituents of the cell, while the crystal grows only in a
solution of its own actual substance[171].

As F. F. Blackman has pointed out, the multiplication of an organism,
for instance the prodigiously rapid increase of a bacterium, {133}
which tends to double its numbers every few minutes, till (were it
not for limiting factors) its numbers would be all but incalculable
in a day[172], is a simple but most striking illustration of the
potentialities of protoplasmic catalysis; and (apart from the large
share taken by mere “turgescence” or imbibition of water) the same is
true of the growth, or cell-multiplication, of a multicellular organism
in its first stage of rapid acceleration.

It is not necessary for us to pursue this subject much further, for it
is sufficiently clear that the normal “curve of growth” of an organism,
in all its general features, very closely resembles the velocity-curve
of chemical autocatalysis. We see in it the first and most typical
phase of greater and greater acceleration; this is followed by a phase
in which limiting conditions (whose details are practically unknown)
lead to a falling off of the former acceleration; and in most cases
we come at length to a third phase, in which retardation of growth
is succeeded by actual diminution of mass. Here we may recognise the
influence of processes, or of products, which have become actually
deleterious; their deleterious influence is staved off for a while,
as the organism draws on its accumulated reserves, but they lead ere
long to the stoppage of all activity, and to the physical phenomenon
of death. But when we have once admitted that the limiting conditions
of growth, which cause a phase of retardation to follow a phase of
acceleration, are very imperfectly known, it is plain that, _ipso
facto_, we must admit that a resemblance rather than an identity
between this phenomenon and that of chemical autocatalysis is all that
we can safely assert meanwhile. Indeed, as Enriques has shewn, points
of contrast between the two phenomena are not lacking; for instance, as
the chemical reaction draws to a close, it is by the gradual attainment
of chemical equilibrium: but when organic growth draws to a close, it
is by reason of a very different kind of equilibrium, due in the main
to the gradual differentiation of the organism into parts, among whose
peculiar {134} and specialised functions that of cell-multiplication
tends to fall into abeyance[173].

It would seem to follow, as a natural consequence, from what has been
said, that we could without much difficulty reduce our curves of
growth to logarithmic formulae[174] akin to those which the physical
chemist finds applicable to his autocatalytic reactions. This has
been diligently attempted by various writers[175]; but the results,
while not destructive of the hypothesis itself, are only partially
successful. The difficulty arises mainly from the fact that, in the
life-history of an organism, we have usually to deal (as indeed we
have seen) with several recurrent periods of relative acceleration
and retardation. It is easy to find a formula which shall satisfy the
conditions during any one of these periodic phases, but it is very
difficult to frame a comprehensive formula which shall apply to the
entire period of growth, or to the whole duration of life.

But if it be meanwhile impossible to formulate or to solve in precise
mathematical terms the equation to the growth of an organism, we have
yet gone a very long way towards the solution of such problems when we
have found a “qualitative expression,” as Poincaré puts it; that is to
say, when we have gained a fair approximate knowledge of the general
curve which represents the unknown function.

――――――――――

As soon as we have touched on such matters as the chemical phenomenon
of catalysis, we are on the threshold of a subject which, if we were
able to pursue it, would soon lead us far into the special domain of
physiology; and there it would be necessary to follow it if we were
dealing with growth as a phenomenon in itself, instead of merely as a
help to our study and comprehension of form. For instance the whole
question of _diet_, of overfeeding and underfeeding, would present
itself for discussion[176]. But without attempting to open up this
large subject, we may say a {135} further passing word upon the
essential fact that certain chemical substances have the power of
accelerating or of retarding, or in some way regulating, growth, and of
so influencing directly the morphological features of the organism.

Thus lecithin has been shewn by Hatai[177], Danilewsky[178] and others
to have a remarkable power of stimulating growth in various animals;
and the so-called “auximones,” which Professor Bottomley prepares
by the action of bacteria upon peat appear to be, after a somewhat
similar fashion, potent accelerators of the growth of plants. But by
much the most interesting cases, from our point of view, are those
where a particular substance appears to exert a differential effect,
stimulating the growth of one part or organ of the body more than
another.

It has been known for a number of years that a diseased condition of
the pituitary body accompanies the phenomenon known as “acromegaly,”
in which the bones are variously enlarged or elongated, and which is
more or less exemplified in every skeleton of a “giant”; while on the
other hand, disease or extirpation of the thyroid causes an arrest of
skeletal development, and, if it take place early, the subject remains
a dwarf. These, then, are well-known illustrations of the regulation of
function by some internal glandular secretion, some enzyme or “hormone”
(as Bayliss and Starling call it), or “harmozone,” as Gley calls it in
the particular case where the function regulated is that of growth,
with its consequent influence on form.

Among other illustrations (which are plentiful) we have, for instance
the growth of the placental decidua, which Loeb has shewn to be due
to a substance given off by the corpus luteum of the ovary, giving to
the uterine tissues an abnormal capacity for growth, which in turn is
called into action by the contact of the ovum, or even of any foreign
body. And various sexual characters, such as the plumage, comb and
spurs of the cock, are believed in like manner to arise in response to
some particular internal secretion. When the source of such a secretion
is removed by castration, well-known morphological changes take place
in various animals; and when a converse change takes place, the female
acquires, in greater or less degree, characters which are {136} proper
to the male, as in certain extreme cases, known from time immemorial,
when late in life a hen assumes the plumage of the cock.

There are some very remarkable experiments by Gudernatsch, in which
he has shewn that by feeding tadpoles (whether of frogs or toads) on
thyroid gland substance, their legs may be made to grow out at any
time, days or weeks before the normal date of their appearance[179]. No
other organic food was found to produce the same effect; but since the
thyroid gland is known to contain iodine[180], Morse experimented with
this latter substance, and found that if the tadpoles were fed with
iodised amino-acids the legs developed precociously, just as when the
thyroid gland itself was used. We may take it, then, as an established
fact, whose full extent and bearings are still awaiting investigation,
that there exist substances both within and without the organism which
have a marvellous power of accelerating growth, and of doing so in such
a way as to affect not only the size but the form or proportions of the
organism.

――――――――――

If we once admit, as we are now bound to do, the existence of such
factors as these, which, by their physiological activity and apart from
any direct action of the nervous system, tend towards the acceleration
of growth and consequent modification of form, we are led into wide
fields of speculation by an easy and a legitimate pathway. Professor
Gley carries such speculations a long, long way: for he says[181] that
by these chemical influences “Toute une partie de la construction des
êtres parait s’expliquer d’une façon toute mécanique. La forteresse,
si longtemps inaccessible, du vitalisme est entamée. Car la notion
morphogénique était, suivant le mot de Dastre[182], comme ‘le dernier
réduit de la force vitale.’ ”

The physiological speculations we need not discuss: but, to take a
single example from morphology, we begin to understand the possibility,
and to comprehend the probable meaning, of the {137} all but sudden
appearance on the earth of such exaggerated and almost monstrous
forms as those of the great secondary reptiles and the great tertiary
mammals[183]. We begin to see that it is in order to account, not for
the appearance, but for the disappearance of such forms as these that
natural selection must be invoked. And we then, I think, draw near to
the conclusion that what is true of these is universally true, and that
the great function of natural selection is not to originate, but to
remove: _donec ad interitum genus id natura redegit_[184].

The world of things living, like the world of things inanimate, grows
of itself, and pursues its ceaseless course of creative evolution.
It has room, wide but not unbounded, for variety of living form and
structure, as these tend towards their seemingly endless, but yet
strictly limited, possibilities of permutation and degree: it has room
for the great and for the small, room for the weak and for the strong.
Environment and circumstance do not always make a prison, wherein
perforce the organism must either live or die; for the ways of life may
be changed, and many a refuge found, before the sentence of unfitness
is pronounced and the penalty of extermination paid. But there comes a
time when “variation,” in form, dimensions, or other qualities of the
organism, goes farther than is compatible with all the means at hand of
health and welfare for the individual and the stock; when, under the
active and creative stimulus of forces from within and from without,
the active and creative energies of growth pass the bounds of physical
and physiological equilibrium: and so reach the limits which, as again
Lucretius tells us, natural law has set between what may and what may
not be,

 “et quid quaeque queant per foedera naturai
  quid porro nequeant.”

Then, at last, we are entitled to use the customary metaphor, and to
see in natural selection an inexorable force, whose function {138} is
not to create but to destroy,—to weed, to prune, to cut down and to
cast into the fire[185].


_Regeneration, or growth and repair._

The phenomenon of regeneration, or the restoration of lost or
amputated parts, is a particular case of growth which deserves
separate consideration. As we are all aware, this property is
manifested in a high degree among invertebrates and many cold-blooded
vertebrates, diminishing as we ascend the scale, until at length, in
the warm-blooded animals, it lessens down to no more than that _vis
medicatrix_ which heals a wound. Ever since the days of Aristotle, and
especially since the experiments of Trembley, Réaumur and Spallanzani
in the middle of the eighteenth century, the physiologist and the
psychologist have alike recognised that the phenomenon is both
perplexing and important. The general phenomenon is amply discussed
elsewhere, and we need only deal with it in its immediate relation to
growth[186].

Regeneration, like growth in other cases, proceeds with a velocity
which varies according to a definite law; the rate varies with the
time, and we may study it as velocity and as acceleration.

Let us take, as an instance, Miss M. L. Durbin’s measurements of the
rate of regeneration of tadpoles’ tails: the rate being here measured
in terms, not of mass, but of length, or longitudinal increment[187].

From a number of tadpoles, whose average length was 34·2 mm., their
tails being on an average 21·2 mm. long, about half the tail {139}
(11·5 mm.) was cut off, and the amounts regenerated in successive
periods are shewn as follows:

 Days after operation               3     7    10    14    18    24    30
 (1)  Amount regenerated in mm.    1·4   3·4   4·3   5·2   5·5   6·2   6·5
 (2)  Increment during each period 1·4   2·0   0·9   0·9   0·3   0·7   0·3
 (3)  (?) Rate per day during
      each period                  0·46  0·50  0·30  0·25  0·07  0·12  0·05

The first line of numbers in this table, if plotted as a curve against
the number of days, will give us a very satisfactory view of the “curve
of growth” within the period of the observations: that is to say, of
the successive relations of length to time, or the _velocity_ of the
process. But the third line is not so satisfactory, and must not be
plotted directly as an acceleration curve. For it is evident that
the “rates” here determined do not correspond to velocities _at_ the
dates to which they are referred, but are the mean velocities over a
preceding period; and moreover the periods over which these means are
taken are here of very unequal length. But we may draw a good deal
more information from this experiment, if we begin by drawing a smooth
curve, as nearly as possible through the points corresponding to the
amounts regenerated (according to the first line of the table); and if
we then interpolate from this smooth curve the actual lengths attained,
day by day, and derive from these, by subtraction, the successive
daily increments, which are the measure of the daily mean _velocities_
(Table, p. 141). (The more accurate and strictly correct method would
be to draw successive tangents to the curve.)

In our curve of growth (Fig. 35) we cannot safely interpolate values
for the first three days, that is to say for the dates between
amputation and the first actual measurement of the regenerated part.
What goes on in these three days is very important; but we know
nothing about it, save that our curve descended to zero somewhere or
other within that period. As we have already learned, we can more or
less safely interpolate between known points, or actual observations;
but here we have no known starting-point. In short, for all that the
observations tell us, and for all that the appearance of the curve
can suggest, the curve of growth may have descended evenly to the
base-line, which it would then have reached about the end of the second
{140} day; or it may have had within the first three days a change of
direction, or “point of inflection,” and may then have sprung at once
from the base-line at zero. That is to say, there may

[Illustration: Fig. 35. Curve of regenerative growth in tadpoles’
tails. (From M. L. Durbin’s data.)]

have been an intervening “latent period,” during which no growth

[Illustration: Fig. 36. Mean daily increments, corresponding to Fig.
35.]

{141}

occurred, between the time of injury and the first measurement
of regenerative growth; or, for all we yet know, regeneration may
have begun at once, but with a velocity much less than that which
it afterwards attained. This apparently trifling difference would
correspond to a very great difference in the nature of the phenomenon,
and would lead to a very striking difference in the curve which we have
next to draw.

The curve already drawn (Fig. 35) illustrates, as we have seen, the
relation of length to time, i.e. _L_/_T_ = _V_. The second (Fig. 36)
represents the rate of change of velocity; it sets _V_ against _T_;

 _The foregoing table, extended by graphic interpolation._

          Total      Daily
 Days   increment  increment  Logs of do.
  1         —
                       —          —
  2         —
                       —          —
  3       1·40
                      ·60       1·78
  4       2·00
                      ·52       1·72
  5       2·52
                      ·45       1·65
  6       2·97
                      ·43       1·63
  7       3·40
                      ·32       1·51
  8       3·72
                      ·30       1·48
  9       4·02
                      ·28       1·45
 10       4·30
                      ·22       1·34
 11       4·52
                      ·21       1·32
 12       4·73
                      ·19       1·28
 13       4·92
                      ·18       1·26
 14       5·10
                      ·17       1·23
 15       5·27
                      ·13       1·11
 16       5·40
                      ·14       1·15
 17       5·54
                      ·13       1·11
 18       5·67
                      ·11       1·04
 19       5·78
                      ·10       1·00
 20       5·88
                      ·10       1·00
 21       5·98
                      ·09        ·95
 22       6·07
                      ·07        ·85
 23       6·14
                      ·07        ·84
 24       6·21
                      ·08        ·90
 25       6·29
                      ·06        ·78
 26       6·35
                      ·06        ·78
 27       6·41
                      ·05        ·70
 28       6·46
                      ·04        ·60
 29       6·50
                      ·03        ·48
 30       6·53

{142}

and _V_/_T_ or _L_/_T_^2, represents (as we have learned) the
_acceleration_ of growth, this being simply the “differential
coefficient,” the first derivative of the former curve.

[Illustration: Fig. 37. Logarithms of values shewn in Fig. 36.]

Now, plotting this acceleration curve from the date of the first
measurement made three days after the amputation of the tail (Fig.
36), we see that it has no point of inflection, but falls steadily,
only more and more slowly, till at last it comes down nearly to the
base-line. The velocities of growth are continually diminishing. As
regards the missing portion at the beginning of the curve, we cannot
be sure whether it bent round and came down to zero, or whether, as
in our ordinary acceleration curves of growth from birth onwards, it
started from a maximum. The former is, in this case, obviously the more
probable, but we cannot be sure.

As regards that large portion of the curve which we are acquainted
with, we see that it resembles the curve known as a rectangular
hyperbola, which is the form assumed when two variables (in this case
_V_ and _T_) vary inversely as one another. If we take the logarithms
of the velocities (as given in the table) and plot them against time
(Fig. 37), we see that they fall, approximately, into a straight line;
and if this curve be plotted on the {143} proper scale we shall find
that the angle which it makes with the base is about 25°, of which the
tangent is ·46, or in round numbers ½.

Had the angle been 45° (tan 45° = 1), the curve would have been
actually a rectangular hyperbola, with _V_ _T_ = constant. As it is,
we may assume, provisionally, that it belongs to the same family
of curves, so that _V_^{_m_} _T_^{_n_}, or _V_^{_m_/_n_} _T_, or
_V_ _T_^{_n_/_m_}, are all severally constant. In other words, the
velocity varies inversely as some power of the time, or _vice versa_.
And in this particular case, the equation _V_ _T_^2 = constant, holds
very nearly true; that is to say the velocity varies, or tends to vary,
inversely as the square of the time. If some general law akin to this
could be established as a general law, or even as a common rule, it
would be of great importance.

[Illustration: Fig. 38. Rate of regenerative growth in larger tadpoles.]

But though neither in this case nor in any other can the minute
increments of growth during the first few hours, or the first couple
of days, after injury, be directly measured, yet the most important
point is quite capable of solution. What the foregoing curve leaves
us in ignorance of, is simply whether growth starts at zero, with
zero velocity, and works up quickly to a maximum velocity from which
it afterwards gradually falls away; or whether after a latent period,
it begins, so to speak, in full force. The answer {144} to this
question-depends on whether, in the days following the first actual
measurement, we can or cannot detect a daily _increment_ in velocity,
before that velocity begins its normal course of diminution. Now
this preliminary ascent to a maximum, or point of inflection of the
curve, though not shewn in the above-quoted experiment, has been often
observed: as for instance, in another similar experiment by the author
of the former, the tadpoles being in this case of larger size (average
49·1 mm.)[188].

 Days         3     5     7    10    12    14    17    24    28    31
 Increment  0·86  2·15  3·66  5·20  5·95  6·38  7·10  7·60  8·20  8·40

Or, by graphic interpolation,

             Total        Daily
 Days      increment       do.
   1          ·23         ·23
   2          ·53         ·30
   3          ·86         ·33
   4         1·30         ·44
   5         2·00         ·70
   6         2·78         ·78
   7         3·58         ·80
   8         4·30         ·72
   9         4·90         ·60
  10         5·29         ·39
  11         5·62         ·33
  12         5·90         ·28
  13         6·13         ·23
  14         6·38         ·25
  15         6·61         ·23
  16         6·81         ·20
  17         7·00         ·19 etc.

The acceleration curve is drawn in Fig. 39.

Here we have just what we lacked in the former case, namely a visible
point of inflection in the curve about the seventh day (Figs. 38,
39), whose existence is confirmed by successive observations on the
3rd, 5th, 7th and 10th days, and which justifies to some extent our
extrapolation for the otherwise unknown period up to and ending with
the third day; but even here there is a short space near the very
beginning during which we are not quite sure of the precise slope of
the curve.

――――――――――

We have now learned that, according to these experiments, with which
many others are in substantial agreement, the rate of growth in the
regenerative process is as follows. After a very short latent period,
not yet actually proved but whose existence is highly probable, growth
commences with a velocity which very {145} rapidly increases to a
maximum. The curve quickly,—almost suddenly,—changes its direction,
as the velocity begins to fall; and the rate of fall, that is, the
negative acceleration, proceeds at a slower and slower rate, which rate
varies inversely as some power of the time, and is found in both of the
above-quoted experiments to be very approximately as 1/_T_^2. But it
is obvious that the value which we have found for the latter portion
of the curve (however closely it be conformed to) is only an empirical
value; it has only a temporary usefulness, and must in time give place
to a formula which shall represent the entire phenomenon, from start to
finish.

[Illustration: Fig. 39. Daily increment, or amount regenerated,
corresponding to Fig. 38.]

While the curve of regenerative growth is apparently different from
the curve of ordinary growth as usually drawn (and while this apparent
difference has been commented on and treated as valid by certain
writers) we are now in a position to see that it only looks different
because we are able to study it, if not from the beginning, at least
very nearly so: while an ordinary curve of growth, as it is usually
presented to us, is one which dates, not {146} from the beginning of
growth, but from the comparatively late, and unimportant, and even
fallacious epoch of birth. A complete curve of growth, starting from
zero, has the same essential characteristics as the regeneration curve.

Indeed the more we consider the phenomenon of regeneration, the more
plainly does it shew itself to us as but a particular case of the
general phenomenon of growth[189], following the same lines, obeying
the same laws, and merely started into activity by the special
stimulus, direct or indirect, caused by the infliction of a wound.
Neither more nor less than in other problems of physiology are we
called upon, in the case of regeneration, to indulge in metaphysical
speculation, or to dwell upon the beneficent purpose which seemingly
underlies this process of healing and restoration.

――――――――――

It is a very general rule, though apparently not a universal one, that
regeneration tends to fall somewhat short of a _complete_ restoration
of the lost part; a certain percentage only of the lost tissues is
restored. This fact was well known to some of those old investigators,
who, like the Abbé Trembley and like Voltaire, found a fascination in
the study of artificial injury and the regeneration which followed
it. Sir John Graham Dalyell, for instance, says, in the course of
an admirable paragraph on regeneration[190]: “The reproductive
faculty ... is not confined to one portion, but may extend over many;
and it may ensue even in relation to the regenerated portion more than
once. Nevertheless, the faculty gradually weakens, so that in general
every successive regeneration is smaller and more imperfect than the
organisation preceding it; and at length it is exhausted.”

In certain minute animals, such as the Infusoria, in which the
capacity for “regeneration” is so great that the entire animal may be
restored from the merest fragment, it becomes of great interest to
discover whether there be some definite size at which the fragment
ceases to display this power. This question has {147} been studied by
Lillie[191], who found that in Stentor, while still smaller fragments
were capable of surviving for days, the smallest portions capable
of regeneration were of a size equal to a sphere of about 80 µ in
diameter, that is to say of a volume equal to about one twenty-seventh
of the average entire animal. He arrives at the remarkable conclusion
that for this, and for all other species of animals, there is a
“minimal organisation mass,” that is to say a “minimal mass of definite
size consisting of nucleus and cytoplasm within which the organisation
of the species can just find its latent expression.” And in like
manner, Boveri[192] has shewn that the fragment of a sea-urchin’s
egg capable of growing up into a new embryo, and so discharging the
complete functions of an entire and uninjured ovum, reaches its limit
at about one-twentieth of the original egg,—other writers having found
a limit at about one-fourth. These magnitudes, small as they are,
represent objects easily visible under a low power of the microscope,
and so stand in a very different category to the minimal magnitudes in
which life itself can be manifested, and which we have discussed in
chapter II.

A number of phenomena connected with the linear rate of regeneration
are illustrated and epitomised in the accompanying diagram (Fig. 40),
which I have constructed from certain data given by Ellis in a paper
on the relation of the amount of tail _regenerated_ to the amount
_removed_, in Tadpoles. These data are summarised in the next Table.
The tadpoles were all very much of a size, about 40 mm.; the average
length of tail was very near to 26 mm., or 65 per cent. of the whole
body-length; and in four series of experiments about 10, 20, 40 and 60
per cent. of the tail were severally removed. The amount regenerated in
successive intervals of three days is shewn in our table. By plotting
the actual amounts regenerated against these three-day intervals of
time, we may interpolate values for the time taken to regenerate
definite percentage amounts, 5 per cent., 10 per cent., etc. of the
{148}

[Illustration: Fig. 40. Relation between the percentage amount of
tail removed, the percentage restored, and the time required for its
restoration. (From M. M. Ellis’s data.)]

 _The Rate of Regenerative Growth in Tadpoles’ Tails._

 (_After M. M. Ellis, J. Exp. Zool._ VII, _p._ 421, 1909.)

             Body    Tail   Amount   Per cent.  % amount regenerated in days
            length  length  removed   of tail   ────────────────────────────
 Series†     mm.     mm.      mm.     removed     3   6   9  12  15  18  32
  _O_       39·575  25·895    3·2      12·36     13  31  44  44  44  44  44
  _P_       40·21   26·13     5·28     20·20     10  29  40  44  44  44  44
  _R_       39·86   25·70    10·4      40·50      6  20  31  40  48  48  48
  _S_       40·34   26·11    14·8      56·7       0  16  33  39  45  48  48

 † Each series gives the mean of 20 experiments.

amount removed; and my diagram is constructed from the four sets of
values thus obtained, that is to say from the four sets of experiments
which differed from one another in the amount of tail amputated.
To these we have to add the general result of a fifth series of
experiments, which shewed that when as much as 75 per cent. of the
tail was cut off, no regeneration took place at all, but the animal
presently died. In our diagram, then, each {149} curve indicates the
time taken to regenerate _n_ per cent. of the amount removed. All the
curves converge towards infinity, when the amount removed (as shewn by
the ordinate) approaches 75 per cent.; and all of the curves start from
zero, for nothing is regenerated where nothing had been removed. Each
curve approximates in form to a cubic parabola.

The amount regenerated varies also with the age of the tadpole and
with other factors, such as temperature; in other words, for any given
age, or size, of tadpole and also for various specific temperatures, a
similar diagram might be constructed.

――――――――――

The power of reproducing, or regenerating, a lost limb is particularly
well developed in arthropod animals, and is sometimes accompanied by
remarkable modification of the form of the regenerated limb. A case in
point, which has attracted much attention, occurs in connection with
the claws of certain Crustacea[193].

In many Crustacea we have an asymmetry of the great claws, one being
larger than the other and also more or less different in form. For
instance, in the common lobster, one claw, the larger of the two, is
provided with a few great “crushing” teeth, while the smaller claw
has more numerous teeth, small and serrated. Though Aristotle thought
otherwise, it appears that the crushing-claw may be on the right or
left side, indifferently; whether it be on one or the other is a
problem of “chance.” It is otherwise in many other Crustacea, where the
larger and more powerful claw is always left or right, as the case may
be, according to the species: where, in other words, the “probability”
of the large or the small claw being left or being right is tantamount
to certainty[194].

The one claw is the larger because it has grown the faster; {150} it
has a higher “coefficient of growth,” and accordingly, as age advances,
the disproportion between the two claws becomes more and more evident.
Moreover, we must assume that the characteristic form of the claw is a
“function” of its magnitude; the knobbiness is a phenomenon coincident
with growth, and we never, under any circumstances, find the smaller
claw with big crushing teeth and the big claw with little serrated
ones. There are many other somewhat similar cases where size and form
are manifestly correlated, and we have already seen, to some extent,
that the phenomenon of growth is accompanied by certain ratios of
velocity that lead inevitably to changes of form. Meanwhile, then, we
must simply assume that the essential difference between the two claws
is one of magnitude, with which a certain differentiation of form is
inseparably associated.

If we amputate a claw, or if, as often happens, the crab “casts it
off,” it undergoes a process of regeneration,—it grows anew, and
evidently does so with an accelerated velocity, which acceleration
will cease when equilibrium of the parts is once more attained: the
accelerated velocity being a case in point to illustrate that _vis
revulsionis_ of Haller, to which we have already referred.

With the help of this principle, Przibram accounts for certain
curious phenomena which accompany the process of regeneration. As his
experiments and those of Morgan shew, if the large or knobby claw (_A_)
be removed, there are certain cases, e.g. the common lobster, where it
is directly regenerated. In other cases, e.g. Alpheus[195], the other
claw (_B_) assumes the size and form of that which was amputated, while
the latter regenerates itself in the form of the other and weaker one;
_A_ and _B_ have apparently changed places. In a third case, as in the
crabs, the _A_-claw regenerates itself as a small or _B_-claw, but the
_B_-claw remains for a time unaltered, though slowly and in the course
of repeated moults it later on assumes the large and heavily toothed
_A_-form.

Much has been written on this phenomenon, but in essence it is
very simple. It depends upon the respective rates of growth, upon a
ratio between the rate of regeneration and the rate of growth of the
uninjured limb: complicated a little, however, by {151} the possibility
of the uninjured limb growing all the faster for a time after the
animal has been relieved of the other. From the time of amputation,
say of _A_, _A_ begins to grow from zero, with a high “regenerative”
velocity; while _B_, starting from a definite magnitude, continues to
increase, with its normal or perhaps somewhat accelerated velocity. The
ratio between the two velocities of growth will determine whether, by a
given time, _A_ has equalled, outstripped, or still fallen short of the
magnitude of _B_.

That this is the gist of the whole problem is confirmed (if
confirmation be necessary) by certain experiments of Wilson’s. It
is known that by section of the nerve to a crab’s claw, its growth
is retarded, and as the general growth of the animal proceeds the
claw comes to appear stunted or dwarfed. Now in such a case as that
of Alpheus, we have seen that the rate of regenerative growth in an
amputated large claw fails to let it reach or overtake the magnitude of
the growing little claw: which latter, in short, now appears as the big
one. But if at the same time as we amputate the big claw we also sever
the nerve to the lesser one, we so far slow down the latter’s growth
that the other is able to make up to it, and in this case the two
claws continue to grow at approximately equal rates, or in other words
continue of coequal size.

――――――――――

The phenomenon of regeneration goes some way towards helping us to
comprehend the phenomenon of “multiplication by fission,” as it is
exemplified at least in its simpler cases in many worms and worm-like
animals. For physical reasons which we shall have to study in another
chapter, there is a natural tendency for any tube, if it have the
properties of a fluid or semi-fluid substance, to break up into
segments after it comes to a certain length; and nothing can prevent
its doing so, except the presence of some controlling force, such for
instance as may be due to the pressure of some external support, or
some superficial thickening or other intrinsic rigidity of its own
substance. If we add to this natural tendency towards fission of a
cylindrical or tubular worm, the ordinary phenomenon of regeneration,
we have all that is essentially implied in “reproduction by fission.”
And in so far {152} as the process rests upon a physical principle, or
natural tendency, we may account for its occurrence in a great variety
of animals, zoologically dissimilar; and also for its presence here
and absence there, in forms which, though materially different in a
physical sense, are zoologically speaking very closely allied.


CONCLUSION AND SUMMARY.

But the phenomena of regeneration, like all the other phenomena
of growth, soon carry us far afield, and we must draw this brief
discussion to a close.

For the main features which appear to be common to all curves of growth
we may hope to have, some day, a physical explanation. In particular we
should like to know the meaning of that point of inflection, or abrupt
change from an increasing to a decreasing velocity of growth which all
our curves, and especially our acceleration curves, demonstrate the
existence of, provided only that they include the initial stages of the
whole phenomenon: just as we should also like to have a full physical
or physiological explanation of the gradually diminishing velocity
of growth which follows, and which (though subject to temporary
interruption or abeyance) is on the whole characteristic of growth in
all cases whatsoever. In short, the characteristic form of the curve
of growth in length (or any other linear dimension) is a phenomenon
which we are at present unable to explain, but which presents us with
a definite and attractive problem for future solution. It would seem
evident that the abrupt change in velocity must be due, either to a
change in that pressure outwards from within, by which the “forces of
growth” make themselves manifest, or to a change in the resistances
against which they act, that is to say the _tension_ of the surface;
and this latter force we do not by any means limit to “surface-tension”
proper, but may extend to the development of a more or less resistant
membrane or “skin,” or even to the resistance of fibres or other
histological elements, binding the boundary layers to the parts within.
I take it that the sudden arrest of velocity is much more likely to be
due to a sudden increase of resistance than to a sudden diminution of
internal energies: in other words, I suspect that it is coincident with
some notable event of histological differentiation, such as {153} the
rapid formation of a comparatively firm skin; and that the dwindling
of velocities, or the negative acceleration, which follows, is the
resultant or composite effect of waning forces of growth on the one
hand, and increasing superficial resistance on the other. This is as
much as to say that growth, while its own energy tends to increase,
leads also, after a while, to the establishment of resistances which
check its own further increase.

Our knowledge of the whole complex phenomenon of growth is so scanty
that it may seem rash to advance even this tentative suggestion. But
yet there are one or two known facts which seem to bear upon the
question, and to indicate at least the manner in which a varying
resistance to expansion may affect the velocity of growth. For
instance, it has been shewn by Frazee[196] that electrical stimulation
of tadpoles, with small current density and low voltage, increases the
rate of regenerative growth. As just such an electrification would tend
to lower the surface-tension, and accordingly decrease the external
resistance, the experiment would seem to support, in some slight
degree, the suggestion which I have made.

 Delage[197] has lately made use of the principle of specific rate of
 growth, in considering the question of heredity itself. We know that
 the chromatin of the fertilised egg comes from the male and female
 parent alike, in equal or nearly equal shares; we know that the
 initial chromatin, so contributed, multiplies many thousand-fold, to
 supply the chromatin for every cell of the offspring’s body; and it
 has, therefore, a high “coefficient of growth.” If we admit, with Van
 Beneden and others, that the initial contributions of male and female
 chromatin continue to be transmitted to the succeeding generations of
 cells, we may then conceive these chromatins to retain each its own
 coefficient of growth; and if these differed ever so little, a gradual
 preponderance of one or other would make itself felt in time, and
 might conceivably explain the preponderating influence of one parent
 or the other upon the characters of the offspring. Indeed O. Hertwig
 is said (according to Delage’s interpretation) to have actually shewn
 that we can artificially modify the rate of growth of one or other
 chromatin, and so increase or diminish the influence of the maternal
 or paternal heredity. This theory of Delage’s has its fascination, but
 it calls for somewhat large assumptions; and in particular, it seems
 (like so many other theories relating to the chromosomes) to rest
 far too much upon material elements, rather than on the imponderable
 dynamic factors of the cell. {154}

We may summarise, as follows, the main results of the foregoing
discussion:

(1) Except in certain minute organisms and minute parts of organisms,
whose form is due to the direct action of molecular forces, we may look
upon the form of the organism as a “function of growth,” or a direct
expression of a rate of growth which varies according to its different
directions.

(2) Rate of growth is subject to definite laws, and the velocities in
different directions tend to maintain a _ratio_ which is more or less
constant for each specific organism; and to this regularity is due the
fact that the form of the organism is in general regular and constant.

(3) Nevertheless, the ratio of velocities in different directions is
not absolutely constant, but tends to alter or fluctuate in a regular
way; and to these progressive changes are due the changes of form which
accompany “development,” and the slower changes of form which continue
perceptibly in after life.

(4) The rate of growth is a function of the age of the organism, it
has a maximum somewhat early in life, after which epoch of maximum it
slowly declines.

(5) The rate of growth is directly affected by temperature, and by
other physical conditions.

(6) It is markedly affected, in the way of acceleration or retardation,
at certain physiological epochs of life, such as birth, puberty, or
metamorphosis.

(7) Under certain circumstances, growth may be _negative_, the organism
growing smaller: and such negative growth is a common accompaniment of
metamorphosis, and a frequent accompaniment of old age.

(8) The phenomenon of regeneration is associated with a large temporary
increase in the rate of growth (or “_acceleration_” of growth) of the
injured surface; in other respects, regenerative growth is similar to
ordinary growth in all its essential phenomena.

――――――――――

In this discussion of growth, we have left out of account a vast number
of processes, or phenomena, by which, in the physiological mechanism
of the body, growth is effected and controlled. We have dealt with
growth in its relation to magnitude, and to {155} that relativity
of magnitudes which constitutes form; and so we have studied it as a
phenomenon which stands at the beginning of a morphological, rather
than at the end of a physiological enquiry. Under these restrictions,
we have treated it as far as possible, or in such fashion as our
present knowledge permits, on strictly physical lines.

In all its aspects, and not least in its relation to form, the growth
of organisms has many analogies, some close and some perhaps more
remote, among inanimate things. As the waves grow when the winds strive
with the other forces which govern the movements of the surface of the
sea, as the heap grows when we pour corn out of a sack, as the crystal
grows when from the surrounding solution the proper molecules fall
into their appropriate places: so in all these cases, very much as in
the organism itself, is growth accompanied by change of form, and by
a development of definite shapes and contours. And in these cases (as
in all other mechanical phenomena), we are led to equate our various
magnitudes with time, and so to recognise that growth is essentially a
question of rate, or of velocity.

The differences of form, and changes of form, which are brought about
by varying rates (or “laws”) of growth, are essentially the same
phenomenon whether they be, so to speak, episodes in the life-history
of the individual, or manifest themselves as the normal and distinctive
characteristics of what we call separate species of the race. From one
form, or ratio of magnitude, to another there is but one straight and
direct road of transformation, be the journey taken fast or slow; and
if the transformation take place at all, it will in all likelihood
proceed in the self-same way, whether it occur within the life-time
of an individual or during the long ancestral history of a race. No
small part of what is known as Wolff’s or von Baer’s law, that the
individual organism tends to pass through the phases characteristic
of its ancestors, or that the life-history of the individual tends to
recapitulate the ancestral history of its race, lies wrapped up in this
simple account of the relation between rate of growth and form.

But enough of this discussion. Let us leave for a while the subject
of the growth of the organism, and attempt to study the conformation,
within and without, of the individual cell.

{156}




CHAPTER IV

ON THE INTERNAL FORM AND STRUCTURE OF THE CELL


In the early days of the cell-theory, more than seventy years ago,
Goodsir was wont to speak of cells as “centres of growth” or “centres
of nutrition,” and to consider them as essentially “centres of force.”
He looked forward to a time when the forces connected with the cell
should be particularly investigated: when, that is to say, minute
anatomy should be studied in its dynamical aspect. “When this branch of
enquiry,” he says “shall have been opened up, we shall expect to have
a science of organic forces, having direct relation to anatomy, the
science of organic forms[198].” And likewise, long afterwards, Giard
contemplated a science of _morphodynamique_,—but still looked upon it
as forming so guarded and hidden a “territoire scientifique, que la
plupart des naturalistes de nos jours ne le verront que comme Moïse vit
la terre promise, seulement de loin et sans pouvoir y entrer[199].”

To the external forms of cells, and to the forces which produce and
modify these forms, we shall pay attention in a later chapter. But
there are forms and configurations of matter within the cell, which
also deserve to be studied with due regard to the forces, known or
unknown, of whose resultant they are the visible expression.

In the long interval since Goodsir’s day, the visible structure, the
conformation and configuration, of the cell, has been studied far
more abundantly than the purely dynamic problems that are associated
therewith. The overwhelming progress of microscopic observation has
multiplied our knowledge of cellular and intracellular structure;
and to the multitude of visible structures it {157} has been often
easier to attribute virtues than to ascribe intelligible functions or
modes of action. But here and there nevertheless, throughout the whole
literature of the subject, we find recognition of the inevitable fact
that dynamical problems lie behind the morphological problems of the
cell.

Bütschli pointed out forty years ago, with emphatic clearness, the
failure of morphological methods, and the need for physical methods, if
we were to penetrate deeper into the essential nature of the cell[200].
And such men as Loeb and Whitman, Driesch and Roux, and not a few
besides, have pursued the same train of thought and similar methods of
enquiry.

Whitman[201], for instance, puts the case in a nutshell when, in
speaking of the so-called “caryokinetic” phenomena of nuclear division,
he reminds us that the leading idea in the term “_caryokinesis_” is
_motion_,—“motion viewed as an exponent of forces residing in, or
acting upon, the nucleus. It regards the nucleus as a _seat of energy,
which displays itself in phenomena of motion_[202].”

In short it would seem evident that, except in relation to a dynamical
investigation, the mere study of cell structure has but little value
of its own. That a given cell, an ovum for instance, contains this
or that visible substance or structure, germinal vesicle or germinal
spot, chromatin or achromatin, chromosomes or centrosomes, obviously
gives no explanation of the _activities_ of the cell. And in all such
hypotheses as that of “pangenesis,” in all the theories which attribute
specific properties to micellae, {158} idioplasts, ids, or other
constituent particles of protoplasm or of the cell, we are apt to fall
into the error of attributing to _matter_ what is due to _energy_ and
is manifested in force: or, more strictly speaking, of attributing to
material particles individually what is due to the energy of their
collocation.

The tendency is a very natural one, as knowledge of structure
increases, to ascribe particular virtues to the material structures
themselves, and the error is one into which the disciple is likely to
fall, but of which we need not suspect the master-mind. The dynamical
aspect of the case was in all probability kept well in view by those
who, like Goodsir himself, first attacked the problem of the cell and
originated our conceptions of its nature and functions.

But if we speak, as Weismann and others speak, of an “hereditary
_substance_,” a substance which is split off from the parent-body, and
which hands on to the new generation the characteristics of the old,
we can only justify our mode of speech by the assumption that that
particular portion of matter is the essential vehicle of a particular
charge or distribution of energy, in which is involved the capability
of producing motion, or of doing “work.”

For, as Newton said, to tell us that a thing “is endowed with an occult
specific quality, by which it acts and produces manifest effects, is
to tell us nothing; but to derive two or three general principles of
motion[203] from phenomena would be a very great step in philosophy,
though the causes of these principles were not yet discovered.” The
_things_ which we see in the cell are less important than the _actions_
which we recognise in the cell; and these latter we must especially
scrutinize, in the hope of discovering how far they may be attributed
to the simple and well-known physical forces, and how far they be
relevant or irrelevant to the phenomena which we associate with, and
deem essential to, the manifestation of _life_. It may be that in this
way we shall in time draw nigh to the recognition of a specific and
ultimate residuum. {159}

And lacking, as we still do lack, direct knowledge of the actual forces
inherent in the cell, we may yet learn something of their distribution,
if not also of their nature, from the outward and inward configuration
of the cell, and from the changes taking place in this configuration;
that is to say from the movements of matter, the kinetic phenomena,
which the forces in action set up.

The fact that the germ-cell develops into a very complex structure,
is no absolute proof that the cell itself is structurally a very
complicated mechanism: nor yet, though this is somewhat less obvious,
is it sufficient to prove that the forces at work, or latent, within it
are especially numerous and complex. If we blow into a bowl of soapsuds
and raise a great mass of many-hued and variously shaped bubbles, if we
explode a rocket and watch the regular and beautiful configuration of
its falling streamers, if we consider the wonders of a limestone cavern
which a filtering stream has filled with stalactites, we soon perceive
that in all these cases we have begun with an initial system of very
slight complexity, whose structure in no way foreshadowed the result,
and whose comparatively simple intrinsic forces only play their part by
complex interaction with the equally simple forces of the surrounding
medium. In an earlier age, men sought for the visible embryo, even for
the _homunculus_, within the reproductive cells; and to this day, we
scrutinize these cells for visible structure, unable to free ourselves
from that old doctrine of “pre-formation[204].”

Moreover, the microscope seemed to substantiate the idea (which we may
trace back to Leibniz[205] and to Hobbes[206]), that there is no limit
to the mechanical complexity which we may postulate in an organism, and
no limit, therefore, to the hypotheses which we may rest thereon.

But no microscopical examination of a stick of sealing-wax, no study
of the material of which it is composed, can enlighten {160} us as to
its electrical manifestations or properties. Matter of itself has no
power to do, to make, or to become: it is in energy that all these
potentialities reside, energy invisibly associated with the material
system, and in interaction with the energies of the surrounding
universe.

That “function presupposes structure” has been declared an accepted
axiom of biology. Who it was that so formulated the aphorism I do
not know; but as regards the structure of the cell it harks back to
Brücke, with whose demand for a mechanism, or organisation, within the
cell histologists have ever since been attempting to comply[207]. But
unless we mean to include thereby invisible, and merely chemical or
molecular, structure, we come at once on dangerous ground. For we have
seen, in a former chapter, that some minute “organisms” are already
known of such all but infinitesimal magnitudes that everything which
the morphologist is accustomed to conceive as “structure” has become
physically impossible; and moreover recent research tends generally
to reduce, rather than to extend, our conceptions of the visible
structure necessarily inherent in living protoplasm. The microscopic
structure which, in the last resort or in the simplest cases, it
seems to shew, is that of a more or less viscous colloid, or rather
mixture of colloids, and nothing more. Now, as Clerk Maxwell puts it,
in discussing this very problem, “one material system can differ from
another only in the configuration and motion which it has at a given
instant[208].” If we cannot assume differences in structure, we must
assume differences in _motion_, that is to say, in _energy_. And if we
cannot do this, then indeed we are thrown back upon modes of reasoning
unauthorised in physical science, and shall find ourselves constrained
to assume, or to “admit, that the properties of a germ are not those of
a purely material system.” {161}

But we are by no means necessarily in this dilemma. For though we
come perilously near to it when we contemplate the lowest orders of
magnitude to which life has been attributed, yet in the case of the
ordinary cell, or ordinary egg or germ which is going to develop into
a complex organism, if we have no reason to assume or to believe that
it comprises an intricate “mechanism,” we may be quite sure, both on
direct and indirect evidence, that, like the powder in our rocket, it
is very heterogeneous in its structure. It is a mixture of substances
of various kinds, more or less fluid, more or less mobile, influenced
in various ways by chemical, electrical, osmotic, and other forces, and
in their admixture separated by a multitude of surfaces, or boundaries,
at which these, or certain of these forces are made manifest.

Indeed, such an arrangement as this is already enough to constitute a
“mechanism”; for we must be very careful not to let our physical or
physiological concept of mechanism be narrowed to an interpretation
of the term derived from the delicate and complicated contrivances
of human skill. From the physical point of view, we understand by a
“mechanism” whatsoever checks or controls, and guides into determinate
paths, the workings of energy; in other words, whatsoever leads in the
degradation of energy to its manifestation in some determinate form
of _work_, at a stage short of that ultimate degradation which lapses
in uniformly diffused heat. This, as Warburg has well explained, is
the general effect or function of the physiological machine, and in
particular of that part of it which we call “cell-structure[209].”
The normal muscle-cell is something which turns energy, derived from
oxidation, into work; it is a mechanism which arrests and utilises the
chemical energy of oxidation in its downward course; but the same cell
when injured or disintegrated, loses its “usefulness,” and sets free a
greatly increased proportion of its energy in the form of heat.

But very great and wonderful things are done after this manner by means
of a mechanism (whether natural or artificial) of extreme simplicity.
A pool of water, by virtue of its surface, {162} is an admirable
mechanism for the making of waves; with a lump of ice in it, it becomes
an efficient and self-contained mechanism for the making of currents.
The great cosmic mechanisms are stupendous in their simplicity; and, in
point of fact, every great or little aggregate of heterogeneous matter
(not identical in “phase”) involves, _ipso facto_, the essentials of a
mechanism. Even a non-living colloid, from its intrinsic heterogeneity,
is in this sense a mechanism, and one in which energy is manifested in
the movement and ceaseless rearrangement of the constituent particles.
For this reason Graham (if I remember rightly) speaks somewhere or
other of the colloid state as “the dynamic state of matter”; or in the
same philosopher’s phrase (of which Mr Hardy[210] has lately reminded
us), it possesses “_energia_[211].”

Let us turn then to consider, briefly and diagrammatically, the
structure of the cell, a fertilised germ-cell or ovum for instance,
not in any vain attempt to correlate this structure with the structure
or properties of the resulting and yet distant organism; but merely
to see how far, by the study of its form and its changing internal
configuration, we may throw light on certain forces which are for the
time being at work within it.

We may say at once that we can scarcely hope to learn more of these
forces, in the first instance, than a few facts regarding their
direction and magnitude; the nature and specific identity of the force
or forces is a very different matter. This latter problem is likely to
be very difficult of elucidation, for the reason, among others, that
very different forces are often very much alike in their outward and
visible manifestations. So it has come to pass that we have a multitude
of discordant hypotheses as to the nature of the forces acting within
the cell, and producing, in cell division, the “caryokinetic” figures
of which we are about to speak. One student may, like Rhumbler, choose
to account for them by an hypothesis of mechanical traction, acting
on a reticular web of protoplasm[212]; another, like Leduc, may shew
us how in {163} many of their most striking features they may be
admirably simulated by the diffusion of salts in a colloid medium;
others again, like Gallardo[213] and Hartog, and Rhumbler (in his
earlier papers)[214], insist on their resemblance to the phenomena of
electricity and magnetism[215]; while Hartog believes that the force
in question is only analogous to these, and has a specific identity of
its own[216]. All these conflicting views are of secondary importance,
so long as we seek only to account for certain _configurations_ which
reveal the direction, rather than the nature, of a force. One and
the same system of lines of force may appear in a field of magnetic
or of electrical energy, of the osmotic energy of diffusion, of the
gravitational energy of a flowing stream. In short, we may expect to
learn something of the pure or abstract dynamics, long before we can
deal with the special physics of the cell. For indeed (as Maillard
has suggested), just as uniform expansion about a single centre, to
whatsoever physical cause it may be due will lead to the configuration
of a sphere, so will any two centres or foci of potential (of
whatsoever kind) lead to the configurations with which Faraday made us
familiar under the name of “lines of force[217]”; and this is as much
as to say that the phenomenon, {164} though physical in the concrete,
is in the abstract purely mathematical, and in its very essence is
neither more nor less than _a property of three-dimensional space_.

But as a matter of fact, in this instance, that is to say in trying
to explain the leading phenomena of the caryokinetic division of the
cell, we shall soon perceive that any explanation which is based, like
Rhumbler’s, on mere mechanical traction, is obviously inadequate, and
we shall find ourselves limited to the hypothesis of some polarised and
polarising force, such as we deal with, for instance, in the phenomena
of magnetism or electricity.

Let us speak first of the cell itself, as it appears in a state of
rest, and let us proceed afterwards to study the more active phenomena
which accompany its division.

――――――――――

Our typical cell is a spherical body; that is to say, the uniform
surface-tension at its boundary is balanced by the outward resistance
of uniform forces within. But at times the surface-tension may be a
fluctuating quantity, as when it produces the rhythmical contractions
or “Ransom’s waves” on the surface of a trout’s egg; or again, while
the egg is in contact with other bodies, the surface-tension may be
locally unequal and variable, giving rise to an amoeboid figure, as in
the egg of Hydra[218].

Within the ovum is a nucleus or germinal vesicle, also spherical, and
consisting as a rule of portions of “chromatin,” aggregated together
within a more fluid drop. The fact has often been commented upon that,
in cells generally, there is no correlation of _form_ (though there
apparently is of _size_) between the nucleus and the “cytoplasm,” or
main body of the cell. So Whitman[219] remarks that “except during
the process of division the nucleus seldom departs from its typical
spherical form. It divides and sub-divides, ever returning to the same
round or oval form .... How different with the cell. It preserves the
spherical form as rarely as the nucleus departs from it. Variation
in form marks the beginning and the end of every important chapter
in its {165} history.” On simple dynamical grounds, the contrast is
easily explained. So long as the fluid substance of the nucleus is
qualitatively different from, and incapable of mixing with, the fluid
or semi-fluid protoplasm which surrounds it, we shall expect it to
be, as it almost always is, of spherical form. For, on the one hand,
it is bounded by a liquid film, whose surface-tension is uniform; and
on the other, it is immersed in a medium which transmits on all sides
a uniform fluid pressure[220]. For a similar reason the contractile
vacuole of a Protozoon is spherical in form: it is just a “drop”
of fluid, bounded by a uniform surface-tension and through whose
boundary-film diffusion is taking place. But here, owning to the small
difference between the fluid constituting, and that surrounding, the
drop, the surface-tension equilibrium is unstable; it is apt to vanish,
and the rounded outline of the drop, like a burst bubble, disappears
in a moment[221]. The case of the spherical nucleus is closely akin
to the spherical form of the yolk within the bird’s egg[222]. But if
the substance of the cell acquire a greater solidity, as for instance
in a muscle {166} cell, or by reason of mucous accumulations in an
epithelium cell, then the laws of fluid pressure no longer apply, the
external pressure on the nucleus tends to become unsymmetrical, and
its shape is modified accordingly. “Amoeboid” movements may be set
up in the nucleus by anything which disturbs the symmetry of its own
surface-tension. And the cases, as in many Rhizopods, where “nuclear
material” is scattered in small portions throughout the cell instead
of being aggregated in a single nucleus, are probably capable of very
simple explanation by supposing that the “phase difference” (as the
chemists say) between the nuclear and the protoplasmic substance is
comparatively slight, and the surface-tension which tends to keep them
separate is correspondingly small[223].

It has been shewn that ordinary nuclei, isolated in a living or fresh
state, easily flow together; and this fact is enough to suggest that
they are aggregations of a particular substance rather than bodies
deserving the name of particular organs. It is by reason of the same
tendency to confluence or aggregation of particles that the ordinary
nucleus is itself formed, until the imposition of a new force leads to
its disruption.

Apart from that invisible or ultra-microscopic heterogeneity which
is inseparable from our notion of a “colloid,” there is a visible
heterogeneity of structure within both the nucleus and the outer
protoplasm. The former, for instance, contains a rounded nucleolus
or “germinal spot,” certain conspicuous granules or strands of the
peculiar substance called chromatin, and a coarse meshwork of a
protoplasmic material known as “linin” or achromatin; the outer
protoplasm, or cytoplasm, is generally believed to consist throughout
of a sponge-work, or rather alveolar meshwork, of more and less fluid
substances; and lastly, there are generally to be detected one or more
very minute bodies, usually in the cytoplasm, sometimes within the
nucleus, known as the centrosome or centrosomes.

The morphologist is accustomed to speak of a “polarity” of {167}
the cell, meaning thereby a symmetry of visible structure about a
particular axis. For instance, whenever we can recognise in a cell
both a nucleus and a centrosome, we may consider a line drawn through
the two as the morphological axis of polarity; in an epithelium cell,
it is obvious that the cell is morphologically symmetrical about a
median axis passing from its free surface to its attached base. Again,
by an extension of the term “polarity,” as is customary in dynamics,
we may have a “radial” polarity, between centre and periphery; and
lastly, we may have several apparently independent centres of polarity
within the single cell. Only in cells of quite irregular, or amoeboid
form, do we fail to recognise a definite and symmetrical “polarity.”
The _morphological_ “polarity” is accompanied by, and is but the
outward expression (or part of it) of a true _dynamical_ polarity, or
distribution of forces; and the “lines of force” are rendered visible
by concatenation of particles of matter, such as come under the
influence of the forces in action.

When the lines of force stream inwards from the periphery towards
a point in the interior of the cell, the particles susceptible of
attraction either crowd towards the surface of the cell, or, when
retarded by friction, are seen forming lines or “fibrillae” which
radiate outwards from the centre and constitute a so-called “aster.” In
the cells of columnar or ciliated epithelium, where the sides of the
cell are symmetrically disposed to their neighbours but the free and
attached surfaces are very diverse from one another in their external
relations, it is these latter surfaces which constitute the opposite
poles; and in accordance with the parallel lines of force so set up,
we very frequently see parallel lines of granules which have ranged
themselves perpendicularly to the free surface of the cell (cf. fig.
97).

A simple manifestation of “polarity” may be well illustrated by the
phenomenon of diffusion, where we may conceive, and may automatically
reproduce, a “field of force,” with its poles and visible lines of
equipotential, very much as in Faraday’s conception of the field of
force of a magnetic system. Thus, in one of Leduc’s experiments[224],
if we spread a layer of salt solution over a level {168} plate of
glass, and let fall into the middle of it a drop of indian ink, or
of blood, we shall find the coloured particles travelling outwards
from the central “pole of concentration” along the lines of diffusive
force, and so mapping out for us a “monopolar field” of diffusion: and
if we set two such drops side by side, their lines of diffusion will
oppose, and repel, one another. Or, instead of the uniform layer of
salt solution, we may place at a little distance from one another a
grain of salt and a drop of blood, representing two opposite poles:
and so obtain a picture of a “bipolar field” of diffusion. In either
case, we obtain results closely analogous to the “morphological,”
but really _dynamical_, polarity of the organic cell. But in all
probability, the dynamical polarity, or asymmetry of the cell is a very
complicated phenomenon: for the obvious reason that, in any system,
one asymmetry will tend to beget another. A chemical asymmetry will
induce an inequality of surface-tension, which will lead directly to a
modification of form; the chemical asymmetry may in turn be due to a
process of electrolysis in a polarised electrical field; and again the
chemical heterogeneity may be intensified into a chemical “polarity,”
by the tendency of certain substances to seek a locus of greater or
less surface-energy. We need not attempt to grapple with a subject
so complicated, and leading to so many problems which lie beyond the
sphere of interest of the morphologist. But yet the morphologist, in
his study of the cell, cannot quite evade these important issues; and
we shall return to them again when we have dealt somewhat with the form
of the cell, and have taken account of some of the simpler phenomena of
surface-tension.

――――――――――

We are now ready, and in some measure prepared, to study the numerous
and complex phenomena which usually accompany the division of the cell,
for instance of the fertilised egg.

Division of the cell is essentially accompanied, and preceded, by a
change from radial or monopolar to a definitely bipolar polarity.

In the hitherto quiescent, or apparently quiescent cell, we perceive
certain movements, which correspond precisely to what must accompany
and result from a “polarisation” of forces within the {169} cell:
of forces which, whatever may be their specific nature, at least are
capable of polarisation, and of producing consequent attraction or
repulsion between charged particles of matter. The opposing forces
which were distributed in equilibrium throughout the substance of
the cell become focussed at two “centrosomes,” which may or may
not be already distinguished as visible portions of matter; in the
egg, one of these is always near to, and the other remote from, the
“animal pole” of the egg, which pole is visibly as well as chemically
different from the other, and is the region in which the more rapid and
conspicuous developmental changes will presently begin. Between the two
centrosomes, a spindle-shaped

[Illustration: Fig. 41. Caryokinetic figure in a dividing cell (or
blastomere) of the Trout’s egg. (After Prenant, from a preparation by
Prof. P. Bouin.)]

figure appears, whose striking resemblance to the lines of force made
visible by iron-filings between the poles of a magnet, was at once
recognised by Hermann Fol, when in 1873 he witnessed for the first time
the phenomenon in question. On the farther side of the centrosomes are
seen star-like figures, or “asters,” in which we can without difficulty
recognise the broken lines of force which run externally to those
stronger lines which lie nearer to the polar axis and which constitute
the “spindle.” The lines of force are rendered visible or “material,”
just as in the experiment of the iron-filings, by the fact that, in the
heterogeneous substance of the cell, certain portions of matter are
more “permeable” to the acting force than the rest, become themselves
polarised after the {170} fashion of a magnetic or “paramagnetic”
body, arrange themselves in an orderly way between the two poles of
the field of force, cling to one another as it were in threads[225],
and are only prevented by the friction of the surrounding medium from
approaching and congregating around the adjacent poles.

As the field of force strengthens, the more will the lines of force
be drawn in towards the interpolar axis, and the less evident will
be those remoter lines which constitute the terminal, or extrapolar,
asters: a clear space, free from materialised lines of force, may
thus tend to be set up on either side of the spindle, the so-called
“Bütschli space” of the histologists[226]. On the other hand, the
lines of force constituting the spindle will be less concentrated if
they find a path of less resistance at the periphery of the cell: as
happens, in our experiment of the iron-filings, when we encircle the
field of force with an iron ring. On this principle, the differences
observed between cells in which the spindle is well developed and the
asters small, and others in which the spindle is weak and the asters
enormously developed, can be easily explained by variations in the
potential of the field, the large, conspicuous asters being probably
correlated with a marked permeability of the surface of the cell.

The visible field of force, though often called the “nuclear spindle,”
is formed outside of, but usually near to, the nucleus. Let us look
a little more closely into the structure of this body, and into the
changes which it presently undergoes.

Within its spherical outline (Fig. 42), it contains an “alveolar”
{171} meshwork (often described, from its appearance in optical
section, as a “reticulum”), consisting of more solid substances, with
more fluid matter filling up the interalveolar meshes. This phenomenon
is nothing else than what we call in ordinary language, a “froth” or
a “foam.” It is a surface-tension phenomenon, due to the interacting
surface-tensions of two intermixed fluids, not very different in
density, as they strive to separate. Of precisely the same kind (as
Bütschli was the first to shew) are the minute alveolar networks which
are to be discerned in the cytoplasm of the cell[227], and which we
now know to be not inherent in the nature of protoplasm, or of living
matter in general, but to be due to various causes, natural as well as
artificial. The microscopic honeycomb structure of cast metal under
various conditions of cooling, even on a grand scale the columnar
structure of basaltic rock, is an example of the same surface-tension
phenomenon. {172}

[Illustration: Fig. 42.]

[Illustration: Fig. 43.]

 But here we touch the brink of a subject so important that we must not
 pass it by without a word, and yet so contentious that we must not
 enter into its details. The question involved is simply whether the
 great mass of recorded observations and accepted beliefs with regard
 to the visible structure of protoplasm and of the cell constitute a
 fair picture of the actual _living cell_, or be based on appearances
 which are incident to death itself and to the artificial treatment
 which the microscopist is accustomed to apply. The great bulk of
 histological work is done by methods which involve the sudden killing
 of the cell or organism by strong reagents, the assumption being that
 death is so rapid that the visible phenomena exhibited during life
 are retained or “fixed” in our preparations. While this assumption
 is reasonable and justified as regards the general outward form of
 small organisms or of individual cells, enough has been done of late
 years to shew that the case is totally different in the case of
 the minute internal networks, granules, etc., which represent the
 alleged _structure_ of protoplasm. For, as Hardy puts it, “It is
 notorious that the various fixing reagents are coagulants of organic
 colloids, and that they produce precipitates which have a certain
 figure or structure, ... and that the figure varies, other things
 being equal, according to the reagent used.” So it comes to pass that
 some writers[228] have altogether denied the existence in the living
 cell-protoplasm of a network or alveolar “foam”; others[229] have
 cast doubts on the main tenets of recent histology regarding nuclear
 structure; and Hardy, discussing the structure of certain gland-cells,
 declares that “there is no evidence that the structure discoverable in
 the cell-substance of these cells after fixation has any counterpart
 in the cell when living.” “A large part of it” he goes on to say “is
 an artefact. The profound difference in the minute structure of a
 secretory cell of a mucous gland according to the reagent which is
 used to fix it would, it seems to me, almost suffice to establish this
 statement in the absence of other evidence.”

 Nevertheless, histological study proceeds, especially on the part of
 the morphologists, with but little change in theory or in method,
 in spite of these and many other warnings. That certain visible
 structures, nucleus, vacuoles, “attraction-spheres” or centrosomes,
 etc., are actually present in the living cell, we know for certain;
 and to this class belong the great majority of structures (including
 the nuclear “spindle” itself) with which we are at present concerned.
 That many other alleged structures are artificial has also been placed
 beyond a doubt; but where to draw the dividing line we often do not
 know[230]. {173}

The following is a brief epitome of the visible changes undergone by a
typical cell, leading up to the act of segmentation, and constituting
the phenomenon of mitosis or caryokinetic division. In the egg of a
sea-urchin, we see with almost diagrammatic completeness what is set
forth here[231].

[Illustration: Fig. 44.]

[Illustration: Fig. 45.]

1. The chromatin, which to begin with was distributed in granules on
the otherwise achromatic reticulum (Fig. 42), concentrates to form a
skein or _spireme_, which may be a continuous thread from the first
(Figs. 43, 44), or from the first segmented. In any case it divides
transversely sooner or later into a number of _chromosomes_ (Fig. 45),
which as a rule have the shape of little rods, straight or curved,
often bent into a V, but which may also be ovoid, or round, or even
annular. Certain deeply staining masses, the nucleoli, which may be
present in the resting nucleus, do not take part in the process of
chromosome formation; they are either cast out of the nucleus and are
dissolved in the cytoplasm, or fade away _in situ_.

2. Meanwhile, the deeply staining granule (here extra-nuclear), known
as the _centrosome_, has divided in two. The two resulting granules
travel to opposite poles of the nucleus, and {174} there each becomes
surrounded by a system of radiating lines, the _asters_; immediately
around the centrosome is a clear space, the _centrosphere_ (Figs.
43–45). Between the two centrosomes with their asters stretches a
bundle of achromatic fibres, the _spindle_.

3. The surface-film bounding the nucleus has broken down, the definite
nuclear boundaries are lost, and the spindle now stretches through the
nuclear material, in which lie the chromosomes (Figs. 45, 46). These
chromosomes now arrange themselves midway between the poles of the
spindle, where they form what is called the _equatorial plate_ (Fig.
47).

[Illustration: Fig. 46.]

[Illustration: Fig. 47.]

4. Each chromosome splits longitudinally into two: usually at this
stage,—but it is to be noticed that the splitting may have taken place
so early as the spireme stage (Fig. 48).

5. The halves of the split chromosomes now separate from one another,
and travel in opposite directions towards the two poles (Fig. 49). As
they move, it becomes apparent that the spindle consists of a median
bundle of “fibres,” the central spindle, running from pole to pole, and
a more superficial sheath of “mantle-fibres,” to which the chromosomes
seem to be attached, and by which they seem to be drawn towards the
asters.

6. The daughter chromosomes, arranged now in two groups, become closely
crowded in a mass near the centre of each aster {175} (Fig. 50).
They fuse together and form once more an alveolar reticulum and may
occasionally at this stage form another spireme.

[Illustration: Fig. 48.]

[Illustration: Fig. 49.]

A boundary or surface wall is now developed round each reconstructed
nuclear mass, and the spindle-fibres disappear (Fig. 51). The
centrosome remains, as a rule, outside the nucleus.

[Illustration: Fig. 50.]

[Illustration: Fig. 51.]

7. On the central spindle, in the position of the equatorial plate,
there has appeared during the migration of the chromosomes, a
“cell-plate” of deeply staining thickenings (Figs. 50, 51). This is
more conspicuous in plant-cells. {176}

8. A constriction has meanwhile appeared in the cytoplasm, and the
cell divides through the equatorial plane. In plant-cells the line
of this division is foreshadowed by the “cell-plate,” which extends
from the spindle across the entire cell, and splits into two layers,
between which appears the membrane by which the daughter cells are
cleft asunder. In animal cells the cell-plate does not attain such
dimensions, and no cell-wall is formed.

――――――――――

The whole, or very nearly the whole of these nuclear phenomena may be
brought into relation with that polarisation of forces, in the cell as
a whole, whose field is made manifest by the “spindle” and “asters” of
which we have already spoken: certain particular phenomena, directly
attributable to surface-tension and diffusion, taking place in more or
less obvious and inevitable dependence upon the polar system†.

† The reference numbers in the following account refer to the
paragraphs and figures of the preceding summary of visible nuclear
phenomena.

At the same time, in attempting to explain the phenomena, we cannot say
too clearly, or too often, that all that we are meanwhile justified
in doing is to try to shew that such and such actions lie _within
the range_ of known physical actions and phenomena, or that known
physical phenomena produce effects similar to them. We want to feel
sure that the whole phenomenon is not _sui generis_, but is somehow or
other capable of being referred to dynamical laws, and to the general
principles of physical science. But when we speak of some particular
force or mode of action, using it as an illustrative hypothesis, we
must stop far short of the implication that this or that force is
necessarily the very one which is actually at work within the living
cell; and certainly we need not attempt the formidable task of trying
to reconcile, or to choose between, the various hypotheses which have
already been enunciated, or the several assumptions on which they
depend.

――――――――――

Any region of space within which action is manifested is a field of
force; and a simple example is a bipolar field, in which the action is
symmetrical with reference to the line joining two points, or poles,
and also with reference to the “equatorial” plane equidistant from
both. We have such a “field of force” in {177} the neighbourhood of
the centrosome of the ripe cell or ovum, when it is about to divide;
and by the time the centrosome has divided, the field is definitely a
bipolar one.

The _quality_ of a medium filling the field of force may be uniform,
or it may vary from point to point. In particular, it may depend upon
the magnitude of the field; and the quality of one medium may differ
from that of another. Such variation of quality, within one medium, or
from one medium to another, is capable of diagrammatic representation
by a variation of the direction or the strength of the field (other
conditions being the same) from the state manifested in some uniform
medium taken as a standard. The medium is said to be _permeable_ to the
force, in greater or less degree than the standard medium, according as
the variation of the density of the lines of force from the standard
case, under otherwise identical conditions, is in excess or defect. _A
body placed in the medium will tend to move towards regions of greater
or less force according as its permeability is greater or less than
that of the surrounding medium_[232]. In the common experiment of
placing iron-filings between the two poles of a magnetic field, the
filings have a very high permeability; and not only do they themselves
become polarised so as to attract one another, but they tend to be
attracted from the weaker to the stronger parts of the field, and as
we have seen, were it not for friction or some other resistance, they
would soon gather together around the nearest pole. But if we repeat
the same experiment with such a metal as bismuth, which is very little
permeable to the magnetic force, then the conditions are reversed, and
the particles, being repelled from the stronger to the weaker parts
of the field, tend to take up their position as far from the poles as
possible. The particles have become polarised, but in a sense opposite
to that of the surrounding, or adjacent, field.

Now, in the field of force whose opposite poles are marked by {178}
the centrosomes the nucleus appears to act as a more or less permeable
body, as a body more permeable than the surrounding medium, that is to
say the “cytoplasm” of the cell. It is accordingly attracted by, and
drawn into, the field of force, and tries, as it were, to set itself
between the poles and as far as possible from both of them. In other
words, the centrosome-foci will be apparently drawn over its surface,
until the nucleus as a whole is involved within the field of force,
which is visibly marked out by the “spindle” (par. 3, Figs. 44, 45).

If the field of force be electrical, or act in a fashion analogous to
an electrical field, the charged nucleus will have its surface-tensions
diminished[233]: with the double result that the inner alveolar
meshwork will be broken up (par. 1), and that the spherical boundary
of the whole nucleus will disappear (par. 2). The break-up of the
alveoli (by thinning and rupture of their partition walls) leads to the
formation of a net, and the further break-up of the net may lead to the
unravelling of a thread or “spireme” (Figs. 43, 44).

Here there comes into play a fundamental principle which, in so far
as we require to understand it, can be explained in simple words.
The effect (and we might even say the _object_) of drawing the more
permeable body in between the poles, is to obtain an “easier path” by
which the lines of force may travel; but it is obvious that a longer
route through the more permeable body may at length be found less
advantageous than a shorter route through the less permeable medium.
That is to say, the more permeable body will only tend to be drawn in
to the field of force until a point is reached where (so to speak)
the way _round_ and the way _through_ are equally advantageous. We
should accordingly expect that (on our hypothesis) there would be
found cases in which the nucleus was wholly, and others in which it
was only partially, and in greater or less degree, drawn in to the
field between the centrosomes. This is precisely what is found to
occur in actual fact. Figs. 44 and 45 represent two so-called “types,”
of a phase which follows that represented in Fig. 43. According to
the usual descriptions (and in particular to Professor {179} E. B.
Wilson’s[234]), we are told that, in such a case as Fig. 44, the
“primary spindle” disappears and the centrosomes diverge to opposite
poles of the nucleus; such a condition being found in many plant-cells,
and in the cleavage-stages of many eggs. In Fig. 45, on the other hand,
the primary spindle persists, and subsequently comes to form the main
or “central” spindle; while at the same time we see the fading away
of the nuclear membrane, the breaking up of the spireme into separate
chromosomes, and an ingrowth into the nuclear area of the “astral
rays,”—all as in Fig. 46, which represents the next succeeding phase
of Fig. 45. This condition, of Fig. 46, occurs in a variety of cases;
it is well seen in the epidermal cells of the salamander, and is also
on the whole characteristic of the mode of formation of the “polar
bodies.” It is clear and obvious that the two “types” correspond to
mere differences of degree, and are such as would naturally be brought
about by differences in the relative permeabilities of the nuclear
mass and of the surrounding cytoplasm, or even by differences in the
magnitude of the former body.

But now an important change takes place, or rather an important
difference appears; for, whereas the nucleus as a whole tended to
be drawn in to the _stronger_ parts of the field, when it comes to
break up we find, on the contrary, that its contained spireme-thread
or separate chromosomes tend to be repelled to the _weaker_ parts.
Whatever this difference may be due to,—whether, for instance, to
actual differences of permeability, or possibly to differences in
“surface-charge,”—the fact is that the chromatin substance now
_behaves_ after the fashion of a “diamagnetic” body, and is repelled
from the stronger to the weaker parts of the field. In other words,
its particles, lying in the inter-polar field, tend to travel towards
the equatorial plane thereof (Figs. 47, 48), and further tend to
move outwards towards the periphery of that plane, towards what the
histologist calls the “mantle-fibres,” or outermost of the lines of
force of which the spindle is made up (par. 5, Fig. 47). And if this
comparatively non-permeable chromatin substance come to consist of
separate portions, more or less elongated in form, these portions, or
separate “chromosomes,” will adjust themselves longitudinally, {180}
in a peripheral equatorial circle (Figs. 48, 49). This is precisely
what actually takes place. Moreover, before the breaking up of the
nucleus, long before the chromatin material has broken up into separate
chromosomes, and at the very time when it is being fashioned into a
“spireme,” this body already lies in a polar field, and must already
have a tendency to set itself in the equatorial plane thereof. But
the long, continuous spireme thread is unable, so long as the nucleus
retains its spherical boundary wall, to adjust itself in a simple
equatorial annulus; in striving to do so, it must tend to coil and
“kink” itself, and in so doing (if all this be so), it must tend to
assume the characteristic convolutions of the “spireme.”

[Illustration: Fig. 52. Chromosomes, undergoing splitting and
separation. (After Hatschek and Flemming, diagrammatised.)]

After the spireme has broken up into separate chromosomes, these
particles come into a position of temporary, and unstable, equilibrium
near the periphery of the equatorial plane, and here they tend to place
themselves in a symmetrical arrangement (Fig. 52). The particles are
rounded, linear, sometimes annular, similar in form and size to one
another; and lying as they do in a fluid, and subject to a symmetrical
system of forces, it is not surprising that they arrange themselves
in a symmetrical manner, the precise arrangement depending on the
form of the particles themselves. This symmetry may perhaps be due,
as has already been suggested, to induced electrical charges. In
discussing Brauer’s observations on the splitting of the chromatic
filament, and the symmetrical arrangement of the separate granules, in
_Ascaris megalocephala_, Lillie[235] {181} remarks: “This behaviour
is strongly suggestive of the division of a colloidal particle under
the influence of its surface electrical charge, and of the effects
of mutual repulsion in keeping the products of division apart.” It
is also probable that surface-tensions between the particles and the
surrounding protoplasm would bring about an identical result, and
would sufficiently account for the obvious, and at first sight, very
curious, symmetry. We know that if we float a couple of matches in
water they tend to approach one another, till they lie close together,
side by side; and, if we lay upon a smooth wet plate four matches, half
broken across, a precisely similar attraction brings the four matches
together in the form of a symmetrical cross. Whether one of these,
or some other, be the actual explanation of the phenomenon, it is at
least plain that by some physical cause, some mutual and symmetrical
attraction or repulsion of the particles, we must seek to account for
the curious symmetry of these so-called “tetrads.” The remarkable
_annular_ chromosomes, shewn in Fig. 53, can also be easily imitated
by means of loops of thread upon a soapy film when the film within the
annulus is broken or its tension reduced.

[Illustration: Fig. 53. Annular chromosomes, formed in the
spermatogenesis of the Mole-cricket. (From Wilson, after Vom Rath.)]

――――――――――

So far as we have now gone, there is no great difficulty in pointing to
simple and familiar phenomena of a field of force which are similar,
or comparable, to the phenomena which we witness within the cell. But
among these latter phenomena there are others for which it is not
so easy to suggest, in accordance with known laws, a simple mode of
physical causation. It is not at once obvious how, in any simple system
of symmetrical forces, {182} the chromosomes, which had at first been
apparently repelled from the poles towards the equatorial plane, should
then be split asunder, and should presently be attracted in opposite
directions, some to one pole and some to the other. Remembering that it
is not our purpose to _assert_ that some one particular mode of action
is at work, but merely to shew that there do exist physical forces, or
distributions of force, which are capable of producing the required
result, I give the following suggestive hypothesis, which I owe to my
colleague Professor W. Peddie.

As we have begun by supposing that the nuclear, or chromosomal
matter differs in _permeability_ from the medium, that is to say the
cytoplasm, in which it lies, let us now make the further assumption
that its permeability is variable, and depends upon the _strength of
the field_.

[Illustration: Fig. 54.]

In Fig. 54, we have a field of force (representing our cell),
consisting of a homogeneous medium, and including two opposite poles:
lines of force are indicated by full lines, and _loci of constant
magnitude of force_ are shewn by dotted lines.

Let us now consider a body whose permeability (µ) depends on the
strength of the field _F_. At two field-strengths, such as _F_{a}_,
_F_{b}_, let the permeability of the body be equal to that of the
{183} medium, and let the curved line in Fig. 55 represent generally
its permeability at other field-strengths; and let the outer and
inner dotted curves in Fig. 54 represent respectively the loci of the
field-strengths _F_{b}_ and _F_{a}_. The body if it be placed in the
medium within either branch of the inner curve, or outside the outer
curve, will tend to move into the neighbourhood of the adjacent pole.
If it be placed in the region intermediate to the two dotted curves, it
will tend to move towards regions of weaker field-strength.

[Illustration: Fig. 55.]

The locus _F_{b}_ is therefore a locus of stable position, towards
which the body tends to move; the locus _F_{a}_ is a locus of unstable
position, from which it tends to move. If the body were placed across
_F_{a}_, it might be torn asunder into two portions, the split
coinciding with the locus _F_{a}_.

Suppose a number of such bodies to be scattered throughout the medium.
Let at first the regions _F_{a}_ and _F_{b}_ be entirely outside the
space where the bodies are situated: and, in making this supposition we
may, if we please, suppose that the loci which we are calling _F_{a}_
and _F_{b}_ are meanwhile situated somewhat farther from the axis than
in our figure, that (for instance) _F_{a}_ is situated where we have
drawn _F_{b}_, and that _F_{b}_ is still further out. The bodies then
tend towards the poles; but the tendency may be very small if, in Fig.
55, the curve and its intersecting straight line do not diverge very
far from one another beyond _F_{a}_; in other {184} words, if, when
situated in this region, the permeability of the bodies is not very
much in excess of that of the medium.

Let the poles now tend to separate farther and farther from one
another, the strength of each pole remaining unaltered; in other words,
let the centrosome-foci recede from one another, as they actually
do, drawing out the spindle-threads between them. The loci _F_{a}_,
_F_{b}_, will close in to nearer relative distances from the poles. In
doing so, when the locus _F_{a}_ crosses one of the bodies, the body
may be torn asunder; if the body be of elongated shape, and be crossed
at more points than one, the forces at work will tend to exaggerate its
foldings, and the tendency to rupture is greatest when _F_{a}_ is in
some median position (Fig. 56).

[Illustration: Fig. 56.]

When the locus _F_{a}_ has passed entirely over the body, the body
tends to move towards regions of weaker force; but when, in turn, the
locus _F_{b}_ has crossed it, then the body again moves towards regions
of stronger force, that is to say, towards the nearest pole. And, in
thus moving towards the pole, it will do so, as appears actually to be
the case in the dividing cell, along the course of the outer lines of
force, the so-called “mantle-fibres” of the histologist[236].

Such considerations as these give general results, easily open to
modification in detail by a change of any of the arbitrary postulates
which have been made for the sake of simplicity. Doubtless there are
many other assumptions which would more or less meet the case; for
instance, that of Ida H. Hyde that, {185} during the active phase
of the chromatin molecule (during which it decomposes and sets free
nucleic acid) it carries a charge opposite to that which it bears
during its resting, or alkaline phase; and that it would accordingly
move towards different poles under the influence of a current,
wandering with its negative charge in an alkaline fluid during its
acid phase to the anode, and to the kathode during its alkaline phase.
A whole field of speculation is opened up when we begin to consider
the cell not merely as a polarised electrical field, but also as an
electrolytic field, full of wandering ions. Indeed it is high time we
reminded ourselves that we have perhaps been dealing too much with
ordinary physical analogies: and that our whole field of force within
the cell is of an order of magnitude where these grosser analogies may
fail to serve us, and might even play us false, or lead us astray.
But our sole object meanwhile, as I have said more than once, is
to demonstrate, by such illustrations as these, that, whatever be
the actual and as yet unknown _modus operandi_, there are physical
conditions and distributions of force which _could_ produce just such
phenomena of movement as we see taking place within the living cell.
This, and no more, is precisely what Descartes is said to have claimed
for his description of the human body as a “mechanism[237].”

――――――――――

The foregoing account is based on the provisional assumption that
the phenomena of caryokinesis are analogous to, if not identical
with those of a bipolar electrical field; and this comparison, in my
opinion, offers without doubt the best available series of analogies.
But we must on no account omit to mention the fact that some of
Leduc’s diffusion-experiments offer very remarkable analogies to
the diagrammatic phenomena of caryokinesis, as shewn in the annexed
figure[238]. Here we have two identical (not opposite) poles of osmotic
concentration, formed by placing a drop of indian ink in salt water,
and then on either side of this central drop, a hypertonic drop of
salt solution more lightly coloured. On either side the pigment of the
central drop has been drawn towards the focus nearest to it; but in
the middle line, the pigment {186} is drawn in opposite directions by
equal forces, and so tends to remain undisturbed, in the form of an
“equatorial plate.”

Nor should we omit to take account (however briefly and inadequately)
of a novel and elegant hypothesis put forward by A. B. Lamb. This
hypothesis makes use of a theorem of Bjerknes, to the effect that
synchronously vibrating or pulsating bodies in a liquid field attract
or repel one another according as their oscillations are identical
or opposite in phase. Under such circumstances, true currents, or
hydrodynamic lines of force, are produced, identical in form with the
lines of force of a magnetic field; and other particles floating,
though not necessarily pulsating, in the liquid field, tend to be
attracted or repelled by the pulsating bodies according as they are
lighter or heavier than the surrounding fluid. Moreover (and this is
the most remarkable point of all), the lines of force set up by the
_oppositely_ pulsating bodies are the same as those which are produced
by _opposite_ magnetic poles: though in the former case repulsion, and
in the latter case attraction, takes place between the two poles[239].

[Illustration: Fig. 57. Artificial caryokinesis (after Leduc), for
comparison with Fig. 41, p. 169.]

――――――――――

But to return to our general discussion.

While it can scarcely be too often repeated that our enquiry is not
directed towards the solution of physiological problems, save {187}
only in so far as they are inseparable from the problems presented by
the visible configurations of form and structure, and while we try, as
far as possible, to evade the difficult question of what particular
forces are at work when the mere visible forms produced are such as
to leave this an open question, yet in this particular case we have
been drawn into the use of electrical analogies, and we are bound to
justify, if possible, our resort to this particular mode of physical
action. There is an important paper by R. S. Lillie, on the “Electrical
Convection of certain Free Cells and Nuclei[240],” which, while I
cannot quote it in direct support of the suggestions which I have made,
yet gives just the evidence we need in order to shew that electrical
forces act upon the constituents of the cell, and that their action
discriminates between the two species of colloids represented by the
cytoplasm and the nuclear chromatin. And the difference is such that,
in the presence of an electrical current, the cell substance and the
nuclei (including sperm-cells) tend to migrate, the former on the whole
with the positive, the latter with the negative stream: a difference
of electrical potential being thus indicated between the particle
and the surrounding medium, just as in the case of minute suspended
particles of various kinds in various feebly conducing media[241]. And
the electrical difference is doubtless greatest, in the case of the
cell constituents, just at the period of mitosis: when the chromatin
is invariably in its most deeply staining, most strongly acid, and
therefore, presumably, in its most electrically negative phase. In
short, {188} Lillie comes easily to the conclusion that “electrical
theories of mitosis are entitled to more careful consideration than
they have hitherto received.”

Among other investigations, all leading towards the same general
conclusion, namely that differences of electric potential play a
great part in the phenomenon of cell division, I would mention a very
noteworthy paper by Ida H. Hyde[242], in which the writer shews (among
other important observations) that not only is there a measurable
difference of potential between the animal and vegetative poles of
a fertilised egg (_Fundulus_, toad, turtle, etc.), but that this
difference is not constant, but fluctuates, or actually reverses
its direction, periodically, at epochs coinciding with successive
acts of segmentation or other important phases in the development of
the egg[243]; just as other physical rhythms, for instance in the
production of CO_{2}, had already been shewn to do. Hence we shall be
by no means surprised to find that the “materialised” lines of force,
which in the earlier stages form the convergent curves of the spindle,
are replaced in the later phases of caryokinesis by divergent curves,
indicating that the two foci, which are marked out within the field by
the divided and reconstituted nuclei, are now alike in their polarity
(Figs. 58, 59).

It is certain, to my mind, that these observations of Miss Hyde’s, and
of Lillie’s, taken together with those of many writers on the behaviour
of colloid particles generally in their relation to an electrical
field, have a close bearing upon the physiological side of our problem,
the full discussion of which lies outside our present field.

――――――――――

The break-up of the nucleus, already referred to and ascribed to
a diminution of its surface-tension, is accompanied by certain
diffusion phenomena which are sometimes visible to the eye; and we
are reminded of Lord Kelvin’s view that diffusion is implicitly {189}
associated with surface-tension changes, of which the first step is a
minute puckering of the surface-skin, a sort of interdigitation with
the surrounding medium. For instance, Schewiakoff has observed in
_Euglypha_[244] that, just before the break-up of the nucleus, a system
of rays appears, concentred about it, but having nothing to do with the
polar asters: and during the existence of this striation, the nucleus
enlarges very considerably, evidently by imbibition of fluid from the
surrounding protoplasm. In short, diffusion is at work, hand in hand
with, and as it were in opposition to, the surface-tensions which
define the nucleus. By diffusion, hand in hand with surface-tension,
the alveoli of the nuclear meshwork are formed, enlarged, and finally
ruptured: diffusion sets up the movements which give rise to the
appearance of rays, or striae, around the nucleus: and through
increasing diffusion, and weakening surface-tension, the rounded
outline of the nucleus finally disappears. {190}

[Illustration: Fig. 58. Final stage in the first segmentation of the
egg of Cerebratulus. (From Prenant, after Coe.)[245]]

[Illustration: Fig. 59. Diagram of field of force with two similar
poles.]

As we study these manifold phenomena, in the individual cases of
particular plants and animals, we recognise a close identity of type,
coupled with almost endless variation of specific detail; and in
particular, the order of succession in which certain of the phenomena
occur is variable and irregular. The precise order of the phenomena,
the time of longitudinal and of transverse fission of the chromatin
thread, of the break-up of the nuclear wall, and so forth, will
depend upon various minor contingencies and “interferences.” And it
is worthy of particular note that these variations, in the order of
events and in other subordinate details, while doubtless attributable
to specific physical conditions, would seem to be without any obvious
classificatory value or other biological significance[246].

――――――――――

As regards the actual mechanical division of the cell into two halves,
we shall see presently that, in certain cases, such as that of a
long cylindrical filament, surface-tension, and what is known as the
principle of “minimal area,” go a long way to explain the mechanical
process of division; and in all cells whatsoever, the process of
division must somehow be explained as the result of a conflict between
surface-tension and its opposing forces. But in such a case as our
spherical cell, it is not very easy to see what physical cause is at
work to disturb its equilibrium and its integrity.

The fact that, when actual division of the cell takes place, it does
so at right angles to the polar axis and precisely in the direction of
the equatorial plane, would lead us to suspect that the new surface
formed in the equatorial plane sets up an annular tension, directed
inwards, where it meets the outer surface layer of the cell itself. But
at this point, the problem becomes more complicated. Before we could
hope to comprehend it, we should have not only to enquire into the
potential distribution at the surface of the cell in relation to that
which we have seen to exist in its interior, but we should probably
also have to take account of the differences of potential which the
material arrangements along the lines of force must themselves tend
to produce. Only {191} thus could we approach a comprehension of the
balance of forces which cohesion, friction, capillarity and electrical
distribution combine to set up.

The manner in which we regard the phenomenon would seem to turn, in
great measure, upon whether or no we are justified in assuming that,
in the liquid surface-film of a minute spherical cell, local, and
symmetrically localised, differences of surface-tension are likely
to occur. If not, then changes in the conformation of the cell
such as lead immediately to its division must be ascribed not to
local changes in its surface-tension, but rather to direct changes
in internal pressure, or to mechanical forces due to an induced
surface-distribution of electrical potential.

It has seemed otherwise to many writers, and we have a number of
theories of cell division which are all based directly on inequalities
or asymmetry of surface-tension. For instance, Bütschli suggested,
some forty years ago[247], that cell division is brought about by an
increase of surface-tension in the equatorial region of the cell.
This explanation, however, can scarcely hold; for it would seem that
such an increase of surface-tension in the equatorial plane would
lead to the cell becoming flattened out into a disc, with a sharply
curved equatorial edge, and to a streaming of material towards the
equator. In 1895, Loeb shewed that the streaming went on from the
equator towards the divided nuclei, and he supposed that the violence
of these streaming movements brought about actual division of the
cell: a hypothesis which was adopted by many other physiologists[248].
This streaming movement would suggest, as Robertson has pointed out,
a _diminution_ of surface-tension in the region of the equator. Now
Quincke has shewn that the formation of soaps at the surface of an
oil-droplet results in a diminution of the surface-tension of the
latter; and that if the saponification be local, that part of the
surface tends to spread. By laying a thread moistened with a dilute
solution of caustic alkali, or even merely smeared with soap, across
a drop of oil, Robertson has further shewn that the drop at once
divides into two: the edges of the drop, that is to say the ends of
the {192} diameter across which the thread lies, recede from the
thread, so forming a notch at each end of the diameter, while violent
streaming motions are set up at the surface, away from the thread in
the direction of the two opposite poles. Robertson[249] suggests,
accordingly, that the division of the cell is actually brought about by
a lowering of the equatorial surface-tension, and that this in turn is
due to a chemical action, such as a liberation of cholin, or of soaps
of cholin, through the splitting of lecithin in nuclear synthesis.

But purely chemical changes are not of necessity the fundamental
cause of alteration in the surface-tension of the egg, for the action
of electrolytes on surface-tension is now well known and easily
demonstrated. So, according to other views than those with which we
have been dealing, electrical charges are sufficient in themselves
to account for alterations of surface-tension; while these in turn
account for that protoplasmic streaming which, as so many investigators
agree, initiates the segmentation of the egg[250]. A great part of our
difficulty arises from the fact that in such a case as this the various
phenomena are so entangled and apparently concurrent that it is hard
to say which initiates another, and to which this or that secondary
phenomenon may be considered due. Of recent years the phenomenon of
_adsorption_ has been adduced (as we have already briefly said) in
order to account for many of the events and appearances which are
associated with the asymmetry, and lead towards the division, of the
cell. But our short discussion of this phenomenon may be reserved for
another chapter.

However, we are not directly concerned here with the phenomena of
segmentation or cell division in themselves, except only in so far as
visible changes of form are capable of easy and obvious correlation
with the play of force. The very fact of “development” indicates that,
while it lasts, the equilibrium of the egg is never complete[251].
And we may simply conclude the {193} matter by saying that, if you
have caryokinetic figures developing inside the cell, that of itself
indicates that the dynamic system and the localised forces arising
from it are in continual alteration; and, consequently, changes in the
outward configuration of the system are bound to take place.

――――――――――

As regards the phenomena of fertilisation,—of the union of the
spermatozoon with the “pronucleus” of the egg,—we might study these
also in illustration, up to a certain point, of the polarised forces
which are manifestly at work. But we shall merely take, as a single
illustration, the paths of the male and female pronuclei, as they
travel to their ultimate meeting place.

The spermatozoon, when within a very short distance of the egg-cell,
is attracted by it. Of the nature of this attractive force we have no
certain knowledge, though we would seem to have a pregnant hint in
Loeb’s discovery that, in the neighbourhood of other substances, such
even as a fragment, or bead, of glass, the spermatozoon undergoes a
similar attraction. But, whatever the force may be, it is one acting
normally to the surface of the ovum, and accordingly, after entry, the
sperm-nucleus points straight towards the centre of the egg; from the
fact that other spermatozoa, subsequent to the first, fail to effect
an entry, we may safely conclude that an immediate consequence of the
entry of the spermatozoon is an increase in the surface-tension of the
egg[252]. Somewhere or other, near or far away, within the egg, lies
its own nuclear body, the so-called female pronucleus, and we find
after a while that this has fused with the head of the spermatozoon
(or male pronucleus), and that the body resulting from their fusion
has come to occupy the centre of the egg. This _must_ be due (as
Whitman pointed out long ago) to a force of attraction acting between
the two bodies, and another force acting upon one or other or both in
the direction of the centre of the cell. Did we know the magnitude
of these several forces, it would be a very easy task to calculate
the precise path which the two pronuclei would follow, leading to
conjugation and the central {194} position. As we do not know the
magnitude, but only the direction, of these forces we can only make a
general statement: (1) the paths of both moving bodies will lie wholly
within a plane triangle drawn between the two bodies and the centre
of the cell; (2) unless the two bodies happen to lie, to begin with,
precisely on a diameter of the cell, their paths until they meet one
another will be curved paths, the convexity of the curve being towards
the straight line joining the two bodies; (3) the two bodies will meet
a little before they reach the centre; and, having met and fused,
will travel on to reach the centre in a straight line. The actual
study and observation of the path followed is not very easy, owing to
the fact that what we usually see is not the path itself, but only a
_projection_ of the path upon the plane of the microscope; but the
curved path is particularly well seen in the frog’s egg, where the path
of the spermatozoon is marked by a little streak of brown pigment, and
the fact of the meeting of the pronuclei before reaching the centre has
been repeatedly seen by many observers.

The problem is nothing else than a particular case of the famous
problem of three bodies, which has so occupied the astronomers; and
it is obvious that the foregoing brief description is very far from
including all possible cases. Many of these are particularly described
in the works of Fol, Roux, Whitman and others[253].

――――――――――

The intracellular phenomena of which we have now spoken have assumed
immense importance in biological literature and discussion during
the last forty years; but it is open to us to doubt whether they
will be found in the end to possess more than a remote and secondary
biological significance. Most, if not all of them, would seem to
follow immediately and inevitably from very simple assumptions as to
the physical constitution of the cell, and from an extremely simple
distribution of polarised forces within it. We have already seen that
how a thing grows, and what it grows into, is a dynamic and not a
merely material problem; so far as the material substance is concerned,
it is so only by reason {195} of the chemical, electrical or other
forces which are associated with it. But there is another consideration
which would lead us to suspect that many features in the structure and
configuration of the cell are of very secondary biological importance;
and that is, the great variation to which these phenomena are subject
in similar or closely related organisms, and the apparent impossibility
of correlating them with the peculiarities of the organism as a whole.
“Comparative study has shewn that almost every detail of the processes
(of mitosis) described above is subject to variation in different
forms of cells[254].” A multitude of cells divide to the accompaniment
of caryokinetic phenomena; but others do so without any visible
caryokinesis at all. Sometimes the polarised field of force is within,
sometimes it is adjacent to, and at other times it lies remote from the
nucleus. The distribution of potential is very often symmetrical and
bipolar, as in the case described; but a less symmetrical distribution
often occurs, with the result that we have, for a time at least,
numerous centres of force, instead of the two main correlated poles:
this is the simple explanation of the numerous stellate figures, or
“Strahlungen,” which have been described in certain eggs, such as
those of _Chaetopterus_. In one and the same species of worm (_Ascaris
megalocephala_), one group or two groups of chromosomes may be present.
And remarkably constant, in general, as the number of chromosomes in
any one species undoubtedly is, yet we must not forget that, in plants
and animals alike, the whole range of observed numbers is but a small
one; for (as regards the germ-nuclei) few organisms have less than six
chromosomes, and fewer still have more than sixteen[255]. In closely
related animals, such as various species of Copepods, and even in the
same species of worm or insect, the form of the chromosomes, and their
arrangement in relation to the nuclear spindle, have been found to
differ in the various ways alluded to above. In short, there seem to be
strong grounds for believing that these and many similar phenomena are
in no way specifically related to the particular organism in which they
have {196} been observed, and are not even specially and indisputably
connected with the organism as such. They include such manifestations
of the physical forces, in their various permutations and combinations,
as may also be witnessed, under appropriate conditions, in non-living
things.

When we attempt to separate our purely morphological or “purely
embryological” studies from physiological and physical investigations,
we tend _ipso facto_ to regard each particular structure and
configuration as an attribute, or a particular “character,” of this or
that particular organism. From this assumption we are apt to go on to
the drawing of new conclusions or the framing of new theories as to the
ancestral history, the classificatory position, the natural affinities
of the several organisms: in fact, to apply our embryological knowledge
mainly, and at times exclusively, to the study of _phylogeny_.
When we find, as we are not long of finding, that our phylogenetic
hypotheses, as drawn from embryology, become complex and unwieldy, we
are nevertheless reluctant to admit that the whole method, with its
fundamental postulates, is at fault. And yet nothing short of this
would seem to be the case, in regard to the earlier phases at least
of embryonic development. All the evidence at hand goes, as it seems
to me, to shew that embryological data, prior to and even long after
the epoch of segmentation, are essentially a subject for physiological
and physical investigation and have but the very slightest link with
the problems of systematic or zoological classification. Comparative
embryology has its own facts to classify, and its own methods and
principles of classification. Thus we may classify eggs according to
the presence or absence, the paucity or abundance, of their associated
food-yolk, the chromosomes according to their form and their number,
the segmentation according to its various “types,” radial, bilateral,
spiral, and so forth. But we have little right to expect, and in
point of fact we shall very seldom and (as it were) only accidentally
find, that these embryological categories coincide with the lines of
“natural” or “phylogenetic” classification which have been arrived at
by the systematic zoologist.

――――――――――

The cell, which Goodsir spoke of as a “centre of force,” is in {197}
reality a “sphere of action” of certain more or less localised
forces; and of these, surface-tension is the particular force which
is especially responsible for giving to the cell its outline and its
morphological individuality. The partially segmented differs from the
totally segmented egg, the unicellular Infusorian from the minute
multicellular Turbellarian, in the intensity and the range of those
surface-tensions which in the one case succeed and in the other fail to
form a visible separation between the “cells.” Adam Sedgwick used to
call attention to the fact that very often, even in eggs that appear
to be totally segmented, it is yet impossible to discover an actual
separation or cleavage, through and through between the cells which
on the surface of the egg are so clearly delimited; so far and no
farther have the physical forces effectuated a visible “cleavage.” The
vacuolation of the protoplasm in _Actinophrys_ or _Actinosphaerium_
is due to localised surface-tensions, quite irrespective of the
multinuclear nature of the latter organism. In short, the boundary
walls due to surface-tension may be present or may be absent with or
without the delimination of the other specific fields of force which
are usually correlated with these boundaries and with the independent
individuality of the cells. What we may safely admit, however, is
that one effect of these circumscribed fields of force is usually
such a separation or segregation of the protoplasmic constituents,
the more fluid from the less fluid and so forth, as to give a field
where surface-tension may do its work and bring a visible boundary
into being. When the formation of a “surface” is once effected, its
physical condition, or phase, will be bound to differ notably from that
of the interior of the cell, and under appropriate chemical conditions
the formation of an actual cell-wall, cellulose or other, is easily
intelligible. To this subject we shall return again, in another chapter.

From the moment that we enter on a dynamical conception of the cell,
we perceive that the old debates were in vain as to what visible
portions of the cell were active or passive, living or non-living.
For the manifestations of force can only be due to the _interaction_
of the various parts, to the transference of energy from one to
another. Certain properties may be manifested, certain functions may
be carried on, by the protoplasm apart {198} from the nucleus; but
the interaction of the two is necessary, that other and more important
properties or functions may be manifested. We know, for instance, that
portions of an Infusorian are incapable of regenerating lost parts
in the absence of a nucleus, while nucleated pieces soon regain the
specific form of the organism: and we are told that reproduction by
fission cannot be _initiated_, though apparently all its later steps
can be carried on, independently of nuclear action. Nor, as Verworn
pointed out, can the nucleus possibly be regarded as the “sole vehicle
of inheritance,” since only in the conjunction of cell and nucleus do
we find the essentials of cell-life. “Kern und Protoplasma sind nur
_vereint_ lebensfähig,” as Nussbaum said. Indeed we may, with E. B.
Wilson, go further, and say that “the terms ‘nucleus’ and ‘cell-body’
should probably be regarded as only topographical expressions denoting
two differentiated areas in a common structural basis.”

Endless discussion has taken place regarding the centrosome, some
holding that it is a specific and essential structure, a permanent
corpuscle derived from a similar pre-existing corpuscle, a “fertilising
element” in the spermatozoon, a special “organ of cell-division,”
a material “dynamic centre” of the cell (as Van Beneden and Boveri
call it); while on the other hand, it is pointed out that many cells
live and multiply without any visible centrosomes, that a centrosome
may disappear and be created anew, and even that under artificial
conditions abnormal chemical stimuli may lead to the formation of
new centrosomes. We may safely take it that the centrosome, or the
“attraction sphere,” is essentially a “centre of force,” and that this
dynamic centre may or may not be constituted by (but will be very apt
to produce) a concrete and visible concentration of matter.

It is far from correct to say, as is often done, that the cell-wall,
or cell-membrane, belongs “to the passive products of protoplasm
rather than to the living cell itself”; or to say that in the animal
cell, the cell-wall, because it is “slightly developed,” is relatively
unimportant compared with the important role which it assumes in
plants. On the contrary, it is quite certain that, whether visibly
differentiated into a semi-permeable membrane, or merely constituted by
a liquid film, the surface of the cell is the seat of {199} important
forces, capillary and electrical, which play an essential part in
the dynamics of the cell. Even in the thickened, largely solidified
cellulose wall of the plant-cell, apart from the mechanical resistances
which it affords, the osmotic forces developed in connection with it
are of essential importance.

But if the cell acts, after this fashion, as a whole, each part
interacting of necessity with the rest, the same is certainly true of
the entire multicellular organism: as Schwann said of old, in very
precise and adequate words, “the whole organism subsists only by means
of the _reciprocal action_ of the single elementary parts[256].”

As Wilson says again, “the physiological autonomy of the individual
cell falls into the background ... and the apparently composite
character which the multicellular organism may exhibit is owing to
a secondary distribution of its energies among local centres of
action[257].”

It is here that the homology breaks down which is so often drawn, and
overdrawn, between the unicellular organism and the individual cell of
the metazoon[258].

Whitman, Adam Sedgwick[259], and others have lost no opportunity of
warning us against a too literal acceptation of the cell-theory,
against the view that the multicellular organism is a colony (or,
as Haeckel called it (in the case of the plant), a “republic”) of
independent units of life[260]. As Goethe said long ago, “Das
lebendige ist zwar in Elemente {200} zerlegt, aber man kann es aus
diesen nicht wieder zusammenstellen und beleben;” the dictum of the
_Cellularpathologie_ being just the opposite, “Jedes Thier erscheint
als eine Summe vitaler Einheiten, von denen _jede den vollen Charakter
des Lebens an sich trägt_.”

Hofmeister and Sachs have taught us that in the plant the growth of
the mass, the growth of the organ, is the primary fact, that “cell
formation is a phenomenon very general in organic life, but still only
of secondary significance.” “Comparative embryology” says Whitman,
“reminds us at every turn that the organism dominates cell-formation,
using for the same purpose one, several, or many cells, massing its
material and directing its movements and shaping its organs, as if
cells did not exist[261].” So Rauber declared that, in the whole world
of organisms, “das Ganze liefert die Theile, nicht die Theile das
Ganze: letzteres setzt die Theile zusammen, nicht diese jenes[262].”
And on the botanical side De Bary has summed up the matter in an
aphorism, “Die Pflanze bildet Zellen, nicht die Zelle bildet Pflanzen.”

Discussed almost wholly from the concrete, or morphological point
of view, the question has for the most part been made to turn on
whether actual protoplasmic continuity can be demonstrated between
one cell and another, whether the organism be an actual reticulum, or
syncytium. But from the dynamical point of view the question is much
simpler. We then deal not with material continuity, not with little
bridges of connecting protoplasm, but with a continuity of forces, a
comprehensive field of force, which runs through and through the entire
organism and is by no means restricted in its passage to a protoplasmic
continuum. And such a continuous field of force, somehow shaping the
whole organism, independently of the number, magnitude and form of the
individual cells, which enter, like a froth, into its fabric, seems to
me certainly and obviously to exist. As Whitman says, “the fact that
physiological unity is not broken by cell-boundaries is confirmed in so
many ways that it must be accepted as one of the fundamental truths of
biology[263].”

{201}




CHAPTER V

THE FORMS OF CELLS


Protoplasm, as we have already said, is a fluid or rather a semifluid
substance, and we need not pause here to attempt to describe the
particular properties of the semifluid, colloid, or jelly-like
substances to which it is allied; we should find it no easy matter. Nor
need we appeal to precise theoretical definitions of fluidity, lest
we come into a debateable land. It is in the most general sense that
protoplasm is “fluid.” As Graham said (of colloid matter in general),
“its softness _partakes of fluidity_, and enables the colloid to become
a vehicle for liquid diffusion, like water itself[264].” When we can
deal with protoplasm in sufficient quantity we see it flow; particles
move freely through it, air-bubbles and liquid droplets shew round
or spherical within it; and we shall have much to say about other
phenomena manifested by its own surface, which are those especially
characteristic of liquids. It may encompass and contain solid bodies,
and it may “secrete” within or around itself solid substances; and very
often in the complex living organism these solid substances formed
by the living protoplasm, like shell or nail or horn or feather, may
remain when the protoplasm which formed them is dead and gone; but the
protoplasm itself is fluid or semifluid, and accordingly permits of
free (though not necessarily rapid) _diffusion_ and easy _convection_
of particles within itself. This simple fact is of elementary
importance in connection with form, and with what appear at first sight
to be common characteristics or peculiarities of the forms of living
things.

The older naturalists, in discussing the differences between inorganic
and organic bodies, laid stress upon the fact or statement that the
former grow by “agglutination,” and the latter by {202} what they
termed “intussusception.” The contrast is true, rather, of solid as
compared with jelly-like bodies of all kinds, living or dead, the great
majority of which as it so happens, but by no means all, are of organic
origin.

A crystal “grows” by deposition of new molecules, one by one and
layer by layer, superimposed or aggregated upon the solid substratum
already formed. Each particle would seem to be influenced, practically
speaking, only by the particles in its immediate neighbourhood, and
to be in a state of freedom and independence from the influence,
either direct or indirect, of its remoter neighbours. As Lord Kelvin
and others have explained the formation and the resulting forms of
crystals, so we believe that each added particle takes up its position
in relation to its immediate neighbours already arranged, generally in
the holes and corners that their arrangement leaves, and in closest
contact with the greatest number[265]. And hence we may repeat or
imitate this process of arrangement, with great or apparently even with
precise accuracy (in the case of the simpler crystalline systems), by
piling up spherical pills or grains of shot. In so doing, we must have
regard to the fact that each particle must drop into the place where it
can go most easily, or where no easier place offers. In more technical
language, each particle is free to take up, and does take up, its
position of least potential energy relative to those already deposited;
in other words, for each particle motion is induced until the energy
of the system is so distributed that no tendency or resultant force
remains to move it more. The application of this principle has been
shewn to lead to the production of _planes_[266] (in all cases where
by the limitation of material, surfaces _must_ occur); and where we
have planes, straight edges and solid angles must obviously also occur;
and, if equilibrium is {203} to follow, must occur symmetrically. Our
piling up of shot, or manufacture of mimic crystals, gives us visible
demonstration that the result is actually to obtain, as in the natural
crystal, plane surfaces and sharp angles, symmetrically disposed.

But the living cell grows in a totally different way, very much
as a piece of glue swells up in water, by “imbibition,” or by
interpenetration into and throughout its entire substance. The
semifluid colloid mass takes up water, partly to combine chemically
with its individual molecules[267], partly by physical diffusion into
the interstices between these molecules, and partly, as it would seem,
in other ways; so that the entire phenomenon is a very complex and
even an obscure one. But, so far as we are concerned, the net result
is a very simple one. For the equilibrium or tendency to equilibrium
of fluid pressure in all parts of its interior while the process of
imbibition is going on, the constant rearrangement of its fluid mass,
the contrast in short with the crystalline method of growth where each
particle comes to rest to move (relatively to the whole) no more, lead
the mass of jelly to swell up, very much as a bladder into which we
blow air, and so, by a _graded_ and harmonious distribution of forces,
to assume everywhere a rounded and more or less bubble-like external
form[268]. So, when the same school of older naturalists called
attention to a new distinction or contrast of form between the organic
and inorganic objects, in that the contours of the former tended to
roundness and curvature, and those of the latter to be bounded by
straight lines, planes and sharp angles, we see that this contrast was
not a new and different one, but only another aspect of their former
statement, and an immediate consequence of the difference between the
processes of agglutination and intussusception.

This common and general contrast between the form of the crystal on
the one hand, and of the colloid or of the organism on the other, must
by no means be pressed too far. For Lehmann, {204} in his great work on
so-called Fluid Crystals[269], to which we shall afterwards return, has
shewn how, under certain circumstances, surface-tension phenomena may
coexist with crystallisation, and produce a form of minimal potential
which is a resultant of both: the fact being that the bonds maintaining
the crystalline arrangement are now so much looser than in the solid
condition that the tendency to least total surface-area is capable
of being satisfied. Thus the phenomenon of “liquid crystallisation”
does not destroy the distinction between crystalline and colloidal
forms, but gives added unity and continuity to the whole series of
phenomena[270]. Lehmann has also demonstrated phenomena within the
crystal, known for instance as transcrystallisation, which shew us that
we must not speak unguardedly of the growth of crystals as limited to
deposition upon a surface, and Bütschli has already pointed out the
possible great importance to the biologist of the various phenomena
which Lehmann has described[271].

So far then, as growth goes on, unaffected by pressure or other
external force, the fluidity of protoplasm, its mobility internal and
external, and the manner in which particles move with comparative
freedom from place to place within, all manifestly tend to the
production of swelling, rounded surfaces, and to their great
predominance over plane surfaces in the contour of the organism. These
rounded contours will tend to be preserved, for a while, in the case of
naked protoplasm by its viscosity, and in the presence of a cell-wall
by its very lack of fluidity. In a general way, the presence of curved
boundary surfaces will be especially obvious in the unicellular
organisms, and still more generally in the _external_ forms of all
organisms; and wherever mutual pressure between adjacent cells, or
other adjacent parts, has not come into play to flatten the rounded
surfaces into planes.

But the rounded contours that are assumed and exhibited by {205} a
piece of hard glue, when we throw it into water and see it expand as it
sucks the water up, are not nearly so regular or so beautiful as are
those which appear when we blow a bubble, or form a drop, or pour water
into a more or less elastic bag. For these curving contours depend upon
the properties of the bag itself, of the film or membrane that contains
the mobile gas, or that contains or bounds the mobile liquid mass. And
hereby, in the case of the fluid or semifluid mass, we are introduced
to the subject of _surface tension_: of which indeed we have spoken in
the preceding chapter, but which we must now examine with greater care.

――――――――――

Among the forces which determine the forms of cells, whether they
be solitary or arranged in contact with one another, this force of
surface-tension is certainly of great, and is probably of paramount
importance. But while we shall try to separate out the phenomena which
are directly due to it, we must not forget that, in each particular
case, the actual conformation which we study may be, and usually is,
the more or less complex resultant of surface tension acting together
with gravity, mechanical pressure, osmosis, or other physical forces.

Surface tension is that force by which we explain the form of a drop
or of a bubble, of the surfaces external and internal of a “froth” or
collocation of bubbles, and of many other things of like nature and in
like circumstances[272]. It is a property of liquids (in the sense at
least with which our subject is concerned), and it is manifested at or
very near the surface, where the liquid comes into contact with another
liquid, a solid or a gas. We note here that the term _surface_ is to
be interpreted in a wide sense; for wherever we have solid particles
imbedded in a fluid, wherever we have a non-homogeneous fluid or
semi-fluid such as a particle {206} of protoplasm, wherever we have
the presence of “impurities,” as in a mass of molten metal, there we
have always to bear in mind the existence of “surfaces” and of surface
tensions, not only on the exterior of the mass but also throughout its
interstices, wherever like meets unlike.

Surface tension is due to molecular force, to force that is to
say arising from the action of one molecule upon another, and it
is accordingly exerted throughout a small thickness of material,
comparable to the range of the molecular forces. We imagine that within
the interior of the liquid mass such molecular interactions negative
one another: but that at and near the free surface, within a layer or
film approximately equal to the range of the molecular force, there
must be a lack of such equilibrium and consequently a manifestation of
force.

The action of the molecular forces has been variously explained. But
one simple explanation (or mode of statement) is that the molecules
of the surface layer (whose thickness is definite and constant) are
being constantly attracted into the interior by those which are more
deeply situated, and that consequently, as molecules keep quitting the
surface for the interior, the bulk of the latter increases while the
surface diminishes; and the process continues till the surface itself
has become a minimum, the _surface-shrinkage_ exhibiting itself as a
_surface-tension_. This is a sufficient description of the phenomenon
in cases where a portion of liquid is subject to no other than _its
own molecular forces_, and (since the sphere has, of all solids, the
smallest surface for a given volume) it accounts for the spherical form
of the raindrop, of the grain of shot, or of the living cell in many
simple organisms. It accounts also, as we shall presently see, for a
great number of much more complicated forms, manifested under less
simple conditions.

Let us here briefly note that surface tension is, in itself, a
comparatively small force, and easily measurable: for instance that
of water is equivalent to but a few grains per linear inch, or a few
grammes per metre. But this small tension, when it exists in a _curved_
surface of very great curvature, gives rise to a very great pressure
directed towards the centre of curvature. We can easily calculate this
pressure, and so satisfy ourselves that, when the radius of curvature
is of molecular dimensions, the {207} pressure is of the magnitude
of thousands of atmospheres,—a conclusion which is supported by other
physical considerations.

The contraction of a liquid surface and other phenomena of surface
tension involve the doing of work, and the power to do work is what
we call energy. It is obvious, in such a simple case as we have just
considered, that the whole energy of the system is diffused throughout
its molecules; but of this whole stock of energy it is only that
part which comes into play at or very near to the surface which
normally manifests itself in work, and hence we may speak (though
the term is open to some objections) of a specific _surface energy_.
The consideration of surface energy, and of the manner in which its
amount is increased and multiplied by the multiplication of surfaces
due to the subdivision of the organism into cells, is of the highest
importance to the physiologist; and even the morphologist cannot wholly
pass it by, if he desires to study the form of the cell in its relation
to the phenomena of surface tension or “capillarity.” The case has
been set forth with the utmost possible lucidity by Tait and by Clerk
Maxwell, on whose teaching the following paragraphs are based: they
having based their teaching upon that of Gauss,—who rested on Laplace.

Let _E_ be the whole potential energy of a mass _M_ of liquid; let
_e__{0} be the energy per unit mass of the interior liquid (we may call
it the _internal energy_); and let _e_ be the energy per unit mass for
a layer of the skin, of surface _S_, of thickness _t_, and density
ρ (_e_ being what we call the _surface energy_). It is obvious that
the total energy consists of the internal _plus_ the surface energy,
and that the former is distributed through the whole mass, minus its
surface layers. That is to say, in mathematical language,

 _E_ = (_M_ − _S_ ⋅ Σ _t_ ρ) _e__{0} + _S_ ⋅ Σ _t_ ρ _e_.

But this is equivalent to writing:

 = _M_ _e__{0} + _S_ ⋅ Σ _t_ ρ(_e_ − _e__{0});

and this is as much as to say that the total energy of the system may
be taken to consist of two portions, one uniform throughout the whole
mass, and another, which is proportional on the one hand to the amount
of surface, and on the other hand is proportional to the difference
between _e_ and _e__{0}, that is to say to the difference between the
unit values of the internal and the surface energy. {208}

It was Gauss who first shewed after this fashion how, from the mutual
attractions between all the particles, we are led to an expression
which is what we now call the _potential energy_ of the system; and we
know, as a fundamental theorem of dynamics, that the potential energy
of the system tends to a minimum, and in that minimum finds, as a
matter of course, its stable equilibrium.

――――――――――

We see in our last equation that the term _M_ _e__{0} is irreducible,
save by a reduction of the mass itself. But the other term may be
diminished (1) by a reduction in the area of surface, _S_, or (2) by
a tendency towards equality of _e_ and _e__{0}, that is to say by a
diminution of the specific surface energy, _e_.

These then are the two methods by which the energy of the system will
manifest itself in work. The one, which is much the more important for
our purposes, leads always to a diminution of surface, to the so-called
“principle of minimal areas”; the other, which leads to the lowering
(under certain circumstances) of surface tension, is the basis of the
theory of Adsorption, to which we shall have some occasion to refer
as the _modus operandi_ in the development of a cell-wall, and in a
variety of other histological phenomena. In the technical phraseology
of the day, the “capacity factor” is involved in the one case, and the
“intensity factor” in the other.

Inasmuch as we are concerned with the form of the cell it is the
former which becomes our main postulate: telling us that the energy
equations of the surface of a cell, or of the free surfaces of cells
partly in contact, or of the partition-surfaces of cells in contact
with one another or with an adjacent solid, all indicate a minimum
of potential energy in the system, by which the system is brought,
_ipso facto_, into equilibrium. And we shall not fail to observe, with
something more than mere historical interest and curiosity, how deeply
and intrinsically there enter into this whole class of problems the
“principle of least action” of Maupertuis, the “_lineae curvae maximi
minimive proprietate gaudentes_” of Euler, by which principles these
old natural philosophers explained correctly a multitude of phenomena,
and drew the lines whereon the foundations of great part of modern
physics are well and truly laid. {209}

In all cases where the principle of maxima and minima comes into play,
as it conspicuously does in the systems of liquid films which are
governed by the laws of surface-tension, the figures and conformations
produced are characterised by obvious and remarkable _symmetry_. Such
symmetry is in a high degree characteristic of organic forms, and is
rarely absent in living things,—save in such cases as amoeba, where
the equilibrium on which symmetry depends is likewise lacking. And if
we ask what physical equilibrium has to do with formal symmetry and
regularity, the reason is not far to seek; nor can it be put better
than in the following words of Mach’s[273]. “In every symmetrical
system every deformation that tends to destroy the symmetry is
complemented by an equal and opposite deformation that tends to restore
it. In each deformation positive and negative work is done. One
condition, therefore, though not an absolutely sufficient one, that a
maximum or minimum of work corresponds to the form of equilibrium, is
thus supplied by symmetry. Regularity is successive symmetry. There is
no reason, therefore, to be astonished that the forms of equilibrium
are often symmetrical and regular.”

――――――――――

As we proceed in our enquiry, and especially when we approach the
subject of _tissues_, or agglomerations of cells, we shall have from
time to time to call in the help of elementary mathematics. But
already, with very little mathematical help, we find ourselves in a
position to deal with some simple examples of organic forms.

When we melt a stick of sealing-wax in the flame, surface tension
(which was ineffectively present in the solid but finds play in the
now fluid mass), rounds off its sharp edges into curves, so striving
towards a surface of minimal area; and in like manner, by melting the
tip of a thin rod of glass, Leeuwenhoek made the little spherical beads
which served him for a microscope[274]. When any drop of protoplasm,
either over all its surface or at some free end, as at the extremity
of the pseudopodium of an amoeba, is {210} seen likewise to “round
itself off,” that is not an effect of “vital contractility,” but (as
Hofmeister shewed so long ago as 1867) a simple consequence of surface
tension; and almost immediately afterwards Engelmann[275] argued on the
same lines, that the forces which cause the contraction of protoplasm
in general may “be just the same as those which tend to make every
non-spherical drop of fluid become spherical!” We are not concerned
here with the many theories and speculations which would connect the
phenomena of surface tension with contractility, muscular movement or
other special _physiological_ functions, but we find ample room to
trace the operation of the same cause in producing, under conditions of
rest and equilibrium, certain definite and inevitable forms of surface.

It is however of great importance to observe that the living cell is
one of those cases where the phenomena of surface tension are by no
means limited to the _outer_ surface; for within the heterogeneous
substance of the cell, between the protoplasm and its nuclear and
other contents, and in the alveolar network of the cytoplasm itself
(so far as that “alveolar structure” is actually present in life), we
have a multitude of interior surfaces; and, especially among plants,
we may have a large inner surface of “interfacial” contact, where the
protoplasm contains cavities or “vacuoles” filled with a different and
more fluid material, the “cell-sap.” Here we have a great field for
the development of surface tension phenomena: and so long ago as 1865,
Nägeli and Schwendener shewed that the streaming currents of plant
cells might be very plausibly explained by this phenomenon. Even ten
years earlier, Weber had remarked upon the resemblance between these
protoplasmic streamings and the streamings to be observed in certain
inanimate drops, for which no cause but surface tension could be
assigned[276].

The case of amoeba, though it is an elementary case, is at the same
time a complicated one. While it remains “amoeboid,” it is never at
rest or in equilibrium; it is always moving, from one to another of its
protean changes of configuration; its surface tension is constantly
varying from point to point. Where the {211} surface tension is
greater, that portion of the surface will contract into spherical or
spheroidal forms; where it is less the surface will correspondingly
extend. While generally speaking the surface energy has a minimal
value, it is not necessarily constant. It may be diminished by a
rise of temperature; it may be altered by contact with adjacent
substances[277], by the transport of constituent materials from the
interior to the surface, or again by actual chemical and fermentative
change. Within the cell, the surface energies developed about its
heterogeneous contents will constantly vary as these contents are
affected by chemical metabolism. As the colloid materials are broken
down and as the particles in suspension are diminished in size the
“free surface energy” will be increased, but the osmotic energy will
be diminished[278]. Thus arise the various fluctuations of surface
tension and the various phenomena of amoeboid form and motion, which
Bütschli and others have reproduced or imitated by means of the fine
emulsions which constitute their “artificial amoebae.” A multitude
of experiments shew how extraordinarily delicate is the adjustment
of the surface tension forces, and how sensitive they are to the
least change of temperature or chemical state. Thus, on a plate
which we have warmed at one side, a drop of alcohol runs towards the
warm area, a drop of oil away from it; and a drop of water on the
glass plate exhibits lively movements when {212} we bring into its
neighbourhood a heated wire, or a glass rod dipped in ether. When we
find that a plasmodium of Aethalium, for instance, creeps towards a
damp spot, or towards a warm spot, or towards substances that happen
to be nutritious, and again creeps away from solutions of sugar or of
salt, we seem to be dealing with phenomena every one of which can be
paralleled by ordinary phenomena of surface tension[279]. Even the
soap-bubble itself is imperfectly in equilibrium, for the reason that
its film, like the protoplasm of amoeba or Aethalium, is an excessively
heterogeneous substance. Its surface tensions vary from point to
point, and chemical changes and changes of temperature increase and
magnify the variation. The whole surface of the bubble is in constant
movement as the concentrated portions of the soapy fluid make their way
outwards from the deeper layers; it thins and it thickens, its colours
change, currents are set up in it, and little bubbles glide over it; it
continues in this state of constant movement, as its parts strive one
with another in all their interactions towards equilibrium[280].

In the case of the naked protoplasmic cell, as the amoeboid phase
is emphatically a phase of freedom and activity, of chemical and
physiological change, so, on the other hand, is the spherical form
indicative of a phase of rest or comparative inactivity. In the one
phase we see unequal surface tensions manifested in the creeping
movements of the amoeboid body, in the rounding off of the ends of the
pseudopodia, in the flowing out of its substance over a particle of
“food,” and in the current-motions in the interior of its mass; till
finally, in the other phase, when internal homogeneity and equilibrium
have been attained and the potential {213} energy of the system is for
the time being at a minimum, the cell assumes a rounded or spherical
form, passing into a state of “rest,” and (for a reason which we shall
presently see) becoming at the same time “encysted.”

[Illustration: Fig. 60.]

In a budding yeast-cell (Fig. 60), we see a more definite and
restricted change of surface tension. When a “bud” appears, whether
with or without actual growth by osmosis or otherwise of the mass,
it does so because at a certain part of the cell-surface the surface
tension has more or less suddenly diminished, and the area of that
portion expands accordingly; but in turn the surface tension of the
expanded area will make itself felt, and the bud will be rounded off
into a more or less spherical form.

The yeast-cell with its bud is a simple example of a principle which
we shall find to be very important. Our whole treatment of cell-form
in relation to surface-tension depends on the fact (which Errera was
the first to point out, or to give clear expression to) that the
_incipient_ cell-wall retains with but little impairment the properties
of a liquid film[281], and that the growing cell, in spite of the
membrane by which it has already begun to be surrounded, behaves very
much like a fluid drop. But even the ordinary yeast-cell shows, by its
ovoid and non-spherical form, that it has acquired its shape under
the influence of some force other than that uniform and symmetrical
surface-tension which would be productive of a sphere; and this or
any other asymmetrical form, once acquired, may be retained by virtue
of the solidification and consequent rigidity of the membranous wall
of the cell. Unless such rigidity ensue, it is plain that such a
conformation as that of the cell with its attached bud could not be
long retained, amidst the constantly varying conditions, as a figure
of even partial equilibrium. But as a matter of fact, the cell in this
case is not in equilibrium at all; it is in _process_ of budding, and
is slowly altering its shape by rounding off the bud. It is plain that
over its surface the surface-energies are unequally distributed, owing
to some heterogeneity of the substance; and to this matter we shall
afterwards return. In like manner the developing egg {214} through all
its successive phases of form is never in complete equilibrium; but
is merely responding to constantly changing conditions, by phases of
partial, transitory, unstable and conditional equilibrium.

It is obvious that there are innumerable solitary plant-cells, and
unicellular organisms in general, which, like the yeast-cell, do not
correspond to any of the simple forms that may be generated under the
influence of simple and homogeneous surface-tension; and in many cases
these forms, which we should expect to be unstable and transitory,
have become fixed and stable by reason of the comparatively sudden or
rapid solidification of the envelope. This is the case, for instance,
in many of the more complicated forms of diatoms or of desmids, where
we are dealing, in a less striking but even more curious way than in
the budding yeast-cell, not with one simple act of formation, but
with a complicated result of successive stages of localised growth,
interrupted by phases of partial consolidation. The original cell has
acquired or assumed a certain form, and then, under altering conditions
and new distributions of energy, has thickened here or weakened there,
and has grown out or tended (as it were) to branch, at particular
points. We can often, or indeed generally, trace in each particular
stage of growth or at each particular temporary growing point, the laws
of surface tension manifesting themselves in what is for the time being
a fluid surface; nay more, even in the adult and completed structure,
we have little difficulty in tracing and recognising (for instance
in the outline of such a desmid as Euastrum) the rounded lobes that
have successively grown or flowed out from the original rounded and
flattened cell. What we see in a many chambered foraminifer, such as
Globigerina or Rotalia, is just the same thing, save that it is carried
out in greater completeness and perfection. The little organism as a
whole is not a figure of equilibrium or of minimal area; but each new
bud or separate chamber is such a figure, conditioned by the forces of
surface tension, and superposed upon the complex aggregate of similar
bubbles after these latter have become consolidated one by one into a
rigid system.

――――――――――

Let us now make some enquiry regarding the various forms {215} which,
under the influence of surface tension, a surface can possibly assume.
In doing so, we are obviously limited to conditions under which other
forces are relatively unimportant, that is to say where the “surface
energy” is a considerable fraction of the whole energy of the system;
and this in general will be the case when we are dealing with portions
of liquid so small that their dimensions come within what we have
called the molecular range, or, more generally, in which the “specific
surface” is large[282]: in other words it will be small or minute
organisms, or the small cellular elements of larger organisms, whose
forms will be governed by surface-tension; while the general forms of
the larger organisms will be due to other and non-molecular forces.
For instance, a large surface of water sets itself level because here
gravity is predominant; but the surface of water in a narrow tube
is manifestly curved, for the reason that we are here dealing with
particles which are mutually within the range of each other’s molecular
forces. The same is the case with the cell-surfaces and cell-partitions
which we are presently to study, and the effect of gravity will
be especially counteracted and concealed when, as in the case of
protoplasm in a watery fluid, the object is immersed in a liquid of
nearly its own specific gravity.

We have already learned, as a fundamental law of surface-tension
phenomena, that a liquid film _in equilibrium_ assumes a form which
gives it a minimal area under the conditions to which it is subject.
And these conditions include (1) the form of the boundary, if such
exist, and (2) the pressure, if any, to which the film is subject;
which pressure is closely related to the volume, of air or of liquid,
which the film (if it be a closed one) may have to contain. In the
simplest of cases, when we take up a soap-film on a plane wire ring,
the film is exposed to equal atmospheric pressure on both sides, and it
obviously has its minimal area in the form of a plane. So long as our
wire ring lies in one plane (however irregular in outline), the film
stretched across it will still be in a plane; but if we bend the ring
so that it lies no longer in a plane, then our film will become curved
into a surface which may be extremely complicated, but is still the
smallest possible {216} surface which can be drawn continuously across
the uneven boundary.

The question of pressure involves not only external pressures acting on
the film, but also that which the film itself is capable of exerting.
For we have seen that the film is always contracting to its smallest
limits; and when the film is curved, this obviously leads to a pressure
directed inwards,—perpendicular, that is to say, to the surface of
the film. In the case of the soap-bubble, the uniform contraction
of whose surface has led to its spherical form, this pressure is
balanced by the pressure of the air within; and if an outlet be given
for this air, then the bubble contracts with perceptible force until
it stretches across the mouth of the tube, for instance the mouth of
the pipe through which we have blown the bubble. A precisely similar
pressure, directed inwards, is exercised by the surface layer of a
drop of water or a globule of mercury, or by the surface pellicle on a
portion or “drop” of protoplasm. Only we must always remember that in
the soap-bubble, or the bubble which a glass-blower blows, there is a
twofold pressure as compared with that which the surface-film exercises
on the drop of liquid of which it is a part; for the bubble consists
(unless it be so thin as to consist of a mere layer of molecules[283])
of a liquid layer, with a free surface within and another without, and
each of these two surfaces exercises its own independent and coequal
tension, and corresponding pressure[284].

If we stretch a tape upon a flat table, whatever be the tension of
the tape it obviously exercises no pressure upon the table below. But
if we stretch it over a _curved_ surface, a cylinder for instance, it
does exercise a downward pressure; and the more curved the surface the
greater is this pressure, that is to say the greater is this share
of the entire force of tension which is resolved in the downward
direction. In mathematical language, the pressure (_p_) varies directly
as the tension (_T_), and inversely as the radius of curvature (_R_):
that is to say, _p_ = _T_/_R_, per unit of surface. {217}

If instead of a cylinder, which is curved only in one direction,
we take a case where there are curvatures in two dimensions (as
for instance a sphere), then the effects of these must be simply
added to one another, and the resulting pressure _p_ is equal to
_T_/_R_ + _T_/_R′_ or _p_ = _T_(1/_R_ + 1/_R′_)[285].

And if in addition to the pressure _p_, which is due to surface
tension, we have to take into account other pressures, _p′_, _p″_,
etc., which are due to gravity or other forces, then we may say that
the _total pressure_, _P_ = _p′_ + _p″_ + _T_(1/_R_ + 1/_R′_). While
in some cases, for instance in speaking of the shape of a bird’s egg,
we shall have to take account of these extraneous pressures, in the
present part of our subject we shall for the most part be able to
neglect them.

Our equation is an equation of equilibrium. The resistance to
compression,—the pressure outwards,—of our fluid mass, is a constant
quantity (_P_); the pressure inwards, _T_(1/_R_ + 1/_R′_), is also
constant; and if (unlike the case of the mobile amoeba) the surface
be homogeneous, so that _T_ is everywhere equal, it follows that
throughout the whole surface 1/_R_ + 1/_R′_ = _C_ (a constant).

Now equilibrium is attained after the surface contraction has done
its utmost, that is to say when it has reduced the surface to the
smallest possible area; and so we arrive, from the physical side, at
the conclusion that a surface such that 1/_R_ + 1/_R′_ = _C_, in other
words a surface which has the same _mean curvature_ at all points, is
equivalent to a surface of minimal area: and to the same conclusion we
may also arrive through purely analytical mathematics. It is obvious
that the plane and the sphere are two examples of such surfaces, for in
both cases the radius of curvature is everywhere constant, being equal
to infinity in the case of the plane, and to some definite magnitude in
the case of the sphere.

From the fact that we may extend a soap-film across a ring of wire
however fantastically the latter may be bent, we realise that there
is no limit to the number of surfaces of minimal area which may be
constructed or may be imagined; and while some of these are very
complicated indeed, some, for instance a spiral helicoid screw, are
relatively very simple. But if we limit ourselves to {218} _surfaces
of revolution_ (that is to say, to surfaces symmetrical about an axis),
we find, as Plateau was the first to shew, that those which meet the
case are very few in number. They are six in all, namely the plane, the
sphere, the cylinder, the catenoid, the unduloid, and a curious surface
which Plateau called the nodoid.

These several surfaces are all closely related, and the passage from
one to another is generally easy. Their mathematical interrelation is
expressed by the fact (first shewn by Delaunay[286], in 1841) that
the plane curves by whose rotation they are generated are themselves
generated as “roulettes” of the conic sections.

Let us imagine a straight line upon which a circle, an ellipse or other
conic section rolls; the focus of the conic section will describe a
line in some relation to the fixed axis, and this line (or roulette),
rotating around the axis, will describe in space one or other of the
six surfaces of revolution with which we are dealing.

[Illustration: Fig. 61.]

If we imagine an ellipse so to roll over a line, either of its
foci will describe a sinuous or wavy line (Fig. 61B) at a distance
alternately maximal and minimal from the axis; and this wavy line,
by rotation about the axis, becomes the meridional line of the
surface which we call the _unduloid_. The more unequal the two axes
are of our ellipse, the more pronounced will be the sinuosity of the
described roulette. If the two axes be equal, then our ellipse becomes
a circle, and the path described by its rolling centre is a straight
line parallel to the axis (A); and obviously the solid of revolution
generated therefrom will be a _cylinder_. If one axis of our ellipse
vanish, while the other remain of finite length, then the ellipse
is reduced to a straight line, and its roulette will appear as a
succession of semicircles touching one another upon the axis (C); the
solid of revolution will be a series of equal _spheres_. If as before
one axis of the ellipse vanish, but the other be infinitely long, then
the curve described by the rotation {219} of this latter will be a
circle of infinite radius, i.e. a straight line infinitely distant
from the axis; and the surface of rotation is now a _plane_. If we
imagine one focus of our ellipse to remain at a given distance from the
axis, but the other to become infinitely remote, that is tantamount to
saying that the ellipse becomes transformed into a parabola; and by the
rolling of this curve along the axis there is described a catenary (D),
whose solid of revolution is the _catenoid_.

Lastly, but this is a little more difficult to imagine, we have the
case of the hyperbola.

We cannot well imagine the hyperbola rolling upon a fixed straight
line so that its focus shall describe a continuous curve. But let
us suppose that the fixed line is, to begin with, asymptotic to one
branch of the hyperbola, and that the rolling proceed until the line
is now asymptotic to the other branch, that is to say touching it at
an infinite distance; there will then be mathematical continuity if
we recommence rolling with this second branch, and so in turn with
the other, when each has run its course. We shall see, on reflection,
that the line traced by one and the same focus will be an “elastic
curve” describing a succession of kinks or knots (E), and the solid
of revolution described by this meridional line about the axis is the
so-called _nodoid_.

The physical transition of one of these surfaces into another can be
experimentally illustrated by means of soap-bubbles, or better still,
after the method of Plateau, by means of a large globule of oil,
supported when necessary by wire rings, within a fluid of specific
gravity equal to its own.

To prepare a mixture of alcohol and water of a density precisely equal
to that of the oil-globule is a troublesome matter, and a method
devised by Mr C. R. Darling is a great improvement on Plateau’s[287].
Mr Darling uses the oily liquid orthotoluidene, which does not mix with
water, has a beautiful and conspicuous red colour, and has precisely
the same density as water when both are kept at a temperature of 24° C.
We have therefore only to run the liquid into water at this temperature
in order to produce beautifully spherical drops of any required size:
and by adding {220} a little salt to the lower layers of water, the
drop may be made to float or rest upon the denser liquid.

We have already seen that the soap-bubble, spherical to begin with,
is transformed into a plane when we relieve its internal pressure and
let the film shrink back upon the orifice of the pipe. If we blow
a small bubble and then catch it up on a second pipe, so that it
stretches between, we may gradually draw the two pipes apart, with
the result that the spheroidal surface will be gradually flattened
in a longitudinal direction, and the bubble will be transformed into
a cylinder. But if we draw the pipes yet farther apart, the cylinder
will narrow in the middle into a sort of hourglass form, the increasing
curvature of its transverse section being balanced by a gradually
increasing _negative_ curvature in the longitudinal section. The
cylinder has, in turn, been converted into an unduloid. When we hold a
portion of a soft glass tube in the flame, and “draw it out,” we are
in the same identical fashion converting a cylinder into an unduloid
(Fig. 62A); when on the other hand we stop the end and blow, we again
convert the cylinder into an unduloid (B), but into one which is now
positively, while the former was negatively curved. The two figures are
essentially the same, save that the two halves of the one are reversed
in the other.

[Illustration: Fig. 62.]

That spheres, cylinders and unduloids are of the commonest occurrence
among the forms of small unicellular organism, or of individual cells
in the simpler aggregates, and that in the processes of growth,
reproduction and development transitions are frequent from one of these
forms to another, is obvious to the naturalist, and we shall deal
presently with a few illustrations of these phenomena.

But before we go further in this enquiry, it will be necessary to
consider, to some small extent at least, the _curvatures_ of the six
different surfaces, that is to say, to determine what modification
{221} is required, in each case, of the general equation which applies
to them all. We shall find that with this question is closely connected
the question of the _pressures_ exercised by, or impinging on the film,
and also the very important question of the limitations which, from the
nature of the case, exist to prevent the extension of certain of the
figures beyond certain bounds. The whole subject is mathematical, and
we shall only deal with it in the most elementary way.

We have seen that, in our general formula, the expression
1/_R_ + 1/_R′_ = _C_, a constant; and that this is, in all cases, the
condition of our surface being one of minimal area. In other words, it
is always true for one and all of the six surfaces which we have to
consider. But the constant _C_ may have any value, positive, negative,
or nil.

In the case of the plane, where _R_ and _R′_ are both infinite, it is
obvious that 1/_R_ + 1/_R′_ = 0. The expression therefore vanishes,
and our dynamical equation of equilibrium becomes _P_ = _p_. In short,
we can only have a plane film, or we shall only find a plane surface
in our cell, when on either side thereof we have equal pressures or no
pressure at all. A simple case is the plane partition between two equal
and similar cells, as in a filament of spirogyra.

In the case of the sphere, the radii are all equal, _R_ = _R′_; they
are also positive, and _T_ (1/_R_ + 1/_R′_), or 2 _T_/_R_, is a
positive quantity, involving a positive pressure _P_, on the other side
of the equation.

In the cylinder, one radius of curvature has the finite and positive
value _R_; but the other is infinite. Our formula becomes _T_/_R_,
to which corresponds a positive pressure _P_, supplied by the
surface-tension as in the case of the sphere, but evidently of just
half the magnitude developed in the latter case for a given value of
the radius _R_.

The catenoid has the remarkable property that its curvature in one
direction is precisely equal and opposite to its curvature in the
other, this property holding good for all points of the surface. That
is to say, _R_ = −_R′_; and the expression becomes

 (1/_R_ + 1/_R′_) = (1/_R_ − 1/_R_) = 0;

in other words, the surface, as in the case of the plane, has _no
{222} curvature_, and exercises no pressure. There are no other
surfaces, save these two, which share this remarkable property; and it
follows, as a simple corollary, that we may expect at times to have
the catenoid and the plane coexisting, as parts of one and the same
boundary system; just as, in a cylindrical drop or cell, the cylinder
is capped by portions of spheres, such that the cylindrical and
spherical portions of the wall exert equal positive pressures.

In the unduloid, unlike the four surfaces which we have just been
considering, it is obvious that the curvatures change from one point
to another. At the middle of one of the swollen portions, or “beads,”
the two curvatures are both positive; the expression (1/_R_ + 1/_R′_)
is therefore positive, and it is also finite. The film, accordingly,
exercises a positive tension inwards, which must be compensated by a
finite and positive outward pressure _P_. At the middle of one of the
narrow necks, between two adjacent beads, there is obviously, in the
transverse direction, a much stronger curvature than in the former
case, and the curvature which balances it is now a negative one. But
the sum of the two must remain positive, as well as constant; and we
therefore see that the convex or positive curvature must always be
greater than the concave or negative curvature at the same point. This
is plainly the case in our figure of the unduloid.

The nodoid is, like the unduloid, a continuous curve which keeps
altering its curvature as it alters its distance from the axis; but
in this case the resultant pressure inwards is negative instead of
positive. But this curve is a complicated one, and a full discussion of
it would carry us beyond our scope.

[Illustration: Fig. 63.]

In one of Plateau’s experiments, a bubble of oil (protected from
gravity by the specific gravity of the surrounding fluid being
identical with its own) is balanced between two annuli. It may then be
brought to assume the form of Fig. 63, that is to say the form of a
cylinder with spherical ends; and there is then everywhere, owing to
the convexity of the surface film, a pressure inwards upon the fluid
contents of the bubble. If the surrounding liquid be ever so little
heavier or lighter than that which constitutes the drop, then the
conditions of equilibrium will be accordingly {223} modified, and the
cylindrical drop will assume the form of an unduloid (Fig. 64 A, B),
with its dilated portion below or above,

[Illustration: Fig. 64.]

as the case may be; and our cylinder may also, of course, be
converted into an unduloid either by elongating it further, or by
abstracting a portion of its oil, until at length rupture ensues and
the cylinder breaks up into two new spherical drops. In all cases
alike, the unduloid, like the original cylinder, will be capped by
spherical ends, which are the sign, and the consequence, of the
positive pressure produced by the curved walls of the unduloid. But
if our initial cylinder, instead of being tall, be a flat or dumpy
one (with certain definite relations of height to breadth), then new
phenomena may be exhibited. For now, if a little oil be cautiously
withdrawn from the mass by help of a small syringe, the cylinder may be
made to flatten down so that its upper and lower surfaces become plane;
which is of itself an indication that the pressure inwards is now
_nil_. But at the very moment when the upper and lower surfaces become
plane, it will be found that the sides curve inwards, in the fashion
shewn in Fig. 65B. This figure is a catenoid, which, as

[Illustration: Fig. 65.]

we have already seen, is, like the plane itself, a surface exercising
no pressure, and which therefore may coexist with the plane as part
of one and the same system. We may continue to withdraw more oil from
our bubble, drop by drop, and now the upper and lower surfaces dimple
down into concave portions of spheres, as the result of the _negative_
internal pressure; and thereupon the peripheral catenoid surface alters
its form (perhaps, on this small scale, imperceptibly), and becomes a
portion of a nodoid (Fig. 65A). {224} It represents, in fact, that
portion of the nodoid, which in Fig. 66 lies between such points as O,
P. While it is easy to

[Illustration: Fig. 66.]

draw the outline, or meridional section, of the nodoid (as in Fig.
66), it is obvious that the solid of revolution to be derived from
it, can never be realised in its entirety: for one part of the solid
figure would cut, or entangle with, another. All that we can ever do,
accordingly, is to realise isolated portions of the nodoid.

If, in a sequel to the preceding experiment of Plateau’s, we use
solid discs instead of annuli, so as to enable us to exert direct
mechanical pressure upon our globule of oil, we again begin by
adjusting the pressure of these discs so that the oil assumes the form
of a cylinder: our discs, that is to say, are adjusted to exercise
a mechanical pressure equal to what in the former case was supplied
by the surface-tension of the spherical caps or ends of the bubble.
If we now increase the pressure slightly, the peripheral walls will
become convexly curved, exercising a precisely corresponding pressure.
Under these circumstances the form assumed by the sides of our figure
will be that of a portion of an unduloid. If we increase the pressure
between the discs, the peripheral surface of oil will bulge out more
and more, and will presently constitute a portion of a sphere. But we
may continue the process yet further, and within certain limits we
shall find that the system remains perfectly stable. What is this new
curved surface which has arisen out of the sphere, as the latter was
produced from the unduloid? It is no other than a portion of a nodoid,
that part which in Fig. 66 lies between such limits as M and N. But
this surface, which is concave in both directions towards the surface
of the oil within, is exerting a pressure upon the latter, just as did
the sphere out of which a moment ago it was transformed; and we had
just stated, in considering the previous experiment, that the pressure
inwards exerted by the nodoid was a negative one. The explanation of
this seeming discrepancy lies in the simple fact that, if we follow the
outline {225} of our nodoid curve in Fig. 66 from O, P, the surface
concerned in the former case, to M, N, that concerned in the present,
we shall see that in the two experiments the surface of the liquid is
not homologous, but lies on the positive side of the curve in the one
case and on the negative side in the other.

――――――――――

Of all the surfaces which we have been describing, the sphere is the
only one which can enclose space; the others can only help to do so, in
combination with one another or with the sphere itself. Thus we have
seen that, in normal equilibrium, the cylindrical vesicle is closed at
either end by a portion of a sphere, and so on. Moreover the sphere is
not only the only one of our figures which can enclose a finite space;
it is also, of all possible figures, that which encloses the greatest
volume with the least area of surface; it is strictly and absolutely
the surface of minimal area, and it is therefore the form which will be
naturally assumed by a unicellular organism (just as by a raindrop),
when it is practically homogeneous and when, like Orbulina floating
in the ocean, its surroundings are likewise practically homogeneous
and symmetrical. It is only relatively speaking that all the rest are
surfaces _minimae areae_; they are so, that is to say, under the given
conditions, which involve various forms of pressure or restraint. Such
restraints are imposed, for instance, by the pipes or annuli with the
help of which we draw out our cylindrical or unduloid oil-globule or
soap-bubble; and in the case of the organic cell, similar restraints
are constantly supplied by solidification, partial or complete, local
or general, of the cell-wall.

Before we pass to biological illustrations of our surface-tension
figures, we have still another preliminary matter to deal with. We have
seen from our description of two of Plateau’s classical experiments,
that at some particular point one type of surface gives place to
another; and again, we know that, when we draw out our soap-bubble into
and then beyond a cylinder, there comes a certain definite point at
which our bubble breaks in two, and leaves us with two bubbles of which
each is a sphere, or a portion of a sphere. In short there are certain
definite limits to the _dimensions_ of our figures, within which limits
equilibrium is stable but at which it becomes unstable, and above which
it {226} breaks down. Moreover in our composite surfaces, when the
cylinder for instance is capped by two spherical cups or lenticular
discs, there is a well-defined ratio which regulates their respective
curvatures, and therefore their respective dimensions. These two
matters we may deal with together.

Let us imagine a liquid drop which by appropriate conditions has
been made to assume the form of a cylinder; we have already seen
that its ends will be terminated by portions of spheres. Since one
and the same liquid film covers the sides and ends of the drop (or
since one and the same delicate membrane encloses the sides and ends
of the cell), we assume the surface-tension (_T_) to be everywhere
identical; and it follows, since the internal fluid-pressure is also
everywhere identical, that the expression (1/_R_ + 1/_R′_) for the
cylinder is equal to the corresponding expression, which we may call
(1/_r_ + 1/_r′_), in the case of the terminal spheres. But in the
cylinder 1/_R′_ = 0, and in the sphere 1/_r_ = 1/_r′_. Therefore our
relation of equality becomes 1/_R_ = 2/_r_, or _r_ = 2 _R_; that is to
say, the sphere in question has just twice the radius of the cylinder
of which it forms a cap.

[Illustration: Fig. 67.]

And if _Ob_, the radius of the sphere, be equal to twice the radius
(_Oa_) of the cylinder, it follows that the angle _aOb_ is an angle of
60°, and _bOc_ is also an angle of 60°; that is to say, the arc _bc_
is equal to (1/3) π. In other words, the spherical disc which (under
the given conditions) caps our cylinder, is not a portion taken at
haphazard, but is neither more nor less than that portion of a sphere
which is subtended by a cone of 60°. Moreover, it is plain that the
height of the spherical cap, _de_,

 = _Ob_ − _ab_ = _R_ (2 − √3) = 0·27 _R_,

where _R_ is the radius of our cylinder, or one-half the radius of
our spherical cap: in other words the normal height of the spherical
cap over the end of the cylindrical cell is just a very little more
than one-eighth of the diameter of the cylinder, or of the radius of
the {227} sphere. And these are the proportions which we recognise,
under normal circumstances, in such a case as the cylindrical cell of
Spirogyra where its free end is capped by a portion of a sphere.

――――――――――

Among the many important theoretical discoveries which we owe to
Plateau, one to which we have just referred is of peculiar importance:
namely that, with the exception of the sphere and the plane, the
surfaces with which we have been dealing are only in complete
equilibrium within certain dimensional limits, or in other words, have
a certain definite limit of stability; only the plane and the sphere,
or any portions of a sphere, are perfectly stable, because they are
perfectly symmetrical, figures. For experimental demonstration, the
case of the cylinder is the simplest. If we produce a liquid film
having the form of a cylinder, either by

[Illustration: Fig. 68.]

drawing out a bubble or by supporting between two rings a globule of
oil, the experiment proceeds easily until the length of the cylinder
becomes just about three times as great as its diameter. But somewhere
about this limit the cylinder alters its form; it begins to narrow at
the waist, so passing into an unduloid, and the deformation progresses
quickly until at last our cylinder breaks in two, and its two halves
assume a spherical form. It is found, by theoretical considerations,
that the precise limit of stability is at the point when the length
of the cylinder is exactly equal to its circumference, that is to
say, when _L_ = 2π_R_, or when the ratio of length to diameter is
represented by π.

In the case of the catenoid, Plateau’s experimental procedure was
as follows. To support his globule of oil (in, as usual, a mixture
of alcohol and water of its own specific gravity), he used {228}
a pair of metal rings, which happened to have a diameter of 71
millimetres; and, in a series of experiments, he set these rings
apart at distances of 55, 49, 47, 45, and 43 mm. successively. In
each case he began by bringing his oil-globule into a cylindrical
form, by sucking superfluous oil out of the drop until this result was
attained; and always, for the reason with which we are now acquainted,
the cylindrical sides were associated with spherical ends to the
cylinder. On continuing to withdraw oil in the hope of converting
these spherical ends into planes, he found, naturally, that the sides
of the cylinder drew in to form a concave surface; but it was by no
means easy to get the extremities actually plane: and unless they
were so, thus indicating that the surface-pressure of the drop was
nil, the curvature of the sides could not be that of a catenoid. For
in the first experiment, when the rings were 55 mm. apart, as soon
as the convexity of the ends was to a certain extent diminished, it
spontaneously increased again; and the transverse constriction of the
globule correspondingly deepened, until at a certain point equilibrium
set in anew. Indeed, the more oil he removed, the more convex became
the ends, until at last the increasing transverse constriction led to
the breaking of the oil-globule into two. In the third experiment,
when the rings were 47 mm. apart, it was easy to obtain end-surfaces
that were actually plane, and they remained so even though more oil
was withdrawn, the transverse constriction deepening accordingly. Only
after a considerable amount of oil had been sucked up did the plane
terminal surface become gradually convex, and presently the narrow
waist, narrowing more and more, broke across in the usual way. Finally
in the fifth experiment, where the rings were still nearer together,
it was again possible to bring the ends of the oil-globule to a plane
surface, as in the third and fourth experiments, and to keep this
surface plane in spite of some continued withdrawal of oil. But very
soon the ends became gradually concave, and the concavity deepened as
more and more oil was withdrawn, until at a certain limit, the whole
oil-globule broke up in general disruption.

We learn from this that the limiting size of the catenoid was reached
when the distance of the supporting rings was to their diameter as 47
to 71, or, as nearly as possible, as two to three; {229} and as a
matter of fact it can be shewn that 2/3 is the true theoretical value.
Above this limit of 2/3, the inevitable convexity of the end-surfaces
shows that a positive pressure inwards is being exerted by the surface
film, and this teaches us that the sides of the figure actually
constitute not a catenoid but an unduloid, whose spontaneous changes
tend to a form of greater stability. Below the 2/3 limit the catenoid
surface is essentially unstable, and the form into which it passes
under certain conditions of disturbance such as that of the excessive
withdrawal of oil, is that of a nodoid (Fig. 65A).

The unduloid has certain peculiar properties as regards its limitations
of stability. But as to these we need mention two facts only: (1)
that when the unduloid, which we produce with our soap-bubble or our
oil-globule, consists of the figure containing a complete constriction,
it has somewhat wide limits of stability; but (2) if it contain the
swollen portion, then equilibrium is limited to the condition that the
figure consists simply of one complete unduloid, that is to say that
its ends are constituted by the narrowest portions, and its middle by
the widest portion of the entire curve. The theoretical proof of this
latter fact is difficult, but if we take the proof for granted, the
fact will serve to throw light on what we have learned regarding the
stability of the cylinder. For, when we remember that the meridional
section of our unduloid is generated by the rolling of an ellipse upon
a straight line in its own plane, we shall easily see that the length
of the entire unduloid is equal to the circumference of the generating
ellipse. As the unduloid becomes less and less sinuous in outline, it
gradually approaches, and in time reaches, the form of a cylinder;
and correspondingly, the ellipse which generated it has its foci more
and more approximated until it passes into a circle. The cylinder
of a length equal to the circumference of its generating circle is
therefore precisely homologous to an unduloid whose length is equal to
the circumference of its generating ellipse; and this is just what we
recognise as constituting one complete segment of the unduloid.

――――――――――

While the figures of equilibrium which are at the same time surfaces
of revolution are only six in number, there is an infinite {230}
number of figures of equilibrium, that is to say of surfaces of
constant mean curvature, which are not surfaces of revolution; and it
can be shewn mathematically that any given contour can be occupied
by a finite portion of some one such surface, in stable equilibrium.
The experimental verification of this theorem lies in the simple fact
(already noted) that however we may bend a wire into a closed curve,
plane or not plane, we may always, under appropriate precautions, fill
the entire area with an unbroken film.

Of the regular figures of equilibrium, that is to say surfaces of
constant mean curvature, apart from the surfaces of revolution which
we have discussed, the helicoid spiral is the most interesting to
the biologist. This is a helicoid generated by a straight line
perpendicular to an axis, about which it turns at a uniform rate while
at the same time it slides, also uniformly, along this same axis. At
any point in this surface, the curvatures are equal and of opposite
sign, and the sum of the curvatures is accordingly nil. Among what are
called “ruled surfaces” (which we may describe as surfaces capable of
being defined by a system of stretched strings), the plane and the
helicoid are the only two whose mean curvature is null, while the
cylinder is the only one whose curvature is finite and constant. As
this simplest of helicoids corresponds, in three dimensions, to what
in two dimensions is merely a plane (the latter being generated by
the rotation of a straight line about an axis without the superadded
gliding motion which generates the helicoid), so there are other and
much more complicated helicoids which correspond to the sphere, the
unduloid and the rest of our figures of revolution, the generating
planes of these latter being supposed to wind spirally about an axis.
In the case of the cylinder it is obvious that the resulting figure is
indistinguishable from the cylinder itself. In the case of the unduloid
we obtain a grooved spiral, such as we may meet with in nature (for
instance in Spirochætes, _Bodo gracilis_, etc.), and which accordingly
it is of interest to us to be able to recognise as a surface of minimal
area or constant curvature.

The foregoing considerations deal with a small part only of the
theory of surface tension, or of capillarity: with that part, namely,
which relates to the forms of surface which are {231} capable of
subsisting in equilibrium under the action of that force, either of
itself or subject to certain simple constraints. And as yet we have
limited ourselves to the case of a single surface, or of a single
drop or bubble, leaving to another occasion a discussion of the forms
assumed when such drops or vesicles meet and combine together. In
short, what we have said may help us to understand the form of a
_cell_,—considered, as with certain limitations we may legitimately
consider it, as a liquid drop or liquid vesicle; the conformation of a
_tissue_ or cell-aggregate must be dealt with in the light of another
series of theoretical considerations. In both cases, we can do no more
than touch upon the fringe of a large and difficult subject. There are
many forms capable of realisation under surface tension, and many of
them doubtless to be recognised among organisms, which we cannot touch
upon in this elementary account. The subject is a very general one;
it is, in its essence, more mathematical than physical; it is part of
the mathematics of surfaces, and only comes into relation with surface
tension, because this physical phenomenon illustrates and exemplifies,
in a concrete way, most of the simple and symmetrical conditions
with which the general mathematical theory is capable of dealing.
And before we pass to illustrate by biological examples the physical
phenomena which we have described, we must be careful to remember
that the physical conditions which we have hitherto presupposed will
never be wholly realised in the organic cell. Its substance will
never be a perfect fluid, and hence equilibrium will be more or less
slowly reached; its surface will seldom be perfectly homogeneous,
and therefore equilibrium will (in the fluid condition) seldom be
perfectly attained; it will very often, or generally, be the seat of
other forces, symmetrical or unsymmetrical; and all these causes will
more or less perturb the effects of surface tension acting by itself.
But we shall find that, on the whole, these effects of surface tension
though modified are not obliterated nor even masked; and accordingly
the phenomena to which I have devoted the foregoing pages will be found
manifestly recurring and repeating themselves among the phenomena of
the organic cell.

――――――――――

In a spider’s web we find exemplified several of the principles {232}
of surface tension which we have now explained. The thread is formed
out of the fluid secretion of a gland, and issues from the body as
a semi-fluid cylinder, that is to say in the form of a surface of
equilibrium, the force of expulsion giving it its elongation and that
of surface tension giving it its circular section. It is prevented,
by almost immediate solidification on exposure to the air, from
breaking up into separate drops or spherules, as it would otherwise
tend to do as soon as the length of the cylinder had passed its limit
of stability. But it is otherwise with the sticky secretion which,
coming from another gland, is simultaneously poured over the issuing
thread when it is to form the spiral portion of the web. This latter
secretion is more fluid than the first, and retains its fluidity for
a very much longer time, finally drying up after several hours. By
capillarity it “wets” the thread, spreading itself over it in an even
film, which film is now itself a cylinder. But this liquid cylinder has
its limit of stability when its length equals its own circumference,
and therefore just at the points so defined it tends to disrupt into
separate segments: or rather, in the actual case, at points somewhat
more distant, owing to the imperfect fluidity of the viscous film, and
still more to the frictional drag upon it of the inner solid cylinder,
or thread, with which it is in contact. The cylinder disrupts in the
usual manner, passing first into the wavy outline of an unduloid, whose
swollen portions swell more and more till the contracted parts break
asunder, and we arrive at a series of spherical drops or beads, of
equal size, strung at equal intervals along the thread. If we try to
spread varnish over a thin stretched wire, we produce automatically
the same identical result[288]; unless our varnish be such as to dry
almost instantaneously, it gathers into beads, and do what we can, we
fail to spread it smooth. It follows that, according to the viscidity
and drying power of the varnish, the process may stop or seem to stop
at any point short of the formation of the perfect spherules; it is
quite possible, therefore, that as our final stage we may only obtain
half-formed beads, or the wavy outline of an unduloid. The formation
of the beads may be facilitated or hastened by jerking the stretched
thread, as the spider actually does: the {233} effect of the jerk
being to disturb and destroy the unstable equilibrium of the viscid
cylinder[289]. Another very curious phenomenon here presents itself.

In Plateau’s experimental separation of a cylinder of oil into two
spherical portions, it was noticed that, when contact was nearly
broken, that is to say when the narrow neck of the unduloid had become
very thin, the two spherical bullae, instead of absorbing the fluid out
of the narrow neck into themselves as they had done with the preceding
portion, drew out this small remaining part of the liquid into a
thin thread as they completed their spherical form and consequently
receded from one another: the reason being that, after the thread or
“neck” has reached a certain tenuity, the internal friction of the
fluid prevents or retards its rapid exit from the little thread to the
adjacent spherule. It is for the same reason that we are able to draw
a glass rod or tube, which we have heated in the middle, into a long
and uniform cylinder or thread, by quickly separating the two ends.
But in the case of the glass rod, the long thin intermediate cylinder
quickly cools and solidifies, while in the ordinary separation of a
liquid cylinder the corresponding intermediate cylinder remains liquid;
and therefore, like any other liquid cylinder, it is liable to break
up, provided that its dimensions exceed the normal limit of stability.
And its length is generally such that it breaks at two points, thus
leaving two terminal portions continuous with the spheres and becoming
confluent with these, and one median portion which resolves itself into
a comparatively tiny spherical drop, midway between the original and
larger two. Occasionally, the same process of formation of a connecting
thread repeats itself a second time, between the small intermediate
spherule and the large spheres; and in this case we obviously obtain
two additional spherules, still smaller in size, and lying one on
either side of our first little one. This whole phenomenon, of equal
and regularly interspaced beads, often with little beads regularly
interspaced between the larger ones, and possibly also even a third
series of still smaller beads regularly intercalated, may be easily
observed in a spider’s web, such as that of _Epeira_, very often with
beautiful regularity,—which {234} naturally, however, is sometimes
interrupted and disturbed owing to a slight want of homogeneity
in the secreted fluid; and the same phenomenon is repeated on a
grosser scale when the web is bespangled with dew, and every thread
bestrung with pearls innumerable. To the older naturalists, these
regularly arranged and beautifully formed globules on the spider’s
web were a cause of great wonder and admiration. Blackwall, counting
some twenty globules in a tenth of an inch, calculated that a large
garden-spider’s web comprised about 120,000 globules; the net was
spun and finished in about forty minutes, and Blackwall was evidently
filled with astonishment at the skill and quickness with which the
spider manufactured these little beads. And no wonder, for according
to the above estimate they had to be made at the rate of about 50 per
second[290].

[Illustration: Fig. 69. Hair of _Trianea_, in glycerine. (After
Berthold.)]

The little delicate beads which stud the long thin pseudopodia of a
foraminifer, such as _Gromia_, or which in like manner appear upon the
cylindrical film of protoplasm which covers the long radiating spicules
of _Globigerina_, represent an identical phenomenon. Indeed there are
many cases, in which we may study in a protoplasmic filament the whole
process of formation of such beads. If we squeeze out on to a slide
the viscid contents of a mistletoe berry, the long sticky threads
into which the substance runs shew the whole phenomenon particularly
well. Another way to demonstrate it was noticed many years ago by
Hofmeister and afterwards explained by Berthold. The hairs of certain
water-plants, such as Hydrocharis or Trianea, constitute very long
cylindrical cells, the protoplasm being supported, and maintained in
equilibrium by its contact with the cell-wall. But if we immerse the
filament in some dense fluid, a little sugar-solution for instance,
or dilute glycerine, the cell-sap tends to diffuse outwards, the
protoplasm parts company with its surrounding and supporting wall,
{235} and lies free as a protoplasmic cylinder in the interior of
the cell. Thereupon it immediately shews signs of instability, and
commences to disrupt. It tends to gather into spheres, which however,
as in our illustration, may be prevented by their narrow quarters from
assuming the complete spherical form; and in between these spheres,
we have more or less regularly alternate ones, of smaller size[291].
Similar, but less regular, beads or droplets may be caused to appear,
under stimulation by an alternating current, in the protoplasmic
threads within the living cells of the hairs of Tradescantia. The
explanation usually given is, that the viscosity of the protoplasm is
reduced, or its fluidity increased; but an increase of the surface
tension would seem a more likely reason[292].

[Illustration: Fig. 70. Phases of a Splash. (From Worthington.)]

――――――――――

We may take note here of a remarkable series of phenomena, which,
though they seem at first sight to be of a very different order, are
closely related to the phenomena which attend and which bring about the
breaking-up of a liquid cylinder or thread.

In some of Mr Worthington’s most beautiful experiments on {236}
splashes, it was found that the fall of a round pebble into water from
a considerable height, caused the rise of a filmy sheet of water in the
form of a cup or cylinder; and the edge of this cylindrical film tended
to be cut up into alternate lobes and notches, and the prominent lobes
or “jets” tended, in more extreme cases, to break off or to break up
into spherical beads (Fig. 70)[293]. A precisely similar appearance is
seen, on a great scale, in the thin edge of a breaking wave: when the
smooth cylindrical edge, at a given moment, shoots out an array of tiny
jets which break up into the droplets which constitute “spray” (Fig.
71, _a_, _b_). We are at once reminded of the beautifully symmetrical
notching on the calycles of many hydroids, which little cups before
they became stiff and rigid had begun their existence as liquid or
semi-liquid films.

[Illustration: Fig. 71. A breaking wave. (From Worthington.)]

The phenomenon is two-fold. In the first place, the edge of our tubular
or crater-like film forms a liquid ring or annulus, which is closely
comparable with the liquid thread or cylinder which we have just been
considering, if only we conceive the thread to be bent round into the
ring. And accordingly, just as the thread spontaneously segments, first
into an unduloid, and then into separate spherical drops, so likewise
will the edge of our annulus tend to do. This phase of notching,
or beading, of the edge of the film is beautifully seen in many of
Worthington’s experiments[294]. In the second place, the very fact of
the rising of the crater means that liquid is flowing up from below
towards the rim; and the segmentation of the rim means that channels
of easier flow are {237} created, along which the liquid is led, or
is driven, into the protuberances: and these are thus exaggerated into
the jets or arms which are sometimes so conspicuous at the edge of the
crater. In short, any film or film-like cup, fluid or semi-fluid in its
consistency, will, like the straight liquid cylinder, be unstable: and
its instability will manifest itself (among other ways) in a tendency
to segmentation or notching of the edge; and just such a peripheral
notching is a conspicuous feature of many minute organic cup-like
structures. In the case of the hydroid calycle (Fig. 72), we are led to
the conclusion that the two common and conspicuous features of notching
or indentation of the cup, and of constriction or annulation of the
long cylindrical stem, are phenomena of the same order and are due to
surface-tension in both cases alike.

[Illustration: Fig. 72. Calycles of Campanularian zoophytes. (A)
_C. integra_; (B) _C. groenlandica_; (C) _C. bispinosa_; (D) _C.
raridentata_.]

Another phenomenon displayed in the same experiments is the formation
of a rope-like or cord-like thickening of the edge of the annulus.
This is due to the more or less sudden checking at the rim of the flow
of liquid rising from below: and a similar peripheral thickening is
frequently seen, not only in some of our hydroid cups, but in many
Vorticellas (cf. Fig. 75), and other organic cup-like conformations. A
perusal of Mr Worthington’s book will soon suggest that these are not
the only manifestations of surface-tension in connection with splashes
which present curious resemblances and analogies to phenomena of
organic form.

The phenomena of an ordinary liquid splash are so swiftly {238}
transitory that their study is only rendered possible by
“instantaneous” photography: but this excessive rapidity is not
an essential part of the phenomenon. For instance, we can repeat
and demonstrate many of the simpler phenomena, in a permanent or
quasi-permanent form, by splashing water on to a surface of dry sand,
or by firing a bullet into a soft metal target. There is nothing,
then, to prevent a slow and lasting manifestation, in a viscous
medium such as a protoplasmic organism, of phenomena which appear
and disappear with prodigious rapidity in a more mobile liquid. Nor
is there anything peculiar in the “splash” itself; it is simply a
convenient method of setting up certain motions or currents, and
producing certain surface-forms, in a liquid medium,—or even in such
an extremely imperfect fluid as is represented (in another series of
experiments) by a bed of sand. Accordingly, we have a large range
of possible conditions under which the organism might conceivably
display configurations analogous to, or identical with, those which Mr
Worthington has shewn us how to exhibit by one particular experimental
method.

To one who has watched the potter at his wheel, it is plain that the
potter’s thumb, like the glass-blower’s blast of air, depends for
its efficacy upon the physical properties of the medium on which it
operates, which for the time being is essentially a fluid. The cup
and the saucer, like the tube and the bulb, display (in their simple
and primitive forms) beautiful surfaces of equilibrium as manifested
under certain limiting conditions. They are neither more nor less than
glorified “splashes,” formed slowly, under conditions of restraint
which enhance or reveal their mathematical symmetry. We have seen, and
we shall see again before we are done, that the art of the glass-blower
is full of lessons for the naturalist as also for the physicist:
illustrating as it does the development of a host of mathematical
configurations and organic conformations which depend essentially on
the establishment of a constant and uniform pressure within a _closed_
elastic shell or fluid envelope. In like manner the potter’s art
illustrates the somewhat obscurer and more complex problems (scarcely
less frequent in biology) of a figure of equilibrium which is an
_open_ surface, or solid, of revolution. It is clear, at the same
time, that the two series of problems are closely akin; for the {239}
glass-blower can make most things that the potter makes, by cutting
off _portions_ of his hollow ware. And besides, when this fails, and
the glass-blower, ceasing to blow, begins to use his rod to trim
the sides or turn the edges of wineglass or of beaker, he is merely
borrowing a trick from the craft of the potter.

It would be venturesome indeed to extend our comparison with these
liquid surface-tension phenomena from the cup or calycle of the
hydrozoon to the little hydroid polype within: and yet I feel convinced
that there is something to be learned by such a comparison, though
not without much detailed consideration and mathematical study of
the surfaces concerned. The cylindrical body of the tiny polype, the
jet-like row of tentacles, the beaded annulations which these tentacles
exhibit, the web-like film which sometimes (when they stand a little
way apart) conjoins their bases, the thin annular film of tissue which
surrounds the little organism’s mouth, and the manner in which this
annular “peristome” contracts[295], like a shrinking soap-bubble, to
close the aperture, are every one of them features to which we may find
a singular and striking parallel in the surface-tension phenomena which
Mr Worthington has illustrated and demonstrated in the case of the
splash.

Here however, we may freely confess that we are for the present on the
uncertain ground of suggestion and conjecture; and so must we remain,
in regard to many other simple and symmetrical organic forms, until
their form and dynamical stability shall have been investigated by the
mathematician: in other words, until the mathematicians shall have
become persuaded that there is an immense unworked field wherein they
may labour, in the detailed study of organic form.

――――――――――

According to Plateau, the viscidity of the liquid, while it helps to
retard the breaking up of the cylinder and so increases the length of
the segments beyond that which theory demands, has nevertheless less
influence in this direction than we might have expected. On the other
hand, any external support or adhesion, such as contact with a solid
body, will be equivalent to a reduction of surface-tension and so will
very greatly increase the {240} stability of our cylinder. It is for
this reason that the mercury in our thermometer tubes does not as a
rule separate into drops, though it occasionally does so, much to our
inconvenience. And again it is for this reason that the protoplasm in
a long and growing tubular or cylindrical cell does not necessarily
divide into separate cells and internodes, until the length of these
far exceeds the theoretic limits. Of course however and whenever it
does so, we must, without ever excluding the agency of surface tension,
remember that there may be other forces affecting the latter, and
accelerating or retarding that manifestation of surface tension by
which the cell is actually rounded off and divided.

In most liquids, Plateau asserts that, on the average, the influence
of viscosity is such as to cause the cylinder to segment when its
length is about four times, or at most from four to six times that
of its diameter: instead of a fraction over three times as, in a
perfect fluid, theory would demand. If we take it at four times, it
may then be shewn that the resulting spheres would have a diameter
of about 1·8 times, and their distance apart would be equal to about
2·2 times the diameter of the original cylinder. The calculation is
not difficult which would shew how these numbers are altered in the
case of a cylinder formed around a solid core, as in the case of
the spider’s web. Plateau has also made the interesting observation
that the _time_ taken in the process of division of the cylinder is
directly proportional to the diameter of the cylinder, while varying
considerably with the nature of the liquid. This question, of the time
occupied in the division of a cell or filament, in relation to the
dimensions of the latter, has not so far as I know been enquired into
by biologists.

――――――――――

From the simple fact that the sphere is of all surfaces that whose
surface-area for a given volume is an absolute minimum, we have already
seen it to be plain that it is the one and only figure of equilibrium
which will be assumed under surface-tension by a drop or vesicle, when
no other disturbing factors are present. One of the most important of
these disturbing factors will be introduced, in the form of complicated
tensions and pressures, when one drop is in contact with another drop
and when a system of intermediate films or partition walls is developed
between them. {241} This subject we shall discuss later, in connection
with cell-aggregates or tissues, and we shall find that further
theoretical considerations are needed as a preliminary to any such
enquiry. Meanwhile let us consider a few cases of the forms of cells,
either solitary, or in such simple aggregates that their individual
form is little disturbed thereby.

Let us clearly understand that the cases we are about to consider
are those cases where the perfect symmetry of the sphere is replaced
by another symmetry, less complete, such as that of an ellipsoidal
or cylindrical cell. The cases of asymmetrical deformation or
displacement, such as is illustrated in the production of a bud or
the development of a lateral branch, are much simpler. For here we
need only assume a slight and localised variation of surface-tension,
such as may be brought about in various ways through the heterogeneous
chemistry of the cell; to this point we shall return in our chapter on
Adsorption. But the diffused and graded asymmetry of the system, which
brings about for instance the ellipsoidal shape of a yeast-cell, is
another matter.

If the sphere be the one surface of complete symmetry and therefore
of independent equilibrium, it follows that in every cell which is
otherwise conformed there must be some definite force to cause its
departure from sphericity; and if this cause be the very simple and
obvious one of the resistance offered by a solidified envelope, such as
an egg-shell or firm cell-wall, we must still seek for the deforming
force which was in action to bring about the given shape, prior to the
assumption of rigidity. Such a cause may be either external to, or may
lie within, the cell itself. On the one hand it may be due to external
pressure or to some form of mechanical restraint: as it is in all our
experiments in which we submit our bubble to the partial restraint of
discs or rings or more complicated cages of wire; and on the other
hand it may be due to intrinsic causes, which must come under the head
either of differences of internal pressure, or of lack of homogeneity
or isotropy in the surface itself[296]. {242}

Our full formula of equilibrium, or equation to an elastic surface,
is _P_ = _p_{e}_ + (_T_/_R_ + _T′_/_R′_), where _P_ is the internal
pressure, _p_{e}_ any extraneous pressure normal to the surface,
_R_, _R′_ the radii of curvature at a point, and _T_, _T′_, the
corresponding tensions, normal to one another, of the envelope.

Now in any given form which we are seeking to account for, _R_, _R′_
are known quantities; but all the other factors of the equation are
unknown and subject to enquiry. And somehow or other, by this formula,
we must account for the form of any solitary cell whatsoever (provided
always that it be not formed by successive stages of solidification),
the cylindrical cell of Spirogyra, the ellipsoidal yeast-cell, or (as
we shall see in another chapter) the shape of the egg of any bird. In
using this formula hitherto, we have taken it in a simplified form,
that is to say we have made several limiting assumptions. We have
assumed that _P_ was simply the uniform hydrostatic pressure, equal in
all directions, of a body of liquid; we have assumed that the tension
_T_ was simply due to surface-tension in a homogeneous liquid film,
and was therefore equal in all directions, so that _T_ = _T′_; and we
have only dealt with surfaces, or parts of a surface, where extraneous
pressure, _p_{n}_, was non-existent. Now in the case of a bird’s egg,
the external pressure _p_{n}_, that is to say the pressure exercised by
the walls of the oviduct, will be found to be a very important factor;
but in the case of the yeast-cell or the Spirogyra, wholly immersed in
water, no such external pressure comes into play. We are accordingly
left, in such cases as these last, with two hypotheses, namely that
the departure from a spherical form is due to inequalities in the
internal pressure _P_, or else to inequalities in the tension _T_,
that is to say to a difference between _T_ and _T′_. In other words,
it is theoretically possible that the oval form of a yeast-cell is due
to a greater internal pressure, a greater “tendency to grow,” in the
direction of the longer axis of the ellipse, or alternatively, that
with equal and symmetrical tendencies to growth there is associated
a difference of external resistance in {243} respect of the tension
of the cell-wall. Now the former hypothesis is not impossible; the
protoplasm is far from being a perfect fluid; it is the seat of various
internal forces, sometimes manifestly polar; and accordingly it is
quite possible that the internal forces, osmotic and other, which
lead to an increase of the content of the cell and are manifested in
pressure outwardly directed upon its wall may be unsymmetrical, and
such as to lead to a deformation of what would otherwise be a simple
sphere. But while this hypothesis is not impossible, it is not very
easy of acceptance. The protoplasm, though not a perfect fluid, has
yet on the whole the properties of a fluid; within the small compass
of the cell there is little room for the development of unsymmetrical
pressures; and, in such a case as Spirogyra, where a large part of the
cavity is filled by a fluid and watery cell-sap, the conditions are
still more obviously those under which a uniform hydrostatic pressure
is to be expected. But in variations of _T_, that is to say of the
specific surface-tension per unit area, we have an ample field for
all the various deformations with which we shall have to deal. Our
condition now is, that (_T_/_R_ + _T′_/_R′_) = a constant; but it
no longer follows, though it may still often be the case, that this
will represent a surface of absolute minimal area. As soon as _T_ and
_T′_ become unequal, it is obvious that we are no longer dealing with
a perfectly liquid surface film; but its departure from a perfect
fluidity may be of all degrees, from that of a slight non-isotropic
viscosity to the state of a firm elastic membrane[297]. And it matters
little whether this viscosity or semi-rigidity be manifested in the
self-same layer which is still a part of the protoplasm of the cell,
or in a layer which is completely differentiated into a distinct and
separate membrane. As soon as, by secretion or “adsorption,” the
molecular constitution of the surface layer is altered, it is clearly
conceivable that the alteration, or the secondary chemical changes
which follow it, may be such as to produce an anisotropy, and to render
the molecular forces less capable in one direction than another of
exerting that contractile force by which they are striving to reduce
to an absolute minimum the {244} surface area of the cell. A slight
inequality in two opposite directions will produce the ellipsoid cell,
and a very great inequality will give rise to the cylindrical cell[298].

I take it therefore, that the cylindrical cell of Spirogyra, or any
other cylindrical cell which grows in freedom from any manifest
external restraint, has assumed that particular form simply by reason
of the molecular constitution of its developing surface-membrane; and
that this molecular constitution was anisotropous, in such a way as to
render extension easier in one direction than another.

Such a lack of homogeneity or of isotropy, in the cell-wall is often
rendered visible, especially in plant-cells, in various ways, in the
form of concentric lamellae, annular and spiral striations, and the
like.

But this phenomenon, while it brings about a certain departure from
complete symmetry, is still compatible with, and coexistent with,
many of the phenomena which we have seen to be associated with
surface-tension. The symmetry of tensions still leaves the cell a solid
of revolution, and its surface is still a surface of equilibrium. The
fluid pressure within the cylinder still causes the film or membrane
which caps its ends to be of a spherical form. And in the young cell,
where the surface pellicle is absent or but little differentiated, as
for instance in the oögonium of Achlya, or in the young zygospore of
Spirogyra, we always see the tendency of the entire structure towards
a spherical form reasserting itself: unless, as in the latter case, it
be overcome by direct compression within the cylindrical mother-cell.
Moreover, in those cases where the adult filament consists of
cylindrical cells, we see that the young, germinating spore, at first
spherical, very soon assumes with growth an elliptical or ovoid form:
the direct result of an incipient anisotropy of its envelope, which
when more developed will convert the ovoid into a cylinder. We may also
notice that a truly cylindrical cell is comparatively rare; for in most
cases, what we call a cylindrical cell shews a distinct bulging of
its sides; it is not truly a cylinder, but a portion of a spheroid or
ellipsoid. {245}

Unicellular organisms in general, including the protozoa, the
unicellular cryptogams, the various bacteria, and the free, isolated
cells, spores, ova, etc. of higher organisms, are referable for the
most part to a very small number of typical forms; but besides a
certain number of others which may be so referable, though obscurely,
there are obviously many others in which either no symmetry is to be
recognized, or in which the form is clearly not one of equilibrium.
Among these latter we have Amoeba itself, and all manner of amoeboid
organisms, and also many curiously shaped cells, such as the
Trypanosomes and various other aberrant Infusoria. We shall return to
the consideration of these; but in the meanwhile it will suffice to
say that, as their surfaces are not equilibrium-surfaces, so neither
are the living cells themselves in any stable equilibrium. On the
contrary, they are in continual flux and movement, each portion of
the surface constantly changing its form, and passing from one phase
to another of an equilibrium which is never stable for more than a
moment. The former class, which rest in stable equilibrium, must fall
(as we have seen) into two classes,—those whose equilibrium arises
from liquid surface-tension alone, and those in whose conformation
some other pressure or restraint has been superimposed upon ordinary
surface-tension.

To the fact that these little organisms belong to an order of magnitude
in which form is mainly, if not wholly, conditioned and controlled
by molecular forces, is due the limited range of forms which they
actually exhibit. These forms vary according to varying physical
conditions. Sometimes they do so in so regular and orderly a way that
we instinctively explain them merely as “phases of a life-history,” and
leave physical properties and physical causation alone: but many of
their variations of form we treat as exceptional, abnormal, decadent
or morbid, and are apt to pass these over in neglect, while we give
our attention to what we suppose to be the typical or “characteristic”
form or attitude. In the case of the smallest organisms, the bacteria,
micrococci, and so forth, the range of form is especially limited,
owing to their minuteness, the powerful pressure which their highly
curved surfaces exert, and the comparatively homogeneous nature of
their substance. But within their narrow range of possible diversity
{246} these minute organisms are protean in their changes of form.
A certain species will not only change its shape from stage to stage
of its little “cycle” of life; but it will be remarkably different
in outward form according to the circumstances under which we find
it, or the histological treatment to which we submit it. Hence the
pathological student, commencing the study of bacteriology, is early
warned to pay little heed to differences of _form_, for purposes of
recognition or specific identification. Whatever grounds we may have
for attributing to these organisms a permanent or stable specific
identity (after the fashion of the higher plants and animals), we can
seldom safely do so on the ground of definite and always recognisable
_form_: we may often be inclined, in short, to ascribe to them a
physiological (sometimes a “pathogenic”), rather than a morphological
specificity.

[Illustration: Fig. 73. A flagellate “monad,” _Distigma proteus_, Ehr.
(After Saville Kent.)]

[Illustration: Fig. 74. _Noctiluca miliaris._]

――――――――――

Among the Infusoria, we have a small number of forms whose symmetry is
distinctly spherical, for instance among the small flagellate monads;
but even these are seldom actually spherical except when we see them
in a non-flagellate and more or less encysted or “resting” stage. In
this condition, it need hardly be remarked that the spherical form is
common and general among a great variety of unicellular organisms.
When our little monad developes a flagellum, that is in itself an
indication of “polarity” or symmetrical non-homogeneity of the cell;
and accordingly, we {247} usually see signs of an unequal tension of
the membrane in the neighbourhood of the base of the flagellum. Here
the tension is usually less than elsewhere, and the radius of curvature
is accordingly less: in other words that end of the cell is drawn out
to a tapering point (Fig. 73). But sometimes it is the other way, as in
Noctiluca, where the large flagellum springs from a depression in the
otherwise uniformly rounded cell. In this case the explanation seems
to lie in the many strands of radiating protoplasm which converge upon
this point, and may be supposed to keep it relatively fixed by their
viscosity, while the rest of the cell-surface is free to expand (Fig.
74).

[Illustration: Fig. 75. Various species of Vorticella. (Mostly after
Saville Kent.)]

A very large number of Infusoria represent unduloids, or portions of
unduloids, and this type of surface appears and reappears in a great
variety of forms. The cups of the various species of Vorticella (Fig.
75) are nothing in the world but a beautiful series of unduloids, or
partial unduloids, in every gradation from a form that is all but
cylindrical to one that is all but a perfect sphere. These unduloids
are not completely symmetrical, but they are such unduloids as develop
themselves when we suspend an oil-globule between two unequal rings,
or blow a soap-bubble between two unequal pipes; for, just as in these
cases, the surface of our Vorticella bell finds its terminal supports,
on the one hand in its attachment to its narrow stalk, and on the other
in the thickened ring from which spring its circumoral cilia. And here
let me say, that a point or zone from which cilia arise would seem
always to have a peculiar relation to the surrounding tensions. It
usually forms a sharp salient, a prominent point or ridge, as in our
little monads of Fig. 73; shewing that, in its formation, the surface
tension had here locally diminished. But if such a ridge or fillet
consolidate in the least degree, it becomes a source of strength, and
a _point d’appui_ for the adjacent film. We shall deal with this point
again in the next chapter. {248}

[Illustration: Fig. 76. Various species of _Salpingoeca_.]

[Illustration: Fig. 77. Various species of _Tintinnus_, _Dinobryon_ and
_Codonella_. (After Saville Kent and others.)]

[Illustration: Fig. 78. _Vaginicola._]

[Illustration: Fig. 79. _Folliculina._]

[Illustration: Fig. 80. _Trachelophyllum._ (After Wreszniowski.)]

Precisely the same series of unduloid forms may be traced in even
greater variety among various other families or genera of the
Infusoria. Sometimes, as in Vorticella itself, the unduloid is seen
merely in the contour of the soft semifluid body of the living
animal. At other times, as in Salpingoeca, Tintinnus, and many other
genera, we have a distinct membranous cup, separate from the animal,
but originally secreted by, and moulded upon, its semifluid living
surface. Here we have an excellent illustration of the contrast
between the different ways in which such a structure may be regarded
and interpreted. The teleological explanation is that it is developed
for the sake of protection, as a domicile and shelter for the little
organism within. The mechanical explanation of the physicist (seeking
only after the “efficient,” and not the “final” cause), is that it is
{249} present, and has its actual conformation, by reason of certain
chemico-physical conditions: that it was inevitable, under the given
conditions, that certain constituent substances actually present in
the protoplasm should be aggregated by molecular forces in its surface
layer; that under this adsorptive process, the conditions continuing
favourable, the particles should accumulate and concentrate till
they formed (with the help of the surrounding medium) a pellicle or
membrane, thicker or thinner as the case might be; that this surface
pellicle or membrane was inevitably bound, by molecular forces, to
become a surface of the least possible area which the circumstances
permitted; that in the present case, the symmetry and “freedom” of
the system permitted, and _ipso facto_ caused, this surface to be a
surface of revolution; and that of the few surfaces of revolution
which, as being also surfaces _minimae areae_, were available, the
unduloid was manifestly the one permitted, and _ipso facto_ caused, by
the dimensions of the organisms and other circumstances of the case.
And just as the thickness or thinness of the pellicle was obviously
a subordinate matter, a mere matter of degree, so we also see that
the actual outline of this or that particular unduloid is also a very
subordinate matter, such as physico-chemical variants of a minute kind
would suffice to bring about; for between the various unduloids which
the various species of Vorticella represent, there is no more real
difference than that difference of ratio or degree which exists between
two circles of different diameter, or two lines of unequal length.
{250}

In very many cases (of which Fig. 80 is an example), we have an
unduloid form exhibited, not by a surrounding pellicle or shell,
but by the soft, protoplasmic body of a ciliated organism. In such
cases the form is mobile, and continually changes from one to another
unduloid contour, according to the movements of the animal. We have
here, apparently, to deal with an unstable equilibrium, and also
sometimes with the more complicated problem of “stream-lines,” as in
the difficult problems suggested by the form of a fish. But this whole
class of cases, and of problems, we can merely take note of in passing,
for their treatment is too hard for us.

――――――――――

In considering such series of forms as the various unduloids which
we have just been regarding, we are brought sharply up (as in the
case of our Bacteria or Micrococci) against the biological concept of
organic _species_. In the intense classificatory activity of the last
hundred years, it has come about that every form which is apparently
characteristic, that is to say which is capable of being described or
portrayed, and capable of being recognised when met with again, has
been recorded as a species,—for we need not concern ourselves with the
occasional discussions, or individual opinions, as to whether such
and such a form deserve “specific rank,” or be “only a variety.” And
this secular labour is pursued in direct obedience to the precept of
the _Systema Naturae_,—“_ut sic in summa confusione rerum apparenti,
summus conspiciatur Naturae ordo_.” In like manner the physicist
records, and is entitled to record, his many hundred “species” of
snow-crystals[299], or of crystals of calcium carbonate. But regarding
these latter species, the physicist makes no assumptions: he records
them _simpliciter_, as specific “forms”; he notes, as best he can, the
circumstances (such as temperature or humidity) under which they occur,
in the hope of elucidating the conditions determining their formation;
but above all, he does not introduce {251} the element of time, and of
succession, or discuss their origin and affiliation as an _historical_
sequence of events. But in biology, the term species carries with it
many large, though often vague assumptions. Though the doctrine or
concept of the “permanence of species” is dead and gone, yet a certain
definite value, or sort of quasi-permanency, is still connoted by the
term. Thus if a tiny foraminiferal shell, a Lagena for instance, be
found living to-day, and a shell indistinguishable from it to the eye
be found fossil in the Chalk or some other remote geological formation,
the assumption is deemed legitimate that that species has “survived,”
and has handed down its minute specific character or characters,
from generation to generation, unchanged for untold myriads of
years[300]. Or if the ancient forms be like to, rather than identical
with the recent, we still assume an unbroken descent, accompanied
by the hereditary transmission of common characters and progressive
variations. And if two identical forms be discovered at the ends of
the earth, still (with occasional slight reservations on the score of
possible “homoplasy”), we build hypotheses on this fact of identity,
taking it for granted that the two appertain to a common stock, whose
dispersal in space must somehow be accounted for, its route traced,
its epoch determined, and its causes discussed or discovered. In
short, the naturalist admits no exception to the rule that a “natural
classification” can only be a _genealogical_ one, nor ever doubts that
“_The fact that we are able to classify organisms at all in accordance
with the structural characteristics which they present, is due to
the fact of their being related by descent_[301].” But this great
generalisation is apt in my opinion, to carry us too far. It may be
safe and sure and helpful and illuminating when we apply it to such
complex entities,—such thousand-fold resultants of the combination
and permutation of many variable characters,—as a horse, a lion or an
eagle; but (to my mind) it has a very different look, and a far less
firm foundation, when we attempt to extend it to minute organisms
whose specific characters are few and simple, whose simplicity {252}
becomes much more manifest when we regard it from the point of view of
physical and mathematical description and analysis, and whose form is
referable, or (to say the least of it) is very largely referable, to
the direct and immediate action of a particular physical force. When we
come to deal with the minute skeletons of the Radiolaria we shall again
find ourselves dealing with endless modifications of form, in which
it becomes still more difficult to discern, or to apply, the guiding
principle of affiliation or _genealogy_.

[Illustration: Fig. 81.]

Among the more aberrant forms of Infusoria is a little species known
as _Trichodina pedicidus_, a parasite on the Hydra, or fresh-water
polype (Fig. 81.) This Trichodina has the form of a more or less
flattened circular disc, with a ring of cilia around both its upper
and lower margins. The salient ridge from which these cilia spring may
be taken, as we have already said, to play the part of a strengthening
“fillet.” The circular base of the animal is flattened, in contact
with the flattened surface of the Hydra over which it creeps, and the
opposite, upper surface may be flattened nearly to a plane, or may at
other times appear slightly convex or slightly concave. The sides of
the little organism are contracted, forming a symmetrical equatorial
groove between the upper and lower discs; and, on account of the minute
size of the animal and its constant movements, we cannot submit the
curvature of this concavity to measurement, nor recognise by the eye
its exact contour. But it is evident that the conditions are precisely
similar to those described on p. 223, where we were considering the
conditions of stability of the catenoid. And it is further evident
that, when the upper disc is actually plane, the equatorial groove is
strictly a catenoid surface of revolution; and when on the other hand
it is depressed, then the equatorial groove will tend to assume the
form of a nodoidal surface.

Another curious type is the flattened spiral of _Dinenympha_[302]
{253} which reminds us of the cylindrical spiral of a Spirillum among
the bacteria. In Dinenympha we have a symmetrical figure, whose two
opposite surfaces each constitute a surface of constant mean curvature;
it is evidently a figure of equilibrium under certain special
conditions of restraint. The cylindrical coil of the Spirillum, on the
other hand, is a surface of constant mean curvature, and therefore of
equilibrium, as truly, and in the same sense, as the cylinder itself.

[Illustration: Fig. 82. _Dinenympha gracilis_, Leidy.]

[Illustration: Fig. 83.]

A very curious conformation is that of the vibratile “collar,” found
in Codosiga and the other “Choanoflagellates,” and which we also
meet with in the “collar-cells” which line the interior cavities of
a sponge. Such collar-cells are always very minute, and the collar
is constituted of a very delicate film, which shews an undulatory or
rippling motion. It is a surface of revolution, and as it maintains
itself in equilibrium (though a somewhat unstable and fluctuating
one), it must be, under the restricted circumstances of its case,
a surface of minimal area. But it is not so easy to see what these
special circumstances are; and it is obvious that the collar, if left
to itself, must at once {254} contract downwards towards its base,
and become confluent with the general surface of the cell; for it has
no longitudinal supports and no strengthening ring at its periphery.
But in all these collar-cells, there stands within the annulus of the
collar a large and powerful cilium or flagellum, in constant movement;
and by the action of this flagellum, and doubtless in part also by the
intrinsic vibrations of the collar itself, there is set up a constant
steady current in the surrounding water, whose direction would seem to
be such that it passes up the outside of the collar, down its inner
side, and out in the middle in the direction of the flagellum; and
there is a distinct eddy, in which foreign particles tend to be caught,
around the peripheral margin of the collar. When the cell dies, that
is to say when motion ceases, the collar immediately shrivels away and
disappears. It is notable, by the way, that the edge of this little
mobile cup is always smooth, never notched or lobed as in the cases we
have discussed on p. 236: this latter condition being the outcome of
a definite instability, marking the close of a period of equilibrium;
while in the vibratile collar of Codosiga the equilibrium, such as
it is, is being constantly renewed and perpetuated like that of a
juggler’s pole, by the motions of the system. I take it that, somehow,
its existence (in a state of partial equilibrium) is due to the current
motions, and to the traction exerted upon it through the friction of
the stream which is constantly passing by. I think, in short, that it
is formed very much in the same way as the cup-like ring of streaming
ribbons, which we see fluttering and vibrating in the air-current of a
ventilating fan.

It is likely enough, however, that a different and much better
explanation may yet be found; and if we turn once more to Mr
Worthington’s _Study of Splashes_, we may find a curious suggestion
of analogy in the beautiful craters encircling a central jet (as the
collar of Codosiga encircles the flagellum), which we see produced in
the later stages of the splash of a pebble[303]. {255}

Among the Foraminifera we have an immense variety of forms, which,
in the light of surface tension and of the principle of minimal
area, are capable of explanation and of reduction to a small number
of characteristic types. Many of the Foraminifera are composite
structures, formed by the successive imposition of cell upon cell, and
these we shall deal with later on; let us glance here at the simpler
conformations exhibited by the single chambered or “monothalamic”
genera, and perhaps one or two of the simplest composites.

We begin with forms, like Astrorhiza (Fig. 219, p. 464), which are in
a high degree irregular, and end with others which manifest a perfect
and mathematical regularity. The broad difference between these two
types is that the former are characterised, like Amoeba, by a variable
surface tension, and consequently by unstable equilibrium; but the
strong contrast between these and the regular forms is bridged over by
various transition-stages, or differences of degree. Indeed, as in all
other Rhizopods, the very fact of the emission of pseudopodia, which
reach their highest development in this group of animals, is a sign
of unstable surface-equilibrium; and we must therefore consider that
those forms which indicate symmetry and equilibrium in their shells
have secreted these during periods when rest and uniformity of surface
conditions alternated with the phases of pseudopodial activity. The
irregular forms are in almost all cases arenaceous, that is to say
they have no solid shells formed by steady adsorptive secretion, but
only a looser covering of sand grains with which the protoplasmic body
has come in contact and cohered. Sometimes, as in Ramulina, we have a
calcareous shell combined with irregularity of form; but here we can
easily see a partial and as it were a broken regularity, the regular
forms of sphere and cylinder being repeated in various parts of the
ramified mass. When we look more closely at the arenaceous forms, we
find that the same thing is true of them; they represent, either in
whole or part, approximations to the form of surfaces of equilibrium,
spheres, cylinders and so forth. In Aschemonella we have a precise
replica of the calcareous Ramulina; and in Astrorhiza itself, in
the forms distinguished by naturalists as _A. crassatina_, what is
described as the “subsegmented interior[304]” {256} seems to shew the
natural, physical tendency of the long semifluid cylinder of protoplasm
to contract, at its limit of stability, into unduloid constrictions, as
a step towards the breaking up into separate spheres: the completion of
which process is restrained or prevented by the rigidity and friction
of the arenaceous covering.

[Illustration: Fig. 84. Various species of _Lagena_. (After Brady.)]

Passing to the typical, calcareous-shelled Foraminifera, we have
the most symmetrical of all possible types in the perfect sphere of
Orbulina; this is a pelagic organism, whose floating habitat places it
in a position of perfect symmetry towards all external forces. Save for
one or two other forms which are also spherical, or approximately so,
like Thurammina, the rest of the monothalamic calcareous Foraminifera
are all comprised by naturalists within the genus Lagena. This large
and varied genus consists of “flask-shaped” shells, whose surface is
simply that of an unduloid, or more frequently, like that of a flask
itself, an unduloid combined with a portion of a sphere. We do not know
the circumstances {257} under which the shell of Lagena is formed, nor
the nature of the force by which, during its formation, the surface is
stretched out into the unduloid form; but we may be pretty sure that
it is suspended vertically in the sea, that is to say in a position of
symmetry as regards its vertical axis, about which the unduloid surface
of revolution is symmetrically formed. At the same time we have other
types of the same shell in which the form is more or less flattened;
and these are doubtless the cases in which such symmetry of position
was not present, or was replaced by a broader, lateral contact with the
surface pellicle[305].

[Illustration: Fig. 85. (After Darling.)]

While Orbulina is a simple spherical drop, Lagena suggests to our
minds a “hanging drop,” drawn out to a long and slender neck by
its own weight, aided by the viscosity of the material. Indeed the
various hanging drops, such as Mr C. R. Darling shews us, are the
most beautiful and perfect unduloids, with spherical ends, that it is
possible to conceive. A suitable liquid, a little denser than water
and incapable of mixing with it (such as ethyl benzoate), is poured on
a surface of water. It spreads {258} over the surface and gradually
forms a hanging drop, approximately hemispherical; but as more liquid
is added the drop sinks or rather grows downwards, still adhering
to the surface film; and the balance of forces between gravity and
surface tension results in the unduloid contour, as the increasing
weight of the drop tends to stretch it out and finally break it in
two. At the moment of rupture, by the way, a tiny droplet is formed in
the attenuated neck, such as we described in the normal division of a
cylindrical thread (p. 233).

 To pass to a much more highly organised class of animals, we find the
 unduloid beautifully exemplified in the little flask-shaped shells
 of certain Pteropod mollusca, e.g. Cuvierina[306]. Here again the
 symmetry of the figure would at once lead us to suspect that the
 creature lived in a position of symmetry to the surrounding forces, as
 for instance if it floated in the ocean in an erect position, that is
 to say with its long axis coincident with the direction of gravity;
 and this we know to be actually the mode of life of the little
 Pteropod.

Many species of Lagena are complicated and beautified by a pattern, and
some by the superaddition to the shell of plane extensions or “wings.”
These latter give a secondary, bilateral symmetry to the little shell,
and are strongly suggestive of a phase or period of growth in which it
lay horizontally on the surface, instead of hanging vertically from
the surface-film: in which, that is to say, it was a floating and not
a hanging drop. The pattern is of two kinds. Sometimes it consists of
a sort of fine reticulation, with rounded or more or less hexagonal
interspaces: in other cases it is produced by a symmetrical series of
ridges or folds, usually longitudinal, on the body of the flask-shaped
cell, but occasionally transversely arranged upon the narrow neck. The
reticulated and folded patterns we may consider separately. The netted
pattern is very similar to the wrinkled surface of a dried pea, or
to the more regular wrinkled patterns upon many other seeds and even
pollen-grains. If a spherical body after developing a “skin” begin
to shrink a little, and if the skin have so far lost its elasticity
as to be unable to keep pace with the shrinkage of the inner mass,
it will tend to fold or wrinkle; and if the shrinkage be uniform,
and the elasticity and flexibility of the skin be also uniform, then
the amount of {259} folding will be uniformly distributed over the
surface. Little concave depressions will appear, regularly interspaced,
and separated by convex folds. The little concavities being of equal
size (unless the system be otherwise perturbed) each one will tend
to be surrounded by six others; and when the process has reached its
limit, the intermediate boundary-walls, or raised folds, will be found
converted into a regular pattern of hexagons.

But the analogy of the mechanical wrinkling of the coat of a seed
is but a rough and distant one; for we are evidently dealing with
molecular rather than with mechanical forces. In one of Darling’s
experiments, a little heavy tar-oil is dropped onto a saucer of water,
over which it spreads in a thin film showing beautiful interference
colours after the fashion of those of a soap-bubble. Presently tiny
holes appear in the film, which gradually increase in size till they
form a cellular pattern or honeycomb, the oil gathering together in the
meshes or walls of the cellular net. Some action of this sort is in
all probability at work in a surface-film of protoplasm covering the
shell. As a physical phenomenon the actions involved are by no means
fully understood, but surface-tension, diffusion and cohesion doubtless
play their respective parts therein[307]. The very perfect cellular
patterns obtained by Leduc (to which we shall have occasion to refer in
a subsequent chapter) are diffusion patterns on a larger scale, but not
essentially different.

[Illustration: Fig. 86.]

The folded or pleated pattern is doubtless to be explained, in a
general way, by the shrinkage of a surface-film under certain {260}
conditions of viscous or frictional restraint. A case which (as it
seems to me) is closely analogous to that of our foraminiferal shells
is described by Quincke[308], who let a film of albumin or of resin set
and harden upon a surface of quicksilver, and found that the little
solid pellicle had been thrown into a pattern of symmetrical folds.
If the surface thus thrown into folds be that of a cylinder, or any
other figure with one principal axis of symmetry, such as an ellipsoid
or unduloid, the direction of the folds will tend to be related to
the axis of symmetry, and we might expect accordingly to find regular
longitudinal, or regular transverse wrinkling. Now as a matter of fact
we almost invariably find in the Lagena the former condition: that is
to say, in our ellipsoid or unduloid cell, the puckering takes the form
of the vertical fluting on a column, rather than that of the transverse
pleating of an accordion. And further, there is often a tendency for
such longitudinal flutings to be more or less localised at the end of
the ellipsoid, or in the region where the unduloid merges into its
spherical base. In this latter region we often meet with a regular
series of short longitudinal folds, as we do in the forms of Lagena
denominated _L. semistriata_. All these various forms of surface can
be imitated, or rather can be precisely reproduced, by the art of the
glass-blower[309].

Furthermore, they remind one, in a striking way, of the regular ribs or
flutings in the film or sheath which splashes up to envelop a smooth
ball which has been dropped into a liquid, as Mr Worthington has so
beautifully shewn[310]. {261}

In Mr Worthington’s experiment, there appears to be something of the
nature of a viscous drag in the surface-pellicle; but whatever be the
actual cause of variation of tension, it is not difficult to see that
there must be in general a tendency towards _longitudinal_ puckering
or “fluting” in the case of a thin-walled cylindrical or other
elongated body, rather than a tendency towards transverse puckering, or
“pleating.” For let us suppose that some change takes place involving
an increase of surface-tension in some small area of the curved wall,
and leading therefore to an increase of pressure: that is to say let
_T_ become _T_ + _t_, and _P_ become _P_ + _p_. Our new equation of
equilibrium, then, in place of _P_ = _T_/_r_ + _T_/_r′_ becomes

 _P_ + _p_ = (_T_ + _t_)/_r_ + (_T_ + _t_)/_r′_,

 and by subtraction,

 _p_ = _t_/_r_ + _t_/_r′_.

 Now if _r_ < _r′_, _t_/_r_ > _t_/_r′_.

Therefore, in order to produce the small increment of pressure _p_,
it is easier to do so by increasing _t_/_r_ than _t_/_r′_; that is
to say, the easier way is to alter, or diminish _r_. And the same
will hold good if the tension and pressure be diminished instead of
increased.

This is as much as to say that, when corrugation or “rippling” of
the walls takes place owing to small changes of surface-tension, and
consequently of pressure, such corrugation is more likely to take
place in the plane of _r_,—that is to say, _in the plane of greatest
curvature_. And it follows that in such a figure as an ellipsoid,
wrinkling will be most likely to take place not only in a longitudinal
direction but near the extremities of the figure, that is to say again
in the region of greatest curvature.

[Illustration: Fig. 87. _Nodosaria scalaris_, Batsch.]

[Illustration: Fig. 88. Gonangia of Campanularians. (_a_) _C.
gracilis_; (_b_) _C. grandis_. (After Allman.)]

The longitudinal wrinkling of the flask-shaped bodies of our Lagenae,
and of the more or less cylindrical cells of many other Foraminifera
(Fig. 87), is in complete accord with the above theoretical
considerations; but nevertheless, we soon find that our result is not
a general one, but is defined by certain limiting conditions, and is
accordingly subject to what are, at first sight, important exceptions.
For instance, when we turn to the narrow neck of the Lagena we see at
once that our theory no longer holds; for {262} the wrinkling which
was invariably longitudinal in the body of the cell is as invariably
transverse in the narrow neck. The reason for the difference is not
far to seek. The conditions in the neck are very different from
those in the expanded portion of the cell: the main difference being
that the thickness of the wall is no longer insignificant, but is of
considerable magnitude as compared with the diameter, or circumference,
of the neck. We must accordingly take it into account in considering
the _bending moments_ at any point in this region of the shell-wall.
And it is at once obvious that, in any portion of the narrow neck,
_flexure_ of a wall in a transverse direction will be very difficult,
while flexure in a longitudinal direction will be comparatively easy;
just as, in the case of a long narrow strip of iron, we may easily
bend it into folds running transversely to its long axis, but not the
other way. The manner in which our little Lagena-shell tends to fold
or wrinkle, longitudinally in its wider part, and transversely or
annularly in its narrow neck, is thus completely and easily explained.

An identical phenomenon is apt to occur in the little flask-shaped
gonangia, or reproductive capsules, of some of the hydroid zoophytes.
In the annexed drawings of these gonangia in two species of
Campanularia, we see that in one case the little vesicle {263} has
the flask-shaped or unduloid configuration of a Lagena; and here the
walls of the flask are longitudinally fluted, just after the manner we
have witnessed in the latter genus. But in the other Campanularian the
vesicles are long, narrow and tubular, and here a transverse folding
or pleating takes the place of the longitudinally fluted pattern. And
the very form of the folds or pleats is enough to suggest that we are
not dealing here with a simple phenomenon of surface-tension, but with
a condition in which surface-tension and _stiffness_ are both present,
and play their parts in the resultant form.

[Illustration: Fig. 89. Various Foraminifera (after Brady), _a_,
_Nodosaria simplex_; _b_, _N. pygmaea_; _c_, _N. costulata_; _e_, _N.
hispida_; _f_, _N. elata_; _d_, _Rheophax_ (_Lituola_) _distans_; _g_,
_Sagrina virgata_.]

Passing from the solitary flask-shaped cell of Lagena, we have, in
another series of forms, a constricted cylinder, or succession of
unduloids; such as are represented in Fig. 89, illustrating certain
species of Nodosaria, Rheophax and Sagrina. In some of these cases,
and certainly in that of the arenaceous genus Rheophax, we have to do
with the ordinary phenomenon of a segmenting or partially segmenting
cylinder. But in others, the structure is not developed out of a
continuous protoplasmic cylinder, but as we can see by examining
the interior of the shell, it has been formed in successive stages,
beginning with a simple unduloid “Lagena,” about whose neck, after its
solidification, another drop of protoplasm accumulated, and in turn
assumed the unduloid, or lagenoid, form. The chains of interconnected
bubbles which {264} Morey and Draper made many years ago of melted
resin are a very similar if not identical phenomenon[311].

――――――――――

There now remain for our consideration, among the Protozoa, the
great oceanic group of the Radiolaria, and the little group of their
freshwater allies, the Heliozoa. In nearly all these forms we have this
specific chemical difference from the Foraminifera, that when they
secrete, as they generally do secrete, a hard skeleton, it is composed
of silica instead of lime. These organisms and the various beautiful
and highly complicated skeletal fabrics which they develop give us
many interesting illustrations of physical phenomena, among which the
manifestations of surface-tension are very prominent. But the chief
phenomena connected with their skeletons we shall deal with in another
place, under the head of spicular concretions.

In a simple and typical Heliozoan, such as the Sun-animalcule,
_Actinophrys sol_, we have a “drop” of protoplasm, contracted by
its surface tension into a spherical form. Within the heterogeneous
protoplasmic mass are more fluid portions, and at the surface which
separates these from the surrounding protoplasm a similar surface
tension causes them also to assume the form of spherical “vacuoles,”
which in reality are little clear drops within the big one; unless
indeed they become numerous and closely packed, in which case, instead
of isolated spheres or droplets they will constitute a “froth,” their
mutual pressures and tensions giving rise to regular configurations
such as we shall study in the next chapter. One or more of such clear
spaces may be what is called a “contractile vacuole”: that is to say,
a droplet whose surface tension is in unstable equilibrium and is apt
to vanish altogether, so that the definite outline of the vacuole
suddenly disappears[312]. Again, within the protoplasm are one or
more nuclei, whose own surface tension (at the surface between the
nucleus and the surrounding protoplasm), has drawn them in turn into
the shape {265} of spheres. Outwards through the protoplasm, and
stretching far beyond the spherical surface of the cell, there run
stiff linear threads of modified or differentiated protoplasm, replaced
or reinforced in some cases by delicate siliceous needles. In either
case we know little or nothing about the forces which lead to their
production, and we do not hide our ignorance when we ascribe their
development to a “radial polarisation” of the cell. In the case of the
protoplasmic filament, we may (if we seek for a hypothesis), suppose
that it is somehow comparable to a viscid stream, or “liquid vein,”
thrust or squirted out from the body of the cell. But when it is once
formed, this long and comparatively rigid filament is separated by a
distinct surface from the neighbouring protoplasm, that is to say from
the more fluid surface-protoplasm of the cell; and the latter begins
to creep up the filament, just as water would creep up the interior of
a glass tube, or the sides of a glass rod immersed in the liquid. It
is the simple case of a balance between three separate tensions: (1)
that between the filament and the adjacent protoplasm, (2) that between
the filament and the adjacent water, and (3) that between the water
and the protoplasm. Calling these tensions respectively _T__{_fp_},
_T__{_fw_}, and _T__{_wp_}, equilibrium will be attained when the angle
of contact between the fluid protoplasm and the filament is such that
cos α = (_T__{_fw_} − _T__{_wp_})/_T__{_fp_}. It is evident in this
case that the angle is a very small one. The precise form of the curve
is somewhat different from that which, under ordinary circumstances,
is assumed by a liquid which creeps up a solid surface, as water in
contact with air creeps up a surface of glass; the difference being due
to the fact that here, owing to the density of the protoplasm being
practically identical with that of the surrounding medium, the whole
system is practically immune from gravity. Under normal circumstances
the curve is part of the “elastic curve” by which that surface of
revolution is generated which we have called, after Plateau, the
nodoid; but in the present case it is apparently a catenary. Whatever
curve it be, it obviously forms a surface of revolution around the
filament.

Since the attraction exercised by this surface tension is symmetrical
around the filament, the latter will be pulled equally {266} in all
directions; in other words it will tend to be set normally to the
surface of the sphere, that is to say radiating directly outwards
from the centre. If the distance between two adjacent filaments be
considerable, the curve will simply meet the filament at the angle α
already referred to; but if they be sufficiently near together, we
shall have a continuous catenary curve forming a hanging loop between
one filament and the other. And when this is so, and the radial
filaments are more or less symmetrically interspaced, we may have a
beautiful system of honeycomb-like depressions over the surface of
the organism, each cell of the honeycomb having a strictly defined
geometric configuration.

[Illustration: Fig. 90. A, _Trypanosoma tineae_ (after Minchin); B,
_Spirochaeta anodontae_ (after Fantham).]

In the simpler Radiolaria, the spherical form of the entire organism is
equally well-marked; and here, as also in the more complicated Heliozoa
(such as Actinosphaerium), the organism is differentiated into several
distinct layers, each boundary surface tending to be spherical, and
so constituting sphere within sphere. One of these layers at least
is close packed with vacuoles, forming an “alveolar meshwork,” with
the configurations of which we shall attempt in another chapter to
correlate the characteristic structure of certain complex types of
skeleton.

――――――――――

An exceptional form of cell, but a beautiful manifestation of
surface-tension (or so I take it to be), occurs in Trypanosomes, those
tiny parasites of the blood that are associated with sleeping-sickness
and many other grave or dire maladies. These tiny organisms consist of
elongated solitary cells down one side of which runs a very delicate
frill, or “undulating membrane,” the free edge of which is seen
to be slightly thickened, and the whole of {267} which undergoes
rhythmical and beautiful wavy movements. When certain Trypanosomes are
artificially cultivated (for instance _T. rotatorium_, from the blood
of the frog), phases of growth are witnessed in which the organism has
no undulating membrane, but possesses a long cilium or “flagellum,”
springing from near the front end, and exceeding the whole body in
length[313]. Again, in _T. lewisii_, when it reproduces by “multiple
fission,” the products of this division are likewise devoid of an
undulating membrane, but are provided with a long free flagellum[314].
It is a plausible assumption to suppose that, as the flagellum waves
about, it comes to lie near and parallel to the body of the cell, and
that the frill or undulating membrane is formed by the clear, fluid
protoplasm of the surface layer springing up in a film to run up and
along the flagellum, just as a soap-film would be formed in similar
circumstances.

[Illustration: Fig. 91. A, _Trichomonas muris_, Hartmann; B,
_Trichomastix serpentis_, Dobell; C, _Trichomonas angusta_, Alexeieff.
(After Kofoid.)]

This mode of formation of the undulating membrane or frill appears to
be confirmed by the appearances shewn in Fig. 91. {268} Here we have
three little organisms closely allied to the ordinary Trypanosomes, of
which one, Trichomastix (_B_), possesses four flagella, and the other
two, Trichomonas, apparently three only: the two latter possess the
frill, which is lacking in the first[315]. But it is impossible to
doubt that when the frill is present (as in _A_ and _C_), its outer
edge is constituted by the apparently missing flagellum (_a_), which
has become _attached_ to the body of the creature at the point _c_,
near its posterior end; and all along its course, the superficial
protoplasm has been drawn out into a film, between the flagellum (_a_)
and the adjacent surface or edge of the body (_b_).

[Illustration: Fig. 92. Herpetomonas assuming the undulatory membrane
of a Trypanosome. (After D. L. Mackinnon.)]

Moreover, this mode of formation has been actually witnessed and
described, though in a somewhat exceptional case. The little flagellate
monad Herpetomonas is normally destitute of an undulating membrane,
but possesses a single long terminal flagellum. According to Dr D. L.
Mackinnon, the cytoplasm in a certain stage of growth becomes somewhat
“sticky,” a phrase which we may in all probability interpret to mean
that its surface tension is being reduced. For this stickiness is shewn
in two ways. In the first place, the long body, in the course of its
various bending movements, is apt to adhere head to tail (so to speak),
giving a rounded or sometimes annular form to the organism, such as
has also been described in certain species or stages of Trypanosomes.
But again, the long flagellum, if it get bent backwards upon the body,
tends to adhere to its surface. “Where the flagellum was pretty long
and active, its efforts to continue movement under these abnormal
conditions resulted in the gradual lifting up from the cytoplasm of the
body of a sort of _pseudo_-undulating membrane (Fig. 92). The movements
of this structure were so exactly those of a true undulating membrane
that it was {269} difficult to believe one was not dealing with a
small, blunt trypanosome[316].” This in short is a precise description
of the mode of development which, from theoretical considerations
alone, we should conceive to be the natural if not the only possible
way in which the undulating membrane could come into existence.

There is a genus closely allied to Trypanosoma, viz. Trypanoplasma,
which possesses one free flagellum, together with an undulating
membrane; and it resembles the neighbouring genus Bodo, save that the
latter has two flagella and no undulating membrane. In like manner,
Trypanosoma so closely resembles Herpetomonas that, when individuals
ascribed to the former genus exhibit a free flagellum only, they are
said to be in the “Herpetomonas stage.” In short all through the
order, we have pairs of genera, which are presumed to be separate
and distinct, viz. Trypanosoma-Herpetomonas, Trypanoplasma-Bodo,
Trichomastix-Trichomonas, in which one differs from the other mainly if
not solely in the fact that a free flagellum in the one is replaced by
an undulating membrane in the other. We can scarcely doubt that the two
structures are essentially one and the same.

The undulating membrane of a Trypanosome, then, according to our
interpretation of it, is a liquid film and must obey the law of
constant mean curvature. It is under curious limitations of freedom:
for by one border it is attached to the comparatively motionless body,
while its free border is constituted by a flagellum which retains its
activity and is being constantly thrown, like the lash of a whip,
into wavy curves. It follows that the membrane, for every alteration
of its longitudinal curvature, must at the same instant become curved
in a direction perpendicular thereto; it bends, not as a tape bends,
but with the accompaniment of beautiful but tiny waves of double
curvature, all tending towards the establishment of an “equipotential
surface”; and its characteristic undulations are not originated by an
active mobility of the membrane but are due to the molecular tensions
which produce the very same result in a soap-film under similar
circumstances.

In certain Spirochaetes, _S. anodontae_ (Fig. 90) and _S. balbiani_
{270} (which we find in oysters), a very similar undulating membrane
exists, but it is coiled in a regular spiral round the body of the
cell. It forms a “screw-surface,” or helicoid, and, though we might
think that nothing could well be more curved, yet its mathematical
properties are such that it constitutes a “ruled surface” whose “mean
curvature” is everywhere _nil_; and this property (as we have seen)
it shares with the plane, and with the plane alone. Precisely such a
surface, and of exquisite beauty, may be produced by bending a wire
upon itself so that part forms an axial rod and part a spiral wrapping
round the axis, and then dipping the whole into a soapy solution.

These undulating and helicoid surfaces are exactly reproduced among
certain forms of spermatozoa. The tail of a spermatozoon consists
normally of an axis surrounded by clearer and more fluid protoplasm,
and the axis sometimes splits up into two or more slender filaments. To
surface tension operating between these and the surface of the fluid
protoplasm (just as in the case of the flagellum of the Trypanosome),
I ascribe the formation of the undulating membrane which we find, for
instance, in the spermatozoa of the newt or salamander; and of the
helicoid membrane, wrapped in a far closer and more beautiful spiral
than that which we saw in Spirochaeta, which is characteristic of the
spermatozoa of many birds.

――――――――――

Before we pass from the subject of the conformation of the solitary
cell we must take some account of certain other exceptional forms,
less easy of explanation, and still less perfectly understood. Such is
the case, for instance, with the red blood-corpuscles of man and other
vertebrates; and among the sperm-cells of the decapod crustacea we find
forms still more aberrant and not less perplexing. These are among the
comparatively few cells or cell-like structures whose form _seems_ to
be incapable of explanation by theories of surface-tension.

In all the mammalia (save a very few) the red blood-corpuscles are
flattened circular discs, dimpled in upon their two opposite sides.
This configuration closely resembles that of an india-rubber ball when
we pinch it tightly between finger and thumb; and we may also compare
it with that experiment of Plateau’s {271} (described on p. 223),
where a flat cylindrical oil-drop, of certain relative dimensions,
can, by sucking away a little of the contained oil, be made to assume
the form of a biconcave disc, whose periphery is part of a nodoidal
surface. From the relation of the nodoid to the “elastic curve,” we
perceive that these two examples are closely akin one to the other.

[Illustration: Fig. 93.]

The form of the corpuscle is symmetrical, and its surface is a surface
of revolution; but it is obviously not a surface of constant mean
curvature, nor of constant pressure. For we see at once that, in the
sectional diagram (Fig. 93), the pressure inwards due to surface
tension is positive at _A_, and negative at _C_; at _B_ there is no
curvature in the plane of the paper, while perpendicular to it the
curvature is negative, and the pressure therefore is also negative.
Accordingly, from the point of view of surface tension alone, the
blood-corpuscle is not a surface of equilibrium; or in other words,
it is not a fluid drop suspended in another liquid. It is obvious
therefore that some other force or forces must be at work, and the
simple effect of mechanical pressure is here excluded, because the
blood-corpuscle exhibits its characteristic shape while floating freely
in the blood. In the lower vertebrates the blood-corpuscles have the
form of a flattened oval disc, with rather sharp edges and ellipsoidal
surfaces, and this again is manifestly not a surface of equilibrium.

Two facts are especially noteworthy in connection with the form of the
blood-corpuscle. In the first place, its form is only maintained, that
is to say it is only in equilibrium, in relation to certain properties
of the medium in which it floats. If we add a little water to the
blood, the corpuscle quickly loses its characteristic shape and becomes
a spherical drop, that is to say a true surface of minimal area and of
stable equilibrium. If on the other hand we add a strong solution of
salt, or a little glycerine, the corpuscle contracts, and its surface
becomes puckered and uneven. In these phenomena it is so far obeying
the laws of diffusion and of surface tension. {272}

In the second place, it can be exactly imitated artificially by means
of other colloid substances. Many years ago Norris made the very
interesting observation that in an emulsion of glue the drops assumed
a biconcave form resembling that of the mammalian corpuscles[317]. The
glue was impure, and doubtless contained lecithin; and it is possible
(as Professor Waymouth Reid tells me) to make a similar emulsion with
cerebrosides and cholesterin oleate, in which the same conformation
of the drops or particles is beautifully shewn. Now such cholesterin
bodies have an important place among those in which Lehmann and others
have shewn and studied the formation of fluid crystals, that is to
say of bodies in which the forces of crystallisation and the forces
of surface tension are battling with one another[318]; and, for want
of a better explanation, we may in the meanwhile suggest that some
such cause is at the bottom of the conformation the explanation of
which presents so many difficulties. But we must not, perhaps, pass
from this subject without adding that the case is a difficult and
complex one from the physiological point of view. For the surface of a
blood-corpuscle consists of a “semi-permeable membrane,” through which
certain substances pass freely and not others (for the most part anions
and not cations), and it may be, accordingly, that we have in life a
continual state of osmotic inequilibrium, of negative osmotic tension
within, to which comparatively simple cause the imperfect distension
of the corpuscle may be also due[319]. The whole phenomenon would
be comparatively easy to understand if we might postulate a stiffer
peripheral region to the corpuscle, in the form for instance of a
peripheral elastic ring. Such an annular thickening or stiffening, like
the “collapse-rings” which an engineer inserts in a boiler, has been
actually asserted to exist, but its presence is not authenticated.

But it is not at all improbable that we have still much to learn about
the phenomena of osmosis itself, as manifested in the case of minute
bodies such as a blood-corpuscle; and (as Professor Peddie suggests to
me) it is by no means impossible that _curvature_ {273} of the surface
may itself modify the osmotic or perhaps the adsorptive action. If it
should be found that osmotic action tended to stop, or to reverse,
on change of curvature, it would follow that this phenomenon would
give rise to internal currents; and the change of pressure consequent
on these would tend to intensify the change of curvature when once
started[320].

[Illustration: Fig. 94. Sperm-cells of Decapod Crustacea (after
Koltzoff). _a_, _Inachus scorpio_; _b_, _Galathea squamifera_; _c_,
_do._ after maceration, to shew spiral fibrillae.]

The sperm-cells of the Decapod crustacea exhibit various singular
shapes. In the Crayfish they are flattened cells with stiff curved
processes radiating outwards like a St Catherine’s wheel; in Inachus
there are two such circles of stiff processes; in Galathea we have a
still more complex form, with long and slightly twisted processes.
In all these cases, just as in the case of the blood-corpuscle, the
structure alters, and finally loses, its characteristic form when the
nature or constitution (or as we may assume in particular—the density)
of the surrounding medium is changed.

Here again, as in the blood-corpuscle, we have to do with a very
important force, which we had not hitherto considered in this
connection,—the force of osmosis, manifested under conditions similar
to those of Pfeffer’s classical experiments on the plant-cell. The
surface of the cell acts as a “semi-permeable membrane,” {274}
permitting the passage of certain dissolved substances (or their
“ions”) and including or excluding others; and thus rendering manifest
and measurable the existence of a definite “osmotic pressure.” In the
case of the sperm-cells of Inachus, certain quantitative experiments
have been performed[321]. The sperm-cell exhibits its characteristic
conformation while lying in the serous fluid of the animal’s body, in
ordinary sea-water, or in a 5 per cent. solution of potassium nitrate;
these three fluids being all “isotonic” with one another. As we alter
the concentration of potassium nitrate, the cell assumes certain
definite forms corresponding to definite concentrations of the salt;
and, as a further and final proof that the phenomenon is entirely
physical, it is found that other salts produce an identical effect
when their concentration is proportionate to their molecular weight,
and whatever identical effect is produced by various salts in their
respective concentrations, a similarly identical effect is produced
when these concentrations are doubled or otherwise proportionately
changed[322].

[Illustration: Fig. 95. Sperm-cells of _Inachus_, as they appear in
saline solutions of varying density. (After Koltzoff.)]

Thus the following table shews the percentage concentrations of certain
salts necessary to bring the cell into the forms _a_ and _c_ of Fig.
95; in each case the quantities are proportional to the molecular
weights, and in each case twice the quantity is necessary to produce
the effect of Fig. 95_c_ compared with that which gives rise to the all
but spherical form of Fig. 95_a_. {275}

                    % concentration of salts
                     in which the sperm-cell
                  of Inachus assumes the form of
                     ─────────────────────
                     fig. _a_     fig. _c_

 Sodium chloride       0·6        1·2
 Sodium nitrate        0·85       1·7
 Potassium nitrate     1·0        2·0
 Acetic acid           2·2        4·5
 Cane sugar            5·0       10·0

[Illustration: Fig. 96. Sperm-cell of _Dromia_. (After Koltzoff.)]

If we look then, upon the spherical form of the cell as its true
condition of symmetry and of equilibrium, we see that what we call
its normal appearance is just one of many intermediate phases of
shrinkage, brought about by the abstraction of fluid from its interior
as the result of an osmotic pressure greater outside than inside the
cell, and where the shrinkage of _volume_ is not kept pace with by a
contraction of the _surface-area_. In the case of the blood-corpuscle,
the shrinkage is of no great amount, and the resulting deformation is
symmetrical; such structural inequality as may be necessary to account
for it need be but small. But in the case of the sperm-cells, we must
have, and we actually do find, a somewhat complicated arrangement of
more or less rigid or elastic structures in the wall of the cell, which
like the wire framework in Plateau’s experiments, restrain and modify
the forces acting on the drop. In one form of Plateau’s experiments,
instead of supporting his drop on rings or frames of wire, he laid
upon its surface one or more elastic coils; and then, on withdrawing
oil from the centre of his globule, he saw its uniform shrinkage
counteracted by the spiral springs, with the result that the centre
of each elastic coil seemed to shoot out into a prominence. Just such
spiral coils are figured (after Koltzoff) in Fig. 96; and they may
be regarded as precisely akin to those local thickenings, spiral and
other, to which we have already ascribed the cylindrical form of the
Spirogyra cell. In all probability we must in like manner attribute the
peculiar spiral and other forms, for instance of many Infusoria, to
the {276} presence, among the multitudinous other differentiations of
their protoplasmic substance, of such more or less elastic fibrillae,
which play as it were the part of a microscopic skeleton[323].

――――――――――

But these cases which we have just dealt with, lead us to another
consideration. In a semi-permeable membrane, through which water
passes freely in and out, the conditions of a liquid surface are
greatly modified; and, in the ideal or ultimate case, there is neither
surface nor surface tension at all. And this would lead us somewhat
to reconsider our position, and to enquire whether the true surface
tension of a liquid film is actually responsible for _all_ that we
have ascribed to it, or whether certain of the phenomena which we have
assigned to that cause may not in part be due to the contractility of
definite and elastic membranes. But to investigate this question, in
particular cases, is rather for the physiologist: and the morphologist
may go on his way, paying little heed to what is no doubt a difficulty.
In surface tension we have the production of a film with the properties
of an elastic membrane, and with the special peculiarity that
contraction continues with the same energy however far the process
may have already gone; while the ordinary elastic membrane contracts
to a certain extent, and contracts no more. But within wide limits
the essential phenomena are the same in both cases. Our fundamental
equations apply to both cases alike. And accordingly, so long as our
purpose is _morphological_, so long as what we seek to explain is
regularity and definiteness of form, it matters little if we should
happen, here or there, to confuse surface tension with elasticity, the
contractile forces manifested at a liquid surface with those which come
into play at the complex internal surfaces of an elastic solid.

{277}




CHAPTER VI

A NOTE ON ADSORPTION


A very important corollary to, or amplification of the theory of
surface tension is to be found in the modern chemico-physical doctrine
of Adsorption[324]. In its full statement this subject soon becomes
complicated, and involves physical conceptions and mathematical
treatment which go beyond our range. But it is necessary for us to take
account of the phenomenon, though it be in the most elementary way.

In the brief account of the theory of surface tension with which our
last chapter began, it was pointed out that, in a drop of liquid,
the potential energy of the system could be diminished, and work
manifested accordingly, in two ways. In the first place we saw that,
at our liquid surface, surface tension tends to set up an equilibrium
of form, in which the surface is reduced or contracted either to the
absolute minimum of a sphere, or at any rate to the least possible
area which is permitted by the various circumstances and conditions;
and if the two bodies which comprise our system, namely the drop of
liquid and its surrounding medium, be simple substances, and the
system be uncomplicated by other distributions of force, then the
energy of the system will have done its work when this equilibrium of
form, this minimal area of surface, is once attained. This phenomenon
of the production of a minimal surface-area we have now seen to be
of fundamental importance in the external morphology of the cell,
and especially (so far as we have yet gone) of the solitary cell or
unicellular organism. {278}

But we also saw, according to Gauss’s equation, that the potential
energy of the system will be diminished (and its diminution will
accordingly be manifested in work) if from any cause the specific
surface energy be diminished, that is to say if it be brought more
nearly to an equality with the specific energy of the molecules in
the interior of the liquid mass. This latter is a phenomenon of great
moment in modern physiology, and, while we need not attempt to deal
with it in detail, it has a bearing on cell-form and cell-structure
which we cannot afford to overlook.

In various ways a diminution of the surface energy may be brought
about. For instance, it is known that every isolated drop of fluid
has, under normal circumstances, a surface-charge of electricity: in
such a way that a positive or negative charge (as the case may be) is
inherent in the surface of the drop, while a corresponding charge,
of contrary sign, is inherent in the immediately adjacent molecular
layer of the surrounding medium. Now the effect of this distribution,
by which all the surface molecules of our drop are similarly charged,
is that by virtue of this charge they tend to repel one another, and
possibly also to draw other molecules, of opposite charge, from the
interior of the mass; the result being in either case to antagonise or
cancel, more or less, that normal tendency of the surface molecules to
attract one another which is manifested in surface tension. In other
words, an increased electrical charge concentrating at the surface of a
drop tends, whether it be positive or negative, to _lower_ the surface
tension.

But a still more important case has next to be considered. Let
us suppose that our drop consists no longer of a single chemical
substance, but contains other substances either in suspension or in
solution. Suppose (as a very simple case) that it be a watery fluid,
exposed to air, and containing droplets of oil: we know that the
specific surface tension of oil in contact with air is much less than
that of water, and it follows that, if the watery surface of our drop
be replaced by an oily surface the specific surface energy of the
system will be notably diminished. Now under these circumstances it is
found that (quite apart from gravity, by which the oil might _float_
to the surface) the oil has a tendency to be _drawn_ to the surface;
and this phenomenon of molecular attraction {279} or “adsorption”
represents the work done, equivalent to the diminished potential energy
of the system[325]. In more general terms, if a liquid (or one or other
of two adjacent liquids) be a chemical mixture, some one constituent
in which, if it entered into or increased in amount in the surface
layer, would have the effect of diminishing its surface tension, then
that constituent will have a tendency to accumulate or concentrate
at the surface: the surface tension may be said, as it were, to
exercise an attraction on this constituent substance, drawing it into
the surface layer, and this tendency will proceed until at a certain
“surface concentration” equilibrium is reached, its opponent being that
osmotic force which tends to keep the substance in uniform solution or
diffusion.

In the complex mixtures which constitute the protoplasm of the living
cell, this phenomenon of “adsorption” has abundant play: for many of
these constituents, such as oils, soaps, albumens, etc. possess the
required property of diminishing surface tension.

Moreover, the more a substance has the power of lowering the surface
tension of the liquid in which it happens to be dissolved, the more
will it tend to displace another and less effective substance from
the surface layer. Thus we know that protoplasm always contains fats
or oils, not only in visible drops, but also in the finest suspension
or “colloidal solution.” If under any impulse, such for instance as
might arise from the Brownian movement, a droplet of oil be brought
close to the surface, it is at once drawn into that surface, and tends
to spread itself in a thin layer over the whole surface of the cell.
But a soapy surface (for instance) would have in contact with the
surrounding water a surface tension even less than that of the film
of oil: and consequently, if soap be present in the water it will in
turn be adsorbed, and will tend to displace the oil from the surface
pellicle[326]. And this is all as {280} much as to say that the
molecules of the dissolved or suspended substance or substances will
so distribute themselves throughout the drop as to lead towards an
equilibrium, for each small unit of volume, between the superficial and
internal energy; or so, in other words, as to lead towards a reduction
to a minimum of the potential energy of the system. This tendency
to concentration at the surface of any substance within the cell by
which the surface tension tends to be diminished, or _vice versa_,
constitutes, then, the phenomenon of _Adsorption_; and the general
statement by which it is defined is known as the Willard-Gibbs, or
Gibbs-Thomson law[327].

Among the many important physical features or concomitants of this
phenomenon, let us take note at present that we need not conceive of a
strictly superficial distribution of the adsorbed substance, that is
to say of its direct association with the surface layer of molecules
such as we imagined in the case of the electrical charge; but rather of
a progressive tendency to concentrate, more and more, as the surface
is nearly approached. Indeed we may conceive the colloid or gelatinous
precipitate in which, in the case of our protoplasmic cell, the
dissolved substance tends often to be thrown down, to constitute one
boundary layer after another, the general effect being intensified and
multiplied by the repeated addition of these new surfaces.

Moreover, it is not less important to observe that the process of
adsorption, in the neighbourhood of the surface of a heterogeneous
liquid mass, is a process which _takes time_; the tendency to surface
concentration is a gradual and progressive one, and will fluctuate with
every minute change in the composition of our substance and with every
change in the area of its surface. In other words, it involves (in
every heterogeneous substance) a continual instability of equilibrium:
and a constant manifestation {281} of motion, sometimes in the mere
invisible transfer of molecules but often in the production of visible
currents of fluid or manifest alterations in the form or outline of the
system.

――――――――――

The physiologist, as we have already remarked, takes account of
the general phenomenon of adsorption in many ways: particularly in
connection with various results and consequences of osmosis, inasmuch
as this process is dependent on the presence of a membrane, or
membranes, such as the phenomenon of adsorption brings into existence.
For instance it plays a leading part in all modern theories of muscular
contraction, in which phenomenon a connection with surface tension
was first indicated by FitzGerald and d’Arsonval nearly forty years
ago[328]. And, as W. Ostwald was the first to shew, it gives us an
entirely new conception of the relation of gases (that is to say, of
oxygen and carbon dioxide) to the red corpuscles of the blood[329].

But restricting ourselves, as much as may be, to our morphological
aspect of the case, there are several ways in which adsorption begins
at once to throw light upon our subject.

In the first place, our preliminary account, such as it is, is
already tantamount to a description of the process of development
of a cell-membrane, or cell-wall. The so-called “secretion” of this
cell-wall is nothing more than a sort of exudation, or striving towards
the surface, of certain constituent molecules or particles within the
cell; and the Gibbs-Thomson law formulates, in part at least, the
conditions under which they do so. The adsorbed material may range
from the almost unrecognisable pellicle of a blood-corpuscle to the
distinctly differentiated “ectosarc” of a protozoan, and again to the
development of a fully formed cell-wall, as in the cellulose partitions
of a vegetable tissue. In such cases, the dissolved and adsorbable
material has not only the property of lowering the surface tension,
and hence {282} of itself accumulating at the surface, but has also
the property of increasing the viscosity and mechanical rigidity
of the material in which it is dissolved or suspended, and so of
constituting a visible and tangible “membrane[330].” The “zoogloea”
around a group of bacteria is probably a phenomenon of the same order.
In the superficial deposition of inorganic materials we see the same
process abundantly exemplified. Not only do we have the simple case of
the building of a shell or “test” upon the outward surface of a living
cell, as for instance in a Foraminifer, but in a subsequent chapter,
when we come to deal with various spicules and spicular skeletons such
as those of the sponges and of the Radiolaria, we shall see that it is
highly characteristic of the whole process of spicule-formation for the
deposits to be laid down just in the “interfacial” boundaries between
cells or vacuoles, and that the form of the spicular structures tends
in many cases to be regulated and determined by the arrangement of
these boundaries.

 In physical chemistry, an important distinction is drawn between
 adsorption and _pseudo-adsorption_[331], the former being a
 _reversible_, the latter an irreversible or permanent phenomenon.
 That is to say, adsorption, strictly speaking, implies the
 surface-concentration of a dissolved substance, under circumstances
 which, if they be altered or reversed, will cause the concentration to
 diminish or disappear. But pseudo-adsorption includes cases, doubtless
 originating in adsorption proper, where subsequent changes leave the
 concentrated substance incapable of re-entering the liquid system. It
 is obvious that many (though not all) of our biological illustrations,
 for instance the formation of spicules or of permanent cell-membranes,
 belong to the class of so-called pseudo-adsorption phenomena. But the
 apparent contrast between the two is in the main a secondary one, and
 however important to the chemist is of little consequence to us. {283}

While this brief sketch of the theory of membrane-formation is cursory
and inadequate, it is enough to shew that the physical theory of
adsorption tends in part to overturn, in part to simplify enormously,
the older histological descriptions. We can no longer be content
with such statements as that of Strasbürger, that membrane-formation
in general is associated with the “activity of the kinoplasm,” or
that of Harper that a certain spore-membrane arises directly from
the astral rays[332]. In short, we have easily reached the general
conclusion that, the formation of a cell-wall or cell-membrane is a
chemico-physical phenomenon, which the purely objective methods of the
biological microscopist do not suffice to interpret.

――――――――――

If the process of adsorption, on which the formation of a membrane
depends, be itself dependent on the power of the adsorbed substance to
lower the surface tension, it is obvious that adsorption can only take
place when the surface tension already present is greater than zero.
It is for this reason that films or threads of creeping protoplasm
shew little tendency, or none, to cover themselves with an encysting
membrane; and that it is only when, in an altered phase, the protoplasm
has developed a positive surface tension, and has accordingly gathered
itself up into a more or less spherical body, that the tendency to
form a membrane is manifested, and the organism develops its “cyst” or
cell-wall.

It is found that a rise of temperature greatly reduces the
adsorbability of a substance, and this doubtless comes, either in part
or whole, from the fact that a rise of temperature is itself a cause
of the lowering of surface tension. We may in all probability ascribe
to this fact and to its converse, or at least associate with it, such
phenomena as the encystment of unicellular organisms at the approach
of winter, or the frequent formation of strong shells or membranous
capsules in “winter-eggs.”

Again, since a film or a froth (which is a system of films) can only
be maintained by virtue of a certain viscosity or rigidity of {284}
the liquid, it may be quickly caused to disappear by the presence in
its neighbourhood of some substance capable of reducing the surface
tension; for this substance, being adsorbed, may displace from the
adsorptive layer a material to which was due the rigidity of the film.
In this way a “bathytonic” substance such as ether causes most foams
to subside, and the pouring oil on troubled waters not only stills the
waves but still more quickly dissipates the foam of the breakers. The
process of breaking up an alveolar network, such as occurs at a certain
stage in the nuclear division of the cell, may perhaps be ascribed in
part to such a cause, as well as to the direct lowering of surface
tension by electrical agency.

Our last illustration has led us back to the subject of a previous
chapter, namely to the visible configuration of the interior of the
cell; and in connection with this wide subject there are many phenomena
on which light is apparently thrown by our knowledge of adsorption, and
of which we took little or no account in our former discussion. One of
these phenomena is that visible or concrete “polarity,” which we have
already seen to be in some way associated with a dynamical polarity of
the cell.

This morphological polarity may be of a very simple kind, as when,
in an epithelial cell, it is manifested by the outward shape of
the elongated or columnar cell itself, by the essential difference
between its free surface and its attached base, or by the presence in
the neighbourhood of the former of mucous or other products of the
cell’s activity. But in a great many cases, this “polarised” symmetry
is supplemented by the presence of various fibrillae, or of linear
arrangements of particles, which in the elongated or “monopolar” cell
run parallel with its axis, and which tend to a radial arrangement in
the more or less rounded or spherical cell. Of late years especially,
an immense importance has been attached to these various linear or
fibrillar arrangements, as they occur (_after staining_) in the
cell-substance of intestinal epithelium, of spermatocytes, of ganglion
cells, and most abundantly and most frequently of all in gland cells.
Various functions, which seem somewhat arbitrarily chosen, have been
assigned, and many hard names given to them; for these structures now
include your mitochondria and your chondriokonts (both of these being
varieties {285} of chondriosomes), your Altmann’s granules, your
microsomes, pseudo-chromosomes, epidermal fibrils and basal filaments,
your archeoplasm and ergastoplasm, and probably your idiozomes,
plasmosomes, and many other histological minutiae[333].

[Illustration: Fig. 97. _A_, _B_, Chondriosomes in kidney-cells, prior
to and during secretory activity (after Barratt); _C_, do. in pancreas
of frog (after Mathews).]

The position of these bodies with regard to the other cell-structures
is carefully described. Sometimes they lie in the neighbourhood of
the nucleus itself, that is to say in proximity to the fluid boundary
surface which separates the nucleus from the cytoplasm; and in this
position they often form a somewhat cloudy sphere which constitutes the
_Nebenkern_. In the majority of cases, as in the epithelial cells, they
form filamentous structures, and rows of granules, whose main direction
is parallel to the axis of the cell, and which may, in some cases,
and in some forms, be conspicuous at the one end, and in some cases
at the other end of the cell. But I do not find that the histologists
attempt to explain, or to correlate with other phenomena, the tendency
of these bodies to lie parallel with the axis, and perpendicular to
the extremities of the cell; it is merely noted as a peculiarity, or
a specific character, of these particular structures. Extraordinarily
complicated and diverse functions have been ascribed to them.
Engelmann’s “Fibrillenkonus,” which was almost certainly another aspect
of the same phenomenon, was held by him and by cytologists like Breda
and Heidenhain, to be an apparatus connected in some {286} unexplained
way with the mechanism of ciliary movement. Meves looked upon the
chondriosomes as the actual carriers or transmitters of heredity[334].
Altmann invented a new aphorism, _Omne granulum e granulo_, as a
refinement of Virchow’s _omnis cellula e cellula_; and many other
histologists, more or less in accord, accepted the chondriosomes as
important entities, _sui generis_, intermediate in grade between
the cell itself and its ultimate molecular components. The extreme
cytologists of the Munich school, Popoff, Goldschmidt and others,
following Richard Hertwig, declaring these structures to be identical
with “chromidia” (under which name Hertwig ranked all extra-nuclear
chromatin), would assign them complex functions in maintaining the
balance between nuclear and cytoplasmic material; and the “chromidial
hypothesis,” as every reader of recent cytological literature knows,
has become a very abstruse and complicated thing[335]. With the help of
the “binuclearity hypothesis” of Schaudinn and his school, it has given
us the chromidial net, the chromidial apparatus, the trophochromidia,
idiochromidia, gametochromidia, the protogonoplasm, and many other
novel and original conceptions. The names are apt to vary somewhat in
significance from one writer to another.

The outstanding fact, as it seems to me, is that physiological science
has been heavily burdened in this matter, with a jargon of names and a
thick cloud of hypotheses; while, from the physical point of view we
are tempted to see but little mystery in the whole phenomenon, and to
ascribe it, in all probability and in general terms, to the gathering
or “clumping” together, under surface tension, of various constituents
of the heterogeneous cell-content, and to the drawing out of these
little clumps along the axis of the cell towards one or other of its
extremities, in relation to osmotic currents, as these in turn are set
up in direct relation {287} to the phenomena of surface energy and of
adsorption[336]. And all this implies that the study of these minute
structures, if it teach us nothing else, at least surely and certainly
reveals to us the presence of a definite “field of force,” and a
dynamical polarity within the cell.

――――――――――

Our next and last illustration of the effects of adsorption, which
we owe to the investigations of Professor Macallum, is of great
importance; for it introduces us to a series of phenomena in regard
to which we seem now to stand on firmer ground than in some of the
foregoing cases, though we cannot yet consider that the whole story
has been told. In our last chapter we were restricted mainly, though
not entirely, to a consideration of figures of equilibrium, such as
the sphere, the cylinder or the unduloid; and we began at once to
find ourselves in difficulties when we were confronted by departures
from symmetry, as for instance in the simple case of the ellipsoidal
yeast-cell and the production of its bud. We found the cylindrical cell
of Spirogyra, with its plane or spherical ends, a comparatively simple
matter to understand; but when this uniform cylinder puts out a lateral
outgrowth, in the act of conjugation, we have a new and very different
system of forces to explain. The analogy of the soap-bubble, or of the
simple liquid drop, was apt to lead us to suppose that the surface
tension was, on the whole, uniform over the surface of our cell; and
that its departures from symmetry of form were therefore likely to be
due to variations in external resistance. But if we have been inclined
to make such an assumption we must now {288} reconsider it, and be
prepared to deal with important localised variations in the surface
tension of the cell. For, as a matter of fact, the simple case of a
perfectly symmetrical drop, with uniform surface, at which adsorption
takes place with similar uniformity, is probably rare in physics, and
rarer still (if it exist at all) in the fluid or fluid-containing
system which we call in biology a cell. We have mostly to do with
cells whose general heterogeneity of substance leads to qualitative
differences of surface, and hence to varying distributions of surface
tension. We must accordingly investigate the case of a cell which
displays some definite and regular heterogeneity of its liquid surface,
just as Amoeba displays a heterogeneity which is complex, irregular and
continually fluctuating in amount and distribution. Such heterogeneity
as we are speaking of must be essentially chemical, and the preliminary
problem is to devise methods of “microchemical” analysis, which shall
reveal _localised_ accumulations of particular substances within the
narrow limits of a cell, in the hope that, their normal effect on
surface tension being ascertained, we may then correlate with their
presence and distribution the actual indications of varying surface
tension which the form or movement of the cell displays. In theory the
method is all that we could wish, but in practice we must be content
with a very limited application of it; for the substances which may
have such action as we are looking for, and which are also actual or
possible constituents of the cell, are very numerous, while the means
are very seldom at hand to demonstrate their precise distribution
and localisation. But in one or two cases we have such means, and
the most notable is in connection with the element potassium. As
Professor Macallum has shewn, this element can be revealed, in very
minute quantities, by means of a certain salt, a nitrite of cobalt
and sodium[337]. This salt penetrates readily into the tissues and
into the interior of the cell; it combines with potassium to form a
sparingly soluble nitrite of cobalt, sodium and potassium; and this,
on subsequent treatment with ammonium sulphide, is converted into a
characteristic black precipitate of cobaltic sulphide[338]. {289}

By this means Macallum demonstrated some years ago the unexpected
presence of accumulations of potassium (i.e. of chloride or other
salts of potassium) localised in particular parts of various cells,
both solitary cells and tissue cells; and he arrived at the conclusion
that the localised accumulations in question were simply evidences
of _concentration_ of the dissolved potassium salts, formed and
localised in accordance with the Gibbs-Thomson law. In other words,
these accumulations, occurring as they actually do in connection with
various boundary surfaces, are evidence, when they appear irregularly
distributed over such a surface, of inequalities in its surface
tension[339]; and we may safely take it that our potassium salts, like
inorganic substances in general, tend to _raise_ the surface tension,
and will therefore be found concentrating at a portion of the surface
whose tension is weak[340].

In Professor Macallum’s figure (Fig. 98, 1) of the little green alga
Pleurocarpus, we see that one side of the cell is beginning to bulge
out in a wide convexity. This bulge is, in the first place, a sign of
weakened surface tension on one side of the cell, which as a whole had
hitherto been a symmetrical cylinder; in the second place, we see that
the bulging area corresponds to the position of a great concentration
of the potassic salt; while in the third place, from the physiological
point of view, we call the phenomenon the first stage in the process of
conjugation. In Fig. 98, 2, of Mesocarpus (a close ally of Spirogyra),
we see the same phenomenon admirably exemplified in a later stage.
From the adjacent cells distinct outgrowths are being emitted, where
the surface tension has been weakened: just as the glass-blower warms
and softens a small part of his tube to blow out the softened area
into a bubble or diverticulum; and in our Mesocarpus cells (besides a
certain amount of potassium rendered visible over the boundary which
{290} separates the green protoplasm from the cell-sap), there is a
very large accumulation precisely at the point where the tension of the
originally cylindrical cell is weakening to produce the bulge. But in a
still later stage, when the boundary between the two conjugating cells
is lost and the cytoplasm of the two cells becomes fused together,
then the signs of potassic concentration quickly disappear, the salt
becoming generally diffused through the now symmetrical and spherical
“zygospore.”

[Illustration: Fig. 98. Adsorptive concentration of potassium salts in
(1) cell of _Pleurocarpus_ about to conjugate; (2) conjugating cells of
_Mesocarpus_; (3) sprouting spores of _Equisetum_. (After Macallum.)]

In a spore of Equisetum (Fig. 98, 3), while it is still a single cell,
no localised concentration of potassium is to be discerned; but as
soon as the spore has divided, by an internal partition, into two
cells, the potassium salt is found to be concentrated in the smaller
one, and especially towards its outer wall, which is marked by a
pronounced convexity. And as this convexity (which corresponds to one
pole of the now asymmetrical, or quasi-ellipsoidal spore) grows out
into the root-hair, the potassium salt accompanies its growth, and is
concentrated under its wall. The concentration is, {291} accordingly,
a concomitant of the diminished surface tension which is manifested in
the altered configuration of the system.

In the case of ciliate or flagellate cells, there is to be found a
characteristic accumulation of potassium at and near the base of the
cilia. The relation of ciliary movement to surface tension lies beyond
our range, but the fact which we have just mentioned throws light
upon the frequent or general presence of a little protuberance of the
cell-surface just where a flagellum is given off (cf. p. 247), and of
a little projecting ridge or fillet at the base of an isolated row of
cilia, such as we find in Vorticella.

Yet another of Professor Macallum’s demonstrations, though its interest
is mainly physiological, will help us somewhat further to comprehend
what is implied in our phenomenon. In a normal cell of Spirogyra, a
concentration of potassium is revealed along the whole surface of the
spiral coil of chlorophyll-bearing, or “chromatophoral,” protoplasm,
the rest of the cell being wholly destitute of the former substance:
the indication being that, at this particular boundary, between
chromatophore and cell-sap, the surface tension is small in comparison
with any other interfacial surface within the system.

Now as Macallum points out, the presence of potassium is known to be
a factor, in connection with the chlorophyll-bearing protoplasm, in
the synthetic production of starch from CO_{2} under the influence of
sunlight. But we are left in some doubt as to the consecutive order
of the phenomena. For the lowered surface tension, indicated by the
presence of the potassium, may be itself a cause of the carbohydrate
synthesis; while on the other hand, this synthesis may be attended
by the production of substances (e.g. formaldehyde) which lower the
surface tension, and so conduce to the concentration of potassium. All
we know for certain is that the several phenomena are associated with
one another, as apparently inseparable parts or inevitable concomitants
of a certain complex action.

――――――――――

And now to return, for a moment, to the question of cell-form. When
we assert that the form of a cell (in the absence of mechanical
pressure) is essentially dependent on surface tension, and even when
we make the preliminary assumption that protoplasm is essentially
{292} a fluid, we are resting our belief on a general consensus of
evidence, rather than on compliance with any one crucial definition.
The simple fact is that the agreement of cell-forms with the forms
which physical experiment and mathematical theory assign to liquids
under the influence of surface tension, is so frequently and often
so typically manifested, that we are led, or driven, to accept the
surface tension hypothesis as generally applicable and as equivalent
to a universal law. The occasional difficulties or apparent exceptions
are such as call for further enquiry, but fall short of throwing doubt
upon our hypothesis. Macallum’s researches introduce a new element
of certainty, a “nail in a sure place,” when they demonstrate that,
in certain movements or changes of form which we should naturally
attribute to weakened surface tension, a chemical concentration which
would naturally accompany such weakening actually takes place. They
further teach us that in the cell a chemical heterogeneity may exist
of a very marked kind, certain substances being accumulated here and
absent there, within the narrow bounds of the system.

Such localised accumulations can as yet only be demonstrated in the
case of a very few substances, and of a single one in particular;
and these are substances whose presence does not produce, but whose
concentration tends to follow, a weakening of surface tension. The
physical cause of the localised inequalities of surface tension remains
unknown. We may assume, if we please, that it is due to the prior
accumulation, or local production, of chemical bodies which would
have this direct effect; though we are by no means limited to this
hypothesis.

But in spite of some remaining difficulties and uncertainties, we have
arrived at the conclusion, as regards unicellular organisms, that
not only their general configuration but also _their departures from
symmetry_ may be correlated with the molecular forces manifested in
their fluid or semi-fluid surfaces.

{293}




CHAPTER VII

THE FORMS OF TISSUES OR CELL-AGGREGATES


We now pass from the consideration of the solitary cell to that of
cells in contact with one another,—to what we may call in the first
instance “cell-aggregates,”—through which we shall be led ultimately to
the study of complex tissues. In this part of our subject, as in the
preceding chapters, we shall have to give some consideration to the
effects of various forces; but, as in the case of the conformation of
the solitary cell, we shall probably find, and we may at least begin
by assuming, that the agency of surface tension is especially manifest
and important. The effect of this surface tension will chiefly manifest
itself in the production of surfaces _minimae areae_: where, as Plateau
was always careful to point out, we must understand by this expression
not an absolute, but a relative minimum, an area, that is to say, which
approximates to an absolute minimum as nearly as circumstances and the
conditions of the case permit.

There are certain fundamental principles, or fundamental equations,
besides those which we have already considered, which we shall need in
our enquiry. For instance the case which we briefly touched upon (on
p. 265) of the angle of contact between the protoplasm and the axial
filament in a Heliozoan we shall now find to be but a particular case
of a general and elementary theorem.

Let us re-state as follows, in terms of _Energy_, the general principle
which underlies the theory of surface tension or capillarity.

When a fluid is in contact with another fluid, or with a solid or a
gas, a portion of the energy of the system (that, namely, which we call
surface energy), is proportional to the area of the surface of contact:
it is also proportional to a coefficient which is specific for each
particular pair of substances, and which is constant for these, save
only in so far as it may be modified by {294} changes of temperature
or of electric charge. The condition of _minimum potential energy_ in
the system, which is the condition of equilibrium, will accordingly be
obtained by the utmost possible diminution in the area of the surfaces
in contact. When we have _three_ bodies in contact, the case becomes
a little more complex. Suppose for instance we have a drop of some
fluid, _A_, floating on another fluid, _B_, and exposed to air, _C_.
The whole surface energy of the system may now be considered as divided
into two parts, one at the surface of the drop, and the other outside
of the same; the latter portion is inherent in the surface _BC_,
between the mass of fluid _B_ and the superincumbent air, _C_; but the
former portion consists of two parts, for it is divided between the two
surfaces _AB_ and _AC_, that namely which separates the drop from the
surrounding fluid and that which separates it from the atmosphere. So
far as

[Illustration: Fig. 99.]

the drop is concerned, then, equilibrium depends on a proper balance
between the energy, per unit area, which is resident in its own two
surfaces, and that which is external thereto: that is to say, if we
call _E__{_bc_} the energy at the surface between the two fluids, and
so on with the other two pairs of surface energies, the condition of
equilibrium, or of maintenance of the drop, is that

 _E__{_bc_} < _E__{_ab_} + _E__{_ac_}.

If, on the other hand, the fluid _A_ happens to be oil and the fluid
_B_, water, then the energy _per unit area_ of the water-air surface
is greater than that of the oil-air surface and that of the oil-water
surface together; i.e.

 _E__{_wa_} > _E__{_oa_} + _E__{_ow_}.

Here there is no equilibrium, and in order to obtain it the water-air
surface must always tend to decrease and the other two interfacial
surfaces to increase; which is as much as to say that the water tends
to become covered by a spreading film of oil, and the water-air surface
to be abolished. {295}

The surface energy of which we have here spoken is manifested in that
contractile force, or “tension,” of which we have already had so much
to say[341]. In any part of the free water surface, for instance, one
surface particle attracts another surface particle, and the resultant
of these multitudinous attractions is an equilibrium of tension
throughout this particular surface. In the case of our three bodies
in contact with one another, and within a small area very near to
the point of contact, a water particle (for instance) will be pulled
outwards by another water particle; but on the opposite side, so to
speak, there will be no water surface, and no water particle, to
furnish the counterbalancing pull; this counterpull,

[Illustration: Fig. 100.]

[Illustration: Fig. 101.]

which is necessary for equilibrium, must therefore be provided by
the tensions existing in the _other two_ surfaces of contact. In
short, if we could imagine a single particle placed at the very point
of contact, it would be drawn upon by three different forces, whose
directions would lie in the three surface planes, and whose magnitude
would be proportional to the specific tensions characteristic of
the two bodies which in each case combine to form the “interfacial”
surface. Now for three forces acting at a point to be in equilibrium,
they must be capable of representation, in magnitude and direction, by
the three sides of a triangle, taken in order, in accordance with the
elementary theorem of the Triangle of Forces. So, if we know the form
of our floating drop (Fig. 100), then by drawing tangents from _O_
(the point of mutual contact), {296} we determine the three angles of
our triangle (Fig. 101), and we therefore know the relative magnitudes
of the three surface tensions, which magnitudes are proportional to
its sides; and conversely, if we know the magnitudes, or relative
magnitudes, of the three sides of the triangle, we also know its
angles, and these determine the form of the section of the drop. It is
scarcely necessary to mention that, since all points on the edge of the
drop are under similar conditions, one with another, the form of the
drop, as we look down upon it from above, must be circular, and the
whole drop must be a solid of revolution.

――――――――――

The principle of the Triangle of Forces is expanded, as follows, by
an old seventeenth-century theorem, called Lami’s Theorem: “_If three
forces acting at a point be in equilibrium, each force is proportional
to the sine of the angle contained between the directions of the other
two._” That is to say

    _P_ : _Q_ : _R_ : = sin _QOR_ : sin _POR_ : sin _POQ_.

 or _P_/sin _QOR_ = _Q_/sin _ROP_ = _R_/sin _POQ_.

And from this, in turn, we derive the equivalent formulae, by which
each force is expressed in terms of the other two, and of the angle
between them:

 _P_^2 = _Q_^2 + _R_^2 + 2_Q_ _R_ cos(_QOR_), etc.

From this and the foregoing, we learn the following important and
useful deductions:

(1) The three forces can only be in equilibrium when any one of them
is less than the sum of the other two: for otherwise, the triangle is
impossible. Now in the case of a drop of olive-oil upon a clean water
surface, the relative magnitudes of the three tensions (at 15° C.) have
been determined as follows:

 Water-air surface   75
 Oil-air surface     32
 Oil-water surface   21

No triangle having sides of these relative magnitudes is possible; and
no such drop therefore can remain in equilibrium. {297}

(2) The three surfaces may be all alike: as when a soap-bubble floats
upon soapy water, or when two soap-bubbles are joined together, on
either side of a partition-film. In this case, the three tensions are
all equal, and therefore the three angles are all equal; that is to
say, when three similar liquid surfaces meet together, they always
do so at an angle of 120°. Whether our two conjoined soap-bubbles
be equal or unequal, this is still the invariable rule; because the
specific tension of a particular surface is unaffected by any changes
of magnitude or form.

(3) If two only of the surfaces be alike, then two of the angles will
be alike, and the other will be unlike; and this last will be the
difference between 360° and the sum of the other two. A particular case
is when a film is stretched between solid and parallel walls, like
a soap-film within a cylindrical tube. Here, so long as there is no
external pressure applied to either side, so long as both ends of the
tube are open or closed, the angles on either side of the film will be
equal, that is to say the film will set itself at right angles to the
sides.

Many years ago Sachs laid it down as a principle, which has become
celebrated in botany under the name of Sachs’s Rule, that one cell-wall
always tends to set itself at right angles to another cell-wall.
This rule applies to the case which we have just illustrated; and
such validity as the rule possesses is due to the fact that among
plant-tissues it very frequently happens that one cell-wall has become
solid and rigid before another and later partition-wall is developed in
connection with it.

(4) There is another important principle which arises not out of our
equations but out of the general considerations by which we were
led to them. We have seen that, at and near the point of contact
between our several surfaces, there is a continued balance of forces,
carried, so to speak, across the interval; in other words, there is
_physical continuity_ between one surface and another. It follows
necessarily from this that the surfaces merge one into another by a
continuous curve. Whatever be the form of our surfaces and whatever
the angle between them, this small intervening surface, approximately
spherical, is always there to bridge over the line of contact[342];
and this little fillet, or “bourrelet,” {298} as Plateau called
it, is large enough to be a common and conspicuous feature in the
microscopy of tissues (Fig. 102). For instance, the so-called
“splitting” of the cell-wall, which is conspicuous at the angles of the
large “parenchymatous” cells in the succulent tissues of all higher
plants (Fig. 103), is nothing more than a manifestation of Plateau’s
“bourrelet,” or surface of continuity[343].

――――――――――

We may now illustrate some of the foregoing principles, before we
proceed to the more complex cases in which more bodies than three are
in mutual contact. But in doing so, we must constantly bear in mind
the principles set forth in our chapter on the forms of cells, and
especially those relating to the pressure exercised by a curved film.

[Illustration: Fig. 102. (After Berthold.)]

[Illustration: Fig. 103. Parenchyma of Maize.]

Let us look for a moment at the case presented by the partition-wall
in a double soap-bubble. As we have just seen, the three films in
contact (viz. the outer walls of the two bubbles and the partition-wall
between) being all composed of the same substance {299} and all alike
in contact with air, the three surface tensions must be equal; and the
three films must therefore, in all cases, meet at an angle of 120°.
But, unless the two bubbles be of precisely equal size (and therefore
of equal curvature) it is obvious that the tangents to the spheres
will not meet the plane of their circle of contact at equal angles,
and therefore that the partition-wall must be a _curved_ surface: it
is only plane when it divides two equal and symmetrical cells. It is
also obvious, from the symmetry of the figure, that the centres of
the spheres, the centre of the partition, and the centres of the two
spherical surfaces are all on one and the same straight line.

[Illustration: Fig. 104.]

Now the surfaces of the two bubbles exert a pressure inwards
which is inversely proportional to their radii: that is to say
_p_ : _p′_ :: 1/_r′_ : 1/_r_; and the partition wall must,
for equilibrium, exert a pressure (_P_) which is equal to the
difference between these two pressures, that is to say, _P_ = 1/_R_
= 1/_r′_ − 1/_r_ = (_r_ − _r′_)/_r_ _r′_. It follows that the curvature
of the partition wall must be just such a curvature as is capable of
exerting this pressure, that is to say, _R_ = _r_ _r′_/(_r_ − _r′_).
The partition wall, then, is always a portion of a spherical surface,
whose radius is equal to the product, divided by the difference, of
the radii of the two vesicles. It follows at once from this that if
the two bubbles be equal, the radius of curvature of the partition is
infinitely great, that is to say the partition is (as we have already
seen) a plane surface.

The geometrical construction by which we obtain the position of the
centres of the two spheres and also of the partition surface is
very simple, always provided that the surface tensions are uniform
throughout the system. If _p_ be a point of contact between the two
spheres, and _cp_ be a radius of one of them, then make the angle _cpm_
= 60°, and mark off on _pm_, _pc′_ equal to the {300} radius of the
other sphere; in like manner, make the angle _c′pn_ = 60°, cutting the
line _cc′_ in _c″_; then _c′_ will be the centre of the second sphere,
and _c″_ that of the spherical partition.

[Illustration: Fig. 105.]

[Illustration: Fig. 106.]

Whether the partition be or be not a plane surface, it is obvious that
its _line of junction_ with the rest of the system lies in a plane, and
is at right angles to the axis of symmetry. The actual curvature of
the partition-wall is easily seen in optical section; but in surface
view, the line of junction is _projected_ as a plane (Fig. 106),
perpendicular to the axis, and this appearance has also helped to lend
support and authority to “Sachs’s Rule.”

――――――――――

[Illustration: Fig. 107. Filaments, or chains of cells, in various
lower Algae. (A) _Nostoc_; (B) _Anabaena_; (C) _Rivularia_; (D)
_Oscillatoria_.]

Many spherical cells, such as Protococcus, divide into two equal
halves, which are therefore separated by a plane partition. Among
the other lower Algae, akin to Protococcus, such as the Nostocs
and Oscillatoriae, in which the cells are imbedded in a gelatinous
matrix, we find a series of forms such as are represented in Fig. 107.
Sometimes the cells are solitary or disunited; sometimes they run in
pairs or in rows, separated one from another by flat partitions; and
sometimes the conjoined cells are approximately hemispherical, but
at other times each half is more than a hemisphere. These various
conditions depend, {301} according to what we have already learned,
upon the relative magnitudes of the tensions at the surface of the
cells and at the boundary between them[344].

In the typical case of an equally divided cell, such as a double and
co-equal soap-bubble, where the partition-wall and the outer walls
are similar to one another and in contact with similar substances, we
can easily determine the form of the system. For, at any point of the
boundary of the partition-wall, _O_, the tensions being equal, the
angles _QOP_, _ROP_, _QOR_ are all equal, and each is, therefore, an
angle of 120°. But _OQ_, _OR_ being tangents, the centres of the two
spheres (or circular arcs in the figure) lie on perpendiculars to them;
therefore the radii _CO_, _C′O_ meet at an

[Illustration: Fig. 108.]

angle of 60°, and _COC′_ is an equilateral triangle. That is to say,
the centre of each circle lies on the circumference of the other; the
partition lies midway between the two centres; and the length (i.e. the
diameter) of the partition-wall, _PO_, is

 2 sin 60° = 1·732

times the radius, or ·866 times the diameter, of each of the cells.
This gives us, then, the _form_ of an aggregate of two equal cells
under uniform conditions.

As soon as the tensions become unequal, whether from changes in their
own substance or from differences in the substances with which they
are in contact, then the form alters. If the tension {302} along
the partition, _P_, diminishes, the partition itself enlarges, and
the angle _QOR_ increases: until, when the tension _P_ is very small
compared to _Q_ or _R_, the whole figure becomes a circle, and the
partition-wall, dividing it into two hemispheres, stands at right
angles to the outer wall. This is the case when the outer wall of the
cell is practically solid. On the other hand, if _P_ begins to increase
relatively to _Q_ and _R_, then the partition-wall contracts, and the
two adjacent cells become larger and larger segments of a sphere, until
at length the system becomes divided into two separate cells.

[Illustration: Fig. 109. Spore of _Pellia_. (After Campbell.)]

In the spores of Liverworts (such as _Pellia_), the first
partition-wall (the equatorial partition in Fig. 109, _a_) divides the
spore into two equal halves, and is therefore a plane surface, normal
to the surface of the cell; but the next partitions arise near to
either end of the original spherical or elliptical cell. Each of these
latter partitions will (like the first) tend to set itself normally to
the cell-wall; at least the angles on either side of the partition will
be identical, and their magnitude will depend upon the tension existing
between the cell-wall and the surrounding medium. They will only be
right angles if the cell-wall is already practically solid, and in all
probability (rigidity of the cell-wall not being quite attained) they
will be somewhat greater. In either case the partition itself will be
a portion of a sphere, whose curvature will now denote a difference of
pressures in the two chambers or cells, which it serves to separate.
(The later stages of cell-division, represented in the figures _b_ and
_c_, we are not yet in a position to deal with.)

We have innumerable cases, near the tip of a growing filament, where
in like manner the partition-wall which cuts off the terminal {303}
cell constitutes a spherical lens-shaped surface, set normally to the
adjacent walls. At the tips of the branches of many Florideae, for
instance, we find such a lenticular partition. In _Dictyota dichotoma_,
as figured by Reinke, we have a succession of such partitions; and,
by the way, in such cases as these, where the tissues are very
transparent, we have often in optical section a puzzling confusion of
lines; one being the optical section of the curved partition-wall, the
other being the straight linear projection of its outer edge to which
we have already referred. In the conical terminal cell of Chara, we
have the same lens-shaped curve, but a little lower down, where the
sides of the shoot are approximately parallel, we have flat transverse
partitions, at the edges of which, however, we recognise a convexity of
the outer cell-wall and a definite angle of contact, equal on the two
sides of the partition.

[Illustration: Fig. 110. Cells of _Dictyota_. (After Reinke.)]

[Illustration: Fig. 111. Terminal and other cells of _Chara_.]

[Illustration: Fig. 112. Young antheridium of _Chara_.]

In the young antheridia of Chara (Fig. 112), and in the not dissimilar
case of the sporangium (or conidiophore) of Mucor, we easily recognise
the hemispherical form of the septum which shuts off the large
spherical cell from the cylindrical filament. Here, in the first phase
of development, we should have to take into consideration the different
pressures exerted by the single curvature of the cylinder and the
double curvature of its spherical cap (p. 221); and we should find
that the partition would have a somewhat low curvature, with a radius
_less_ than the diameter of the cylinder; which it would have exactly
equalled but for the additional pressure inwards which it receives
{304} from the curvature of the large surrounding sphere. But as the
latter continues to grow, its curvature decreases, and so likewise does
the inward pressure of its surface; and accordingly the little convex
partition bulges out more and more.

――――――――――

In order to epitomise the foregoing facts let the annexed diagrams
(Fig. 113) represent a system of three films, of which one is a
partition-wall between the other two; and let the tensions at the
three surfaces, or the tractions exercised upon a point at their
meeting-place, be proportional to _T_, _T′_ and _t_. Let α, β, γ be, as
in the figure, the opposite angles. Then:

(1) If _T_ be equal to _T′_, and _t_ be relatively insignificant, the
angles α, β will be of 90°.

[Illustration: Fig. 113.]

(2) If _T_ = _T′_, but be a little greater than _t_, then _t_ will
exert an appreciable traction, and α, β will be more than 90°, say, for
instance, 100°.

(3) If _T_ = _T′_ = _t_, then α, β, γ will all equal 120°.

The more complicated cases, when _t_, _T_ and _T′_ are all unequal, are
already sufficiently explained.

――――――――――

The biological facts which the foregoing considerations go a long way
to explain and account for have been the subject of much argument
and discussion, especially on the part of the botanists. Let me
recapitulate, in a very few words, the history of this long discussion.

Some fifty years ago, Hofmeister laid it down as a general law that
“The partition-wall stands always perpendicular to what was previously
the principal direction of growth in the cell,”—or, in other words,
perpendicular to the long axis of the cell[345]. Ten {305} years
later, Sachs formulated his rule, or principle, of “rectangular
section,” declaring that in all tissues, however complex, the
cell-walls cut one another (at the time of their formation) at right
angles[346]. Years before, Schwendener had found, in the final results
of cell-division, a universal system of “orthogonal trajectories[347]”;
and this idea Sachs further developed, introducing complicated systems
of confocal ellipses and hyperbolæ, and distinguishing between
periclinal walls, whose curves approximate to the peripheral contours,
radial partitions, which cut these at an angle of 90°, and finally
anticlines, which stand at right angles to the other two.

Reinke, in 1880, was the first to throw some doubt upon this
explanation. He pointed out various cases where the angle was not
a right angle, but was very definitely an acute one; and he saw,
apparently, in the more common rectangular symmetry merely what he
calls a necessary, but _secondary_, result of growth[348].

Within the next few years, a number of botanical writers were content
to point out further exceptions to Sachs’s Rule[349]; and in some cases
to show that the _curvatures_ of the partition-walls, especially such
cases of lenticular curvature as we have described, were by no means
accounted for by either Hofmeister or Sachs; while within the same
period, Sachs himself, and also Rauber, attempted to extend the main
generalisation to animal tissues[350].

While these writers regarded the form and arrangement of the
cell-walls as a biological phenomenon, with little if any direct
relation to ordinary physical laws, or with but a vague reference to
“mechanical conditions,” the physical side of the case was soon urged
by others, with more or less force and cogency. Indeed the general
resemblance between a cellular tissue and a “froth” {306} had been
pointed out long before, by Melsens, who had made an “artificial
tissue” by blowing into a solution of white of egg[351].

In 1886, Berthold published his _Protoplasmamechanik_, in which he
definitely adopted the principle of “minimal areas,” and, following
on the lines of Plateau, compared the forms of many cell-surfaces and
the arrangement of their partitions with those assumed under surface
tension by a system of “weightless films.” But, as Klebs[352] points
out in reviewing Berthold’s book, Berthold was careful to stop short of
attributing the biological phenomena to a definite mechanical cause.
They remained for him, as they had done for Sachs, so many “phenomena
of growth,” or “properties of protoplasm.”

In the same year, but while still apparently unacquainted with
Berthold’s work, Errera[353] published a short but very lucid article,
in which he definitely ascribed to the cell-wall (as Hofmeister had
already done) the properties of a semi-liquid film and drew from
this as a logical consequence the deduction that it _must_ assume
the various configurations which the law of minimal areas imposes on
the soap-bubble. So what we may call _Errera’s Law_ is formulated as
follows: A cellular membrane, at the moment of its formation, tends to
assume the form which would be assumed, under the same conditions, by a
liquid film destitute of weight.

Soon afterwards Chabry, in discussing the embryology of the Ascidians,
indicated many of the points in which the contacts between cells repeat
the surface-tension phenomena of the soap-bubble, and came to the
conclusion that part, at least, of the embryological phenomena were
purely physical[354]; and the same line of investigation and thought
were pursued and developed by Robert, in connection with the embryology
of the Mollusca[355]. Driesch again, in a series of papers, continued
to draw attention to the presence of capillary phenomena in the
segmenting cells {307} of various embryos, and came to the conclusion
that the mode of segmentation was of little importance as regards the
final result[356].

Lastly de Wildeman[357], in a somewhat wider, but also vaguer
generalisation than Errera’s, declared that “The form of the cellular
framework of vegetables, and also of animals, in its essential
features, depends upon the forces of molecular physics.”

――――――――――

[Illustration: Fig. 114.]

Let us return to our problem of the arrangement of partition films.
When we have three bubbles in contact, instead of two as in the case
already considered, the phenomenon is strictly analogous to our former
case. The three bubbles will be separated by three partition surfaces,
whose curvature will depend upon the relative size of the spheres, and
which will be plane if the latter are all of the same dimensions; but
whether plane or curved, the three partitions will meet one another
at an angle of 120°, in an axial line. Various pretty geometrical
corollaries accompany this arrangement. For instance, if Fig. 114
represent the three associated bubbles in a plane drawn through their
centres, _c_, _c′_, _c″_ (or what is the same thing, if it represent
the base of three bubbles resting on a plane), then the lines _uc_,
_uc″_, or _sc_, _sc′_, etc., drawn to the {308} centres from the
points of intersection of the circular arcs, will always enclose an
angle of 60°. Again (Fig. 115), if we make the angle _c″uf_ equal to
60°, and produce _uf_ to meet _cc″_ in _f_, _f_ will be the centre of
the circular arc which constitutes the partition _Ou_; and further, the
three points _f_, _g_, _h_, successively determined in this

[Illustration: Fig. 115.]

manner, will lie on one and the same straight line. In the case
of coequal bubbles or cells (as in Fig. 114, B), it is obvious that
the lines joining their centres form an equilateral triangle; and
consequently, that the centre of each circle (or sphere) lies on the
circumference of the other two; it is also obvious that _uf_ is now
{309} parallel to _cc″_, and accordingly that the centre of curvature
of the partition is now infinitely distant, or (as we have already
said), that the partition itself is plane.

[Illustration: Fig. 116.]

When we have four bubbles in conjunction, they would seem to be capable
of arrangement in two symmetrical ways: either, as in Fig. 116 (A),
with the four partition-walls meeting at right angles, or, as in (B),
with _five_ partitions meeting, three and three, at angles of 120°.
This latter arrangement is strictly analogous to the arrangement of
three bubbles in Fig. 114. Now, though both of these figures, from
their symmetry, are apparently figures of equilibrium, yet, physically,
the former turns out to be of unstable and the latter of stable
equilibrium. If we try to bring our four bubbles into the form of Fig.
116, A, such an arrangement endures only for an instant; the partitions
glide upon each other, a median wall springs into existence, and the
system at once assumes the form of our second figure (B). This is a
direct consequence of the law of minimal areas: for it can be shewn, by
somewhat difficult mathematics (as was first done by Lamarle), that,
in dividing a closed space into a given number of chambers by means
of partition-walls, the least possible area of these partition-walls,
taken together, can only be attained when they meet together in groups
of three, at equal angles, that is to say at angles of 120°. {310}

Wherever we have a true cellular complex, an arrangement of cells in
actual physical contact by means of a boundary film, we find this
general principle in force; we must only bear in mind that, for its
perfect recognition, we must be able to view the object in a plane
at right angles to the boundary walls. For instance, in any ordinary
section of a vegetable parenchyma, we recognise the appearance of
a “froth,” precisely resembling that which we can construct by
imprisoning a mass of soap-bubbles in a narrow vessel with flat sides
of glass; in both cases we see the cell-walls everywhere meeting, by
threes, at angles of 120°, irrespective of the size of the individual
cells: whose relative size, on the other hand, determines the
_curvature_ of the partition-walls. On the surface of a honey-comb we
have precisely the same conjunction, between cell and cell, of three
boundary walls, meeting at 120°. In embryology, when we examine a
segmenting egg, of four (or more) segments, we find in like manner, in
the great majority of cases, if not in all, that the same principle
is still exemplified; the four segments do not meet in a common
centre, but each cell is in contact with two others, and the three,
and only three, common boundary walls meet at the normal angle of
120°. A so-called _polar furrow_[358], the visible edge of a vertical
partition-wall, joins (or separates) the two triple contacts, precisely
as in Fig. 116, B.

In the four-celled stage of the frog’s egg, Rauber (an exceptionally
careful observer) shews us three alternative modes in which the
four cells may be found to be conjoined (Fig. 117). In (A) we have
the commonest arrangement, which is that which we have just studied
and found to be the simplest theoretical one; that namely where a
straight “polar furrow” intervenes, and where, at its extremities,
the partition-walls are conjoined three by three. In (B), we have
again a polar furrow, which is now seen to be a portion of the first
“segmentation-furrow” (cf. Fig. 155 etc.) by which the egg was
originally divided into two; the four-celled stage being reached by
the appearance of the transverse furrows {311} and their corresponding
partitions. In this case, the polar furrow is seen to be sinuously
curved, and Rauber tells us that its curvature gradually alters: as a
matter of fact, it (or rather the partition-wall corresponding to it)
is gradually setting itself into a position of equilibrium, that is
to say of equiangular contact with its neighbours, which position of
equilibrium is already attained or nearly so in Fig. 117, A. In Fig.
117, C, we have a very different condition, with which we shall deal in
a moment.

[Illustration: Fig. 117. Various ways in which the four cells are
co-arranged in the four-celled stage of the frog’s egg. (After Rauber.)]

According to the relative magnitude of the bodies in contact, this
“polar furrow” may be longer or shorter, and it may be so minute as to
be not easily discernible; but it is quite certain that no simple and
homogeneous system of fluid films such as we are dealing with is in
equilibrium without its presence. In the accounts given, however, by
embryologists of the segmentation of the egg, while the polar furrow
is depicted in the great majority of cases, there are others in which
it has not been seen and some in which its absence is definitely
asserted[359]. The cases where four cells, lying in one plane, meet _in
a point_, such as were frequently figured by the older embryologists,
are very difficult to verify, and I have not come across a single
clear case in recent literature. Considering the physical stability
of the other arrangement, the great preponderance of cases in which
it is known to occur, the difficulty of recognising the polar furrow
in cases where it is very small and unless it be specially looked
for, and the natural tendency of the draughtsman to make an all but
symmetrical structure appear wholly so, I am much inclined to attribute
to {312} error or imperfect observation all those cases where the
junction-lines of four cells are represented (after the manner of Fig.
116, A) as a simple cross[360].

But while a true four-rayed intersection, or simple cross, is
theoretically impossible (save as a transitory and highly unstable
condition), there is another condition which may closely simulate
it, and which is common enough. There are plenty of representations
of segmenting eggs, in which, instead of the triple junction and
polar furrow, the four cells (and in like manner their more numerous
successors) are represented as _rounded off_, and separated from one
another by an empty space, or by a little drop of an extraneous fluid,
evidently not directly miscible with the fluid surfaces of the cells.
Such is the case in the obviously accurate figure which Rauber gives
(Fig. 117, C) of the third mode of conjunction in the four-celled stage
of the frog’s egg. Here Rauber is most careful to point out that the
furrows do not simply “cross,” or meet in a point, but are separated
by a little space, which he calls the _Polgrübchen_, and asserts to be
constantly present whensoever the polar furrow, or _Brechungslinie_, is
not to be discerned. This little interposed space, with its contained
drop of fluid, materially alters the case, and implies a new condition
of theoretical and actual equilibrium. For, on the one hand, we see
that now the four intercellular partitions do not meet _one another
at all_; but really impinge upon four new and separate partitions,
which constitute interfacial contacts, not between cell and cell, but
between the respective cells and the intercalated drop. And secondly,
the angles at which these four little surfaces will meet the four
cell-partitions, will be determined, in the usual way, by the balance
between the respective tensions of these several surfaces. In an
extreme case (as in some pollen-grains) it may be found that the cells
under the observed circumstances are not truly in surface contact:
that they are so many drops which touch but do not “wet” one another,
and which are merely held together by the pressure of the surrounding
envelope. But even supposing, {313} as is in all probability the
actual case, that they are in actual fluid contact, the case from the
point of view of surface tension presents no difficulty. In the case of
the conjoined soap-bubbles, we were dealing with _similar_ contacts and
with _equal_ surface tensions throughout the system; but in the system
of protoplasmic cells which constitute the segmenting egg we must make
allowance for _an inequality_ of tensions, between the surfaces where
cell meets cell, and where on the other hand cell-surface is in contact
with the surrounding medium,—in this case generally water or one of the
fluids of the body. Remember that our general condition is that, in our

[Illustration: Fig. 118.]

entire system, the _sum of the surface energies_ is a minimum; and,
while this is attained by the _sum of the surfaces_ being a minimum
in the case where the energy is uniformly distributed, it is not
necessarily so under non-uniform conditions. In the diagram (Fig. 118)
if the energy per unit area be greater along the contact surface _cc′_,
where cell meets cell, than along _ca_ or _cb_, where cell-surface is
in contact with the surrounding medium, these latter surfaces will
tend to increase and the surface of cell-contact to diminish. In short
there will be the usual balance of forces between the tension along
the surface _cc′_, and the two opposing tensions along _ca_ and _cb_.
If the former be greater than either of the other two, the outside
angle will be less than 120°; and if the tension along the surface
_cc′_ be as much or more than the sum of the other two, then the drops
will stand in contact only, save for the possible effect of external
pressure, at a point. This is the explanation, in general terms, of
the peculiar conditions obtaining in Nostoc and its allies (p. 300),
and it also leads us to a consideration of the general properties and
characters of an “epidermal” layer.

――――――――――

While the inner cells of the honey-comb are symmetrically situated,
sharing with their neighbours in equally distributed pressures or
tensions, and therefore all tending with great accuracy {314} to
identity of form, the case is obviously different with the cells at
the borders of the system. So it is, in like manner, with our froth of
soap-bubbles. The bubbles, or cells, in the interior of the mass are
all alike in general character, and if they be equal in size are alike
in every respect: their sides are uniformly flattened[361], and tend
to meet at equal angles of 120°. But the bubbles which constitute the
outer layer retain their spherical surfaces, which however still tend
to meet the partition-walls connected with them at constant angles
of 120°. This outer layer of bubbles, which forms the surface of our
froth, constitutes after a fashion what we should call in botany an
“epidermal” layer. But in our froth of soap-bubbles we have, as a rule,
the same kind of contact (that is to say, contact with _air_) both
within and without the bubbles; while in our living cell, the outer
wall of the epidermal cell is exposed to air on the one side, but is in
contact with the

[Illustration: Fig. 119.]

protoplasm of the cell on the other: and this involves a difference
of tensions, so that the outer walls and their adjacent partitions
are no longer likely to meet at equal angles of 120°. Moreover, a
chemical change, due for instance to oxidation or possibly also to
adsorption, is very likely to affect the external wall, and may tend
to its consolidation; and this process, as we have seen, is tantamount
to a large increase, and at the same time an equalisation, of tension
in that outer wall, and will lead the adjacent partitions to impinge
upon it at angles more and more nearly approximating to 90°: the
bubble-like, or spherical, surfaces of the individual cells being
more and more flattened in consequence. Lastly, the chemical changes
which affect the outer walls of the superficial cells may extend, in
greater or less degree, to their inner walls also: with the result
that these {315} cells will tend to become more or less rectangular
throughout, and will cease to dovetail into the interstices of the
next subjacent layer. These then are the general characters which
we recognise in an epidermis; and we perceive that the fundamental
character of an epidermis simply is that it lies on the outside, and
that its main physical characteristics follow, as a matter of course,
from the position which it occupies and from the various consequences
which that situation entails. We have however by no means exhausted
the subject in this short account; for the botanist is accustomed to
draw a sharp distinction between a true epidermis and what is called
epidermal tissue. The latter, which is found in such a sea-weed as
Laminaria and in very many other cryptogamic plants, consists, as in
the hypothetical case we have described, of a more or less simple and
direct modification of the general or fundamental tissue. But a “true
epidermis,” such as we have it in the higher plants, is something
with a long morphological history, something which has been laid down
or differentiated in an early stage of the plant’s growth, and which
afterwards retains its separate and independent character. We shall
see presently that a physical reason is again at hand to account,
under certain circumstances, for the early partitioning off, from a
mass of embryonic tissue, of an outer layer of cells which from their
first appearance are marked off from the rest by their rectangular and
flattened form.

――――――――――

We have hitherto considered our cells, or bubbles, as lying in a plane
of symmetry, and further, we have only considered the appearance which
they present as projected on that plane: in simpler words, we have been
considering their appearance in surface or in sectional view. But we
have further to consider them as solids, whether they be still grouped
in relation to a single plane (like the four cells in Fig. 116) or
heaped upon one another, as for instance in a tetrahedral form like
four cannon-balls; and in either case we have to pass from the problems
of plane to those of solid geometry. In short, the further development
of our theme must lead us along two paths of enquiry, which continually
intercross, namely (1) the study of more complex cases of partition and
of contact in a plane, and (2) the whole question of the surfaces {316}
and angles presented by solid figures in symmetrical juxtaposition.
Let us take a simple case of the latter kind, and again afterwards, so
far as possible, let us try to keep the two themes separate.

Where we have three spheres in contact, as in Fig. 114 or in either
half of Fig. 116, B, let us consider the point of contact (_O_, Fig.
114) not as a point in the plane section of the diagram, but as a point
where three _furrows_ meet on the surface of the system. At this point,
_three cells_ meet; but it is also obvious that there meet here _six
surfaces_, namely the outer, spherical walls of the three bubbles,
and the three partition-walls which divide them, two and two. Also,
_four_ lines or _edges_ meet here; viz. the three external arcs which
form the outer boundaries of the partition-walls (and which correspond
to what we commonly call the “furrows” in the segmenting egg); and
as a fourth edge, the “arris” or junction of the three partitions
(perpendicular to the plane of the paper), where they all three meet
together, as we have seen, at equal angles of 120°. Lastly, there meet
at the point _four solid angles_, each bounded by three surfaces: to
wit, within each bubble a solid angle bounded by two partition-walls
and by the surface wall; and (fourthly) an external solid angle bounded
by the outer surfaces of all three bubbles. Now in the case of the
soap-bubbles (whose surfaces are all in contact with air, both outside
and in), the six films meeting at the point, whether surface films
or partition films, are all similar, with similar tensions. In other
words the tensions, or forces, acting at the point are all similar
and symmetrically arranged, and it at once follows from this that the
angles, solid as well as plane, are all equal. It is also obvious that,
as regards the point of contact, the system will still be symmetrical,
and its symmetry will be quite unchanged, if we add a fourth bubble in
contact with the other three: that is to say, if where we had merely
the outer air before, we now replace it by the air in the interior of
another bubble. The only difference will be that the pressure exercised
by the walls of this fourth bubble will alter the curvature of the
surfaces of the others, so far as it encloses them; and, if all four
bubbles be identical in size, these surfaces which formerly we called
external and which have now come to be internal partitions, will,
like the others, be flattened by equal and opposite pressure, into
planes. We are now dealing, in short, {317} with six planes, meeting
symmetrically in a point, and constituting there four equal solid
angles.

[Illustration: Fig. 120.]

If we make a wire cage, in the form of a regular tetrahedron, and dip
it into soap-solution, then when we withdraw it we see that to each
one of the six edges of the tetrahedron, i.e. to each one of the six
wires which constitute the little cage, a film has attached itself; and
these six films meet internally at a point, and constitute in every
respect the symmetrical figure which we have just been describing. In
short, the system of films we have hereby automatically produced is
precisely the system of partition-walls which exist in our tetrahedral
aggregation of four spherical bubbles:—precisely the same, that is to
say, in the neighbourhood of the meeting-point, and only differing in
that we have made the wires of our tetrahedron straight, instead of
imitating the circular arcs which actually form the intersections of
our bubbles. This detail we can easily introduce in our wire model if
we please.

Let us look for a moment at the geometry of our figure. Let _o_ (Fig.
120) be the centre of the tetrahedron, i.e. the centre of symmetry
where our films meet; and let _oa_, _ob_, _oc_, _od_, be lines drawn
to the four corners of the tetrahedron. Produce _ao_ to meet the base
in _p_; then _apd_ is a right-angled triangle. It is not difficult to
prove that in such a figure, _o_ (the centre of gravity of the system)
{318} lies just three-quarters of the way between an apex, _a_, and
a point, _p_, which is the centre of gravity of the opposite base.
Therefore

 _op_ = _oa_/3 = _od_/3.

 Therefore cos _dop_ = 1/3 and cos _aod_ = − 1/3.

That is to say, the angle _aod_ is just, as nearly as possible,
109° 28′ 16″. This angle, then, of 109° 28′ 16″, or very nearly 109
degrees and a half, is the angle at which, in this and _every other
solid system_ of liquid films, the edges of the partition-walls meet
one another at a point. It is the fundamental angle in the solid
geometry of our systems, just as 120° was the fundamental angle of
symmetry so long as we considered only the plane projection, or plane
section, of three films meeting in an edge.

――――――――――

Out of these two angles, we may construct a great variety of figures,
plane and solid, which become all the more varied and complex when, by
considering the case of unequal as well as equal cells, we admit curved
(e.g. spherical) as well as plane boundary surfaces. Let us consider
some examples and illustrations of these, beginning with those which we
need only consider in reference to a plane.

Let us imagine a system of equal cylinders, or equal spheres, in
contact with one another in a plane, and represented in section by
the equal and contiguous circles of Fig. 121. I borrow my figure, by
the way, from an old Italian naturalist, Bonanni (a contemporary of
Borelli, of Hay and Willoughby and of Martin Lister), who dealt with
this matter in a book chiefly devoted to molluscan shells[362].

It is obvious, as a simple geometrical fact, that each of these equal
circles is in contact with six surrounding circles. Imagine now that
the whole system comes under some uniform stress. It may be of uniform
surface tension at the boundaries of all the cells; it may be of
pressure caused by uniform growth or expansion within the cells; or
it may be due to some uniformly applied constricting pressure from
without. In all of these cases the _points_ of contact between the
circles in the diagram will be extended into {319} _lines_ of contact,
representing _surfaces_ of contact in the actual spheres or cylinders;
and the equal circles of our diagram will be converted into regular and
equal hexagons. The angles of these hexagons, at each of which three
hexagons meet, are of course angles of 120°. So far as the form is
concerned, so long as we are concerned only with a morphological result
and not with a physiological process, the result is precisely the same
whatever be the force which brings the bodies together in symmetrical
apposition; it is by no means necessary for us, in the first instance,
even to enquire whether it be surface tension or mechanical pressure or
some other physical force which is the cause, or the main cause, of the
phenomenon.

[Illustration: Fig. 121. Diagram of hexagonal cells. (After Bonanni.)]

The production by mutual interaction of polyhedral cells, which, under
conditions of perfect symmetry, become regular hexagons, is very
beautifully illustrated by Prof. Bénard’s “_tourbillons cellulaires_”
(cf. p. 259), and also in some of Leduc’s diffusion experiments. A weak
(5 per cent.) solution of gelatine is allowed to set on a plate of
glass, and little drops of a 5 or 10 per cent. solution of ferrocyanide
of potassium are then placed at regular intervals upon the gelatine.
Immediately each little drop becomes the centre, or pole, of a system
of diffusion currents, {320} and the several systems conflict with and
repel one another, so that presently each little area becomes the seat
of a double current system, from its centre outwards and back again;
until at length the concentration of the field becomes equalised and
the currents {321}

[Illustration: Fig. 122. An “artificial tissue,” formed by coloured
drops of sodium chloride solution diffusing in a less dense solution of
the same salt. (After Leduc.)]

[Illustration: Fig. 123. An artificial cellular tissue, formed by the
diffusion in gelatine of drops of a solution of potassium ferrocyanide.
(After Leduc.)]

cease. After equilibrium is attained, and when the gelatinous mass
is permitted to dry, we have an artificial tissue of more or less
regularly hexagonal “cells,” which simulate in the closest way an
organic parenchyma. And by varying the experiment, in ways which Leduc
describes, we may simulate various forms of tissue, and produce cells
with thick walls or with thin, cells in close contact or with wide
intercellular spaces, cells with plane or with curved partitions, and
so forth.

――――――――――

[Illustration: Fig. 124. Epidermis of _Girardia_. (After Goebel.)]

The hexagonal pattern is illustrated among organisms in countless
cases, but those in which the pattern is perfectly regular, by
reason of perfect uniformity of force and perfect equality of the
individual cells, are not so numerous. The hexagonal epithelium-cells
of the pigment layer of the eye, external to the retina, are a good
example. Here we have a single layer of uniform cells, reposing on
the one hand upon a basement membrane, supported behind by the solid
wall of the sclerotic, and exposed on the other hand to the uniform
fluid pressure of the vitreous humour. The conditions all point, and
lead, to a perfectly symmetrical result: that is to say, the cells,
uniform in size, are flattened out to a uniform thickness by the fluid
pressure acting radially; and their reaction on each other converts
the flattened discs into regular hexagons. In an ordinary columnar
epithelium, such as that of the intestine, we see again that the
columnar cells have been compressed into hexagonal prisms; but here as
a rule the cells are less uniform in size, small cells are apt to be
intercalated among the larger, and the perfect symmetry is accordingly
lost. The same is true of ordinary vegetable parenchyma; the
originally spherical cells are approximately equal in size, but only
approximately; and there are accordingly all degrees in the regularity
and symmetry of the resulting tissue. But obviously, wherever we {322}
have, in addition to the forces which tend to produce the regular
hexagonal symmetry, some other asymmetrical component arising from
growth or traction, then our regular hexagons will be distorted in
various simple ways. This condition is illustrated in the accompanying
diagram of the epidermis of Girardia; it also accounts for the more or
less pointed or fusiform cells, each still in contact (as a rule) with
six others, which form the epithelial lining of the blood-vessels: and
other similar, or analogous, instances are very common.

[Illustration: Fig. 125. Soap-froth under pressure. (After Rhumbler.)]

In a soap-froth imprisoned between two glass plates, we have a
symmetrical system of cells, which appear in optical section (as in
Fig. 125, B) as regular hexagons; but if we press the plates a little
closer together, the hexagons become deformed or flattened (Fig. 125,
A). In this case, however, if we cease to apply further pressure, the
tension of the films throughout the system soon adjusts itself again,
and in a short time the system has regained the former symmetry of Fig.
125, B.

[Illustration: Fig. 126. From leaf of _Elodea canadensis_. (After
Berthold.)]

In the growth of an ordinary dicotyledonous leaf, we once more see
reflected in the form of its epidermal cells the tractions, irregular
but on the whole longitudinal, which growth has superposed on the
tensions of the partition-walls (Fig. 126). In the narrow elongated
leaf of a Monocotyledon, such as a hyacinth, the elongated, apparently
quadrangular {323} cells of the epidermis appear as a necessary
consequence of the simpler laws of growth which gave its simple form to
the leaf as a whole. In this last case, however, as in all the others,
the rule still holds that only three partitions (in surface view) meet
in a point; and at their point of meeting the walls are for a short
distance manifestly curved, so as to permit the junction to take place
at or nearly at the normal angle of 120°.

Briefly speaking, wherever we have a system of cylinders or spheres,
associated together with sufficient mutual interaction to bring them
into complete surface contact, there, in section or in surface view, we
tend to get a pattern of hexagons.

 While the formation of an hexagonal pattern on the basis of
 ready-formed and symmetrically arranged material units is a very
 common, and indeed the general way, it does not follow that there are
 not others by which such a pattern can be obtained. For instance,
 if we take a little triangular dish of mercury and set it vibrating
 (either by help of a tuning-fork, or by simply tapping on the sides)
 we shall have a series of little waves or ripples starting inwards
 from each of the three faces; and the intercrossing, or interference
 of these three sets of waves produces crests and hollows, and
 intermediate points of no disturbance, _whose loci are seen_ as a
 beautiful pattern of minute hexagons. It is possible that the very
 minute and astonishingly regular pattern of hexagons which we see,
 for instance, on the surface of many diatoms, may be a phenomenon
 of this order[363]. The same may be the case also in Arcella, where
 an apparently hexagonal pattern is found not to consist of simple
 hexagons, but of “straight lines in three sets of parallels, the lines
 of each set making an angle of sixty degrees with those of the other
 two sets[364].” We must also bear in mind, in the case of the minuter
 forms, the large possibilities of optical illusion. For instance, in
 one of Abbe’s “diffraction-plates,” a pattern of dots, set at equal
 interspaces, is reproduced on a very minute scale by photography; but
 under certain conditions of microscopic illumination and focussing,
 these isolated dots appear as a pattern of hexagons.

 ――――――――――

 A symmetrical arrangement of hexagons, such as we have just been
 studying, suggests various simple geometrical corollaries, of which
 the following may perhaps be a useful one.

 We may sometimes desire to estimate the number of hexagonal areas or
 facets in some structure where these are numerous, such for instance
 as the {324} cornea of an insect’s eye, or in the minute pattern of
 hexagons on many diatoms. An approximate enumeration is easily made as
 follows.

 For the area of a hexagon (if we call δ the short diameter, that
 namely which bisects two of the opposite sides) is δ^2 × (√3)/2,
 the area of a circle being _d_^2 ⋅ π/4. Then, if the diameter (_d_)
 of a circular area include _n_ hexagons, the area of that circle
 equals (_n_ ⋅ δ)^2 × π/4. And, dividing this number by the area of
 a single hexagon, we obtain for the number of areas in the circle,
 each equal to a hexagonal facet, the expression _n_^2 × π/4 × 2/(√3)
 = 0·907_n_^2, or (9/10) ⋅ _n_^2, nearly.

 This calculation deals, not only with the complete facets, but with
 the areas of the broken hexagons at the periphery of the circle. If
 we neglect these latter, and consider our whole field as consisting
 of successive rings of hexagons about a central one, we may obtain a
 still simpler rule[365]. For obviously, around our central hexagon
 there stands a zone of six, and around these a zone of twelve, and
 around these a zone of eighteen, and so on. And the total number,
 excluding the central hexagon, is accordingly:

 For one zone      6 = 2 × 3    = 3 × 1 × 2,
 For two zones    18 = 3 × 6    = 3 × 2 × 3,
 For three zones  36 = 4 × 9    = 3 × 3 × 4,
 For four zones   60 = 5 × 12   = 3 × 4 × 5,
 For five zones   90 = 6 x 15   = 3 × 5 × 6,

 and so forth. If _N_ be the number of zones, and if we add one to
 the above numbers for the odd central hexagon, the rule evidently
 is, that the total number, _H_, = 3_N_(_N_ + 1) + 1. Thus, if in a
 preparation of a fly’s cornea, I can count twenty-five facets in a
 line from a central one, the total number in the entire circular field
 is (3 × 25 × 26) + 1 = 1951[366].

――――――――――

The same principles which account for the development of hexagonal
symmetry hold true, as a matter of course, not only of individual
_cells_ (in the biological sense), but of any close-packed bodies
of uniform size and originally circular outline; and the hexagonal
pattern is therefore of very common occurrence, under widely different
circumstances. The curious reader may consult Sir Thomas Browne’s
quaint and beautiful account, in the _Garden of Cyrus_, of hexagonal
(and also of quincuncial) symmetry in plants and animals, which “doth
neatly declare how nature Geometrizeth, and observeth order in all
things.” {325}

We have many varied examples of this principle among corals, wherever
the polypes are in close juxtaposition, with neither empty space nor
accumulations of matrix between their adjacent walls. _Favosites
gothlandica_, for instance, furnishes us with an excellent example. In
the great genus Lithostrotion we have some species that are “massive”
and others that are “fasciculate”; in other words in some the long
cylindrical corallites are in close contact with one another, and in
others they are separate and loosely bundled (Fig. 127). Accordingly in
the former the corallites are

[Illustration: Fig. 127. _Lithostrotion Martini._ (After Nicholson.)]

[Illustration: Fig. 128. _Cyathophyllum hexagonum._ (From Nicholson,
after Zittel.)]

squeezed into hexagonal prisms, while in the latter they retain
their cylindrical form. Where the polypes are comparatively few, and
so have room to spread, the mutual pressure ceases to work or only
tends to push them asunder, letting them remain circular in outline
(e.g. Thecosmilia). Where they vary gradually in size, as for instance
in _Cyathophyllum hexagonum_, they are more or less hexagonal but are
not regular hexagons; and where there is greater and more irregular
variation in size, the cells will be _on the average_ hexagonal, but
some will have fewer and some more sides than six, as in the annexed
figure of Arachnophyllum (Fig. 129). {326} Where larger and smaller
cells, corresponding to two different kinds of zooids, are mixed
together, we may get various results. If the larger cells are numerous
enough to be more or less in contact with one another (e.g. various
Monticuliporae) they will be irregular hexagons, while the smaller
cells between them will be crushed into all manner of irregular angular
forms. If on the other hand the large cells are comparatively few and
are large and strong-walled compared with their smaller neighbours,
then the latter alone will be squeezed into hexagons, while the larger
ones will tend to retain their circular outline undisturbed (e.g.
Heliopora, Heliolites, etc.).

[Illustration: Fig. 129. _Arachnophyllum pentagonum._ (After
Nicholson.)]

[Illustration: Fig. 130. _Heliolites._ (After Woods.)]

When, as happens in certain corals, the peripheral walls or “thecae”
of the individual polypes remain undeveloped but the radiating
septa are formed and calcified, then we obtain new and beautiful
mathematical configurations (Fig. 131). For the radiating septa are
no longer confined to the circular or hexagonal bounds of a polypite,
but tend to meet and become confluent with their neighbours on every
side; and, tending to assume positions of equilibrium, or of minimal
area, under the restraints to which they are subject, they fall into
congruent curves; and these correspond, in a striking manner, to the
lines of force running, in a common field of force, between a number
of secondary centres. Similar patterns may be produced in various
ways, by the play of osmotic or magnetic forces; and a particular
and very curious case is to be found in those complicated forms of
nuclear division {327} known as triasters, polyasters, etc., whose
relation to a field of force Hartog has explained[367]. It is obvious
that, in our corals, these curving septa are all orthogonal to the
non-existent hexagonal boundaries. As the phenomenon is wholly due to
the imperfect development or non-existence of a thecal wall, it is
not surprising that we find identical configurations among various
corals, or families of corals, not otherwise related to one another;
we find the same or very similar patterns displayed, for instance, in
Synhelia (_Oculinidae_), in Phillipsastraea (_Rugosa_), in Thamnastraea
(_Fungida_), and in many more.

[Illustration: Fig. 131. Surface-views of Corals with undeveloped
thecae and confluent septa. A, _Thamnastraea_; B, _Comoseris_. (From
Nicholson, after Zittel.)]

――――――――――

The most famous of all hexagonal conformations and perhaps the most
beautiful is that of the bee’s cell. Here we have, as in our last
examples, a series of equal cylinders, compressed by symmetrical forces
into regular hexagonal prisms. But in this case we have two rows of
such cylinders, set opposite to one another, end to end; and we have
accordingly to consider also the conformation of their ends. We may
suppose our original cylindrical cells to have spherical ends, which
is their normal and symmetrical mode of termination; and, for closest
packing, it is obvious that the end of any one cylinder will touch, and
fit in between, the ends of three cylinders in the opposite row. It is
just as when we pile round-shot in a heap; each sphere that we {328}
set down fits into its nest between three others, and the four form a
regular tetrahedral arrangement. Just as it was obvious, then, that by
mutual pressure from the six _laterally_ adjacent cells, any one cell
would be squeezed into a hexagonal prism, so is it also obvious that,
by mutual pressure against the three _terminal_ neighbours, the end
of any one cell will be compressed into a solid trihedral angle whose
edges will meet, as in the analogous case already described of a system
of soap-bubbles, at a plane angle of 109° and so many minutes and
seconds. What we have to comprehend, then, is how the _six_ sides of
the cell are to be combined with its _three_ terminal facets. This is
done by bevelling off three alternate angles of the prism, in a uniform
manner, until we have tapered the prism to a point; and by so doing,
we evidently produce three _rhombic_ surfaces, each of which is double
of the triangle formed by joining the apex to the three untouched
angles of the prism. If we experiment, not with cylinders, but with
spheres, if for instance we pile together a mass of bread-pills (or
pills of plasticine), and then submit the whole to a uniform pressure,
it is obvious that each ball (like the seeds in a pomegranate, as
Kepler said), will be in contact with _twelve_ others,—six in its
own plane, three below and three above, and in compression it will
therefore develop twelve plane surfaces. It will in short repeat,
above and below, the conditions to which the bee’s cell is subject at
one end only; and, since the sphere is symmetrically situated towards
its neighbours on all sides, it follows that the twelve plane sides
to which its surface has been reduced will be all similar, equal and
similarly situated. Moreover, since we have produced this result by
squeezing our original spheres close together, it is evident that the
bodies so formed completely fill space. The regular solid which fulfils
all these conditions is the _rhombic dodecahedron_. The bee’s cell,
then, is this figure incompletely formed: it is a hexagonal prism
with one open or unfinished end, and one trihedral apex of a rhombic
dodecahedron.

The geometrical form of the bee’s cell must have attracted the
attention and excited the admiration of mathematicians from time
immemorial. Pappus the Alexandrine has left us (in the introduction to
the Fifth Book of his _Collections_) an account of its hexagonal plan,
and he drew from its mathematical symmetry the {329} conclusion that
the bees were endowed with reason: “There being, then, three figures
which of themselves can fill up the space round a point, viz. the
triangle, the square and the hexagon, the bees have wisely selected
for their structure that which contains most angles, suspecting indeed
that it could hold more honey than either of the other two.” Erasmus
Bartholinus was apparently the first to suggest that this hypothesis
was not warranted, and that the hexagonal form was no more than the
necessary result of equal pressures, each bee striving to make its own
little circle as large as possible.

The investigation of the ends of the cell was a more difficult
matter, and came later, than that of its sides. In general terms this
arrangement was doubtless often studied and described: as for instance,
in the _Garden of Cyrus_: “And the Combes themselves so regularly
contrived that their mutual intersections make three Lozenges at the
bottom of every Cell; which severally regarded make three Rows of
neat Rhomboidall Figures, connected at the angles, and so continue
three several chains throughout the whole comb.” But Maraldi[368]
(Cassini’s nephew) was the first to measure the terminal solid angle or
determine the form of the rhombs in the pyramidal ending of the cell.
He tells us that the angles of the rhomb are 110° and 70°: “Chaque
base d’alvéole est formée par trois rhombes presque toujours égaux et
semblables, qui, suivant les mesures que nous avons prises, ont les
deux angles obtus chacun de 110 degrés, et par conséquent les deux
aigus chacun de 70°.” He also stated that the angles of the trapeziums
which form the sides of the body of the cell were identical angles, of
110° and 70°; but in the same paper he speaks of the angles as being,
respectively, 109° 28′ and 70° 32′. Here a singular confusion at once
arose, and has been perpetuated in the books[369]. “Unfortunately
Réaumur chose to look upon this second determination of Maraldi’s as
being, as well as the first, a direct result of measurement, whereas
it is in reality theoretical. He speaks of it as Maraldi’s more
precise measurement, and this error has been repeated in spite of its
absurdity to the present day; nobody {330} appears to have thought of
the impossibility of measuring such a thing as the end of a bee’s cell
to the nearest minute.” At any rate, it now occurred to Réaumur (as
curiously enough, it had not done to Maraldi) that, just as the closely
packed hexagons gave the minimal extent of boundary in a plane, so the
actual solid figure, as determined by Maraldi, might be that which,
for a given solid content, gives the minimum of surface: or which, in
other words, would hold the most honey for the least wax. He set this
problem before Koenig, and the geometer confirmed his conjecture, the
result of his calculations agreeing within two minutes (109° 26′ and
70° 34′) with Maraldi’s determination. But again, Maclaurin[370] and
Lhuilier[371], by different methods, obtained a result identical with
Maraldi’s; and were able to shew that the discrepancy of 2′ was due to
an error in Koenig’s calculation (of tan θ = √2),—that is to say to the
imperfection of his logarithmic tables,—not (as the books say[372]) “to
a mistake on the part of the Bee.” “Not to a mistake on the part of
Maraldi” is, of course, all that we are entitled to say.

[Illustration: Fig. 132.]

The theorem may be proved as follows:

_ABCDEF_, _abcdef_, is a right prism upon a regular hexagonal base. The
corners _BDF_ are cut off by planes through the lines _AC_, _CE_, _EA_,
meeting in a point _V_ on the axis _VN_ of the prism, and intersecting
_Bb_, _Dd_, _Ff_, at _X_, _Y_, _Z_. It is evident that the volume of
the figure thus formed is the same as that of the original prism with
hexagonal ends. For, if the axis cut the hexagon _ABCDEF_ in _N_, the
volumes _ACVN_, _ACBX_ are equal. {331}

It is required to find the inclination of the faces forming the
trihedral angle at _V_ to the axis, such that the surface of the figure
may be a minimum.

Let the angle _NVX_, which is half the solid angle of the prism, = θ;
the side of the hexagon, as _AB_, = _a_; and the height, as _Aa_, = _h_.

 Then, _AC_ = 2_a_ cos 30° = _a_√3.

 And _VX_ = _a_/sin θ (from inspection of the triangle _LXB_)

 Therefore the area of the rhombus _VAXC_ = (_a_^2 √3)/(2 sin θ).

 And the area of _AabX_ = (_a_/2)(2_h_ − ½_VX_ cos θ)

   = (_a_/2)(2_h_ − _a_/2 ⋅ cot θ).

 Therefore the total area of the figure

   = hexagon _abcdef_ + 3_a_(2_h_ − (_a_/2) cot θ)
     + 3((_a_^2 √3)/(2 sin θ)).

 Therefore _d_(Area)/_d_θ = (3_a_^2/2)((1/sin^2 θ)
     − (√3 cos θ)/(sin^2 θ)).

But this expression vanishes, that is to say, _d_(Area)/_d_θ = 0,
when cos θ = 1/√3, that is when θ = 54° 44′ 8″ = ½(109° 28′ 16″).

This then is the condition under which the total area of the figure has
its minimal value.

――――――――――

That the beautiful regularity of the bee’s architecture is due to some
automatic play of the physical forces, and that it were fantastic to
assume (with Pappus and Réaumur) that the bee intentionally seeks for a
method of economising wax, is certain, but the precise manner of this
automatic action is not so clear. When the hive-bee builds a solitary
cell, or a small cluster of cells, as it does for those eggs which are
to develop into queens, it makes but a rude production. The queen-cells
are lumps of coarse wax hollowed out and roughly bitten into shape,
bearing the marks of the bee’s jaws, like the marks of a blunt adze on
a rough-hewn log. Omitting the simplest of all cases, when (as among
some humble-bees) the old cocoons are used to hold honey, the cells
built by the “solitary” wasps and bees are of various kinds. They may
be formed by partitioning off little chambers in a hollow stem; {332}
they may be rounded or oval capsules, often very neatly constructed,
out of mud, or vegetable _fibre_ or little stones, agglutinated
together with a salivary glue; but they shew, except for their rounded
or tubular form, no mathematical symmetry. The social wasps and many
bees build, usually out of vegetable matter chewed into a paste with
saliva, very beautiful nests of “combs”; and the close-set papery
cells which constitute these combs are just as regularly hexagonal as
are the waxen cells of the hive-bee. But in these cases (or nearly
all of them) the cells are in a single row; their sides are regularly
hexagonal, but their ends, from the want of opponent forces, remain
simply spherical. In _Melipona domestica_ (of which Darwin epitomises
Pierre Huber’s description) “the large waxen honey-cells are nearly
spherical, nearly equal in size, and are aggregated into an irregular
mass.” But the spherical form is only seen on the outside of the mass;
for inwardly each cell is flattened into “two, three or more flat
surfaces, according as the cell adjoins two, three or more other cells.
When one cell rests on three other cells, which from the spheres being
nearly of the same size is very frequently and necessarily the case,
the three flat surfaces are united into a pyramid; and this pyramid, as
Huber has remarked, is manifestly a gross imitation of the three-sided
pyramidal base of the cell of the hive-bee[373].” The question is, to
what particular force are we to ascribe the plane surfaces and definite
angles which define the sides of the cell in all these cases, and the
ends of the cell in cases where one row meets and opposes another. We
have seen that Bartholin suggested, and it is still commonly believed,
that this result is due to simple physical pressure, each bee enlarging
as much as it can the cell which it is a-building, and nudging its wall
outwards till it fills every intervening gap and presses hard against
the similar efforts of its neighbour in the cell next door[374].
But it is very doubtful {333} whether such physical or mechanical
pressure, more or less intermittently exercised, could produce the
all but perfectly smooth, plane surfaces and the all but perfectly
definite and constant angles which characterise the cell, whether it
be constructed of wax or papery pulp. It seems more likely that we
have to do with a true surface-tension effect; in other words, that
the walls assume their configuration when in a semi-fluid state, while
the papery pulp is still liquid, or while the wax is warm under the
high temperature of the crowded hive[375]. Under these circumstances,
the direct efforts of the wasp or bee may be supposed to be limited to
the making of a tubular cell, as thin as the nature of the material
permits, and packing these little cells as close as possible together.
It is then easily conceivable that the symmetrical tensions of the
adjacent films (though somewhat retarded by viscosity) should suffice
to bring the whole system into equilibrium, that is to say into the
precise configuration which the comb actually presents. In short, the
Maraldi pyramids which terminate the bee’s cell are precisely identical
with the facets of a rhombic dodecahedron, such as we have assumed to
constitute (and which doubtless under certain conditions do constitute)
the surfaces of contact in the interior of a mass of soap-bubbles or of
uniform parenchymatous cells; and there is every reason to believe that
the physical explanation is identical, and not merely mathematically
analogous.

The remarkable passage in which Buffon discusses the bee’s cell and the
hexagonal configuration in general is of such historical importance,
and tallies so closely with the whole trend of our enquiry, that I
will quote it in full: “Dirai-je encore un mot; ces cellules des
abeilles, tant vantées, tant admirées, me fournissent une preuve
de plus contre l’enthousiasme et l’admiration; cette figure, toute
géométrique et toute régulière qu’elle nous paraît, et qu’elle est
en effet dans la spéculation, n’est ici qu’un résultat mécanique et
assez imparfait qui se trouve souvent dans la nature, {334} et que
l’on remarque même dans les productions les plus brutes; les cristaux
et plusieurs autres pierres, quelques sels, etc., prennent constamment
cette figure dans leur formation. Qu’on observe les petites écailles
de la peau d’une roussette, on verra qu’elles sont hexagones, parce
que chaque écaille croissant en même temps se fait obstacle, et tend à
occuper le plus d’espace qu’il est possible dans un espace donné: on
voit ces mêmes hexagones dans le second estomac des animaux ruminans,
on les trouve dans les graines, dans leurs capsules, dans certaines
fleurs, etc. Qu’on remplisse un vaisseau de pois, ou plûtot de
quelque autre graine cylindrique, et qu’on le ferme exactement après
y avoir versé autant d’eau que les intervalles qui restent entre ces
graines peuvent en recevoir; qu’on fasse bouillir cette eau, tous
ces cylindres deviendront de colonnes à six pans[376]. On y voit
clairement la raison, qui est purement mécanique; chaque graine, dont
la figure est cylindrique, tend par son renflement à occuper le plus
d’espace possible dans un espace donné, elles deviennent donc toutes
nécessairement hexagones par la compression réciproque. Chaque abeille
cherche à occuper de même le plus d’espace possible dans un espace
donné, il est donc nécessaire aussi, puisque le corps des abeilles est
cylindrique, que leurs cellules sont hexagones,—par la même raison
des obstacles réciproques. On donne plus d’esprit aux mouches dont
les ouvrages sont les plus réguliers; les abeilles sont, dit-on, plus
ingénieuses que les guêpes, que les frélons, etc., qui savent aussi
l’architecture, mais dont les constructions sont plus grossières et
plus irrégulières que celles des abeilles: on ne veut pas voir, ou l’on
ne se doute pas que cette régularité, plus ou moins grande, dépend
uniquement du nombre et de la figure, et nullement de l’intelligence
de ces petites bêtes; plus elles sont nombreuses, plus il y a des
forces qui agissent également et s’opposent de même, plus il y a
par conséquent de contrainte mécanique, de régularité forcée, et de
perfection apparente dans leurs productions[377].” {335}

A very beautiful hexagonal symmetry, as seen in section, or
dodecahedral, as viewed in the solid, is presented by the cells which
form the pith of certain rushes (e.g. _Juncus effusus_), and somewhat
less diagrammatically by those which make the pith of the banana.
These cells are stellate in form, and the tissue presents in section
the appearance of a network of six-rayed stars (Fig. 133, _c_), linked
together by the tips of the rays, and separated by symmetrical,
air-filled, intercellular spaces. In thick sections, the solid
twelve-rayed stars may be very beautifully seen under the binocular
microscope.

[Illustration: Fig. 133. Diagram of development of “stellate cells,” in
pith of _Juncus_. (The dark, or shaded, areas represent the cells; the
light areas being the gradually enlarging “intercellular spaces.”)]

What has happened here is not difficult to understand. Imagine, as
before, a system of equal spheres all in contact, each one therefore
touching six others in an equatorial plane; and let the cells be not
only in contact, but become attached at the points of contact. Then
instead of each cell expanding, so as to encroach on and fill up the
intercellular spaces, let each cell tend to contract or shrivel up,
by the withdrawal of fluid from its interior. The {336} result will
obviously be that the intercellular spaces will increase; the six
equatorial attachments of each cell (Fig. 133, _a_) (or its twelve
attachments in all, to adjacent cells) will remain fixed, and the
portions of cell-wall between these points of attachment will be
withdrawn in a symmetrical fashion (_b_) towards the centre. As the
final result (_c_) we shall have a “dodecahedral star” or star-polygon,
which appears in section as a six-rayed figure. It is obviously
necessary that the pith-cells should not only be attached to one
another, but that the outermost layer should be firmly attached to
a boundary wall, so as to preserve the symmetry of the system. What
actually occurs in the rush is tantamount to this, but not absolutely
identical. Here it is not so much the pith-cells which tend to shrivel
within a boundary of constant size, but rather the boundary wall (that
is, the peripheral ring of woody and other tissues) which continues to
expand after the pith-cells which it encloses have ceased to grow or
to multiply. The twelve points of attachment on the spherical surface
of each little pith-cell are uniformly drawn asunder; but the content,
or volume, of the cell does not increase correspondingly; and the
remaining portions of the surface, accordingly, shrink inwards and
gradually constitute the complicated surface of a twelve-pointed star,
which is still a symmetrical figure and is still also a surface of
minimal area under the new conditions.

――――――――――

A few years after the publication of Plateau’s book, Lord Kelvin
shewed, in a short but very beautiful paper[378], that we must
not hastily assume from such arguments as the foregoing, that a
close-packed assemblage of rhombic dodecahedra will be the true and
general solution of the problem of dividing space with a minimum
partitional area, or will be present in a cellular liquid “foam,” in
which it is manifest that the problem is actually and automatically
solved. The general mathematical solution of the problem (as we have
already indicated) is, that every interface or partition-wall must
have constant curvature throughout; that where such partitions meet
in an edge, they must intersect at angles such that equal forces, in
planes perpendicular to the line {337} of intersection, shall balance;
and finally, that no more than three such interfaces may meet in a
line or edge, whence it follows that the angle of intersection of the
film-surfaces must be exactly 120°. An assemblage of equal and similar
rhombic dodecahedra goes far to meet the case: it completely fills
up space; all its surfaces or interfaces are planes, that is to say,
surfaces of constant curvature throughout; and these surfaces all meet
together at angles of 120°. Nevertheless, the proof that our rhombic
dodecahedron (such as we find exemplified in the bee’s cell) is a
surface of minimal area, is not a comprehensive proof; it is limited to
certain conditions, and practically amounts to no more than this, that
of the regular solids, with all sides plane and similar, this one has
the least surface for its solid content.

[Illustration: Fig. 134.]

The rhombic dodecahedron has six tetrahedral angles, and eight
trihedral angles; and it is obvious, on consideration, that at each of
the former six dodecahedra meet in a point, and that, where the four
tetrahedral facets of each coalesce with their neighbours, we have
twelve plane films, or interfaces, meeting in a point. In a precisely
similar fashion, we may imagine twelve plane films, drawn inwards
from the twelve edges of a cube, to meet at a point in the centre of
the cube. But, as Plateau discovered[379], when we dip a cubical wire
skeleton into soap-solution and take it out again, the twelve films
which are thus generated do _not_ meet in a point, but are grouped
around a small central, plane, quadrilateral film (Fig. 134). In
other words, twelve plane films, meeting in a point, are _essentially
unstable_. If we blow upon our artificial film-system, the little
quadrilateral alters its place, setting itself parallel now to one and
now to another of the paired faces of the cube; but we never get rid
of it. Moreover, the size and shape of the quadrilateral, as of all
the other films in the system, are perfectly definite. Of the twelve
films (which we had {338} expected to find all plane and all similar)
four are plane isosceles triangles, and eight are slightly curved
quadrilateral figures. The former have two curved sides, meeting at an
angle of 109° 28′, and their apices coincide with the corners of the
central quadrilateral, whose sides are also curved, and also meet at
this identical angle;—which (as we observe) is likewise an angle which
we have been dealing with in the simpler case of the bee’s cell, and
indeed in all the regular solids of which we have yet treated.

By completing the assemblage of polyhedra of which Plateau’s
skeleton-cube gives a part, Lord Kelvin shewed that we should
obtain a set of equal and similar fourteen-sided figures, or
“tetrakaidecahedra”; and that by means of an assemblage of these
figures space is homogeneously partitioned—that is to say, into equal,
similar and similarly situated cells—with an economy of surface
in relation to area even greater than in an assemblage of rhombic
dodecahedra.

In the most generalised case, the tetrakaidecahedron is bounded by
three pairs of equal and parallel quadrilateral faces, and four pairs
of equal and parallel hexagonal faces, neither the quadrilaterals nor
the hexagons being necessarily plane. In a certain particular case, the
quadrilaterals are plane surfaces, but the hexagons slightly curved
“anticlastic” surfaces; and these latter have at every point equal
and opposite curvatures, and are surfaces of minimal curvature for a
boundary of six curved edges. The figure has the remarkable property
that, like the plane rhombic dodecahedron, it so partitions space that
three faces meeting in an edge do so everywhere at equal angles of
120° [380].

We may take it as certain that, in a system of _perfectly_ fluid
films, like the interior of a mass of soap-bubbles, where the films
are perfectly free to glide or to rotate over one another, the mass
is actually divided into cells of this remarkable conformation. {339}
And it is quite possible, also, that in the cells of a vegetable
parenchyma, by carefully macerating them apart, the same conformation
may yet be demonstrated under suitable conditions; that is to say when
the whole tissue is highly symmetrical, and the individual cells are
as nearly as possible equal in size. But in an ordinary microscopic
_section_, it would seem practically impossible to distinguish the
fourteen-sided figure from the twelve-sided. Moreover, if we have
anything whatsoever interposed so as to prevent our twelve films
meeting in a point, and (so to speak) to take the place of our little
central quadrilateral,—if we have, for instance, a tiny bead or droplet
in the centre of our artificial system, or even a little thickening,
or “bourrelet” as Plateau called it, of the cell-wall, then it is
no longer necessary that the tetrakaidecahedron should be formed.
Accordingly, it is very probably the case that, in the parenchymatous
tissue, under the actual conditions of restraint and of very imperfect
fluidity, it is after all the rhombic dodecahedral configuration which,
even under perfectly symmetrical conditions, is generally assumed.

――――――――――

It follows from all that we have said, that the problems connected
with the conformation of cells, and with the manner in which a given
space is partitioned by them, soon become exceedingly complex. And
while this is so even when all our cells are equal and symmetrically
placed, it becomes vastly more so when cells varying even slightly in
size, in hardness, rigidity or other qualities, are packed together.
The mathematics of the case very soon become too hard for us; but in
its essence, the phenomenon remains the same. We have little reason to
doubt, and no just cause to disbelieve, that the whole configuration,
for instance of an egg in the advanced stages of segmentation, is
accurately determined by simple physical laws, just as much as in
the early stages of two or four cells, during which early stages we
are able to recognise and demonstrate the forces and their resultant
effects. But when mathematical investigation has become too difficult,
it often happens that physical experiment can reproduce for us the
phenomena which Nature exhibits to us, and which we are striving to
comprehend. For instance, in an admirable research, M. Robert shewed,
some years ago, not only that the early segmentation of {340} the
egg of _Trochus_ (a marine univalve mollusc) proceeded in accordance
with the laws of surface tension, but he also succeeded in imitating
by means of soap-bubbles, several stages, one after another, of the
developing egg.

[Illustration: Fig. 135. Aggregations of four soap-bubbles, to shew
various arrangements of the intermediate partition and polar furrows.
(After Robert.)]

M. Robert carried his experiments as far as the stage of sixteen cells,
or bubbles. It is not easy to carry the artificial system quite so far,
but in the earlier stages the experiment is easy; we have merely to
blow our bubbles in a little dish, adding one to another, and adjusting
their sizes to produce a symmetrical system. One of the simplest and
prettiest parts of his investigation concerned the “polar furrow” of
which we have spoken on p. 310. On blowing four little contiguous
bubbles he found (as we may all find with the greatest ease) that they
form a symmetrical system, two in contact with one another by a laminar
film, and two, which are elevated a little above the others, and which
are separated by the length of the aforesaid lamina. The bubbles are
thus in contact three by three, their partition-walls making with
one another equal angles of 120°. The upper and lower edges of the
intermediate lamina (the lower one visible through the transparent
system) constitute the two polar furrows of the embryologist (Fig.
135, 1–3). The lamina itself is plane when the system is symmetrical,
but it responds by a corresponding curvature to the least inequality
of the bubbles on either side. In the experiment, the upper polar
furrow is usually a little shorter than the lower, but parallel to
it; that is to say, the lamina is of trapezoidal form: this lack of
perfect symmetry being due (in the experimental case) to the lower
portion of the bubbles being somewhat drawn asunder by the tension of
their attachments to the sides of the dish (Fig. 135, 4). A similar
phenomenon is usually found in Trochus, according to Robert, and many
other observers have likewise found the upper furrow to be shorter
than the one below. In the various species of the genus Crepidula,
Conklin asserts that the two furrows are equal in _C. convexa_, that
the upper one is the shorter in _C. fornicata_, and that the upper
one all but disappears in _C. plana_; but we may well be permitted to
doubt, without the evidence of very special investigations, whether
these slight physical differences are actually characteristic of, and
constant in, particular allied _species_. {341} Returning to the
experimental case, Robert found that by withdrawing a little air from,
and so diminishing the bulk of the two terminal bubbles (i.e. those
at the ends of the intermediate lamina), the upper polar furrow was
caused to elongate, till it became equal in length to the lower; and
by continuing the process it became the longer in its turn. These two
conditions have again been described by investigators as characteristic
of this embryo or that; for instance in Unio, Lillie has described the
two furrows as gradually altering their respective lengths[381]; and
Wilson (as Lillie remarks) had already pointed out that “the reduction
of the apical cross-furrow, as compared with that at the vegetative
pole {342} in molluscs and annelids ‘stands in obvious relation to the
different size of the cells produced at the two poles[382].’ ”

When the two lateral bubbles are gradually reduced in size, or the two
terminal ones enlarged, the upper furrow becomes shorter and shorter;
and at the moment when it is about to vanish, a new furrow makes its
instantaneous appearance in a direction perpendicular to the old one;
but the inferior furrow, constrained by its attachment to the base,
remains unchanged, and accordingly our two polar furrows, which were
formerly parallel, are now at right angles to one another. Instead of
a single plane quadrilateral partition, we have now two triangular
ones, meeting in the middle of the system by their apices, and lying in
planes at right angles to one another (Fig. 135, 5–7)[383]. Two such
polar furrows, equal in length and arranged in a cross, have again
been frequently described by the embryologists. Robert himself found
this condition in Trochus, as an occasional or exceptional occurrence:
it has been described as normal in Asterina by Ludwig, in Branchipus
by Spangenberg, and in Podocoryne and Hydractinia by Bunting. It is
evident that it represents a state of unstable equilibrium, only to be
maintained under certain conditions of restraint within the system.

So, by slight and delicate modifications in the relative size of the
cells, we may pass through all the possible arrangements of the median
partition, and of the “furrows” which correspond to its upper and
lower edges; and every one of these arrangements has been frequently
observed in the four-celled stage of various embryos. As the phases
pass one into the other, they are accompanied by changes in the
curvature of the partition, which in like manner correspond precisely
to phenomena which the embryologists have witnessed and described.
And all these configurations belong to that large class of phenomena
whose distribution among embryos, or among organisms in general,
bears no relation to the boundaries of zoological classification;
through molluscs, worms, {343} coelenterates, vertebrates and what
not, we meet with now one and now another, in a medley which defies
classification. They are not “vital phenomena,” or “functions” of the
organism, or special characteristics of this or that organism, but
purely physical phenomena. The kindred but more complicated phenomena
which correspond to the polar furrow when a larger number of cells than
four are associated together, we shall deal with in the next chapter.

Having shewn that the capillary phenomena are patent and unmistakable
during the earlier stages of embryonic development, but soon become
more obscure and incapable of experimental reproduction in the later
stages, when the cells have increased in number, various writers
including Robert himself have been inclined to argue that the physical
phenomena die away, and are overpowered and cancelled by agencies
of a very different order. Here we pass into a region where direct
observation and experiment are not at hand to guide us, and where a
man’s trend of thought, and way of judging the whole evidence in the
case, must shape his philosophy. We must remember that, even in a froth
of soap-bubbles, we can apply an exact analysis only to the simplest
cases and conditions of the phenomenon; we cannot describe, but can
only imagine, the forces which in such a froth control the respective
sizes, positions and curvatures of the innumerable bubbles and films
of which it consists; but our knowledge is enough to leave us assured
that what we have learned by investigation of the simplest cases
includes the principles which determine the most complex. In the case
of the growing embryo we know from the beginning that surface tension
is only one of the physical forces at work; and that other forces,
including those displayed within the interior of each living cell,
play their part in the determination of the system. But we have no
evidence whatsoever that at this point, or that point, or at any, the
dominion of the physical forces over the material system gives place
to a new condition where agencies at present unknown to the physicist
impose themselves on the living matter, and become responsible for the
conformation of its material fabric.

――――――――――

Before we leave for the present the subject of the segmenting {344}
egg, we must take brief note of two associated problems: viz. (1)
the formation and enlargement of the segmentation cavity, or central
interspace around which the cells tend to group themselves in a single
layer, and (2) the formation of the gastrula, that is to say (in a
typical case) the conversion “by invagination,” of the one-layered ball
into a two-layered cup. Neither problem is free from difficulty, and
all we can do meanwhile is to state them in general terms, introducing
some more or less plausible assumptions.

The former problem is comparatively easy, as regards the tendency of a
segmentation cavity to _enlarge_, when once it has been established.
We may then assume that subdivision of the cells is due to the
appearance of a new-formed septum within each cell, that this septum
has a tendency to shrink under surface tension, and that these changes
will be accompanied on the whole by a diminution of surface energy
in the system. This being so, it may be shewn that the volume of the
divided cells must be less than it was prior to division, or in other
words that part of their contents must exude during the process of
segmentation[384]. Accordingly, the case where the segmentation cavity
enlarges and the embryo developes into a hollow blastosphere may, under
the circumstances, be simply described as the case where that outflow
or exudation from the cells of the blastoderm is directed on the whole
inwards.

The physical forces involved in the invagination of the cell-layer to
form the gastrula have been repeatedly discussed[385], but the true
explanation seems as yet to be by no means clear. The case, however,
is probably not a very difficult one, provided that we may assume a
difference of osmotic pressure at the two poles of the blastosphere,
that is to say between the cells which are being differentiated into
outer and inner, into epiblast and hypoblast. It is plain that a
blastosphere, or hollow vesicle bounded by a layer of vesicles, is
under very different physical conditions from a single, simple vesicle
or bubble. The blastosphere has no effective surface tension of its
own, such as to exert pressure on {345} its contents or bring the
whole into a spherical form; nor will local variations of surface
energy be directly capable of affecting the form of the system. But if
the substance of our blastosphere be sufficiently viscous, then osmotic
forces may set up currents which, reacting on the external fluid
pressure, may easily cause modifications of shape; and the particular
case of invagination itself will not be difficult to account for on
this assumption of non-uniform exudation and imbibition.

{346}




CHAPTER VIII

THE FORMS OF TISSUES OR CELL-AGGREGATES (_continued_)


The problems which we have been considering, and especially that of
the bee’s cell, belong to a class of “isoperimetrical” problems, which
deal with figures whose surface is a minimum for a definite content or
volume. Such problems soon become difficult, but we may find many easy
examples which lead us towards the explanation of biological phenomena;
and the particular subject which we shall find most easy of approach
is that of the division, in definite proportions, of some definite
portion of space, by a partition-wall of minimal area. The theoretical
principles so arrived at we shall then attempt to apply, after the
manner of Berthold and Errera, to the actual biological phenomena of
cell-division.

This investigation we may approach in two ways: by considering, namely,
the partitioning off from some given space or area of one-half (or some
other fraction) of its content; or again, by dealing simultaneously
with the partitions necessary for the breaking up of a given space into
a definite number of compartments.

If we take, to begin with, the simple case of a cubical cell, it is
obvious that, to divide it into two halves, the smallest possible
partition-wall is one which runs parallel to, and midway between, two
of its opposite sides. If we call _a_ the length of one of the edges of
the cube, then _a_^2 is the area, alike of one of its sides, and of the
partition which we have interposed parallel, or normal, thereto. But if
we now consider the bisected cube, and wish to divide the one-half of
it again, it is obvious that another partition parallel to the first,
so far from being the smallest possible, is precisely twice the size of
a cross-partition perpendicular to it; {347} for the area of this new
partition is _a_ × _a_/2. And again, for a third bisection, our next
partition must be perpendicular to the other two, and it is obviously a
little square, with an area of (½_a_)^2 = ¼(_a_^2).

From this we may draw the simple rule that, for a rectangular body or
parallelopiped to be divided equally by means of a partition of minimal
area, (1) the partition must cut across the longest axis of the figure;
and (2) in the event of successive bisections, each partition must run
at right angles to its immediate predecessor.

[Illustration: Fig. 136. (After Berthold.)]

We have already spoken of “Sachs’s Rules,” which are an empirical
statement of the method of cell-division in plant-tissues; and we may
now set them forth in full.

(1) The cell typically tends to divide into two co-equal parts.

(2) Each new plane of division tends to intersect at right angles the
preceding plane of division.

The first of these rules is a statement of physiological fact,
not without its exceptions, but so generally true that it will
justify us in limiting our enquiry, for the most part, to cases
of equal subdivision. That it is by no means universally true for
cells generally is shewn, for instance, by such well-known cases
{348} as the unequal segmentation of the frog’s egg. It is true
when the dividing cell is homogeneous, and under the influence of
symmetrical forces; but it ceases to be true when the field is no
longer dynamically symmetrical, for instance, when the parts differ
in surface tension or internal pressure. This latter condition, of
asymmetry of field, is frequent in segmenting eggs[386], and is then
equivalent to the principle upon which Balfour laid stress, as leading
to “unequal” or to “partial” segmentation of the egg,—viz. the unequal
or asymmetrical distribution of protoplasm and of food-yolk.

The second rule, which also has its exceptions, is true in a large
number of cases; and it owes its validity, as we may judge from the
illustration of the repeatedly bisected cube, solely to the guiding
principle of minimal areas. It is in short subordinate to, and covers
certain cases included under, a much more important and fundamental
rule, due not to Sachs but to Errera; that (3) the incipient
partition-wall of a dividing cell tends to be such that its area is the
least possible by which the given space-content can be enclosed.

――――――――――

Let us return to the case of our cube, and let us suppose that, instead
of bisecting it, we desire to shut off some small portion only of its
volume. It is found in the course of experiments upon soap-films, that
if we try to bring a partition-film too near to one side of a cubical
(or rectangular) space, it becomes unstable; and is easily shifted to
a totally new position, in which it constitutes a curved cylindrical
wall, cutting off one corner of the cube. It meets the sides of the
cube at right angles (for reasons which we have already considered);
and, as we may see from the symmetry {349} of the case, it constitutes
precisely one-quarter of a cylinder. Our plane transverse partition,
wherever it was placed, had always the same area, viz. _a_^2; and it
is obvious that a cylindrical wall, if it cut off a small corner, may
be much less than this. We want, accordingly, to determine what is the
particular volume which might be partitioned off with equal economy
of wall-space in one way as the other, that is to say, what area of
cylindrical wall would be neither more nor less than the area _a_^2.
The calculation is very easy.

The _surface-area_ of a cylinder of length _a_ is 2π_r_ ⋅ _a_, and that
of our quarter-cylinder is, therefore, _a_ ⋅ π_r_/2; and this being, by
hypothesis, = _a_^2, we have _a_ = π_r_/2, or _r_ = 2_a_/π.

The _volume_ of a cylinder, of length _a_, is _a_π_r_^2, and that of
our quarter-cylinder is (_a_ ⋅ π_r_^2)/4, which (by substituting the
value of _r_) is equal to (_a_^3)/π.

Now precisely this same volume is, obviously, shut off by a transverse
partition of area _a_^2, if the third side of the rectangular space
be equal to _a_/π. And this fraction, if we take _a_ = 1, is equal to
0·318..., or rather less than one-third. And, as we have just seen, the
radius, or side, of the corresponding quarter-cylinder will be twice
that fraction, or equal to ·636 times the side of the cubical cell.

[Illustration: Fig. 137.]

If then, in the process of division of a cubical cell, it so divide
that the two portions be not equal in volume but that one portion by
anything less than about three-tenths of the whole, or three-sevenths
of the other portion, there will be a tendency for the cell to divide,
not by means of a plane transverse partition, but by means of a curved,
cylindrical wall cutting off one corner of the original cell; and the
part so cut off will be one-quarter of a cylinder.

By a similar calculation we can shew that a _spherical_ wall, cutting
off one solid angle of the cube, and constituting an octant of a
sphere, would likewise be of less area than a plane partition as
soon as the volume to be enclosed was not greater than about {350}
one-quarter of the original cell[387]. But while both the cylindrical
wall and the spherical wall would be of less area than the plane
transverse partition after that limit (of one-quarter volume) was
passed, the cylindrical would still be the better of the two up to a
further limit. It is only when the volume to be partitioned off {351}
is no greater than about 0·15, or somewhere about one-seventh, of the
whole, that the spherical cell-wall in an angle of the cubical cell,
that is to say the octant of a sphere, is definitely of less area
than the quarter-cylinder. In the accompanying diagram (Fig. 138) the
relative areas of the three partitions are shewn for all fractions,
less than one-half, of the divided cell.

[Illustration: Fig. 138.]

 In this figure, we see that the plane transverse partition, whatever
 fraction of the cube it cut off, is always of the same dimensions,
 that is to say is always equal to _a_^2, or = 1. If one-half of the
 cube have to be cut off, this plane transverse partition is much the
 best, for we see by the diagram that a cylindrical partition cutting
 off an equal volume would have an area about 25%, and a spherical
 partition would have an area about 50% greater. The point _A_ in the
 diagram corresponds to the point where the cylindrical partition
 would begin to have an advantage over the plane, that is to say (as
 we have seen) when the fraction to be cut off is about one-third, or
 ·318 of the whole. In like manner, at _B_ the spherical octant begins
 to have an advantage over the plane; and it is not till we reach the
 point _C_ that the spherical octant becomes of less area than the
 quarter-cylinder.

[Illustration: Fig. 139.]

The case we have dealt with is of little practical importance to the
biologist, because the cases in which a cubical, or rectangular,
cell divides unequally, and unsymmetrically, are apparently few; but
we can find, as Berthold pointed out, a few examples, for instance
in the hairs within the reproductive “conceptacles” of certain Fuci
(Sphacelaria, etc., Fig. 139), or in the “paraphyses” of mosses
(Fig. 142). But it is of great theoretical importance: as serving to
introduce us to a large class of cases, in which the shape and the
relative dimensions of the original cavity lead, according to the
principle of minimal areas, to cell-division in very definite and
sometimes unexpected ways. It is not easy, nor indeed possible, to
give a generalised account of these cases, for the limiting conditions
are somewhat complex, and the mathematical treatment soon becomes
difficult. But it is easy to comprehend a few simple cases, which of
themselves will carry us a good long way; and which will go far to
convince the student that, in other cases {352} which we cannot fully
master, the same guiding principle is at the root of the matter.

――――――――――

The bisection of a solid (or the subdivision of its volume in other
definite proportions) soon leads us into a geometry which, if not
necessarily difficult, is apt to be unfamiliar; but in such problems
we can go a long way, and often far enough for our particular purpose,
if we merely consider the plane geometry of a side or section of our
figure. For instance, in the case of the cube which we have been just
considering, and in the case of the plane and cylindrical partitions
by which it has been divided, it is obvious that, since these two
partitions extend symmetrically from top to bottom of our cube, that
we need only consider (so far as they are concerned) the manner in
which they subdivide the _base_ of the cube. The whole problem of the
solid, up to a certain point, is contained in our plane diagram of
Fig. 138. And when our particular solid is a solid of revolution, then
it is obvious that a study of its plane of symmetry (that is to say
any plane passing through its axis of rotation) gives us the solution
of the whole problem. The right cone is a case in point, for here the
investigation of its modes of symmetrical subdivision is completely met
by an examination of the isosceles triangle which constitutes its plane
of symmetry.

The bisection of an isosceles triangle by a line which shall be the
shortest possible is a very easy problem. Let _ABC_ be such a triangle
of which _A_ is the apex; it may be shewn that, for its shortest line
of bisection, we are limited to three cases: viz. to a vertical line
_AD_, bisecting the angle at _A_ and the side _BC_; to a transverse
line parallel to the base _BC_; or to an oblique line parallel to _AB_
or to _AC_. The respective magnitudes, or lengths, of these partition
lines follow at once from the magnitudes of the angles of our triangle.
For we know, to begin with, since the areas of similar figures vary as
the squares of their linear dimensions, that, in order to bisect the
area, a line parallel to one side of our triangle must always have a
length equal to 1/√2 of that side. If then, we take our base, _BC_, in
all cases of a length = 2, the transverse partition drawn parallel to
it will always have a length equal to 2/√2, or = √2. The vertical {353}
partition, _AD_, since _BD_ = 1, will always equal tan β (β being the
angle _ABC_). And the oblique partition, _GH_, being equal to _AB_/√2
= 1/(√2 cos β). If then we call our vertical, transverse

[Illustration: Fig. 140.]

and oblique partitions, _V_, _T_, and _O_, we have _V_ = tan β; _T_
= √2; and _O_ = 1/(√2 cos β), or

 _V_ : _T_ : _O_ = tan β/√2 : 1 : 1/(2 cos β).

And, working out these equations for various values of β, we very
soon see that the vertical partition (_V_) is the least of the three
until β = 45°, at which limit _V_ and _O_ are each equal to 1/√2
= ·707; and that again, when β = 60°, _O_ and _T_ are each = 1, after
which _T_ (whose value always = 1) is the shortest of the three
partitions. And, as we have seen, these results are at once applicable,
not only to the case of the plane triangle, but also to that of the
conical cell.

[Illustration: Fig. 141.]

In like manner, if we have a spheroidal body, less than a hemisphere,
such for instance as a low, watch-glass shaped cell (Fig. 141, _a_),
it is obvious that the smallest possible partition by which we can
divide it into two equal halves {354} is (as in our flattened disc)
a median vertical one. And likewise, the hemisphere itself can be
bisected by no smaller partition meeting the walls at right angles
than that median one which divides it into two similar quadrants of a
sphere. But if we produce our hemisphere into a more elevated, conical
body, or into a cylinder with spherical cap, it is obvious that there
comes a point where a transverse, horizontal partition will bisect
the figure with less area of partition-wall than a median vertical
one (_c_). And furthermore, there will be an intermediate region, a
region where height and base have their relative dimensions nearly
equal (as in _b_), where an oblique partition will be better than
either the vertical or the transverse, though here the analogy of our
triangle does not suffice to give us the precise limiting values. We
need not examine these limitations in detail, but we must look at the
curvatures which accompany the several conditions. We have seen that a
film tends to set itself at equal angles to the surface which it meets,
and therefore, when that surface is a solid, to meet it (or its tangent
if it be a curved surface) at right angles. Our _vertical_ partition
is, therefore, everywhere normal to the original cell-walls, and
constitutes a plane surface.

But in the taller, conical cell with transverse partition, the latter
still meets the opposite sides of the cell at right angles, and it
follows that it must itself be curved; moreover, since the tension,
and therefore the curvature, of the partition is everywhere uniform,
it follows that its curved surface must be a portion of a sphere,
concave towards the apex of the original, now divided, cell. In the
intermediate case, where we have an oblique partition, meeting both
the base and the curved sides of the mother-cell, the contact must
still be everywhere at right angles: provided we continue to suppose
that the walls of the mother-cell (like those of our diagrammatic
cube) have become practically rigid before the partition appears,
and are therefore not affected and deformed by the tension of the
latter. In such a case, and especially when the cell is elliptical in
cross-section, or is still more complicated in form, it is evident that
the partition, in adapting itself to circumstances and in maintaining
itself as a surface of minimal area subject to all the conditions of
the case, may have to assume a complex curvature. {355}

[Illustration: Fig. 142. S-shaped partitions: _A_, from _Taonia
atomaria_ (after Reinke); _B_, from paraphyses of _Fucus_; _C_, from
rhizoids of Moss; _D_, from paraphyses of _Polytrichum_.]

While in very many cases the partitions (like the walls of the original
cell) will be either plane or spherical, a more complex curvature will
be assumed under a variety of conditions. It will be apt to occur,
for instance, when the mother-cell is irregular in shape, and one
particular case of such asymmetry will be that in which (as in Fig.
143) the cell has begun to branch, or give off a diverticulum, before
division takes place. A very complicated case of a different kind,
though not without its analogies to the cases we are considering, will
occur in the partitions of minimal area which subdivide the spiral
tube of a nautilus, as we shall presently see. And again, whenever we
have a marked internal asymmetry of the cell, leading to irregular
and anomalous modes of division, in which the cell is not necessarily
divided into two equal halves and in which the partition-wall may
assume an oblique position, then apparently anomalous curvatures will
tend to make their appearance[388].

Suppose that a more or less oblong cell have a tendency to divide by
means of an oblique partition (as may happen through various causes
or conditions of asymmetry), such a partition will still have a
tendency to set itself at right angles to the rigid walls {356} of the
mother-cell: and it will at once follow that our oblique partition,
throughout its whole extent, will assume the form of a complex,
saddle-shaped or anticlastic surface.

[Illustration: Fig. 143. Diagrammatic explanation of S-shaped
partition.]

Many such cases of partitions with complex or double curvature exist,
but they are not always easy of recognition, nor is the particular
case where they appear in a _terminal_ cell a common one. We may see
them, for instance, in the roots (or rhizoids) of Mosses, especially
at the point of development of a new rootlet (Fig. 142, C); and again
among Mosses, in the “paraphyses” of the male prothalli (e.g. in
_Polytrichum_), we find more or less similar partitions (D). They are
frequent also among many Fuci, as in the hairs or paraphyses of Fucus
itself (B). In _Taonia atomaria_, as figured in Reinke’s memoir on
the Dictyotaceae of the Gulf of Naples[389], we see, in like manner,
_oblique_ partitions, which on more careful examination are seen to be
curves of double curvature (Fig. 142, A).

The physical cause and origin of these S-shaped partitions is somewhat
obscure, but we may attempt a tentative explanation. When we assert
a tendency for the cell to divide transversely to its long axis, we
are not only stating empirically that the partition tends to appear
in a small, rather than a large cross-section of the cell: but we
are also implicitly ascribing to the cell a longitudinal _polarity_
(Fig. 143, A), and implicitly asserting that it tends to {357}
divide (just as the segmenting egg does), by a partition transverse
to its polar axis. Such a polarity may conceivably be due to a
chemical asymmetry, or anisotropy, such as we have learned of (from
Professor Macallum’s experiments) in our chapter on Adsorption. Now
if the chemical concentration, on which this anisotropy or polarity
(by hypothesis) depends, be unsymmetrical, one of its poles being
as it were deflected to one side, where a little branch or bud is
being (or about to be) given off,—all in precise accordance with the
adsorption phenomena described on p. 289,—then our “polar axis” would
necessarily be a curved axis, and the partition, being constrained
(again _ex hypothesi_) to arise transversely to the polar axis, would
lie obliquely to the _apparent_ axis of the cell (Fig. 143, B, C).
And if the oblique partition be so situated that it has to meet the
_opposite_ walls (as in C), then, in order to do so symmetrically (i.e.
either perpendicularly, as when the cell-wall is already solidified,
or at least at equal angles on either side), it is evident that the
partition, in its course from one side of the cell to the other, must
necessarily assume a more or less S-shaped curvature (Fig. 143, D).

As a matter of fact, while we have abundant simple illustrations of
the principles which we have now begun to study, apparent exceptions
to this simplicity, due to an asymmetry of the cell itself, or of the
system of which the single cell is but a part, are by no means rare.
For example, we know that in cambium-cells, division frequently takes
place parallel to the long axis of the cell, when a partition of much
less area would suffice if it were set cross-ways: and it is only
when a considerable disproportion has been set up between the length
and breadth of the cell, that the balance is in part redressed by the
appearance of a transverse partition. It was owing to such exceptions
that Berthold was led to qualify and even to depreciate the importance
of the law of minimal areas as a factor in cell-division, after he
himself had done so much to demonstrate and elucidate it[390]. He was
deeply and rightly impressed by the fact that other forces besides
surface {358} tension, both external and internal to the cell, play
their part in the determination of its partitions, and that the
answer to our problem is not to be given in a word. How fundamentally
important it is, however, in spite of all conflicting tendencies and
apparent exceptions, we shall see better and better as we proceed.

――――――――――

But let us leave the exceptions and return to a consideration of the
simpler and more general phenomena. And in so doing, let us leave the
case of the cubical, quadrangular or cylindrical cell, and examine the
case of a spherical cell and of its successive divisions, or the still
simpler case of a circular, discoidal cell.

When we attempt to investigate mathematically the position and form
of a partition of minimal area, it is plain that we shall be dealing
with comparatively simple cases wherever even one dimension of the
cell is much less than the other two. Where two dimensions are small
compared with the third, as in a thin cylindrical filament like that
of Spirogyra, we have the problem at its simplest; for it is at once
obvious, then, that the partition must lie transversely to the long
axis of the thread. But even where one dimension only is relatively
small, as for instance in a flattened plate, our problem is so far
simplified that we see at once that the partition cannot be parallel to
the extended plane, but must cut the cell, somehow, at right angles to
that plane. In short, the problem of dividing a much flattened solid
becomes identical with that of dividing a simple _surface_ of the same
form.

There are a number of small Algae, growing in the form of small
flattened discs, consisting (for a time at any rate) of but a single
layer of cells, which, as Berthold shewed, exemplify this comparatively
simple problem; and we shall find presently that it is also admirably
illustrated in the cell-divisions which occur in the egg of a frog or
a sea-urchin, when the egg for the sake of experiment is flattened out
under artificial pressure.

[Illustration: Fig. 144. Development of _Erythrotrichia_. (After
Berthold.)]

[Illustration: Fig. 145.]

Fig. 144 (taken from Berthold’s _Monograph of the Naples Bangiaciae_)
represents younger and older discs of the little alga _Erythrotrichia
discigera_; and it will be seen that, in all stages save the first, we
have an arrangement of cell-partitions which looks somewhat complex,
but into which we must attempt to throw some light and order. Starting
with the original single, and flattened, {359} cell, we have no
difficulty with the first two cell-divisions; for we know that no
bisecting partitions can possibly be shorter than the two diameters,
which divide the cell into halves and into quarters. We have only to
remember that, for the sum total of partitions to be a minimum, three
only must meet in a point; and therefore, the four quadrantal walls
must shift a little, producing the usual little median partition, or
cross-furrow, instead of one common, central point of junction. This
little intermediate wall, however, will be very small, and to all
intents and purposes we may deal with the case as though we had now
to do with four equal cells, each one of them a perfect quadrant.
And so our problem is, to find the shortest line which shall divide
the quadrant of a circle into two halves of equal area. A radial
partition (Fig. 145, A), starting from the apex of the quadrant, is
at once excluded, for a reason similar to that just referred to; our
choice must lie therefore between two modes of division such as are
illustrated in Fig. 145, where the partition is either (as in B) {360}
concentric with the outer border of the cell, or else (as in C) cuts
that outer border; in other words, our partition may (B) cut _both_
radial walls, or (C) may cut _one_ radial wall and the periphery. These
are the two methods of division which Sachs called, respectively, (B)
_periclinal_, and (C) _anticlinal_[391]. We may either treat the walls
of the dividing quadrant as already solidified, or at least as having
a tension compared with which that of the incipient partition film is
inconsiderable. In either case the partition must meet the cell-wall,
on either side, at right angles, and (its own tension and curvature
being everywhere uniform) it must take the form of a circular arc.

Now we find that a flattened cell which is approximately a quadrant of
a circle invariably divides after the manner of Fig. 145, C, that is to
say, by an approximately circular, _anticlinal_ wall, such as we now
recognise in the eight-celled stage of Erythrotrichia (Fig. 144); let
us then consider that Nature has solved our problem for us, and let us
work out the actual geometric conditions.

Let the quadrant _OAB_ (in Fig. 146) be divided into two parts of equal
area, by the circular arc _MP_. It is required to determine (1) the
position of _P_ upon the arc of the quadrant, that is to say the angle
_BOP_; (2) the position of the point _M_ on the side _OA_; and (3) the
length of the arc _MP_ in terms of a radius of the quadrant.

(1) Draw _OP_; also _PC_ a tangent, meeting _OA_ in _C_; and _PN_,
perpendicular to _OA_. Let us call _a_ a radius; and θ the angle at
_C_, which is obviously equal to _OPN_, or _POB_. Then

 _CP_ = _a_ cot θ; _PN_ = _a_ cos θ; _NC_ = _CP_ cos θ = _a_ ⋅ (cos^2 θ)/(sin θ).

The area of the portion _PMN_

 = ½_C_(_P_^2)θ − ½_PN_ ⋅ _NC_

 = ½_a_^2(cot^2 θ) − ½_a_(cos θ) ⋅ _a_(cos^2 θ)/(sin θ)

 = ½_a_^2(cot^2 θ − (cos^3 θ)/(sin θ)).

{361}

And the area of the portion _PNA_

 = ½_a_^2(π/2 − θ) − ½_ON_ ⋅ _NP_

 = ½_a_^2(π/2 − θ) − ½_a_(sin θ) ⋅ _a_(cos θ)

 = ½_a_^2(π/2 − θ − sin θ ⋅ cos θ).

Therefore the area of the whole portion _PMA_

 = (_a_^2)/2 (π/2 − θ + θ cot^2 θ − (cos^3 θ)/sin θ − sin θ ⋅ cos θ)

 = (_a_^2)/2 (π/2 − θ + θ cot^2 θ − cot θ),

and also, by hypothesis, = ½ ⋅ area of the quadrant, = (π_a_^2)/8.

[Illustration: Fig. 146.]

Hence θ is defined by the equation

 _a_^2/2 (π/2 − θ + θ cot^2 θ − cot θ) = (π_a_^2)/8,

 or π/4 − θ + θ cot^2 θ − cot θ = 0.

We may solve this equation by constructing a table (of which the
following is a small portion) for various values of θ.

   θ        π/4     − θ   − cot θ + θ cot^2 θ = _x_
 34° 34′   ·7854 − ·6033 − 1·4514  + 1·2709   = ·0016
     35′   ·7854   ·6036   1·4505    1·2700     ·0013
     36′   ·7854   ·6039   1·4496    1·2690     ·0009
     37′   ·7854   ·6042   1·4487    1·2680     ·0005
     38′   ·7854   ·6045   1·4478    1·2671     ·0002
     39′   ·7854   ·6048   1·4469    1·2661    −·0002
     40′   ·7854   ·6051   1·4460    1·2652    −·0005

{362}

We see accordingly that the equation is solved (as accurately as need
be) when θ is an angle somewhat over 34° 38′, or say 34° 38½′. That is
to say, a quadrant of a circle is bisected by a circular arc cutting
the side and the periphery of the quadrant at right angles, when
the arc is such as to include (90° − 34° 38′), i.e. 55° 22′ of the
quadrantal arc.

This determination of ours is practically identical with that which
Berthold arrived at by a rough and ready method, without the use of
mathematics. He simply tried various ways of dividing a quadrant of
paper by means of a circular arc, and went on doing so till he got the
weights of his two pieces of paper approximately equal. The angle, as
he thus determined it, was 34·6°, or say 34° 36′.

(2) The position of _M_ on the side of the quadrant _OA_ is given
by the equation _OM_ = _a_ cosec θ − _a_ cot θ; the value of which
expression, for the angle which we have just discovered, is ·3028. That
is to say, the radius (or side) of the quadrant will be divided by the
new partition into two parts, in the proportions of nearly three to
seven.

(3) The length of the arc _MP_ is equal to _a_ θ cot θ; and the value
of this for the given angle is ·8751. This is as much as to say that
the curved partition-wall which we are considering is shorter than a
radial partition in the proportion of 8¾ to 10, or seven-eights almost
exactly.

But we must also compare the length of this curved “anticlinal”
partition-wall (_MP_) with that of the concentric, or periclinal, one
(_RS_, Fig. 147) by which the quadrant might also be bisected. The
length of this partition is obviously equal to the arc of the quadrant
(i.e. the peripheral wall of the cell) divided by √2; or, in terms
of the radius, = π/2√2 = 1·111. So that, not only is the anticlinal
partition (such as we actually find in nature) notably the best, but
the periclinal one, when it comes to dividing an entire quadrant, is
very considerably larger even than a radial partition.

[Illustration: Fig. 147.]

The two cells into which our original quadrant is now divided, while
they are equal in volume, are of very different shapes; the {363} one
is a triangle (_MAP_) with two sides formed of circular arcs, and the
other is a four-sided figure (_MOBP_), which we may call approximately
oblong. We cannot say as yet how the triangular portion ought to
divide; but it is obvious that the least possible partition-wall
which shall bisect the other must run across the long axis of the
oblong, that is to say periclinally. This, also, is precisely what
tends actually to take place. In the following diagrams (Fig. 148)
of a frog’s egg dividing under pressure, that is to say when reduced
to the form of a flattened plate, we see, firstly, the division into
four quadrants (by the partitions 1, 2); secondly, the division of
each quadrant by means of an anticlinal circular arc (3, 3), cutting
the peripheral wall of the quadrant approximately in the proportions
of three to seven; and thirdly, we see that of the eight cells (four
triangular and four oblong) into which the whole egg is now divided,
the four which we have called oblong now proceed to divide by
partitions transverse to their long axes, or roughly parallel to the
periphery of the egg.

[Illustration: Fig. 148. Segmentation of frog’s egg, under artificial
compression. (After Roux.)]

――――――――――

The question how the other, or triangular, portion of the divided
quadrant will next divide leads us to another well-defined problem,
which is only a slight extension, making allowance for the circular
arcs, of that elementary problem of the triangle we have already
considered. We know now that an entire quadrant must divide (so that
its bisecting wall shall have the least possible area) by means of an
anticlinal partition, but how about any smaller sectors of circles?
It is obvious in the case of a small prismatic {364} sector, such as
that shewn in Fig. 149, that a _periclinal_ partition is the smallest
by which we can possibly bisect the cell; we want, accordingly, to know
the limits below which the periclinal partition is always the best, and
above which the anticlinal arc, as in the case of the whole quadrant,
has the advantage in regard to smallness of surface area.

This may be easily determined; for the preceding investigation is a
perfectly general one, and the results hold good for sectors of any
other arc, as well as for the quadrant, or arc of 90°. That is to say,
the length of the partition-wall _MP_ is always determined by the angle
θ, according to our equation _MP_ = _a_θ cot θ; and the angle θ has a
definite relation to α, the angle of arc.

[Illustration: Fig. 149.]

Moreover, in the case of the periclinal boundary, _RS_ (Fig. 147) (or
_ab_, Fig. 149), we know that, if it bisect the cell,

 _RS_ = _a_ ⋅ α/√2.

Accordingly, the arc _RS_ will be just equal to the arc _MP_ when

 θ cot θ = α/√2.

 When      θ cot θ > α/√2 or _MP_ > _RS_,

 then division will take place as in _RS_.

 When      θ cot θ < α/√2, or _MP_ < _RS_,

 then division will take place as in _MP_.

In the accompanying diagram (Fig. 150), I have plotted the various
magnitudes with which we are concerned, in order to exhibit the several
limiting values. Here we see, in the first place, the curve marked α,
which shews on the (left-hand) vertical scale the various possible
magnitudes of that angle (viz. the angle {365} of arc of the whole
sector which we wish to divide), and on the horizontal scale the
corresponding values of θ, or the angle which

[Illustration: Fig. 150.]

determines the point on the periphery where it is cut by the
partition-wall, _MP_. Two limiting cases are to be noticed here: (1)
at 90° (point _A_ in diagram), because we are at present only {366}
dealing with arcs no greater than a quadrant; and (2), the point (_B_)
where the angle θ comes to equal the angle α, for after that point
the construction becomes impossible, since an anticlinal bisecting
partition-wall would be partly outside the cell. The only partition
which, after the point, can possibly exist, is a periclinal one. This
point, as our diagram shews us, occurs when the angles (α and θ) are
each rather under 52°.

Next I have plotted, on the same diagram, and in relation to the same
scales of angles, the corresponding lengths of the two partitions, viz.
_RS_ and _MP_, their lengths being expressed (on the right-hand side of
the diagram) in relation to the radius of the circle (_a_), that is to
say the side wall, _OA_, of our cell.

The limiting values here are (1), _C_, _C′_, where the angle of arc
is 90°, and where, as we have already seen, the two partition-walls
have the relative magnitudes of _MP_ : _RS_ = 0·875 : 1·111; (2) the
point _D_, where _RS_ equals unity, that is to say where the periclinal
partition has the same length as a radial one; this occurs when α is
rather under 82° (cf. the points _D_, _D′_); (3) the point _E_, where
_RS_ and _MP_ intersect; that is to say the point at which the two
partitions, periclinal and anticlinal, are of the same magnitude;
this is the case, according to our diagram, when the angle of arc is
just over 62½°. We see from this, then, that what we have called an
anticlinal partition, as _MP_, is only likely to occur in a triangular
or prismatic cell whose angle of arc lies between 90° and 62½°. In all
narrower or more tapering cells, the periclinal partition will be of
less area, and will therefore be more and more likely to occur.

The case (_F_) where the angle α is just 60° is of some interest. Here,
owing to the curvature of the peripheral border, and the consequent
fact that the peripheral angles are somewhat greater than the apical
angle α, the periclinal partition has a very slight and almost
imperceptible advantage over the anticlinal, the relative proportions
being about as _MP_ : _RS_ = 0·73 : 0·72. But if the equilateral
triangle be a plane spherical triangle, i.e. a plane triangle bounded
by circular arcs, then we see that there is no longer any distinction
at all between our two partitions; _MP_ and _RS_ are now identical.

On the same diagram, I have inserted the curve for values of {367}
cosec θ − cot θ = _OM_, that is to say the distances from the centre,
along the side of the cell, of the starting-point (_M_) of the
anticlinal partition. The point _C″_ represents its position in the
case of a quadrant, and shews it to be (as we have already said) about
3/10 of the length of the radius from the centre. If, on the other
hand, our cell be an equilateral triangle, then we have to read off the
point on this curve corresponding to α = 60°, and we find it at the
point _F‴_ (vertically under _F_), which tells us that the partition
now starts 4·5/10, or nearly halfway, along the radial wall.

――――――――――

The foregoing considerations carry us a long way in our investigations
of many of the simpler forms of cell-division. Strictly speaking they
are limited to the case of flattened cells, in which we can treat the
problem as though we were simply partitioning a plane surface. But it
is obvious that, though they do not teach us the whole conformation of
the partition which divides a more complicated solid into two halves,
yet they do, even in such a case, enlighten us so far, that they tell
us the appearance presented in one plane of the actual solid. And as
this is all that we see in a microscopic section, it follows that the
results we have arrived at will greatly help us in the interpretation
of microscopic appearances, even in comparatively complex cases of
cell-division.

[Illustration: Fig. 151.]

Let us now return to our quadrant cell (_OAPB_), which we have found
to be divided into a triangular and a quadrilateral portion, as in
Fig. 147 or Fig. 151; and let us now suppose the whole system to
grow, in a uniform fashion, as a prelude to further subdivision. The
whole quadrant, growing uniformly (or with equal radial increments),
will still remain a quadrant, and it is obvious, therefore, that
for every new increment of size, more will be added to the margin
of its triangular portion than to the {368} narrower margin of its
quadrilateral portion; and these increments will be in proportion to
the angles of arc, viz. 55° 22′ : 34° 38′, or as ·96 : ·60, i.e. as
8 : 5. And accordingly, if we may assume (and the assumption is a
very plausible one), that, just as the quadrant itself divided into
two halves after it got to a certain size, so each of its two halves
will reach the same size before again dividing, it is obvious that
the triangular portion will be doubled in size, and therefore ready
to divide, a considerable time before the quadrilateral part. To work
out the problem in detail would lead us into troublesome mathematics;
but if we simply assume that the increments are proportional to the
increasing radii of the circle, we have the following equations:―

Let us call the triangular cell _T_, and the quadrilateral, _Q_ (Fig.
151); let the radius, _OA_, of the original quadrantal cell = _a_ = 1;
and let the increment which is required to add on a portion equal to
_T_ (such as _PP′A′A_) be called _x_, and let that required, similarly,
for the doubling of _Q_ be called _x′_.

Then we see that the area of the original quadrant

               = _T_ + _Q_ = ¼π_a_^2 = ·7854_a_^2,

 while the area of _T_ = _Q_ = ·3927_a_^2.

The area of the enlarged sector, _p′OA′_,

 = (_a_ + _x_)^2 × (55° 22′) ÷ 2 = ·4831(_a_ + _x_)^2,

 and the area _OPA_

 = _a_^2 × (55° 22′) ÷ 2  = ·4831_a_^2.

 Therefore the area of the added portion, _T′_,

 = ·4831 ((_a_ + _x_)^2 − _a_^2).

 And this, by hypothesis,

 = _T_ = ·3927_a_^2.

We get, accordingly, since _a_ = 1,

 _x_^2 + 2_x_ = ·3927/·4831 = ·810,

 and, solving,

 _x_ + 1 = √1·81 = 1·345, or _x_ = 0·345.

Working out _x′_ in the same way, we arrive at the approximate value,
_x′_ + 1 = 1·517. {369}

This is as much as to say that, supposing each cell tends to divide
into two halves when (and not before) its original size is doubled,
then, in our flattened disc, the triangular cell _T_ will tend to
divide when the radius of the disc has increased by about a third (from
1 to 1·345), but the quadrilateral cell, _Q_, will not tend to divide
until the linear dimensions of the disc have increased by about a half
(from 1 to 1·517).

The case here illustrated is of no small general importance. For
it shews us that a uniform and symmetrical growth of the organism
(symmetrical, that is to say, under the limitations of a plane surface,
or plane section) by no means involves a uniform or symmetrical growth
of the individual cells, but may, under certain conditions, actually
lead to inequality among these; and this inequality may be further
emphasised by differences which arise out of it, in regard to the
order of frequency of further subdivision. This phenomenon (or to be
quite candid, this hypothesis, which is due to Berthold) is entirely
independent of any change or variation in individual surface tensions;
and accordingly it is essentially different from the phenomenon of
unequal segmentation (as studied by Balfour), to which we have referred
on p. 348.

In this fashion, we might go on to consider the manner, and the
order of succession, in which the subsequent cell-divisions would
tend to take place, as governed by the principle of minimal areas.
But the calculations would grow more difficult, or the results got
by simple methods would grow less and less exact. At the same time,
some of these results would be of great interest, and well worth the
trouble of obtaining. For instance, the precise manner in which our
triangular cell, _T_, would next divide would be interesting to know,
and a general solution of this problem is certainly troublesome to
calculate. But in this particular case we can see that the width of the
triangular cell near _P_ is so obviously less than that near either of
the other two angles, that a circular arc cutting off that angle is
bound to be the shortest possible bisecting line; and that, in short,
our triangular cell will tend to subdivide, just like the original
quadrant, into a triangular and a quadrilateral portion.

But the case will be different next time, because in this new {370}
triangle, _PRQ_, the least width is near the innermost angle, that at
_Q_; and the bisecting circular arc will therefore be opposite to _Q_,
or (approximately) parallel to _PR_. The importance of this fact is at
once evident; for it means to say that there soon comes a time when,
whether by the division of triangles or of quadrilaterals, we find only
quadrilateral cells adjoining the periphery of our circular disc. In
the subsequent division of these quadrilaterals, the partitions will
arise transversely to their long axes, that is to say, _radially_ (as
_U_, _V_); and we shall consequently have a superficial or peripheral
layer of quadrilateral cells, with sides approximately parallel, that
is to say what we are accustomed to call _an epidermis_. And this
epidermis or superficial layer will be in clear contrast with the more
irregularly shaped cells, the products of triangles and quadrilaterals,
which make up the deeper, underlying layers of tissue.

[Illustration: Fig. 152.]

In following out these theoretic principles and others like to them,
in the actual division of living cells, we must always bear in mind
certain conditions and qualifications. In the first place, the law
of minimal area and the other rules which we have arrived at are not
absolute but relative: they are links, and very important links, in a
chain of physical causation; they are always at work, but their effects
may be overridden and concealed by the operation of other forces.
Secondly, we must remember that, in the great majority of cases, the
cell-system which we have in view is constantly increasing in magnitude
by active growth; and by this means the form and also the proportions
of the cells are continually liable to alteration, of which phenomenon
we have already had an example. Thirdly, we must carefully remember
that, until our cell-walls become absolutely solid and rigid, they are
always apt to be modified in form owing to the tension of the adjacent
{371} walls; and again, that so long as our partition films are fluid
or semifluid, their points and lines of contact with one another
may shift, like the shifting outlines of a system of soap-bubbles.
This is the physical cause of the movements frequently seen among
segmenting cells, like those to which Rauber called attention in the
segmenting ovum of the frog, and like those more striking movements
or accommodations which give rise to a so-called “spiral” type of
segmentation.

――――――――――

[Illustration: Fig. 153. Diagram of flattened or discoid cell
dividing into octants: to shew gradual tendency towards a position of
equilibrium.]

Bearing in mind, then, these considerations, let us see what our
flattened disc is likely to look like, after a few successive divisions
into component cells. In Fig. 153, _a_, we have a diagrammatic
representation of our disc, after it has divided into four quadrants,
and each of these in turn into a triangular and a quadrilateral
portion; but as yet, this figure scarcely suggests to us anything like
the normal look of an aggregate of living cells. But let us go a little
further, still limiting ourselves, however, to the consideration of the
eight-celled stage. Wherever one of our radiating partitions meets the
peripheral wall, there will (as we know) be a mutual tension between
the three convergent films, which will tend to set their edges at equal
angles to one another, angles that is to say of 120°. In consequence of
this, the outer wall of each individual cell will (in this surface view
of our disc) {372} be an arc of a circle of which we can determine the
centre by the method used on p. 307; and, furthermore, the narrower
cells, that is to say the quadrilaterals, will have this outer border
somewhat more curved than their broader neighbours. We arrive, then, at
the condition shewn in Fig. 153, _b_. Within the cell, also, wherever
wall meets wall, the angle of contact must tend, in every case, to be
an angle of 120°; and in no case may more than three films (as seen in
section) meet in a point (_c_); and this condition, of the partitions
meeting three by three, and at co-equal angles, will obviously involve
the curvature of some, if not all, of the partitions (_d_) which in our
preliminary investigation we treated as plane. To solve this problem
in a general way is no easy matter; but it is a problem which Nature
solves in every case where, as in the case we are considering, eight
bubbles, or eight cells, meet together in a (plane or curved) surface.
An approximate solution has been given in Fig. 153, _d_; and it will
now at once be recognised that this figure has vastly more resemblance
to an aggregate of living cells than had the diagram of Fig. 153, _a_
with which we began.

[Illustration: Fig. 154.]

Just as we have constructed in this case a series of purely
diagrammatic or schematic figures, so it will be as a rule possible to
diagrammatise, with but little alteration, the complicated appearances
presented by any ordinary aggregate of cells. The accompanying little
figure (Fig. 154), of a germinating spore of a Liverwort (Riccia),
after a drawing of Professor Campbell’s, scarcely needs further
explanation: for it is well-nigh a typical diagram of the method of
space-partitioning which we are now considering. Let us look again
at our figures (on p. 359) of the disc of Erythrotrichia, from
Berthold’s _Monograph of the Bangiaceae_ and redraw the earlier stages
in diagrammatic fashion. In the following series of diagrams the new
partitions, or those just about to form, are in each case outlined;
and in the next succeeding stage they are shewn after settling down
into position, and after exercising their respective tractions on the
walls previously laid down. It is clear, I think, that these four
diagrammatic figures represent all that is shewn in the first five
stages drawn by Berthold from the plant itself; but the correspondence
cannot {373} in this case be precisely accurate, for the simple reason
that Berthold’s figures are taken from different individuals, and are
therefore only approximately consecutive and not strictly continuous.
The last of the six drawings in Fig. 144 is already too

[Illustration: Fig. 155. Theoretical arrangement of successive
partitions in a discoid cell; for comparison with Fig. 144.]

complicated for diagrammatisation, that is to say it is too
complicated for us to decipher with certainty the precise order of
appearance of the numerous partitions which it contains. But in Fig.
156 I shew one more diagrammatic figure, of a disc which

[Illustration: Fig. 156. Theoretical division of a discoid cell into
sixty-four chambers: no allowance being made for the mutual tractions
of the cell-walls.]

has divided, according to the theoretical plan, into about sixty-four
cells; and making due allowance for the successive changes which the
mutual tensions and tractions of the partitions must {374} bring
about, increasing in complexity with each succeeding stage, we can
see, even at this advanced and complicated stage, a very considerable
resemblance between the actual picture (Fig. 144) and the diagram which
we have here constructed in obedience to a few simple rules.

In like manner, in the annexed figures, representing sections through
a young embryo of a Moss, we have very little difficulty in discerning
the successive stages that must have intervened between the two stages
shewn: so as to lead from the just divided quadrants (one of which, by
the way, has not yet divided in our figure (_a_)) to the stage (_b_)
in which a well-marked epidermal layer surrounds an at first sight
irregular agglomeration of “fundamental” tissue.

[Illustration: Fig. 157. Sections of embryo of a moss. (After
Kienitz-Gerloff.)]

In the last paragraph but one, I have spoken of the difficulty of so
arranging the meeting-places of a number of cells that at each junction
only three cell-walls shall meet in a line, and all three shall meet it
at equal angles of 120°. As a matter of fact, the problem is soluble in
a number of ways; that is to say, when we have a number of cells, say
eight as in the case considered, enclosed in a common boundary, there
are various ways in which their walls can be made to meet internally,
three by three, at equal angles; and these differences will entail
differences also in the curvature of the walls, and consequently in the
shape of the cells. The question is somewhat complex; it has been dealt
with by Plateau, and treated mathematically by M. Van Rees[392].

[Illustration: Fig. 158. Various possible arrangements of intermediate
partitions, in groups of 4, 5, 6, 7 or 8 cells.]

If within our boundary we have three cells all meeting {375}
internally, they must meet in a point; furthermore, they tend to do so
at equal angles of 120°, and there is an end of the matter. If we have
four cells, then, as we have already seen, the conditions are satisfied
by interposing a little intermediate wall, the two extremities of
which constitute the meeting-points of three cells each, and the
upper edge of which marks the “polar furrow.” Similarly, in the case
of five cells, we require _two_ little intermediate walls, and two
polar furrows; and we soon arrive at the rule that, for _n_ cells,
we require _n_ − 3 little longitudinal partitions (and corresponding
polar furrows), connecting the triple junctions of the cells; and these
little walls, like all the rest within the system, must be inclined
to one another at angles of 120°. Where we have only one such wall
(as in the case of four cells), or only two (as in the case of five
cells), there is no room for ambiguity. But where we have three little
connecting-walls, as in the case of six cells, it is obvious that we
can arrange them in three different ways, as in the annexed Fig. 159.
In the system of seven cells, the four partitions can be arranged in
four ways; and the five partitions required in the case of eight cells
can be arranged in no less than thirteen different ways, of which
Fig. 158 shews some half-dozen only. It does not follow that, so to
speak, these various {376} arrangements are all equally good; some are
known to be much more stable than others, and some have never yet been
realised in actual experiment.

The conditions which lead to the presence of any one of them, in
preference to another, are as yet, so far as I am aware, undetermined,
but to this point we shall return.

――――――――――

Examples of these various arrangements meet us at every turn, and
not only in cell-aggregates, but in various cases where non-rigid
and semi-fluid partitions (or partitions that were so to begin
with) meet together. And it is a necessary consequence of this
physical phenomenon, and of the limited and very small number of
possible arrangements, that we get similar appearances, capable of
representation by the same diagram, in the most diverse fields of
biology[393].

[Illustration: Fig. 159.]

Among the published figures of embryonic stages and other cell
aggregates, we only discern these little intermediate partitions in
cases where the investigator has drawn carefully just what lay before
him, without any preconceived notions as to radial or other symmetry;
but even in other cases we can generally recognise, without much
difficulty, what the actual arrangement was whereby the cell-walls
met together in equilibrium. I have a strong suspicion that a leaning
towards Sachs’s Rule, that one cell-wall tends to set itself at right
angles to another cell-wall (a rule whose strict limitations, and
narrow range of application, we have already {377} considered) is
responsible for many inaccurate or incomplete representations of the
mutual arrangement of aggregated cells.

[Illustration: Fig. 160. Segmenting egg of _Trochus_. (After Robert.)]

[Illustration: Fig. 161. Two views of segmenting egg of _Cynthia
partita_. (After Conklin.)]

[Illustration: Fig. 162. (_a_) Section of apical cone of _Salvinia_.
(After Pringsheim[394].) (_b_) Diagram of probable actual arrangement.]

[Illustration: Fig. 163. Egg of _Pyrosoma_. (After Korotneff).]

[Illustration: Fig. 164. Egg of _Echinus_, segmenting under pressure.
(After Driesch.)]

In the accompanying series of figures (Figs. 160–167) I have {378}
set forth a few aggregates of eight cells, mostly from drawings of
segmenting eggs. In some cases they shew clearly the manner in which
the cells meet one another, always at angles of 120°, and always with
the help of five intermediate boundary walls within the eight-celled
system; in other cases I have added a slightly altered drawing, so as
to shew, with as little change as {379} possible, the arrangement
of boundaries which probably actually existed, and gave rise to the
appearance which the observer drew. These drawings may be compared
with the various diagrams of Fig. 158, in which some seven out of the
possible thirteen arrangements of five intermediate partitions (for a
system of eight cells) have been already set forth.

[Illustration: Fig. 165. (_a_) Part of segmenting egg of Cephalopod
(after Watase); (_b_) probable actual arrangement.]

[Illustration: Fig. 166. (_a_) Egg of _Echinus_; (_b_) do. of _Nereis_,
under pressure. (After Driesch).]

[Illustration: Fig. 167. (_a_) Egg of frog, under pressure (after
Roux); (_b_) probable actual arrangement.]

It will be seen that M. Robert-Tornow’s figure of the segmenting egg of
Trochus (Fig. 160) clearly shews the cells grouped after the fashion of
Fig. 158, _a_. In like manner, Mr Conklin’s figure of the ascidian egg
(_Cynthia_) shews equally clearly the arrangement _g_.

A sea-urchin egg, segmenting under pressure, as figured by Driesch,
scarcely requires any modification of the drawing to appear as
a diagram of the type _d_. Turning for a moment to a botanical
illustration, we have a figure of Pringsheim’s shewing an eight-celled
stage in the apex of the young cone of Salvinia; it is in all
probability referable, as in my modified diagram, to type _c_. Beside
it is figured a very different object, a segmenting egg of the Ascidian
_Pyrosoma_, after Korotneff; it may be that this also is to be referred
to type _c_, but I think it is more easily referable to type _b_. For
there is a difference between this diagram and that of Salvinia, in
that here apparently, of the pairs of lateral cells, the upper and the
lower cell are alternately the larger, while in the diagram of Salvinia
the lower lateral cells both appear much larger than the upper ones;
and this difference tallies with the appearance produced if we fill
in the eight cells according to the type _b_ or the type _c_. In the
segmenting cuttlefish egg, there is again a slight dubiety as to which
type it should be referred to, but it is in all probability referable,
like Driesch’s Echinus egg, to _d_. Lastly, I have copied from Roux a
curious figure of the egg of _Rana esculenta_, viewed from the animal
pole, which appears to me referable, in all probability, to type _g_.
Of type _f_, in which the five partitions form a figure with four
re-entrant angles, that is to say a figure representing the five sides
of a hexagon, I have found no examples among segmenting eggs, and that
arrangement in all probability is a very unstable one.

――――――――――

It is obvious enough, without more ado, that these phenomena are in
the strictest and completest way common to both plants {380} and
animals. In other words they tally with, and they further extend,
the general and fundamental conclusions laid down by Schwann, in
his _Mikroskopische Untersuchungen über die Uebereinstimmung in der
Struktur und dem Wachsthum der Thiere und Pflanzen_.

But now that we have seen how a certain limited number of types of
eight-celled segmentation (or of arrangements of eight cell-partitions)
appear and reappear, here and there, throughout the whole world of
organisms, there still remains the very important question, whether
_in each particular organism_ the conditions are such as to lead to
one particular arrangement being predominant, characteristic, or even
invariable. In short, is a particular arrangement of cell-partitions to
be looked upon (as the published figures of the embryologist are apt to
suggest) as a _specific character_, or at least a constant or normal
character, of the particular organism? The answer to this question
is a direct negative, but it is only in the work of the most careful
and accurate observers that we find it revealed. Rauber (whom we have
more than once had occasion to quote) was one of those embryologists
who recorded just what he saw, without prejudice or preconception; as
Boerhaave said of Swammerdam, _quod vidit id asseruit_. Now Rauber has
put on record a considerable number of variations in the arrangement
of the first eight cells, which form a discoid surface about the
dorsal (or “animal”) pole of the frog’s egg. In a certain number of
cases these figures are identical with one another in type, identical
(that is to say) save for slight differences in magnitude, relative
proportions, or orientation. But I have selected (Fig. 168) six
diagrammatic figures, which are all _essentially different_, and these
diagrams seem to me to bear intrinsic evidence of their accuracy: the
curvatures of the partition-walls, and the angles at which they meet
agree closely with the requirements of theory, and when they depart
from theoretical symmetry they do so only to the slight extent which
we should naturally expect in a material and imperfectly homogeneous
system[395]. {381}

[Illustration: Fig. 168. Various modes of grouping of eight cells, at
the dorsal or epiblastic pole of the frog’s egg. (After Rauber.)]

Of these six illustrations, two are exceptional. In Fig. 168, 5, we
observe that one of the eight cells is surrounded on all sides by the
other seven. This is a perfectly natural condition, and represents,
like the rest, a phase of partial or conditional equilibrium. But it is
not included in the series we are now considering, which is restricted
to the case of eight cells extending outwards to a common boundary.
The condition shewn in Fig. 168, 6, is again peculiar, and is probably
rare; but it is included under the cases considered on p. 312, in
which the cells are not in complete fluid contact, but are separated
by little droplets of extraneous matter; it needs no further comment.
But the other four cases are beautiful diagrams of space-partitioning,
similar to those we have just been considering, but so exquisitely
clear that they need no modification, no “touching-up,” to exhibit
their mathematical regularity. It will easily be recognised that in
Fig. 168, 1 and 2, we have the arrangements corresponding to _a_
and _d_ of our diagram (Fig. 158): but the other two (i.e. 3 and 4)
represent other of the thirteen possible arrangements, which are not
included in that {382} diagram. It would be a curious and interesting
investigation to ascertain, in a large number of frogs’ eggs, all at
this stage of development, the percentage of cases in which these
various arrangements occur, with a view of correlating their frequency
with the theoretical conditions (so far as they are known, or can
be ascertained) of relative stability. One thing stands out as very
certain indeed: that the elementary diagram of the frog’s egg commonly
given in text-books of embryology,—in which the cells are depicted as
uniformly symmetrical quadrangular bodies,—is entirely inaccurate and
grossly misleading[396].

We now begin to realise the remarkable fact, which may even appear
a startling one to the biologist, that all possible groupings or
arrangements whatsoever of eight cells (where all take part in the
_surface_ of the group, none being submerged or wholly enveloped by
the rest) are referable to some one or other of _thirteen_ types or
forms. And that all the thousands and thousands of drawings which
diligent observers have made of such eight-celled structures, animal
or vegetable, anatomical, histological or embryological, are one and
all representations of some one or another of these thirteen types:—or
rather indeed of somewhat less than the whole thirteen, for there is
reason to believe that, out of the total number of possible groupings,
a certain small number are essentially unstable, and have at best, in
the concrete, but a transitory and evanescent existence.

――――――――――

Before we leave this subject, on which a vast deal more might be said,
there are one or two points which we must not omit to consider. Let us
note, in the first place, that the appearance which our plane diagrams
suggest of inequality of the several cells is apt to be deceptive; for
the differences of magnitude apparent in one plane may well be, and
probably generally are, balanced by equal and opposite differences in
another. Secondly, let us remark that the rule which we are considering
refers only {383} to angles, and to the number, not to the length of
the intermediate partitions; it is to a great extent by variations in
the length of these that the magnitudes of the cells may be equalised,
or otherwise balanced, and the whole system brought into equilibrium.
Lastly, there is a curious point to consider, in regard to the number
of actual contacts, in the various cases, between cell and cell. If
we inspect the diagrams in Fig. 169 (which represent three out of our
thirteen possible arrangements of eight cells) we shall see that, in
the case of type _b_, two cells are each in contact with two others,
two cells with three others, and four cells each with four other cells.
In type _a_ four cells are each in contact with two, two with four,
and two with five. In type _f_, two are in contact with two, four with
three, and one with no less than seven. In all cases the

[Illustration: Fig. 169.]

number of contacts is twenty-six in all; or, in other words, there
are thirteen internal partitions, besides the eight peripheral walls.
For it is easy to see that, in all cases of _n_ cells with a common
external boundary, the number of internal partitions is 2_n_ − 3; or
the number of what we call the internal or interfacial contacts is
2(2_n_ − 3). But it would appear that the most stable arrangements are
those in which the total number of contacts is most evenly divided,
and the least stable are those in which some one cell has, as in
type _f_, a predominant number of contacts. In a well-known series
of experiments, Roux has shewn how, by means of oil-drops, various
arrangements, or aggregations, of cells can be simulated; and in Fig.
170 I shew a number of Roux’s figures, and have ascribed them to what
seem to be their appropriate “types” among those which we have just
been considering; but {384} it will be observed that in these figures
of Roux’s the drops are not always in complete contact, a little
air-bubble often keeping them apart at their apical junctions, so that
we see the configuration towards which the system is _tending_ rather
than that which it has fully attained[397]. The type which we have
called _f_ was found by Roux to be unstable, the large (or apparently
large) drop _a″_ quickly passing into the centre of the system, and
here taking up a position of equilibrium in which, as usual, three
cells meet throughout in a point, at equal angles, and in which, in
this case, all the cells have an equal number of “interfacial” contacts.

[Illustration: Fig. 170. Aggregations of oil-drops. (After Roux.) Figs.
4–6 represent successive changes in a single system.]

We need by no means be surprised to find that, in such arrangements,
the commonest and most stable distributions are those in which the
cell-contacts are distributed as uniformly as possible between the
several cells. We always expect to find some such tendency to equality
in cases where we have to do with small oscillations on either side of
a symmetrical condition. {385}

The rules and principles which we have arrived at from the point of
view of surface tension have a much wider bearing than is at once
suggested by the problems to which we have applied them; for in this
elementary study of the cell-boundaries in a segmenting egg or tissue
we are on the verge of a difficult and important subject in pure
mathematics. It is a subject adumbrated by Leibniz, studied somewhat
more deeply by Euler, and greatly developed of recent years. It is the
_Geometria Situs_ of Gauss, the _Analysis Situs_ of Riemann, the Theory
of Partitions of Cayley, and of Spatial Complexes of Listing[398]. The
crucial point for the biologist to comprehend is, that in a closed
surface divided into a number of faces, the arrangement of all the
faces, lines and points in the system is capable of analysis, and that,
when the number of faces or areas is small, the number of possible
arrangements is small also. This is the simple reason why we meet in
such a case as we have been discussing (viz. the arrangement of a group
or system of eight cells) with the same few types recurring again and
again in all sorts of organisms, plants as well as animals, and with no
relation to the lines of biological classification: and why, further,
we find similar configurations occurring to mark the symmetry, not
of cells merely, but of the parts and organs of entire animals. The
phenomena are not “functions,” or specific characters, of this or that
tissue or organism, but involve general principles which lie within the
province of the mathematician.

――――――――――

The theory of space-partitioning, to which the segmentation of the egg
gives us an easy practical introduction, is illustrated in much more
complex ways in other fields of natural history. A very beautiful but
immensely complicated case is furnished by the “venation” of the wings
of insects. Here we have sometimes (as in the dragon-flies), a general
reticulum of small, more or less hexagonal “cells”: but in most other
cases, in flies, bees, butterflies, etc., we have a moderate number of
cells, whose partitions always impinge upon one another three by three,
and whose arrangement, therefore, includes of necessity a number of
small intermediate partitions, analogous to our polar furrows. I think
{386} that a mathematical study of these, including an investigation
of the “deformation” of the wing (that is to say, of the changes in
shape and changes in the form of its “cells” which it undergoes during
the life of the individual, and from one species to another) would be
of great interest. In very many cases, the entomologist relies upon
this venation, and upon the occurrence of this or that intermediate
vein, for his classification, and therefore for his hypothetical
phylogeny of particular groups; which latter procedure hardly commends
itself to the physicist or the mathematician.

[Illustration: Fig. 171. (A) _Asterolampra marylandica_, Ehr.; (B, C)
_A. variabilis_, Grev. (After Greville.)]

Another case, geometrically akin but biologically very different, is
to be found in the little diatoms of the genus Asterolampra, and their
immediate congeners[399]. In Asterolampra we have a little disc, in
which we see (as it were) radiating spokes of one material, alternating
with intervals occupied on the flattened wheel-like disc by another
(Fig. 171). The spokes vary in number, but the general appearance is
in a high degree suggestive of the Chladni figures produced by the
vibration of a circular plate. The spokes broaden out towards the
centre, and interlock by visible junctions, which obey the rule of
triple intersection, and accordingly exemplify the partition-figures
with which we are dealing. But whereas we have found the particular
arrangement in which one cell is in contact with all the rest to
be unstable, according to Roux’s oil-drop experiments, and to be
conspicuous {387} by its absence from our diagrams of segmenting
eggs, here in Asterolampra, on the other hand, it occurs frequently,
and is indeed the commonest arrangement[400] (Fig. 171, B). In all
probability, we are entitled to consider this marked difference natural
enough. For we may suppose that in Asterolampra (unlike the case of the
segmenting egg) the tendency is to perfect radial symmetry, all the
spokes emanating from a point in the centre: such a condition would be
eminently unstable, and would break down under the least asymmetry. A
very simple, perhaps the simplest case, would be that one single spoke
should differ slightly from the rest, and should so tend to be drawn in
amid the others, these latter remaining similar and symmetrical among
themselves. Such a configuration would be vastly less unstable than
the original one in which all the boundaries meet in a point; and the
fact that further progress is not made towards other configurations of
still greater stability may be sufficiently accounted for by viscosity,
rapid solidification, or other conditions of restraint. A perfectly
stable condition would of course be obtained if, as in the case of
Roux’s oil-drop (Fig. 170, 6), one of the cellular spaces passed into
the centre of the system, the other partitions radiating outwards from
its circular wall to the periphery of the whole system. Precisely
such a condition occurs among our diatoms; but when it does so, it is
looked upon as the mark and characterisation of the _allied genus_
Arachnoidiscus.

――――――――――

[Illustration: Fig. 172. Section of Alcyonarian polype.]

In a diagrammatic section of an Alcyonarian polype (Fig. 172), we have
eight chambers set, symmetrically, about a ninth, which constitutes
the “stomach.” In this arrangement there is no difficulty, for it is
obvious that, throughout the system, three boundaries meet (in plane
section) in a point. In many corals we have as {388} simple, or even
simpler conditions, for the radiating calcified partitions either
converge upon a central chamber, or fail to meet it and end freely.
But in a few cases, the partitions or “septa” converge to meet _one
another_, there being no central chamber on which they may impinge; and
here the manner in which contact is effected becomes complicated, and
involves problems identical with those which we are now studying.

[Illustration: Fig. 173. _Heterophyllia angulata_. (After Nicholson.)]

In the great majority of corals we have as simple or even simpler
conditions than those of Alcyonium; for as a rule the calcified
partitions or septa of the coral either converge upon a central chamber
(or central “columella”), or else fail to meet it and end freely. In
the latter case the problem of space-partitioning does not arise; in
the former, however numerous the septa be, their separate contacts
with the wall of the central chamber comply with our fundamental rule
according to which three lines and no more meet in a point, and from
this simple and symmetrical arrangement there is little tendency to
variation. But in a few cases, the septal partitions converge to
meet _one another_, there being no central chamber on which they may
impinge; and here the manner in which contact is effected becomes
complicated, and involves problems of space-partitioning identical
with those which we are now studying. In the genus Heterophyllia and
in a few allied forms we have such conditions, and students of the
Coelenterata have found them very puzzling. McCoy[401], their first
discoverer, pronounced these corals to be “totally unlike” any other
group, recent or fossil; and Professor Martin Duncan, writing a memoir
on Heterophyllia and its allies[402], described them as “paradoxical in
their anatomy.”

[Illustration: Fig. 174. _Heterophyllia_ sp. (After Martin Duncan.)]

The simplest or youngest Heterophylliae known have six septa (as in
Fig. 174, _a_); in the case figured, four of these septa are conjoined
two and two, thus forming the usual triple junctions together with
their intermediate partition-walls: and in the {389} case of the other
two we may fairly assume that their proper and original arrangement
was that of our type 6_b_ (Fig. 158), though the central intermediate
partition has been crowded out by partial coalescence. When with
increasing age the septa become more numerous, their arrangement
becomes exceedingly variable; for the simple reason that, from the
mathematical point of view, the number of possible arrangements, of 10,
12 or more cellular partitions in triple contact, tends to increase
with great rapidity, and there is little to choose between many of
them in regard to symmetry and equilibrium. But while, mathematically
speaking, each particular case among the multitude of possible cases
is an orderly and definite arrangement, from the purely biological
point of view on the other hand no law or order is recognisable; and so
McCoy described the genus as being characterised by the possession of
septa “destitute of any order of arrangement, but irregularly branching
and coalescing in their passage from the solid external walls towards
some indefinite point near the centre where the few main lamellae
irregularly anastomose.” {390}

In the two examples figured (Fig. 174), both comparatively simple ones,
it will be seen that, of the main chambers, one is in each case an
unsymmetrical one; that is to say, there is one chamber which is in
contact with a greater number of its neighbours than any other, and
which at an earlier stage must have had contact with them all; this was
the case of our type _f_, in the eight-celled system (Fig. 158). Such
an asymmetrical chamber (which may occur in a system of any number of
cells greater than six), constitutes what is known to students of the
Coelenterata as a “fossula”; and we may recognise it not only here,
but also in Zaphrentis and its allies, and in a good many other corals
besides. Moreover certain corals are described as having more than one
fossula: this appearance being naturally produced under certain of
the other asymmetrical variations of normal space-partitioning. Where
a single fossula occurs, we are usually told that it is a symptom of
“bilaterality”; and this is in turn interpreted as an indication of
a higher grade of organisation than is implied in the purely “radial
symmetry” of the commoner types of coral. The mathematical aspect of
the case gives no warrant for this interpretation.

Let us carefully notice (lest we run the risk of confusing two
distinct problems) that the space-partitioning of Heterophyllia by
no means agrees with the details of that which we have studied in
(for instance) the case of the developing disc of Erythrotrichia: the
difference simply being that Heterophyllia illustrates the general
case of cell-partitioning as Plateau and Van Rees studied it, while
in Erythrotrichia, and in our other embryological and histological
instances, we have found ourselves justified in making the additional
assumption that each new partition divided a cell into _co-equal
parts_. No such law holds in Heterophyllia, whose case is essentially
different from the others: inasmuch as the chambers whose partition
we are discussing in the coral are mere empty spaces (empty save
for the mere access of sea-water); while in our histological and
embryological instances, we were speaking of the division of a cellular
unit of living protoplasm. Accordingly, among other differences, the
“transverse” or “periclinal” partitions, which were bound to appear at
regular intervals and in definite positions, when co-equal bisection
was a feature of the {391} case, are comparatively few and irregular
in the earlier stages of Heterophyllia, though they begin to appear in
numbers after the main, more or less radial, partitions have become
numerous, and when accordingly these radiating partitions come to
bound narrow and almost parallel-sided interspaces; then it is that
the transverse or periclinal partitions begin to come in, and form
what the student of the Coelenterata calls the “dissepiments” of the
coral. We need go no further into the configuration and anatomy of the
corals; but it seems to me beyond a doubt that the whole question of
the complicated arrangement of septa and dissepiments throughout the
group (including the curious vesicular or bubble-like tissue of the
Cyathophyllidae and the general structural plan of the Tetracoralla,
such as Streptoplasma and its allies) is well worth investigation from
the physical and mathematical point of view, after the fashion which is
here slightly adumbrated.

[Illustration: Fig. 175. Diagrammatic section of a Ctenophore
(_Eucharis_).]

――――――――――

The method of dividing a circular, or spherical, system into eight
parts, equal as to their areas but unequal in their peripheral
boundaries, is probably of wide biological application; that is to say,
without necessarily supposing it to be rigorously followed, the typical
configuration which it yields seems to recur again and again, with
more or less approximation to precision, and under widely different
circumstances. I am inclined to think, for instance, that the unequal
division of the surface of a Ctenophore by its {392} meridian-like
ciliated bands is a case in point (Fig. 175). Here, if we imagine
each quadrant to be twice bisected by a curved anticline, we shall
get what is apparently a close approximation to the actual position
of the ciliated bands. The case however is complicated by the fact
that the sectional plan of the organism is never quite circular, but
always more or less elliptical. One point, at least, is clearly seen
in the symmetry of the Ctenophores; and that is that the radiating
canals which pass outwards to correspond in position with the ciliated
bands, have no common centre, but diverge from one another by repeated
bifurcations, in a manner comparable to the conjunctions of our
cell-walls.

In like manner I am inclined to suggest that the same principle may
help us to understand the apparently complex arrangement of the
skeletal rods of a larval Echinoderm, and the very complex conformation
of the larva which is brought about by the presence of these long,
slender skeletal radii.

[Illustration: Fig. 176. Diagrammatic arrangement of partitions,
represented by skeletal rods, in larval Echinoderm (_Ophiura_).]

In Fig. 176 I have divided a circle into its four quadrants, and have
bisected each quadrant by a circular arc (_BC_), passing from radius to
periphery, as in the foregoing cases of cell-division; and I have again
bisected, in a similar way, the triangular halves of each quadrant
(_DD_). I have also inserted a small circle in the middle of the
figure, concentric with the large one. If now we imagine those lines
in the figure which I have drawn black to be replaced by solid rods we
shall have at once the frame-work of an Ophiurid (Pluteus) larva. Let
us imagine all these arms to be {393} bent symmetrically downwards, so
that the plane of the paper is transformed into a spheroidal surface,
such as that of a hemisphere, or that of a tall conical figure with
curved sides; let a membrane be spread, umbrella-like, between the
outstretched skeletal rods, and let its margin loop from rod to rod in
curves which are possibly catenaries, but are more probably portions
of an “elastic curve,” and the outward resemblance to a Pluteus
larva is now complete. By various slight modifications, by altering
the relative lengths of the rods, by modifying their curvature or
by replacing the curved rod by a tangent to itself, we can ring the
changes which lead us from one known type of Pluteus to another. The
case of the Bipinnaria larvae of Echinids is certainly analogous,
but it becomes very much more complicated; we have to do with a more
complex partitioning of space, and I confess that I am not yet able to
represent the more complicated forms in so simple a way.

[Illustration: Fig. 177. Pluteus-larva of Ophiurid.]

――――――――――

[Illustration: Fig. 178. Diagrammatic development of Stomata in
_Sedum_. (Cf. fig. in Sachs’s _Botany_, 1882, p. 103.)]

There are a few notable exceptions (besides the various unequally
segmenting eggs) to the general rule that in cell-division the
mother-cell tends to divide into equal halves; and one of these
exceptional cases is to be found in connection with the development of
“stomata” in the leaves of plants. The epidermal cells by which the
leaf is covered may be of various shapes; sometimes, as in a hyacinth,
they are oblong, but more often they have an irregular shape in which
we can recognise, more or less clearly, a distorted or imperfect
hexagon. In the case of the oblong cells, a transverse partition
will be the least possible, whether the cell be equally or unequally
divided, unless (as we have already seen) {394} the space to be cut
off be a very small one, not more than about three-tenths the area of
a square based on the _short_ side of the original rectangular cell.
As the portion usually cut off is not nearly so small as this, we
get the form of partition shewn in Fig. 179, and the cell so cut off
is next bisected by a partition at right angles to the first; this
latter partition splits, and the two last-formed cells constitute the
so-called “guard-cells” of the

[Illustration: Fig. 179. Diagrammatic development of stomata in
Hyacinth.]

stoma. In other cases, as in Fig. 178, there will come a point where
the minimal partition necessary to cut off the required fraction of
the cell-content is no longer a transverse one, but is a portion of a
cylindrical wall (2) cutting off one corner of the mother-cell. The
cell so cut off is now a certain segment of a circle, with an arc of
approximately 120°; and its next division will be by means of a curved
wall cutting it into a triangular and a quadrangular portion (3). The
triangular portion will continue to divide in a similar way (4, 5),
and at length (for a reason which is not yet clear) the partition wall
{395} between the new-formed cells splits, and again we have the
phenomenon of a “stoma” with its attendant guard-cells. In Fig. 179 are
shewn the successive stages of division, and the changing curvatures
of the various walls which ensue as each subsequent partition appears,
introducing a new tension into the system.

It is obvious that in the case of the oblong cells of the epidermis in
the hyacinth the stomata will be found arranged in regular rows, while
they will be irregularly distributed over the surface of the leaf in
such a case as we have depicted in Sedum.

While, as I have said, the mechanical cause of the split which
constitutes the orifice of the stoma is not quite clear, yet there
can be little or no doubt that it, like the rest of the phenomenon,
is related to surface tension. It might well be that it is directly
due to the presence underneath this portion of epidermis of the hollow
air-space which the stoma is apparently developed “for the purpose”
of communicating with; this air-surface on both sides of the delicate
epidermis might well cause such an alteration of tensions that the
two halves of the dividing cell would tend to part company. In short,
if the surface-energy in a cell-air contact were half or less than
half that in a contact between cell and cell, then it is obvious that
our partition would tend to split, and give us a two-fold surface
in contact with air, instead of the original boundary or interface
between one cell and the other. In Professor Macallum’s experiments,
which we have briefly discussed in our short chapter on Adsorption, it
was found that large quantities of potassium gathered together along
the outer walls of the guard-cells of the stoma, thereby indicating
a low surface-tension along these outer walls. The tendency of the
guard-cells to bulge outwards is so far explained, and it is possible
that, under the existing conditions of restraint, we may have here a
force tending, or helping, to split the two cells asunder. It is clear
enough, however, that the last stage in the development of a stoma, is,
from the physical point of view, not yet properly understood.

――――――――――

[Illustration: Fig. 180. Various pollen-grains and spores (after
Berthold, Campbell, Goebel and others). (1) _Epilobium_; (2)
_Passiflora_; (3) _Neottia_; (4) _Periploca graeca_; (5) _Apocynum_;
(6) _Erica_; (7) Spore of _Osmunda_; (8) Tetraspore of _Callithamnion_.]

[Illustration: Fig. 181. Dividing spore of _Anthoceros_. (After
Campbell.)]

In all our foregoing examples of the development of a “tissue” we
have seen that the process consists in the _successive_ division of
cells, each act of division being accompanied by the formation {396}
of a boundary-surface, which, whether it become at once a solid or
semi-solid partition or whether it remain semi-fluid, exercises in all
cases an effect on the position and the form of the boundary which
comes into being with the next act of division. In contrast to this
general process stands the phenomenon known as “free cell-formation,”
in which, out of a common mass of protoplasm, a number of separate
cells are _simultaneously_, or all but simultaneously, differentiated.
In a number of cases it happens that, to begin with, a number of
“mother-cells” are formed simultaneously, and each of these divides,
by two successive divisions, into four “daughter-cells.” These
daughter-cells will tend to group themselves, just as would four
soap-bubbles, into a “tetrad,” the four cells corresponding to the
angles of a regular tetrahedron. For the system of four bodies is
evidently here in perfect symmetry; the partition-walls and their
respective edges meet at equal angles: three walls everywhere meeting
in an edge, and the four edges converging to a point in the geometrical
centre of the system. This is the typical mode of development of
pollen-grains, common among Monocotyledons and all but universal among
Dicotyledonous plants. By a loosening of the surrounding tissue and
an expansion of the cavity, or anther-cell, in which {397} they lie,
the pollen-grains afterwards fall apart, and their individual form
will depend upon whether or no their walls have solidified before
this liberation takes place. For if not, then the separate grains
will be free to assume a spherical form as a consequence of their
own individual and unrestricted growth; but if they become solid or
rigid prior to the separation of the tetrad, then they will conserve
more or less completely the plane interfaces and sharp angles of the
elements of the tetrahedron. The latter is the case, for instance, in
the pollen-grains of Epilobium (Fig. 180, 1) and in many others. In
the Passion-flower (2) we have an intermediate condition: where we
can still see an indication of the facets where the grains abutted on
one another in the tetrad, but the plane faces have been swollen by
growth into spheroidal or spherical surfaces. It is obvious that there
may easily be cases where the tetrads of daughter-cells are prevented
from assuming the tetrahedral form: cases, that is to say, where the
four cells are forced and crushed into one plane. The figures given by
Goebel of the development of the pollen of Neottia (3, _a_–_e_: all the
figures referring to grains taken from a single anther), illustrate
this to perfection; and it will be seen that, when the four cells lie
in a plane, they conform exactly to our typical diagram of the first
four cells in a segmenting ovum. Occasionally, though the four cells
lie in a plane, the diagram seems to fail us, for the cells appear to
meet in a simple cross (as in 5); but here we soon perceive that the
cells are not in complete interfacial contact, but are kept apart by a
little intervening drop of fluid or bubble of air. The spores of ferns
(7) develop in very much the same way as pollen-grains; and they also
very often retain traces of the shape which they assumed as members of
a tetrahedral figure. Among the “tetraspores” (8) of the Florideae, or
Red Seaweeds, we have a phenomenon which is in every respect analogous.

Here again it is obvious that, apart from differences in actual
magnitude, and apart from superficial or “accidental” differences
(referable to other physical phenomena) in the way of colour, {398}
texture and minute sculpture or pattern, it comes to pass, through the
laws of surface-tension and the principles of the geometry of position,
that a very small number of diagrammatic figures will sufficiently
represent the outward forms of all the tetraspores, four-celled
pollen-grains, and other four-celled aggregates which are known or are
even capable of existence.

――――――――――

We have been dealing hitherto (save for some slight exceptions) with
the partitioning of cells on the assumption that the system either
remains unaltered in size or else that growth has proceeded uniformly
in all directions. But we extend the scope of our enquiry very greatly
when we begin to deal with _unequal growth_, with growth, that is
to say, which produces a greater extension along some one axis than
another. And here we come close in touch with that great and still (as
I think) insufficiently appreciated generalisation of Sachs, that the
manner in which the cells divide is _the result_, and not the cause, of
the form of the dividing structure: that the form of the mass is caused
by its growth as a whole, and is not a resultant of the growth of the
cells individually considered[403]. Such asymmetry of growth may be
easily imagined, and may conceivably arise from a variety of causes.
In any individual cell, for instance, it may arise from molecular
asymmetry of the structure of the cell-wall, giving it greater rigidity
in one direction than another, while all the while the hydrostatic
pressure within the cell remains constant and uniform. In an aggregate
of cells, it may very well arise from a greater chemical, or osmotic,
activity in one than another, leading to a localised increase in the
fluid pressure, and to a corresponding bulge over a certain area of
the external surface. It might conceivably occur as a direct result
of the preceding cell-divisions, when these are such as to produce
many peripheral or concentric walls in one part and few or none in
another, with the obvious result of strengthening the common boundary
wall and resisting the outward pressure of growth in parts where the
former is the case; that is to say, in our dividing quadrant, if {399}
its quadrangular portion subdivide by periclines, and the triangular
portion by oblique anticlines (as we have seen to be the natural
tendency), then we might expect that external growth would be more
manifest over the latter than over the former areas. As a direct and
immediate consequence of this we might expect a tendency for special
outgrowths, or “buds,” to arise from the triangular rather than from
the quadrangular cells; and this turns out to be not merely a tendency
towards which theoretical considerations point, but a widespread and
important factor in the morphology of the cryptogams. But meanwhile,
without enquiring further into this complicated question, let us simply
take it that, if we start from such a simple case as a round cell which
has divided into two halves, or four quarters (as the case may be),
we shall at once get bilateral symmetry about a main axis, and other
secondary results arising therefrom, as soon as one of the halves, or
one of the quarters, begins to shew a rate of growth in advance of
the others; for the more rapidly growing cell, or the peripheral wall
common to two or more such rapidly growing cells, will bulge out into
an ellipsoid form, and may finally extend into a cylinder with rounded
or ellipsoid end.

This latter very simple case is illustrated in the development of a
pollen-tube, where the rapidly growing cell develops into the elongated
cylindrical tube, and the slow-growing or quiescent part remains behind
as the so-called “vegetative” cell or cells.

Just as we have found it easier to study the segmentation of a circular
disc than that of a spherical cell, so let us begin in the same way, by
enquiring into the divisions which will ensue if the disc tend to grow,
or elongate, in some one particular direction, instead of in radial
symmetry. The figures which we shall then obtain will not only apply
to the disc, but will also represent, in all essential features, a
projection or longitudinal section of a solid body, spherical to begin
with, preserving its symmetry as a solid of revolution, and subject to
the same general laws as we have studied in the disc[404]. {400}

(1) Suppose, in the first place, that the axis of growth lies
symmetrically in one of the original quadrantal cells of a segmenting
disc; and let this growing cell elongate with comparative rapidity
before it subdivides. When it does divide, it will necessarily do so by
a transverse partition, concave towards the apex of the cell: and, as
further elongation takes place, the cylindrical structure which will be
developed thereby will tend to be again and again subdivided by similar
concave transverse partitions. If at any time, through this process
of concurrent elongation and subdivision, the apical cell become
equivalent to, or less than, a hemisphere, it will next divide by means
of a longitudinal, or vertical partition; and similar longitudinal
partitions will arise in the other segments of the cylinder, as soon as
it comes about that their length (in the direction of the axis) is less
than their breadth.

[Illustration: Fig. 182.]

But when we think of this structure in the solid, we at once perceive
that each of these flattened segments of the cylinder, into which our
cylinder has divided, is equivalent to a flattened circular disc;
and its further division will accordingly tend to proceed like any
other flattened disc, namely into four quadrants, and afterwards by
anticlines and periclines in the usual way. {401} A section across the
cylinder, then, will tend to shew us precisely the same arrangements
as we have already so fully studied in connection with the typical
division of a circular cell into quadrants, and of these quadrants into
triangular and quadrangular portions, and so on.

But there are other possibilities to be considered, in regard to the
mode of division of the elongating quasi-cylindrical portion, as it
gradually develops out of the growing and bulging quadrantal cell; for
the manner in which this latter cell divides will simply depend upon
the form it has assumed before each successive act of division takes
place, that is to say upon the ratio between its rate of growth and
the frequency of its successive divisions. For, as we have already
seen, if the growing cell attain a markedly oblong or cylindrical form
before division ensues, then the partition will arise transversely to
the long axis; if it be but a little more than a hemisphere, it will
divide by an oblique partition; and if it be less than a hemisphere
(as it may come to be after successive transverse divisions) it will
divide by a vertical partition, that is to say by one coinciding with
its axis of growth. An immense number of permutations and combinations
may arise in this way, and we must confine our illustrations to a small
number of cases. The important thing is not so much to trace out the
various conformations which may arise, but to grasp the fundamental
principle: which is, that the forces which dominate the _form_ of each
cell regulate the manner of its subdivision, that is to say the form of
the new cells into which it subdivides; or in other words, the form of
the growing organism regulates the form and number of the cells which
eventually constitute it. The complex cell-network is not the cause but
the result of the general configuration, which latter has its essential
cause in whatsoever physical and chemical processes have led to a
varying velocity of growth in one direction as compared with another.

[Illustration: Fig. 183. Development of _Sphagnum_. (After Campbell.)]

In the annexed figure of an embryo of Sphagnum we see a mode of
development almost precisely corresponding to the hypothetical case
which we have just described,—the case, that is to say, where one of
the four original quadrants of the mother-cell is the chief agent in
future growth and development. We see at the base of our first figure
(_a_), the three stationary, or {402} undivided quadrants, one of
which has further slowly divided in the stage _b_. The active quadrant
has grown quickly into a cylindrical structure, which inevitably
divides, in the next place, into a series of transverse partitions; and
accordingly, this mode of development carries with it the presence of a
single “apical cell,” whose lower wall is a spherical surface with its
convexity downwards. Each cell of the subdivided cylinder now appears
as a more or less flattened disc, whose mode of further sub-division
we may prognosticate according to our former investigation, to which
subject we shall presently return.

[Illustration: Fig. 184.]

(2) In the next place, still keeping to the case where only one of the
original quadrant-cells continues to grow and develop, let us suppose
that this growing cell falls to be divided when by growth it has
become just a little greater than a hemisphere; it will then divide,
as in Fig. 184, 2, by an oblique partition, in the usual way, whose
precise position and inclination to the base will depend entirely on
the configuration of the cell itself, save only, of course, that we
may have also to take into account the possibility of the division
being into two unequal halves. By our hypothesis, {403} the growth
of the whole system is mainly in a vertical direction, which is as
much as to say that the more actively growing protoplasm, or at least
the strongest osmotic force, will be found near the apex; where
indeed there is obviously more external surface for osmotic action.
It will therefore be that one of the two cells which contains, or
constitutes, the apex which will grow more rapidly than the other,
and which therefore will be the first to divide, and indeed in any
case, it will usually be this one of the two which will tend to
divide first, inasmuch as the triangular and not the quadrangular
half is bound to constitute the apex[405]. It is obvious that (unless
the act of division be so long postponed that the cell has become
quasi-cylindrical) it will divide by another oblique partition,
starting from, and running at right angles to, the first. And so
division will proceed,

[Illustration: Fig. 185. Gemma of Moss. (After Campbell.)]

by oblique alternate partitions, each one tending to be, at first,
perpendicular to that on which it is based and also to the peripheral
wall; but all these points of contact soon tending, by reason of the
equal tensions of the three films or surfaces which meet there, to
form angles of 120°. There will always be, in such a case, a single
apical cell, of a more or less distinctly triangular form. The annexed
figure of the developing antheridium of a Liverwort (Riccia) is a
typical example of such a case. In Fig. 185 which represents a “gemma”
of a Moss, we see just the same thing; with this addition, that here
the lower of the two original cells has grown even more quickly than
the other, constituting a long cylindrical stalk, and dividing in
accordance with its shape, by means of transverse septa.

In all such cases as these, the cells whose development we have studied
will in turn tend to subdivide, and the manner in which they will do so
must depend upon their own proportions; and in all cases, as we have
already seen, there will sooner or later be a tendency to the formation
of periclinal walls, cutting off an “epidermal layer of cells,” as Fig.
186 illustrates very well.

[Illustration: Fig. 186. Development of antheridium of _Riccia_. (After
Campbell.)]

[Illustration: Fig. 187. Section of growing shoot of Selaginella,
diagrammatic.]

[Illustration: Fig. 188. Embryo of Jungermannia. (After
Kienitz-Gerloff.)]

The method of division by means of oblique partitions is a common one
in the case of ‘growing points’; for it evidently {404} includes all
cases in which the act of cell-division does not lag far behind that
elongation which is determined by the specific rate of growth. And it
is also obvious that, under a common type, there must here be included
a variety of cases which will, at first sight, present a very different
appearance one from another. For instance, in Fig. 187 which represents
a growing shoot of Selaginella, and somewhat less diagrammatically in
the young embryo of Jungermannia (Fig. 188), we have the appearance of
an almost straight vertical partition running up in the axis of the
system, and the primary cell-walls are set almost at right angles to
it,—almost transversely, that is to say to the outer walls and to the
long axis of the structure. We soon recognise, however, {405} that
the difference is merely a difference of degree. The more remote the
partitions are, that is to say the greater the velocity of growth
relatively to division, the less abrupt will be the alternate kinks or
curvatures of the portions which lie in the neighbourhood of the axis,
and the more will these portions appear to constitute a single unbroken
wall.

[Illustration: Fig. 189.]

(3) But an appearance nearly, if not quite, indistinguishable from
this may be got in another way, namely, when the original growing cell
is so nearly hemispherical that it is actually divided by a vertical
partition, into two quadrants; and from this vertical partition, as it
elongates, lateral partition-walls will arise on either side. And by
the tensions exercised by these, the vertical partition will be bent
into little portions set at 120° one to another, and the whole will
come to look just like that which, in the former case, was made up of
portions of many successive oblique partitions.

――――――――――

Let us now, in one or two cases, follow out a little further the
stages of cell-division whose beginning we have studied in the last
paragraphs. In the antheridium of Riccia, after the successive oblique
partitions have produced the longitudinal series of cells shewn in Fig.
186, it is plain that the next partitions will arise periclinally, that
is to say parallel to the outer wall, which in this particular case
represents the short axis of the oblong cells. The effect is at once to
produce an epidermal layer, whose cells will tend to subdivide further
by means of partitions perpendicular to the free surface, that is to
say crossing the flattened cells by their shortest diameter. The inner
mass, beneath the epidermis, consists of cells which are still more or
less oblong, or which become {406} definitely so in process of growth;
and these again divide, parallel to their short axes, into squarish
cells, which as usual, by the mutual tension of their walls, become
hexagonal, as seen in a plane section. There is a clear distinction,
then, in form as well as in position, between the outer covering-cells
and those which lie within this envelope; the latter are reduced to a
condition which merely fulfils the mechanical function of a protective
coat, while the former undergo less modification, and give rise to the
actively living, reproductive elements.

[Illustration: Fig. 190. Development of sporangium of _Osmunda_. (After
Bower.)]

In Fig. 190 is shewn the development of the sporangium of a fern
(Osmunda). We may trace here the common phenomenon of a series of
oblique partitions, built alternately on one another, and cutting off a
conspicuous triangular apical cell. Over the whole system an epidermal
layer has been formed, in the manner we have described; and in this
case it covers the apical cell also, owing to the fact that it was of
such dimensions that, at one stage of growth, a periclinal partition
wall, cutting off its outer end, was indicated as of less area than
an anticlinal one. This periclinal wall cuts down the apical cell to
the proportions, very nearly, of an equilateral triangle, but the
solid form of the cell is obviously that of a tetrahedron with curved
faces; and accordingly, the least possible partitions by which further
subdivision can be effected will run successively parallel to its four
sides (or its three sides when we confine ourselves to the appearances
as seen in {407} section). The effect, as seen in section, is to
cut off on each side a characteristically flattened cell, oblong as
seen in section, still leaving a triangular (or strictly speaking, a
tetrahedral) one in the centre. The former cells, which constitute no
specific structure or perform no specific physiological function, but
which merely represent certain directions in space towards which the
whole system of partitioning has gradually led, are called by botanists
the “tapetum.” The active growing tetrahedral cell which lies between
them, and from which in a sense every other cell in the system has
been either directly or indirectly segmented off, still manifests, as
it were, its vigour and activity, and now, by internal subdivision,
becomes the mother-cell of the spores.

――――――――――

In all these cases, for simplicity’s sake, we have merely considered
the appearances presented in a single, longitudinal, plane of optical
section. But it is not difficult to interpret from these appearances
what would be seen in another plane, for instance in a transverse
section. In our first example, for instance, that of the developing
embryo of Sphagnum (Fig. 183), we can see that, at appropriate levels,
the cells of the original cylindrical row have divided into transverse
rows of four, and then of eight cells. We may be sure that the four
cells represent, approximately, quadrants of a cylindrical disc, the
four cells, as usual, not meeting in a point, but intercepted by a
small intermediate partition. Again, where we have a plate of eight
cells, we may well imagine that the eight octants are arranged in what
we have found to be the way naturally resulting from the division
of four quadrants, that is to say into alternately triangular and
quadrangular portions; and this is found by means of sections to be
the case. The accompanying figure is precisely comparable to our
previous diagrams of the arrangement of an aggregate of eight cells in
a dividing disc, save only that, in two cases, the cells have already
undergone a further subdivision.

[Illustration: Fig. 191. (A, B,) Sections of younger and older embryos
of _Phascum_; (C) do. of _Adiantum_. (After Kienitz-Gerloff.)]

[Illustration: Fig. 192. Section through frond of _Girardia
sphacelaria_. (After Goebel.)]

It follows in like manner, that in a host of cases we meet with this
characteristic figure, in one or other of its possible, and strictly
limited, variations,—in the cross sections of growing embryonic
structures, just as we have already seen that it appears in a host of
cases where the entire system (or a portion of its {408} surface)
consists of eight cells only. For example, in Fig. 191, we have it
again, in a section of a young embryo of a moss (Phascum), and in
a section of an embryo of a fern (Adiantum). In Fig. 192 shewing a
section through a growing frond of a sea-weed (Girardia) we have a
case where the partitions forming the eight octants have conformed to
the usual type; but instead of the usual division by periclines of the
four quadrangular spaces, these latter are dividing by means of oblique
septa, apparently owing to the fact that the cell is not dividing into
two equal, but into two unequal portions. In this last figure we have
a peculiar look of stiffness or formality, such that it appears at
first to bear little resemblance to the rest. The explanation is of
the simplest. The mode of partitioning differs little (except to some
slight extent in the way already mentioned) from the normal type; but
in this case the partition walls are so thick and become so quickly
comparatively solid and rigid, that the secondary curvatures due to
their successive mutual tractions are here imperceptible.

[Illustration: Fig. 193. Development of antheridium of _Pteris_. (After
Strasbürger.)]

A curious and beautiful case, apparently aberrant but which would
doubtless be found conforming strictly to physical laws, if {409}
only we clearly understood the actual conditions, is indicated
in the development of the antheridium of a fern, as described by
Strasbürger. Here the antheridium develops from a single cell, whose
form has grown to be something more than a hemisphere; and the first
partition, instead of stretching transversely across the cell, as
we should expect it to do if the cell were actually spherical, has
as it were sagged down to come in contact with the base, and so to
develop into an annular partition, running round the lower margin of
the cell. The phenomenon is akin to that cutting off of the corner
of a cubical cell by a spherical partition, of which we have spoken
on p. 349, and the annular film is very easy to reproduce by means
of a soap-bubble in the bottom of a cylindrical dish or beaker. The
next partition is a periclinal one, concentric with the outer surface
of the young antheridium; and this in turn is followed by a concave
partition which cuts off the apex of the original cell: but which
becomes connected with the second, or periclinal partition in precisely
the same annular fashion as the first partition did with the base of
the little antheridium. The result is that, at this stage, we have
four cell-cavities in the little antheridium: (1) a central cavity;
(2) an annular space around the lower margin; (3) a narrow annular or
cylindrical space around the sides of the antheridium; and (4) a small
terminal or apical cell. It is evident that the tendency, in the next
place, will be to subdivide the flattened external cells by means of
anticlinal partitions, and so to convert the whole structure into a
single layer of epidermal cells, surrounding a central cell within
which, in course of time, the antherozoids are developed.

――――――――――

The foregoing account deals only with a few elementary phenomena,
and may seem to fall far short of an attempt to deal in general
with “the forms of tissues.” But it is the principle involved, and
not its ultimate and very complex results, that we can alone {410}
attempt to grapple with. The stock-in-trade of mathematical physics,
in all the subjects with which that science deals, is for the most
part made up of simple, or simplified, cases of phenomena which in
their actual and concrete manifestations are usually too complex
for mathematical analysis; and when we attempt to apply its methods
to our biological and histological phenomena, in a preliminary and
elementary way, we need not wonder if we be limited to illustrations
which are obviously of a simple kind, and which cover but a small part
of the phenomena with which the histologist has become familiar. But
it is only relatively that these phenomena to which we have found the
method applicable are to be deemed simple and few. They go already far
beyond the simplest phenomena of all, such as we see in the dividing
Protococcus, and in the first stages, two-celled or four-celled, of the
segmenting egg. They carry us into stages where the cells are already
numerous, and where the whole conformation has become by no means
easy to depict or visualise, without the help and guidance which the
phenomena of surface-tension, the laws of equilibrium and the principle
of minimal areas are at hand to supply. And so far as we have gone,
and so far as we can discern, we see no sign of the guiding principles
failing us, or of the simple laws ceasing to hold good.

{411}




CHAPTER IX

ON CONCRETIONS, SPICULES, AND SPICULAR SKELETONS


The deposition of inorganic material in the living body, usually in the
form of calcium salts or of silica, is a very common and wide-spread
phenomenon. It begins in simple ways, by the appearance of small
isolated particles, crystalline or non-crystalline, whose form has
little relation or sometimes none to the structure of the organism; it
culminates in the complex skeletons of the vertebrate animals, in the
massive skeletons of the corals, or in the polished, sculptured and
mathematically regular molluscan shells. Even among many very simple
organisms, such as the Diatoms, the Radiolarians, the Foraminifera,
or the Sponges, the skeleton displays extraordinary variety and
beauty, whether by reason of the intrinsic form of its elementary
constituents or the geometric symmetry with which these are arranged
and interconnected.

With regard to the form of these various structures (and this is
all that immediately concerns us here), it is plain that we have to
do with two distinct problems, which however, though theoretically
distinct, may merge with one another. For the form of the spicule or
other skeletal element may depend simply upon its chemical nature, as
for instance, to take a simple but not the only case, when the form is
purely crystalline; or the inorganic solid material may be laid down
in conformity with the shapes assumed by the cells, tissues or organs,
and so be, as it were, moulded to the shape of the living organism; and
again, there may well be intermediate stages in which both phenomena
may be simultaneously recognised, the molecular forces playing their
part in conjunction with, and under the restraint of, the other forces
inherent in the system. {412}

So far as the problem is a purely chemical one, we must deal with it
very briefly indeed; and all the more because special investigations
regarding it have as yet been few, and even the main facts of the case
are very imperfectly known. This at least is evident, that the whole
series of phenomena with which we are about to deal go deep into the
subject of colloid chemistry, and especially with that branch of the
science which deals with the properties of colloids in connection with
capillary or surface phenomena. It is to the special student of colloid
chemistry that we must ultimately and chiefly look for the elucidation
of our problem[406].

In the first and simplest part of our subject, the essential problem
is the problem of crystallisation in presence of colloids. In the
cells of plants, true crystals are found in comparative abundance,
and they consist, in the great majority of cases, of calcium oxalate.
In the stem and root of the rhubarb, for instance, in the leaf-stalk
of Begonia, and in countless other cases, sometimes within the cell,
sometimes in the substance of the cell-wall, we find large and
well-formed crystals of this salt; their varieties of form, which are
extremely numerous, are simply the crystalline forms proper to the salt
itself, and belong to the two systems, cubic and monoclinic, in one or
other of which, according to the amount of water of crystallisation,
this salt is known to crystallise. When calcium oxalate crystallises
according to the latter system (as it does when its molecule is
combined with two molecules of water of crystallisation), the
microscopic crystals have the form of fine needles, or “raphides,” such
as are very common in plants; and it has been found that these are
artificially produced when the salt is crystallised out in presence of
glucose or of dextrin[407].

[Illustration: Fig. 194. Alcyonarian spicules: _Siphonogorgia_ and
_Anthogorgia_. (After Studer.)]

Calcium carbonate, on the other hand, when it occurs in plant-cells (as
it does abundantly, for instance in the “cystoliths” of the Urticaceae
and Acanthaceae, and in great quantities in Melobesia {413} and the
other calcareous or “stony” algae), appears in the form of fine rounded
granules, whose inherent crystalline structure is not outwardly
visible, but is only revealed (like that of a molluscan shell) under
polarised light. Among animals, a skeleton of carbonate of lime occurs
under a multitude of forms, of which we need only mention now a very
few of the most conspicuous. The spicules of the calcareous sponges
are triradiate, occasionally quadriradiate, bodies, with pointed rays,
not crystalline in outward form but with a definitely crystalline
internal structure. We shall return again to these, and find for them
what would seem to be a satisfactory explanation of their form. Among
the Alcyonarian zoophytes we have a great variety of spicules[408],
which are sometimes straight and slender rods, sometimes flattened and
more or less striated plates, and still more often rounded or branched
concretions with rough or knobby surfaces (Figs. 194, 200). A third
type, presented by several very different things, such as a pearl, or
the ear-bone of a bony fish, consists of a more or less {414} rounded
body, sometimes spherical, sometimes flattened, in which the calcareous
matter is laid down in concentric zones, denser and clearer layers
alternating with one another. In the development of the molluscan shell
and in the calcification of a bird’s egg or the shell of a crab, for
instance, spheroidal bodies with similar concentric striation make
their appearance; but instead of remaining separate they become crowded
together, and as they coalesce they combine to form a pattern of
hexagons. In some cases, the carbonate of lime on being dissolved away
by acid leaves behind it a certain small amount of organic residue; in
most cases other salts, such as phosphates of lime, ammonia or magnesia
are present in small quantities; and in most cases if not all the
developing spicule or concretion is somehow or other so associated with
living cells that we are apt to take it for granted that it owes its
peculiarities of form to the constructive or plastic agency of these.

The appearance of direct association with living cells, however, is
apt to be fallacious; for the actual _precipitation_ takes place,
as a rule, not in actively living, but in dead or at least inactive
tissue[409]: that is to say in the “formed material” or matrix which
(as for instance in cartilage) accumulates round the living cells, in
the interspaces between these latter, or at least, as often happens,
in connection with the cell-wall or cell-membrane rather than within
the substance of the protoplasm itself. We need not go the length of
asserting that this is a rule without exception; but, so far as it
goes, it is of great importance and to its consideration we shall
presently return[410].

Cognate with this is the fact that it is known, at least in some
cases, that the organism can go on living and multiplying with
apparently unimpaired health, when stinted or even wholly deprived
of the material of which it is wont to make its spicules {415} or
its shell. Thus, Pouchet and Chabry[411] have shown that the eggs of
sea-urchins reared in lime-free water develop in apparent health, into
larvae entirely destitute of the usual skeleton of calcareous rods,
and in which, accordingly, the long arms of the Pluteus larva, which
the rods support and distend, are entirely suppressed. And again,
when Foraminifera are kept for generations in water from which they
gradually exhaust the lime, their shells grow hyaline and transparent,
and seem to consist only of chitinous material. On the other hand,
in the presence of excess of lime, the shells become much altered,
strengthened with various “ornaments,” and assuming characters
described as proper to other varieties and even species[412].

The crucial experiment, then, is to attempt the formation of similar
structures or forms, apart from the living organism: but, however
feasible the attempt may be in theory, we shall be prepared from the
first to encounter difficulties, and to realise that, though the
actions involved may be wholly within the range of chemistry and
physics, yet the actual conditions of the case may be so complex,
subtle and delicate, that only now and then, and in the simplest of
cases, shall we find ourselves in a position to imitate them completely
and successfully. Such an investigation is only part of that much
wider field of enquiry through which Stephane Leduc and many other
workers[413] have sought to produce, by synthetic means, forms similar
to those of living things; but it is a well-defined and circumscribed
part of that wider investigation. When by chemical or physical
experiment we obtain configurations similar, for instance, to the
phenomena of nuclear division, or conformations similar to a pattern of
hexagonal cells, or a group of vesicles which resemble some particular
tissue or cell-aggregate, we indeed prove what it is the main object of
this book to illustrate, namely, that the physical forces are capable
of producing particular organic forms. But it is by no means always
that we can feel perfectly assured that the physical forces which
we deal with in our experiment are identical with, and not merely
analogous to, {416} the physical forces which, at work in nature, are
bringing about the result which we have succeeded in imitating. In
the present case, however, our enquiry is restricted and apparently
simplified; we are seeking in the first instance to obtain by purely
chemical means a purely chemical result, and there is little room for
ambiguity in our interpretation of the experiment.

――――――――――

When we find ourselves investigating the forms assumed by chemical
compounds under the peculiar circumstances of association with a
living body, and when we find these forms to be characteristic or
recognisable, and somehow different from those which, under other
circumstances, the same substance is wont to assume, an analogy
presents itself to our minds, captivating though perhaps somewhat
remote, between this subject of ours and certain synthetic problems of
the organic chemist. There is doubtless an essential difference, as
well as a difference of scale, between the visible form of a spicule
or concretion and the hypothetical form of an individual molecule;
but molecular form is a very important concept; and the chemist has
not only succeeded, since the days of Wöhler, in synthesising many
substances which are characteristically associated with living matter,
but his task has included the attempt to account for the molecular
_forms_ of certain “asymmetric” substances, glucose, malic acid and
many more, as they occur in nature. These are bodies which, when
artificially synthesised, have no optical activity, but which, as we
actually find them in organisms, turn (when _in solution_) the plane
of polarised light in one direction or the other; thus dextro-glucose
and laevomalic acid are common products of plant metabolism; but
dextromalic acid and laevo-glucose do not occur in nature at all. The
optical activity of these bodies depends, as Pasteur shewed more than
fifty years ago[414], upon the form, right-handed or left-handed,
of their molecules, which molecular asymmetry further gives rise to
a corresponding right or left-handedness (or enantiomorphism) in
the crystalline aggregates. It is a distinct problem in organic or
physiological chemistry, {417} and by no means without its interest
for the morphologist, to discover how it is that nature, for each
particular substance, habitually builds up, or at least selects, its
molecules in a one-sided fashion, right-handed or left-handed as the
case may be. It will serve us no better to assert that this phenomenon
has its origin in “fortuity,” than to repeat the Abbé Galiani’s saying,
“_les dés de la nature sont pipés._”

The problem is not so closely related to our immediate subject that
we need discuss it at length; but at the same time it has its clear
relation to the general question of _form_ in relation to vital
phenomena, and moreover it has acquired interest as a theme of
long-continued discussion and new importance from some comparatively
recent discoveries.

According to Pasteur, there lay in the molecular asymmetry of the
natural bodies and the symmetry of the artificial products, one of the
most deep-seated differences between vital and non-vital phenomena:
he went further, and declared that “this was perhaps the _only_
well-marked line of demarcation that can at present [1860] be drawn
between the chemistry of dead and of living matter.” Nearly forty
years afterwards the same theme was pursued and elaborated by Japp in
a celebrated lecture[415], and the distinction still has its weight, I
believe, in the minds of many if not most chemists.

“We arrive at the conclusion,” said Professor Japp, “that the
production of single asymmetric compounds, or their isolation from
the mixture of their enantiomorphs, is, as Pasteur firmly held,
the prerogative of life. Only the living organism, or the living
intelligence with its conception of asymmetry, can produce this
result. Only asymmetry can beget asymmetry.” In these last words
(which, so far as the chemist and the biologist are concerned, we
may acknowledge to be perfectly true[416]) lies the {418} crux of
the difficulty; for they at once bid us enquire whether in nature,
external to and antecedent to life, there be not some asymmetry to
which we may refer the further propagation or “begetting” of the new
asymmetries: or whether in default thereof, we be rigorously confined
to the conclusion, from which Japp “saw no escape,” that “at the moment
when life first arose, a directive force came into play,—a force of
precisely the same character as that which enables the intelligent
operator, by the exercise of his will, to select one crystallised
enantiomorph and reject its asymmetric opposite[417].”

Observe that it is only the first beginnings of chemical asymmetry
that we need to discover; for when asymmetry is once manifested, it is
not disputed that it will continue “to beget asymmetry.” A plausible
suggestion is now at hand, which if it be confirmed and extended will
supply or at least sufficiently illustrate the kind of explanation
which is required[418].

We know in the first place that in cases where ordinary non-polarised
light acts upon a chemical substance, the amount of chemical action
is proportionate to the amount of light absorbed. We know in the
second place[419], in certain cases, that light circularly polarised
is absorbed in different amounts by the right-handed or left-handed
varieties, as the case may be, of an asymmetric substance. And thirdly,
we know that a portion of the light which comes to us from the sun
is already plane-polarised light, which becomes in part circularly
polarised, by reflection (according to Jamin) at the surface of the
sea, and then rotated in a particular direction under the influence of
terrestrial magnetism. We only require to be assured that the relation
between absorption of light and chemical activity will continue to hold
good in the case of circularly polarised light; that is to say {419}
that the formation of some new substance or other, under the influence
of light so polarised, will proceed asymmetrically in consonance
with the asymmetry of the light itself; or conversely, that the
asymmetrically polarised light will tend to more rapid decomposition
of those molecules by which it is chiefly absorbed. This latter proof
is now said to be furnished by Byk[420], who asserts that certain
tartrates become unsymmetrical under the continued influence of the
asymmetric rays. Here then we seem to have an example, of a particular
kind and in a particular instance, an example limited but yet crucial
(_if confirmed_), of an asymmetric force, non-vital in its origin,
which might conceivably be the starting-point of that asymmetry which
is characteristic of so many organic products.

The mysteries of organic chemistry are great, and the differences
between its processes or reactions as they are carried out in the
organism and in the laboratory are many[421]. The actions, catalytic
and other, which go on in the living cell are of extraordinary
complexity. But the contention that they are different in kind from
what we term ordinary chemical operations, or that in the production
of single asymmetric compounds there is actually to be witnessed, as
Pasteur maintained, a “prerogative of life,” would seem to be no longer
safely tenable. And furthermore, it behoves us to remember that, even
though failure continued to attend all artificial attempts to originate
the asymmetric or optically active compounds which organic nature
produces in abundance, this would only prove that a certain _physical
force_, or mode of _physical action_, is at work among living things
though unknown elsewhere. It is a mode of action which we can easily
imagine, though the actual mechanism we cannot set agoing when we
please. And it follows that such a difference between living matter and
dead would carry us but a little way, for it would still be confined
strictly to the physical or mechanical plane.

Our historic interest in the whole question is increased by the
{420} fact, or the great probability, that “the tenacity with which
Pasteur fought against the doctrine of spontaneous generation was
not unconnected with his belief that chemical compounds of one-sided
symmetry could not arise save under the influence of life[422].” But
the question whether spontaneous generation be a fact or not does
not depend upon theoretical considerations; our negative response is
based, and is so far soundly based, on repeated failures to demonstrate
its occurrence. Many a great law of physical science, not excepting
gravitation itself, has no higher claim on our acceptance.

――――――――――

Let us return then, after this digression, to the general subject
of the forms assumed by certain chemical bodies when deposited or
precipitated within the organism, and to the question of how far these
forms may be artificially imitated or theoretically explained.

Mr George Rainey, of St Bartholomew’s Hospital (to whom we have already
referred), and Professor P. Harting, of Utrecht, were the first to
deal with this specific problem. Mr Rainey published, between 1857 and
1861, a series of valuable and thoughtful papers to shew that shell and
bone and certain other organic structures were formed “by a process
of molecular coalescence, demonstrable in certain artificially-formed
products[423].” Professor Harting, after thirty years of experimental
work, published in 1872 a paper, which has become classical, entitled
_Recherches de Morphologie Synthétique, sur la production artificielle
de quelques formations calcaires organiques_; his aim was to pave the
way for a “morphologie synthétique,” as Wöhler had laid the foundations
of a “chimie synthétique,” by his classical discovery forty years
before. {421}

[Illustration: Fig. 195. Calcospherites, or concretions of calcium
carbonate, deposited in white of egg. (After Harting.)]

[Illustration: Fig. 196. A single calcospherite, with central
“nucleus,” and striated, iridescent border. (After Harting.)]

[Illustration: Fig. 197. Later stages in the same experiment.]

[Illustration: Fig. 198, A. Section of shell of Mya; B. Section of
hinge-tooth of do. (After Carpenter.)]

Rainey and Harting used similar methods, and these were such as many
other workers have continued to employ,—partly with the direct object
of explaining the genesis of organic forms and partly as an integral
part of what is now known as Colloid Chemistry. The whole gist of the
method was to bring some soluble salt of lime, such as the chloride
or nitrate, into solution within a colloid medium, such as gum,
gelatine or albumin; and then to precipitate it out in the form of
some insoluble compound, such as the carbonate or oxalate. Harting
found that, when he added a little sodium or potassium carbonate to a
concentrated solution of calcium chloride in albumin, he got at first
a gelatinous mass, or “colloid precipitate”: which slowly transformed
by the appearance of tiny microscopic particles, at first motionless,
but afterwards as they grew larger shewing the typical Brownian
movement. So far, very much the same phenomena were witnessed whether
the solution were albuminous or not, and similar appearances indeed had
been witnessed and recorded by Gustav Rose, so far back as 1837[424];
but in the later stages the presence of albuminoid matter made a great
difference. Now, after a few days, the calcium carbonate was seen to
be deposited in the form of large rounded concretions, with a more or
less distinct central nucleus, and with a surrounding structure at once
radiate and {422} concentric; the presence of concentric zones or
lamellae, alternately dark and clear, was especially characteristic.
These round “calcospherites” shewed a tendency to aggregate together in
layers, and then to assume polyhedral, or often regularly hexagonal,
outlines. In this latter condition they closely resemble the early
stages of calcification in a molluscan (Fig. 198), or still more in a
crustacean shell[425]; while in their isolated condition {423} they
very closely resemble the little calcareous bodies in the tissues of
a trematode or a cestode worm, or in the oesophageal glands of an
earthworm[426].

[Illustration: Fig. 199. Large irregular calcareous concretions, or
spicules, deposited in a piece of dead cartilage, in presence of
calcium phosphate. (After Harting.)]

When the albumin was somewhat scanty, or when it was mixed with
gelatine, and especially when a little phosphate of lime was {424}
added to the mixture, the spheroidal globules tended to become rough,
by an outgrowth of spinous or digitiform projections; and in some
cases, but not without the presence of the phosphate, the result was an
irregularly shaped knobby spicule, precisely similar to those which are
characteristic of the Alcyonaria[427].

[Illustration: Fig. 200. Additional illustrations of Alcyonarian
spicules: _Eunicea_. (After Studer.)]

 The rough spicules of the Alcyonaria are extraordinarily variable
 in shape and size, as, looking at them from the chemist’s or the
 physicist’s point of view, we should expect them to be. Partly upon
 the form of these spicules, and partly on the general form or mode of
 branching of the entire colony of polypes, a vast number of separate
 “species” have been based by systematic zoologists. But it is now
 admitted that even in specimens of a single species, from one and the
 same locality, the spicules may vary immensely in shape and size: and
 Professor Hickson declares (in a paper published while these sheets
 are passing through the press) that after many years of laborious work
 in striving to determine species of these animal colonies, he feels
 “quite convinced that we have been engaged in a more or less fruitless
 task[428]”.

 The formation of a tooth has very lately been shown to be a phenomenon
 of the same order. That is to say, “calcification in both dentine
 and enamel {425} is in great part a physical phenomenon; the actual
 deposit in both tissues occurs in the form of calcospherites, and the
 process in mammalian tissue is identical in every point with the same
 process occurring in lower organisms[429].” The ossification of bone,
 we may be sure, is in the same sense and to the same extent a physical
 phenomenon.

The typical structure of a calcospherite is no other than that of
a pearl, nor does it differ essentially from that of the otolith
of a mollusc or of a bony fish. (The otoliths, by the way, of the
elasmobranch fishes, like those of reptiles and birds, are not
developed after this fashion, but are true crystals of calc-spar.)

[Illustration: Fig. 201. A “crust” of close-packed calcareous
concretions, precipitated at the surface of an albuminous solution.
(After Harting.)]

Throughout these phenomena, the effect of surface-tension is manifest.
It is by surface-tension that ultra-microscopic particles are brought
together in the first floccular precipitate or coagulum; by the same
agency, the coarser particles are in turn agglutinated into visible
lumps; and the form of the calcospherites, whether it be that of the
solitary spheres or that assumed in various stages of aggregation (e.g.
Fig. 202)[430], is likewise due to the same agency.

[Illustration: Fig. 202. Aggregated calcospherites. (After Harting.)]

From the point of view of colloid chemistry the whole phenomenon is
very important and significant; and not the least significant part
is this tendency of the solidified deposits to assume the form of
“spherulites,” and other rounded contours. In the phraseology of that
science, we are dealing with a _two-phase_ system, which finally
consists of solid particles in suspension in a liquid (the former
being styled the _disperse phase_, the latter the {426} _dispersion
medium_). In accordance with a rule first recognised by Ostwald[431],
when a substance begins to separate out from a solution, so making
its appearance as a _new phase_, it always makes its appearance first
as a liquid[432]. Here is a case in point. The minute quantities
of material, on their way from a state of solution to a state of
“suspension,” pass through a liquid to a solid form; and their
temporary sojourn in the former leaves its impress in the rounded
contours which surface-tension brought about while the little aggregate
was still labile or fluid: while coincidently with this surface-tension
effect upon the surface, crystallisation tended to take place
throughout the little liquid mass, or in such portion of it as had not
yet consolidated and crystallised.

[Illustration: Fig. 203. (After Harting.)]

Where we have simple aggregates of two or three calcospherites, the
resulting figure is precisely that of so many contiguous soap-bubbles.
In other cases, composite forms result which are not so easily
explained, but which, if we could only account for them, would be
of very great interest to the biologist. For instance, when smaller
calcospheres seem, as it were, to invade the substance of a larger one,
we get curious conformations which in the closest possible way resemble
the outlines of certain of the Diatoms (Fig. 203). Another very
curious formation, which Harting calls a “conostat,” is of frequent
occurrence, and in it we see at least a suggestion of analogy with
the configuration which, in a protoplasmic structure, we have spoken
of as a “collar-cell.” The {427} conostats, which are formed in the
surface layer of the solution, consist of a portion of a spheroidal
calcospherite, whose upper part is continued into a thin spheroidal
collar, of somewhat larger radius than the solid sphere; but the
precise manner in which the collar is formed, possibly around a bubble
of gas, possibly about a vortex-like diffusion-current[433] is not
obvious.

――――――――――

Among these various phenomena, the concentric striation observed in
the calcospherite has acquired a special interest and importance[434].
It is part of a phenomenon now widely known, and recognised as an
important factor in colloid chemistry, under the name of “Liesegang’s
Rings[435].”

[Illustration: Fig. 204. Conostats. (After Harting.)]

If we dissolve, for instance, a little bichromate of potash in
gelatine, pour it on to a glass plate, and after it is set place upon
it a drop of silver nitrate solution, there appears in the course of
a few hours the phenomenon of Liesegang’s rings. At first the silver
forms a central patch of abundant reddish brown chromate precipitate;
but around this, as the silver nitrate diffuses slowly through the
gelatine, the precipitate no longer comes down in a continuous,
uniform layer, but forms a series of zones, beautifully regular, which
alternate with clear interspaces of jelly, and which stand farther and
farther apart, in logarithmic ratio, as they recede from the centre.
For a discussion of the _raison d’être_ of {428} this phenomenon,
still somewhat problematic, the student must consult the text-books of
physical and colloid chemistry[436].

But, speaking very generally, we may say the appearance of Liesegang’s
rings is but a particular and striking case of a more general
phenomenon, namely the influence on crystallisation of the presence of
foreign bodies or “impurities,” represented in this case by the “gel”
or colloid matrix[437]. Faraday shewed long ago that to the presence
of slight impurities might be ascribed the banded structure of ice,
of banded quartz or agate, onyx, etc.; and Quincke and Tomlinson have
added to our scanty knowledge of the same phenomenon[438].

[Illustration: Fig. 205. Liesegang’s Rings. (After Leduc.)]

Besides the tendency to rhythmic action, as manifested in Liesegang’s
rings, the association of colloid matter with a crystalloid in solution
may lead to other well-marked effects. These, according to Professor
J. H. Bowman[439], may be grouped somewhat as follows: (1) total
prevention of crystallisation; (2) suppression of certain of the
lines of crystalline growth; (3) extension of the crystal to abnormal
proportions, with a tendency for it to become a compound crystal; (4) a
curving or gyrating of the crystal or its parts. {429}

[Illustration: Fig. 206. Relay-crystals of common salt. (After
Bowman.)]

[Illustration: Fig. 207. Wheel-like crystals in a colloid. (After
Bowman.)]

For instance, it would seem that, if the supply of material to the
growing crystal be not forthcoming in sufficient quantity (as may well
happen in a colloid medium, for lack of convection-currents), then
growth will follow only the strongest lines of crystallising force,
and will be suppressed or partially suppressed along other axes.
The crystal will have a tendency to become filiform, or “fibrous”;
and the raphides of our plant-cells are a case in point. Again, the
long slender crystal so formed, pushing its way into new material,
may initiate a new centre of crystallisation: we get the phenomenon
known as a “relay,” along the principal lines of force, and sometimes
along subordinate axes as well. This phenomenon is illustrated in the
accompanying figure of crystallisation in a colloid medium of common
salt; and it may possibly be that we have here an explanation, or
part of an explanation, of the compound siliceous spicules of the
Hexactinellid sponges. Lastly, when the crystallising force is nearly
equalled by the resistance of the viscous medium, the crystal takes
the line of least resistance, with very various results. One of these
results would seem to be a gyratory course, giving to the crystal a
curious wheel-like shape, as in Fig. 207; and other results are the
feathery, fern-like {430} or arborescent shapes so frequently seen in
microscopic crystallisation.

To return to Liesegang’s rings, the typical appearance of concentric
rings upon a gelatinous plate may be modified in various experimental
ways. For instance, our gelatinous medium may be placed in a capillary
tube immersed in a solution of the precipitating salt, and in this
case we shall obtain a vertical succession of bands or zones regularly
interspaced: the result being very closely comparable to the banded
pigmentation which we see in the hair of a rabbit or a rat. In the
ordinary plate preparation, the free surface of the gelatine is under
different conditions to the lower layers and especially to the lowest
layer in contact with the glass; and therefore it often happens that we
obtain a double series of rings, one deep and the other superficial,
which by occasional blending or interlacing, may produce a netted
pattern. In some cases, as when only the inner surface of our capillary
tube is covered with a layer of gelatine, there is a tendency for the
deposit to take place in a continuous spiral line, rather than in
concentric and separate zones. By such means, according to Küster[440]
various forms of annular, spiral and reticulated thickenings in
the vascular tissue of plants may be closely imitated; and he and
certain other writers have of late been inclined to carry the same
chemico-physical phenomenon a very long way, in the explanation of
various banded, striped, and other rhythmically successional types of
structure or pigmentation. For example, the striped pigmentation of
the leaves in many plants (such as _Eulalia japonica_), the striped or
clouded colouring of many feathers or of a cat’s skin, the patterns
of many fishes, such for instance as the brightly coloured tropical
Chaetodonts and the like, are all regarded by him as so many instances
of “diffusion-figures” closely related to the typical Liesegang
phenomenon. Gebhardt has made a particular study of the same subject in
the case of insects[441]. He declares, for instance, that the banded
wings of _Papilio podalirius_ are precisely imitated in Liesegang’s
experiments; that the finer markings on the wings of the Goatmoth
(_Cossus ligniperda_) shew the double arrangement of larger and of
{431} smaller intermediate rhythms, likewise manifested in certain
cases of the same kind; that the alternate banding of the antennae (for
instance in _Sesia spheciformis_), a pigmentation not concurrent with
the segmented structure of the antenna, is explicable in the same way;
and that the “ocelli,” for instance of the Emperor moth, are typical
illustrations of the common concentric type. Darwin’s well-known
disquisition[442] on the ocellar pattern of the feathers of the Argus
Pheasant, as a result of sexual selection, will occur to the reader’s
mind, in striking contrast to this or to any other direct physical
explanation[443]. To turn from the distribution of pigment to more
deeply seated structural characters, Leduc has shewn how, for instance,
the laminar structure of the cornea or the lens is again, apparently,
a similar phenomenon. In the lens of the fish’s eye, we have a very
curious appearance, the consecutive lamellae being roughened or
notched by close-set, interlocking sinuosities; and precisely the same
appearance, save that it is not quite so regular, is presented in one
of Küster’s figures as the effect of precipitating a little sodium
phosphate in a gelatinous medium. Biedermann has studied, from the
same point of view, the structure and development of the molluscan
shell, the problem which Rainey had first attacked more than fifty
years before[444]; and Liesegang himself has applied his results to the
formation of pearls, and to the development of bone[445]. {432}

Among all the many cases where this phenomenon of Liesegang’s
comes to the naturalist’s aid in explanation of rhythmic or zonary
configurations in organic forms, it has a special interest where the
presence of concentric zones or rings appears, at first sight, as
a sure and certain sign of periodicity of growth, depending on the
seasons, and capable therefore of serving as a mark and record of the
creature’s age. This is the case, for instance, with the scales, bones
and otoliths of fishes; and a kindred phenomena in starch-grains has
given rise, in like manner, to the belief that they indicate a diurnal
and nocturnal periodicity of activity and rest[446].

[Illustration: Fig. 208.]

That this is actually the case in growing starch-grains is generally
believed, on the authority of Meyer[447]; but while under certain
circumstances a marked alternation of growing and resting periods
may occur, and may leave its impress on the structure of the grain,
there is now great reason to believe that, apart from such external
influences, the internal phenomena of diffusion may, just as in the
typical Liesegang experiment, produce the well-known concentric
rings. The spherocrystals of inulin, in like manner, shew, like the
“calcospherites” of Harting (Fig. 208), a concentric structure which in
all likelihood has had no causative impulse save from within.

[Illustration: Fig. 209. Otoliths of Plaice, showing four zones or
“age-rings.” (After Wallace.)]

The striation, or concentric lamellation, of the scales and otoliths
of fishes has been much employed of recent years as a trustworthy and
unmistakeable mark of the fish’s age. There are difficulties in the
way of accepting this hypothesis, not the least of which is the fact
that the otolith-zones, for instance, are extremely well marked even in
the case of some fishes which spend their lives in deep water, {433}
where the temperature and other physical conditions shew little or
no appreciable fluctuation with the seasons of the year. There are,
on the other hand, phenomena which seem strongly confirmatory of the
hypothesis: for instance the fact (if it be fully established) that
in such a fish as the cod, zones of growth, _identical in number_,
are found both on the scales and in the otoliths[448]. The subject
has become a much debated one, and this is not the place for its
discussion; but it is at least obvious, with the Liesegang phenomenon
in view, that we have no right to _assume_ that an appearance of rhythm
and periodicity in structure and growth is necessarily bound up with,
and indubitably brought about by, a periodic recurrence of particular
_external_ conditions.

But while in the Liesegang phenomenon we have rhythmic precipitation
which depends only on forces intrinsic to the system, and is
independent of any corresponding rhythmic changes in temperature or
other external conditions, we have not far to seek for instances of
chemico-physical phenomena where rhythmic alternations of appearance
or structure are produced in close relation to periodic fluctuations
of temperature. A well-known instance is that of the Stassfurt
deposits, where the rock-salt alternates regularly with thin layers of
“anhydrite,” or (in another series of beds) with “polyhalite[449]”: and
where these zones are commonly regarded as marking years, and their
alternate bands as having been formed in connection with the seasons.
A discussion, however, of this remarkable and significant phenomenon,
and of how the chemist explains it, by help of the “phase-rule,” in
connection with temperature conditions, would lead us far beyond our
scope[450].

――――――――――

We now see that the methods by which we attempt to study the chemical
or chemico-physical phenomena which accompany the development of an
inorganic concretion or spicule within the {434} body of an organism
soon introduce us to a multitude of kindred phenomena, of which our
knowledge is still scanty, and which we must not attempt to discuss
at greater length. As regards our main point, namely the formation
of spicules and other elementary skeletal forms, we have seen that
certain of them may be safely ascribed to simple precipitation or
crystallisation of inorganic materials, in ways more or less modified
by the presence of albuminous or other colloid substances. The effect
of these latter is found to be much greater in the case of some
crystallisable bodies than in others. For instance, Harting, and Rainey
also, found as a rule that calcium oxalate was much less affected by a
colloid medium than was calcium carbonate; it shewed in their hands no
tendency to form rounded concretions or “calcospherites” in presence
of a colloid, but continued to crystallise, either normally, or with a
tendency to form needles or raphides. It is doubtless for this reason
that, as we have seen, _crystals_ of calcium oxalate are so common in
the tissues of plants, while those of other calcium salts are rare. But
true calcospherites, or spherocrystals, of the oxalate are occasionally
found, for instance in certain Cacti, and Bütschli[451] has succeeded
in making them artificially in Harting’s usual way, that is to say by
crystallisation in a colloid medium.

There link on to these latter observations, and to the statement
already quoted that calcareous deposits are associated with the dead
products rather than with the living cells of the organism, certain
very interesting facts in regard to the _solubility_ of salts in
colloid media, which have been made known to us of late, and which go
far to account for the presence (apart from the form) of calcareous
precipitates within the organism[452]. It has been shewn, in the
first place, that the presence of albumin has a notable effect on
the solubility in a watery solution of calcium salts, increasing
the solubility of the phosphate in a marked degree, and that of the
carbonate in still greater proportion; but the {435} sulphate is only
very little more soluble in presence of albumin than in pure water, and
the rarity of its occurrence within the organism is so far accounted
for. On the other hand, the bodies derived from the breaking down of
the albumins, their “catabolic” products, such as the peptones, etc.,
dissolve the calcium salts to a much less degree than albumin itself;
and in the case of the phosphate, its solubility in them is scarcely
greater than in water. The probability is, therefore, that the actual
precipitation of the calcium salts is not due to the direct action
of carbonic acid, etc. on a more soluble salt (as was at one time
believed); but to catabolic changes in the proteids of the organism,
which tend to throw down the salts already formed, which had remained
hitherto in albuminous solution. The very slight solubility of calcium
phosphate under such circumstances accounts for its predominance in,
for instance, mammalian bone[453]; and wherever, in short, the supply
of this salt has been available to the organism.

To sum up, we see that, whether from food or from sea-water, calcium
sulphate will tend to pass but little into solution in the albuminoid
substances of the body: calcium carbonate will enter more freely, but a
considerable part of it will tend to remain in solution: while calcium
phosphate will pass into solution in considerable amount, but will be
almost wholly precipitated again, as the albumin becomes broken down in
the normal process of metabolism.

We have still to wait for a similar and equally illuminating study of
the solution and precipitation of _silica_, in presence of organic
colloids.

――――――――――

From the comparatively small group of inorganic formations which,
arising within living organisms, owe their form solely to precipitation
or to crystallisation, that is to say to chemical or other molecular
forces, we shall presently pass to that other and larger group which
appear to be conformed in direct relation to the forms and the
arrangement of the cells or other protoplasmic elements[454]. {436}
The two principles of conformation are both illustrated in the
spicular skeletons of the Sponges.

[Illustration: Fig. 210. Close-packed calcospherites, or so-called
“spicules,” of Astrosclera. (After Lister.)]

In a considerable number, but withal a minority of cases, the form
of the sponge-spicule may be deemed sufficiently explained on the
lines of Harting’s and Rainey’s experiments, that is to say as the
direct result of chemical or physical phenomena associated with the
deposition of lime or of silica in presence of colloids[455]. This is
the case, for instance, with various small spicules of a globular or
spheroidal form, formed of amorphous silica, concentrically striated
within, and often developing irregular knobs or tiny tubercles over
their surfaces. In the aberrant sponge _Astrosclera_[456], we have,
to begin with, rounded, striated discs or globules, which in like
manner are nothing more or less than the {437} “calcospherites” of
Harting’s experiments; and as these grow they become closely aggregated
together (Fig. 210), and assume an angular, polyhedral form, once more
in complete accordance with the results of experiment[457]. Again,
in many Monaxonid sponges, we have irregularly shaped, or branched
spicules, roughened or tuberculated by secondary superficial deposits,
and reminding one of the spicules of some Alcyonaria. These also must
be looked upon as the simple result of chemical deposition, the form of
the deposit being somewhat modified in conformity with the surrounding
tissues, just as in the simple experiment the form of the concretionary
precipitate is affected by the heterogeneity, visible or invisible,
of the matrix. Lastly, the simple needles of amorphous silica, which
constitute one of the commonest types of spicule, call for little
in the way of explanation; they are accretions or deposits about a
linear axis, or fine thread of organic material, just as the ordinary
rounded calcospherite is deposited about some minute point or centre of
crystallisation, and as ordinary crystallisation is often started by a
particle of atmospheric dust; in some cases they also, like the others,
are apt to be roughened by more irregular secondary deposits, which
probably, as in Harting’s experiments, appear in this irregular form
when the supply of material has become relatively scanty.

――――――――――

Our few foregoing examples, diverse as they are in look and kind and
ranging from the spicules of Astrosclera or Alcyonium to the otoliths
of a fish, seem all to have their free origin in some larger or smaller
fluid-containing space, or cavity of the body: pretty much as Harting’s
calcospheres made their appearance in the albuminous content of a dish.
But we now come at last to a much larger class of spicular and skeletal
structures, for whose regular and often complex forms some other
explanation than the intrinsic forces of crystallisation or molecular
adhesion is manifestly necessary. As we enter on this subject, which
is certainly no small or easy one, it may conduce to simplicity, and
to brevity, {438} if we try to make a rough classification, by way of
forecast, of the chief conditions which we are likely to meet with.

Just as we look upon animals as constituted, some of a vast number of
cells, and others of a single cell or of a very few, and just as the
shape of the former has no longer a visible relation to the individual
shapes of its constituent cells, while in the latter it is cell-form
which dominates or is actually equivalent to the form of the organism,
so shall we find it to be, with more or less exact analogy, in the
case of the skeleton. For example, our own skeleton consists of bones,
in the formation of each of which a vast number of minute living
cellular elements are necessarily concerned; but the form and even the
arrangement of these bone-forming cells or corpuscles are monotonously
simple, and we cannot find in these a physical explanation of the
outward and visible configuration of the bone. It is as part of a far
larger field of force,—in which we must consider gravity, the action of
various muscles, the compressions, tensions and bending moments due to
variously distributed loads, the whole interaction of a very complex
mechanical system,—that we must explain (if we are to explain at all)
the configuration of a bone.

In contrast to these massive skeletons, or constituents of a skeleton,
we have other skeletal elements whose whole magnitude, or whose
magnitude in some dimension or another, is commensurate with the
magnitude of a single living cell, or (as comes to very much the same
thing) is comparable to the range of action of the molecular forces.
Such is the case with the ordinary spicules of a sponge, with the
delicate skeleton of a Radiolarian, or with the denser and robuster
shells of the Foraminifera. The effect of _scale_, then, of which
we had so much to say in our introductory chapter on Magnitude, is
bound to be apparent in the study of skeletal fabrics, and to lead to
essential differences between the big and the little, the massive and
the minute, in regard to their controlling forces and their resultant
forms. And if all this be so, and if the range of action of the
molecular forces be in truth the important and fundamental thing, then
we may somewhat extend our statement of the case, and include in it not
only association with the living cellular elements of the body, but
also association with any bubbles, drops, vacuoles or vesicles which
{439} may be comprised within the bounds of the organism, and which
are (as their names and characters connote) of the order of magnitude
of which we are speaking.

Proceeding a little farther in our classification, we may conceive
each little skeletal element to be associated, in one case, with a
single cell or vesicle, and in another with a cluster or “system” of
consociated cells. In either case there are various possibilities.
For instance, the calcified or other skeletal material may tend to
overspread the entire outer surface of the cell or cluster of cells,
and so tend accordingly to assume some configuration comparable to
that of a fluid drop or of an aggregation of drops; this, in brief, is
the gist and essence of our story of the foraminiferal shell. Another
common, but very different condition will arise if, in the case of
the cell-aggregates, the skeletal material tends to accumulate in the
interstices _between_ the cells, in the partition-walls which separate
them, or in the still more restricted distribution indicated by the
_lines_ of junction between these partition-walls. Conditions such as
these will go a very long way to help us in our understanding of many
sponge-spicules and of an immense variety of radiolarian skeletons. And
lastly (for the present), there is a possible and very interesting case
of a skeletal element associated with the surface of a cell, not so as
to cover it like a shell, but only so as to pursue a course of its own
within it, and subject to the restraints imposed by such confinement to
a curved and limited surface. With this curious condition we shall deal
immediately.

This preliminary and much simplified classification of skeletal forms
(as is evident enough) does not pretend to completeness. It leaves out
of account some kinds of conformation and configuration with which
we shall attempt to deal, and others which we must perforce omit.
But nevertheless it may help to clear or to mark our way towards the
subjects which this chapter has to consider, and the conditions by
which they are at least partially defined.

――――――――――

Among the several possible, or conceivable, types of microscopic
skeletons let us choose, to begin with, the case of a spicule, more
or less simply linear as far as its _intrinsic_ powers of growth are
{440} concerned, but which owes its now somewhat complicated form to
a restraint imposed by the individual cell to which it is confined,
and within whose bounds it is generated. The conception of a spicule
developed under such conditions we owe to a distinguished physicist,
the late Professor G. F. FitzGerald.

Many years ago, Sollas pointed out that if a spicule begin to grow in
some particular way, presumably under the control or constraint imposed
by the organism, it continues to grow by further chemical deposition in
the same form or direction even after it has got beyond the boundaries
of the organism or its cells. This phenomenon is what we see in, and
this imperfect explanation goes so far to account for, the continued
growth in straight lines of the long calcareous spines of Globigerina
or Hastigerina, or the similarly radiating but siliceous spicules of
many Radiolaria. In physical language, if our crystalline structure has
once begun to be laid down in a definite orientation, further additions
tend to accrue in a like regular fashion and in an identical direction;
and this corresponds to the phenomenon of so-called “orientirte
Adsorption,” as described by Lehmann.

In Globigerina or in Acanthocystis the long needles grow out freely
into the surrounding medium, with nothing to impede their rectilinear
growth and their approximately radiate distribution. But let us
consider some simple cases to illustrate the forms which a spicule will
tend to assume when, striving (as it were) to grow straight, it comes
under the influence of some simple and constant restraint or compulsion.

If we take any two points on some curved surface, such as that of a
sphere or an ellipsoid, and imagine a string stretched between them,
we obtain what is known in mathematics as a “geodetic” curve. It is
the shortest line which can be traced between the two points, upon the
surface itself; and the most familiar of all cases, from which the name
is derived, is that curve upon the earth’s surface which the navigator
learns to follow in the practice of “great-circle sailing.” Where
the surface is spherical, the geodetic is always literally a “great
circle,” a circle, that is to say, whose centre is the centre of the
sphere. If instead of a sphere we be dealing with an ellipsoid, the
geodetic becomes a variable figure, according to the position of our
two points. {441} For obviously, if they lie in a line perpendicular
to the long axis of the ellipsoid, the geodetic which connects them is
a circle, also perpendicular to that axis; and if they lie in a line
parallel to the axis, their geodetic is a portion of that ellipse about
which the whole figure is a solid of revolution. But if our two points
lie, relatively to one another, in any other direction, then their
geodetic is part of a spiral curve in space, winding over the surface
of the ellipsoid.

To say, as we have done, that the geodetic is the shortest line
between two points upon the surface, is as much as to say that it is
a _projection_ of some particular straight line upon the surface in
question; and it follows that, if any linear body be confined to that
surface, while retaining a tendency to grow by successive increments
always (save only for its confinement to that surface) in a straight
line, the resultant form which it will assume will be that of a
geodetic. In mathematical language, it is a property of a geodetic that
the plane of any two consecutive elements is a plane perpendicular
to that in which the geodetic lies; or, in simpler words, any two
consecutive elements lie in a straight line _in the plane of the
surface_, and only diverge from a straight line in space by the actual
curvature of the surface to which they are restrained.

[Illustration: Fig. 211. Sponge and Holothurian spicules.]

[Illustration: Fig. 212.]

[Illustration: Fig. 213. An “amphidisc” of Hyalonema.]

Let us now imagine a spicule, whose natural tendency is to grow into
a straight linear element, either by reason of its own molecular
anisotropy, or because it is deposited about a thread-like axis; and
let us suppose that it is confined either within a cell-wall or in
adhesion thereto; it at once follows that its line of growth will be
simply a geodetic to the surface of the cell. And if the cell be an
imperfect sphere, or a more or less regular ellipsoid, the spicule will
tend to grow into one or other of three forms: either a plane curve
of circular arc; or, more commonly, a plane curve which is a portion
of an ellipse; or, most commonly of all, a curve which is a portion
of a spiral in space. In the latter case, the number of turns of the
spiral will depend, not only on the length of the spicule, but on
the relative dimensions of the ellipsoidal cell, as well as upon the
angle by which the spicule is inclined to the ellipsoid axes; but a
very common case will probably be that in which the spicule looks at
first sight to be {442} a plane C-shaped figure, but is discovered,
on more careful inspection, to lie not in one plane but in a more
complicated spiral twist. This investigation includes a series of
forms which are abundantly represented among actual sponge-spicules,
as illustrated in Figs. 211 and 212. If the spicule be not restricted
to linear growth, but have a tendency to expand, or to branch out from
a main axis, we shall obtain a series of more complex figures, all
related to the geodetic system of curves. A very simple case will arise
where the spicule occupies, in the first instance, the axis of the
containing cell, and then, on reaching its boundary, tends to branch
or spread outwards. We shall now get various figures, in some of which
the spicule will appear as an axis expanding into a disc or wheel at
either end; and in other cases, the terminal disc will be replaced, or
represented, by a series of rays or spokes, with a reflex curvature,
corresponding to the spherical or ellipsoid curvature of the surface
of the cell. Such spicules as these are again exceedingly common among
various sponges (Fig. 213).

Furthermore, if these mechanical methods of conformation, and others
like to these, be the true cause of the shapes which the spicules
assume, it is plain that the production of these spicular shapes is not
a specific function of sponges or of any particular sponge, but that
we should expect {443} the same or very similar phenomena to occur
in other organisms, wherever the conditions of inorganic secretion
within closed cells was very much the same. As a matter of fact, in the
group of Holothuroidea, where the formation of intracellular spicules
is a characteristic feature of the group, all the principal types of
conformation which we have just described can be closely paralleled.
Indeed in many cases, the forms of the Holothurian spicules are
identical and indistinguishable from those of the sponges[458]. But
the Holothurian spicules are composed of calcium carbonate while those
which we have just described in the case of sponges are usually, if not
always, siliceous: this being just another proof of the fact that in
such cases the form of the spicule is not due to its chemical nature or
molecular structure, but to the external forces to which, during its
growth, the spicule is submitted.

――――――――――

So much for that comparatively limited class of sponge-spicules
whose forms seem capable of explanation on the hypothesis that
they are developed within, or under the restraint imposed by, the
surface of a cell or vesicle. Such spicules are usually of small
size, as well as of comparatively simple form; and they are greatly
outstripped in number, in size, and in supposed importance as guides
to zoological classification, by another class of spicules. This new
class includes such as we have supposed to be capable of explanation
on the assumption that they develop in association (of some sort
or another) with the _lines of junction_ of contiguous cells. They
include the triradiate spicules of the calcareous sponges, the
quadriradiate or “tetractinellid” spicules which occur in the same
group, but more characteristically in certain siliceous sponges known
as the Tetractinellidae, and lastly perhaps (though these last are
admittedly somewhat harder to understand) the six-rayed spicules of the
Hexactinellids.

The spicules of the calcareous sponges are commonly triradiate, and the
three radii are usually inclined to one another at equal, or nearly
equal angles; in certain cases, two of the three rays are nearly in
a straight line, and at right angles to the {444} third[459]. They
are seldom in a plane, but are usually inclined to one another in a
solid, trihedral angle, not easy of precise measurement under the
microscope. The three rays are very often supplemented by a fourth,
which is set tetrahedrally, making, that is to say, coequal angles with
the other three. The calcareous spicule consists mainly of carbonate
of lime, in the form of calcite, with (according to von Ebner) some
admixture of soda and magnesia, of sulphates and of water. According
to the same writer (but the fact, though it would seem easy to test,
is still disputed) there is no organic matter in the spicule, either
in the form of an axial filament or otherwise, and the appearance
of stratification, often simulating the presence of an axial fibre,
is due to “mixed crystallisation” of the various constituents. The
spicule is a true crystal, and therefore its existence and its form are
_primarily_ due to the molecular forces of crystallisation; moreover
it is a single crystal and not a group of crystals, as is at once seen
by its behaviour in polarised light. But its axes are not crystalline
axes, and its form neither agrees with, nor in any way resembles,
any one of the many polymorphic forms in which calcite is capable of
crystallising. It is as though it were carved out of a solid crystal;
it is, in fact, a crystal under restraint, a crystal growing, as it
were, in an artificial mould; and this mould is constituted by the
surrounding cells, or structural vesicles of the sponge.

[Illustration: Fig. 214. Spicules of Grantia and other calcareous
sponges. (After Haeckel.)]

We have already studied in an elementary way, but amply for our
present purpose, the manner in which three or more cells, or bubbles,
tend to meet together under the influence of surface-tension, and
also the outwardly similar phenomena which may be brought about by a
uniform distribution of mechanical pressure. We have seen that when we
confine ourselves to a plane assemblage of such bodies, we find them
meeting one another in threes; that in a section or plane projection
of such an assemblage we see the partition-walls meeting one another
at equal angles of 120°; that when the bodies are uniform in size, the
partitions are straight lines, which combine to form regular hexagons;
and that when {445} the bodies are unequal in size, the partitions
are curved, and combine to form other and less regular polygons. It
is plain, accordingly, that in any flattened or stratified assemblage
of such cells, a solidified skeletal deposit which originates or
accumulates either between the cells or within the thickness of their
mutual partitions, will tend to take the form of triradiate bodies,
whose rays (in a typical case) will be set at equal angles of 120°
(Fig. 214, _F_). And this latter condition of equality will be open to
modification in various ways. It will be modified by any inequality
in the specific tensions of adjacent cells; as a special case, it
will be apt to be greatly modified at the surface of the system,
where a spicule happens to be formed in a plane perpendicular to the
cell-layer, so that one of its three rays lies between two adjacent
cells and the other two are associated with the surface of contact
between the cells and the surrounding medium; in such a case (as in the
cases considered in connection with the forms of the cells themselves
{446} on p. 314), we shall tend to obtain a spicule with two equal
angles and one unequal (Fig. 214, _A_, _C_). In the last case, the two
outer, or superficial rays, will tend to be markedly curved. Again, the
equiangular condition will be departed from, and more or less curvature
will be imparted to the rays, wherever the cells of the system cease
to be uniform in size, and when the hexagonal symmetry of the system
is lost accordingly. Lastly, although we speak of the rays as meeting
at certain definite angles, this statement applies to their _axes_,
rather than to the rays themselves. For, if the triradiate spicule be
developed in the _interspace_ between three juxtaposed cells, it is
obvious that its sides will tend to be concave, for the interspace
between our three contiguous equal circles is an equilateral,
curvilinear triangle; and even if our spicule be deposited, not in the
space between our three cells, but in the thickness of the intervening
wall, then we may recollect (from p. 297) that the several partitions
never actually meet at sharp angles, but the angle of contact is always
bridged over by a small accumulation of material (varying in amount
according to its fluidity) whose boundary takes the form of a circular
arc, and which constitutes the “bourrelet” of Plateau.

In any sample of the triradiate spicules of Grantia, or in any series
of careful drawings, such as those of Haeckel among others, we shall
find that all these various configurations are precisely and completely
illustrated.

The tetrahedral, or rather tetractinellid, spicule needs no explanation
in detail (Fig. 214, _D_, _E_). For just as a triradiate spicule
corresponds to the case of three cells in mutual contact, so does the
four-rayed spicule to that of a solid aggregate of four cells: these
latter tending to meet one another in a tetrahedral system, shewing
four edges, at each of which four surfaces meet, the edges being
inclined to one another at equal angles of about 109°. And even in the
case of a single layer, or superficial layer, of cells, if the skeleton
originate in connection with all the edges of mutual contact, we shall,
in complete and typical cases, have a four-rayed spicule, of which one
straight limb will correspond to the line of junction between the three
cells, and the other three limbs (which will then be curved limbs) will
correspond to the edges where two cells meet one another on the surface
of the system. {447}

But if such a physical explanation of the forms of our spicules is
to be accepted, we must seek at once for some physical agency by
which we may explain the presence of the solid material just at the
junctions or interfaces of the cells, and for the forces by which
it is confined to, and moulded to the form of, these intercellular
or interfacial contacts. It is to Dreyer that we chiefly owe the
physical or mechanical theory of spicular conformation which I have
just described,—a theory which ultimately rests on the form assumed,
under surface-tension, by an aggregation of cells or vesicles. But
this fundamental point being granted, we have still several possible
alternatives by which to explain the details of the phenomenon.

Dreyer, if I understand him aright, was content to assume that the
solid material, secreted or excreted by the organism, accumulated
in the interstices between the cells, and was there subjected to
mechanical pressure or constraint as the cells got more and more
crowded together by their own growth and that of the system generally.
As far as the general form of the spicules goes, such explanation is
not inadequate, though under it we may have to renounce some of our
assumptions as to what takes place at the outer surface of the system.

But in all (or most) cases where, but a few years ago, the concepts
of secretion or excretion seemed precise enough, we are now-a-days
inclined to turn to the phenomenon of adsorption as a further stage
towards the elucidation of our facts. Here we have a case in point.
In the tissues of our sponge, wherever two cells meet, there we
have a definite _surface_ of contact, and there accordingly we
have a manifestation of surface-energy; and the concentration of
surface-energy will tend to be a maximum at the _lines_ or edges
whereby the three, or four, such surfaces are conjoined. Of the
micro-chemistry of the sponge-cells our ignorance is great; but
(without venturing on any hypothesis involving the chemical details of
the process) we may safely assert that there is an inherent probability
that certain substances will tend to be concentrated and ultimately
deposited just in these lines of intercellular contact and conjunction.
In other words, adsorptive concentration, under osmotic pressure, at
and in the surface-film which constitutes the mutual boundary between
contiguous {448} cells, emerges as an alternative (and, as it seems
to me, a highly preferable alternative) to Dreyer’s conception of
an accumulation under mechanical pressure in the vacant spaces left
between one cell and another.

But a purely chemical, or purely molecular adsorption, is not the
only form of the hypothesis on which we may rely. For from the purely
physical point of view, angles and edges of contact between adjacent
cells will be _loci_ in the field of distribution of surface-energy,
and any material particles whatsoever will tend to undergo a diminution
of freedom on entering one of those boundary regions. In a very
simple case, let us imagine a couple of soap bubbles in contact with
one another. Over the surface of each bubble there glide in every
direction, as usual, a multitude of tiny bubbles and droplets; but
as soon as these find their way into the groove or re-entrant angle
between the two bubbles, there their freedom of movement is so far
restrained, and out of that groove they have little or no tendency to
emerge. A cognate phenomenon is to be witnessed in microscopic sections
of steel or other metals. Here, amid the “crystalline” structure of
the metal (where in cooling its imperfectly homogeneous material has
developed a cellular structure, shewing (in section) hexagonal or
polygonal contours), we can easily observe, as Professor Peddie has
shewn me, that the little particles of graphite and other foreign
bodies common in the matrix, have tended to aggregate themselves
in the walls and at the angles of the polygonal cells—this being a
direct result of the diminished freedom which the particles undergo on
entering one of these boundary regions[460].

It is by a combination of these two principles, chemical adsorption
on the one hand, and physical quasi-adsorption or concentration of
grosser particles on the other, that I conceive the substance of
the sponge-spicule to be concentrated and aggregated at the cell
boundaries; and the forms of the triradiate and tetractinellid spicules
are in precise conformity with this hypothesis. A few general matters,
and a few particular cases, remain to be considered.

It matters little or not at all, for the phenomenon in question, {449}
what is the histological nature or “grade” of the vesicular structures
on which it depends. In some cases (apart from sponges), they may be no
more than the little alveoli of the intracellular protoplasmic network,
and this would seem to be the case at least in one known case, that
of the protozoan _Entosolenia aspera_, in which, within the vesicular
protoplasm of the single cell, Möbius has described tiny spicules in
the shape of little tetrahedra with concave sides. It is probably
also the case in the small beginnings of the Echinoderm spicules,
which are likewise intracellular, and are of similar shape. In the
case of our sponges we have many varying conditions, which we need
not attempt to examine in detail. In some cases there is evidence for
believing that the spicule is formed at the boundaries of true cells
or histological units. But in the case of the larger triradiate or
tetractinellid spicules of the sponge-body, they far surpass in size
the actual “cells”; we find them lying, regularly and symmetrically
arranged, between the “pore-canals” or “ciliated chambers,” and it
is in conformity with the shape and arrangement of these rounded or
spheroidal structures that their shape is assumed.

Again, it is not necessarily at variance with our hypothesis to find
that, in the adult sponge, the larger spicules may greatly outgrow the
bounds not only of actual cells but also of the ciliated chambers, and
may even appear to project freely from the surface of the sponge. For
we have already seen that the spicule is capable of growing, without
marked change of form, by further deposition, or crystallisation,
of layer upon layer of calcareous molecules, even in an artificial
solution; and we are entitled to believe that the same process may
be carried on in the tissues of the sponge, without greatly altering
the symmetry of the spicule, long after it has established its
characteristic form of a system of slender trihedral or tetrahedral
rays.

Neither is it of great importance to our hypothesis whether the rayed
spicule necessarily arises as a single structure, or does so from
separate minute centres of aggregation. Minchin has shewn that, in
some cases at least, the latter is the case; the spicule begins, he
tells us, as three tiny rods, separate from one another, each developed
in the interspace between two sister-cells, which are themselves the
results of the division of one of a {450} little trio of cells; and
the little rods meet and fuse together while still very minute, when
the whole spicule is only about 1/200 of a millimetre long. At this
stage, it is interesting to learn that the spicule is non-crystalline;
but the new accretions of calcareous matter are soon deposited in
crystalline form.

This observation threw considerable difficulties in the way of former
mechanical theories of the conformation of the spicule, and was quite
at variance with Dreyer’s theory, according to which the spicule was
bound to begin from a central nucleus coinciding with the meeting-place
of the three contiguous cells, or rather the interspace between them.
But the difficulty is removed when we import the concept of adsorption;
for by this agency it is natural enough, or conceivable enough, that
the process of deposition should go on at separate parts of a common
system of surfaces; and if the cells tend to meet one another by their
interfaces before these interfaces extend to the angles and so complete
the polygonal cell, it is again conceivable and natural that the
spicule should first arise in the form of separate and detached limbs
or rays.

[Illustration: Fig. 215. Spicules of tetractinellid sponges (after
Sollas). _a_–_e_, anatriaenes; _d_–_f_, protriaenes.]

Among the tetractinellid sponges, whose spicules are composed of
amorphous silica or opal, all or most of the above-described main
types of spicule occur, and, as the name of the group implies, the
four-rayed, tetrahedral spicules are especially represented. A
somewhat frequent type of spicule is one in which one of the four
rays is greatly developed, and the other three constitute small
prongs diverging at equal angles from the main or axial ray. In all
probability, as Dreyer suggests, we have here had to do with a group of
four vesicles, of which three were large and co-equal, while a fourth
and very much smaller one lay above and between the other three. In
certain cases where we have likewise one large and three much smaller
{451} rays, the latter are recurved, as in Fig. 215. This type, save
for the constancy of the number of rays, and the limitation of the
terminal ones to three, and save also for the more important difference
that they occur only at one and not at both ends of the long axis, is
similar to the type of spicule illustrated in Fig. 213, which we have
explained as being probably developed within an oval cell, by whose
walls its branches have been conformed to geodetic curves. But it is
much more probable that we have here to do with a spicule developed
in the midst of a group of three coequal and more or less elongated
or cylindrical cells or vesicles, the long axial ray corresponding to
their common line of contact, and the three short rays having each lain
in the surface furrow between two out of the three adjacent cells.

[Illustration: Fig. 216. Various holothurian spicules. (After Théel.)]

Just as in the case of the little curved or S-shaped spicules, formed
apparently within the bounds of a single cell, so also in the case of
the larger tetractinellid and analogous types do we find among the
Holothuroidea the same configurations reproduced as we have dealt with
in the sponges. The holothurian spicules are a little less neatly
formed, a little rougher, than the sponge-spicules; and certain forms
occur among the former group which do not present themselves among
the latter; but for the most part a community of type is obvious and
striking (Fig. 216).

A curious and, physically speaking, strictly analogous formation to
the tetrahedral spicules of the sponges is found in the {452} spores
of a certain little group of parasitic protozoa, the Actinomyxidia.
These spores are formed from clusters of six cells, of which three
come to constitute the capsule of the spore; and this capsule, always
triradiate in its symmetry, is in some species drawn out into long
rays, of which one constitutes a straight central axis, while the
others, coming off from it at equal angles, are recurved in wide
circular arcs. The account given of the development of this structure
by its discoverers[461] is somewhat obscure to me, but I think that, on
physical grounds, there can be no doubt whatever that the quadriradiate
capsule has been somehow modelled upon a group of three surrounding
cells, its axis lying between the three, and its three radial arcs
occupying the furrows between adjacent pairs.

[Illustration: Fig. 217. Spicules of hexactinellid sponges. (After F.
E. Schultze.)]

The typically six-rayed siliceous spicules of the hexactinellid
sponges, while they are perhaps the most regular and beautifully formed
spicules to be found within the entire group, have been found very
difficult to explain, and Dreyer has confessed his complete inability
to account for their conformation. But, though it is doubtless only
throwing the difficulty a little further back, we may so far account
for them by considering that the cells or vesicles by which they
are conformed are not arranged in {453} what is known as “closest
packing,” but in linear series; so that in their arrangement, and by
their mutual compression, we tend to get a pattern, not of hexagons,
but of squares: or, looking to the solid, not of dodecahedra but of
cubes or parallelopipeda. This indeed appears to be the case, not with
the individual cells (in the histological sense), but with the larger
units or vesicles which make up the body of the hexactinellid. And this
being so, the spicules formed between the linear, or cubical series of
vesicles, will have the same tendency towards a “hexactinellid” shape,
corresponding to the angles and adjacent edges of a system of cubes, as
in our former case they had to a triradiate or a tetractinellid form,
when developed in connection with the angles and edges of a system of
hexagons, or a system of dodecahedra.

Histologically, the case is illustrated by a well-known phenomenon in
embryology. In the segmenting ovum, there is a tendency for the cells
to be budded off in linear series; and so they often remain, in rows
side by side, at least for a considerable time and during the course
of several consecutive cell divisions. Such an arrangement constitutes
what the embryologists call the “radial type” of segmentation[462]. But
in what is described as the “spiral type” of segmentation, it is stated
that, as soon as the first horizontal furrow has divided the cells into
an upper and a lower layer, those of “the upper layer are shifted in
respect to the lower layer, by means of a rotation about the vertical
axis[463].” It is, of course, evident that the whole process is merely
that which is familiar to physicists as “close packing.” It is a very
simple case of what Lord Kelvin used to call “a problem in tactics.”
It is a mere question of the rigidity of the system, of the freedom of
movement on the part of its constituent cells, whether or at what stage
this tendency to slip into the closest propinquity, or position of
minimum potential, will be found to manifest itself.

However the hexactinellid spicules be arranged (and this is {454} not
at all easy to determine) in relation to the tissues and chambers of
the sponge, it is at least clear that, whether they be separate or be
fused together (as often happens) in a composite skeleton, they effect
a symmetrical partitioning of space according to the cubical system, in
contrast to that closer packing which is represented and effected by
the tetrahedral system[464].

――――――――――

This question of the origin and causation of the forms of
sponge-spicules, with which we have now briefly dealt, is all the
more important and all the more interesting because it has been
discussed time and again, from points of view which are characteristic
of very different schools of thought in biology. Haeckel found in
the form of the sponge-spicule a typical illustration of his theory
of “bio-crystallisation”; he considered that these “biocrystals”
represented “something midway—_ein Mittelding_—between an inorganic
crystal and an organic secretion”; that there was a “compromise
between the crystallising efforts of the calcium carbonate and the
formative activity of the fused cells of the syncytium”; and that
the semi-crystalline secretions of calcium carbonate “were utilised
by natural selection as ‘spicules’ for building up a skeleton, and
afterwards, by the interaction of adaptation and heredity, became
modified in form and differentiated in a vast variety of ways in the
struggle for existence[465].” What Haeckel precisely signified by these
words is not clear to me.

F. E. Schultze, perceiving that identical forms of spicule were
developed whether the material were crystalline or non-crystalline,
abandoned all theories based upon crystallisation; he simply saw in the
form and arrangement of the spicules something which was “best fitted”
for its purpose, that is to say for the support and strengthening of
the porous walls of the sponge, and found clear evidence of “utility”
in the specific structure of these skeletal elements. {455}

Sollas and Dreyer, as we have seen, introduced in various ways the
conception of physical causation,—as indeed Haeckel himself had done
in regard to one particular, when he supposed the _position_ of the
spicules to be due to the constant passage of the water-currents.
Though even here, by the way, if I understand Haeckel aright, he was
thinking not merely of a direct or immediate physical causation, but of
one manifesting itself through the agency of natural selection[466].
Sollas laid stress upon the “path of least resistance” as determining
the direction of growth; while Dreyer dealt in greater detail with
the various tensions and pressures to which the growing spicule was
exposed, amid the alveolar or vesicular structure which was represented
alike by the chambers of the sponge, by the reticulum of constituent
cells, or by the minute structure of the intracellular protoplasm. But
neither of these writers, so far as I can discover, was inclined to
doubt for a moment the received canon of biology, which sees in such
structures as these the characteristics of true organic species, and
the indications of an hereditary affinity by which blood-relationship
and the succession of evolutionary descent throughout geologic time can
be ultimately deduced.

Lastly, Minchin, in a well-known paper[467], took sides with Schultze,
and gave reasons for dissenting from such mechanical theories as those
of Sollas and of Dreyer. For example, after pointing out that all
protoplasm contains a number of “granules” or microsomes, contained in
the alveolar framework and lodged at the nodes of the reticulum, he
argued that these also ought to acquire a form such as the spicules
possess, if it were the case that these latter owed their form to their
very similar or identical position. “If vesicular tension cannot in any
other instance cause the granules at the nodes to assume a tetraxon
form, why should it do so for the sclerites?” In all probability the
answer to this question is not far to seek. If the force which the
“mechanical” hypothesis has in view were simply that of mechanical
_pressure_, {456} as between solid bodies, then indeed we should
expect that any substances whatsoever, lying between the impinging
spheres, would tend (unless they were infinitely hard) to assume the
quadriradiate or “tetraxon” form; but this conclusion does not follow
at all, in so far as it is to _surface-energy_ that we ascribe the
phenomenon. Here the specific nature of the substances involved makes
all the difference. We cannot argue from one substance to another;
adsorptive attraction shews its effect on one and not on another; and
we have not the least reason to be surprised if we find that the little
granules of protoplasmic material, which as they lie bathed in the
more fluid protoplasm have (presumably, and as their shape indicates)
a strong surface-tension of their own, behave towards the adjacent
vesicles in a very different fashion to the incipient aggregations
of calcareous or siliceous matter in a colloid medium. “The ontogeny
of the spicules,” says Professor Minchin, “points clearly to their
regular form being a _phylogenetic adaptation, which has become fixed
and handed on by heredity, appearing in the ontogeny as a prophetic
adaptation_.” And again, “The forms of the spicules are the result of
adaptation to the requirements of the sponge as a whole, produced by
_the action of natural selection upon variation in every direction_.”
It would scarcely be possible to illustrate more briefly and more
cogently than by these few words (or the similar words of Haeckel
quoted on p. 454), the fundamental difference between the Darwinian
conception of the causation and determination of Form, and that which
is characteristic of the physical sciences.

――――――――――

If I have dealt comparatively briefly with the inorganic skeleton of
sponges, in spite of the obvious importance of this part of our subject
from the physical or mechanical point of view, it has been owing to
several reasons. In the first place, though the general trend of the
phenomena is clear, it must be at once admitted that many points are
obscure, and could only be discussed at the cost of a long argument.
In the second place, the physical theory is (as I have shewn) in
manifest conflict with the accounts given by various embryologists of
the development of the spicules, and of the current biological theories
which their descriptions embody; it is beyond our scope to deal with
such descriptions {457} in detail. Lastly, we find ourselves able to
illustrate the same physical principles with greater clearness and
greater certitude in another group of animals, namely the Radiolaria.
In our description of the skeletons occurring within this group we
shall by no means abandon the preliminary classification of microscopic
skeletons which we have laid down; but we shall have occasion to blend
with it the consideration of certain other more or less correlated
phenomena.

The group of microscopic organisms known as the Radiolaria is
extraordinarily rich in diverse forms, or “species.” I do not know how
many of such species have been described and defined by naturalists,
but some thirty years ago the number was said to be over four thousand,
arranged in more than seven hundred genera[468]. Of late years there
has been a tendency to reduce the number, it being found that some
of the earlier species and even genera are but growth-stages of one
and the same form, sometimes mere fragments or “fission-products”
common to several species, or sometimes forms so similar and so
interconnected by intermediate forms that the naturalist denominates
them not “species” but “varieties.” It has to be admitted, in short,
that the conception of species among the Radiolaria has not hitherto
been, and is not yet, on the same footing as that among most other
groups of animals. But apart from the extraordinary multiplicity of
forms among the Radiolaria, there are certain other features in this
multiplicity which arrest our attention. For instance, the distribution
of species in space is curious and vague; many species are found all
over the world, or at least every here and there, with no evidence
of specific limitations of geographical habitat; others occur in the
neighbourhood of the two poles; some are confined to warm and others
to cold currents of the ocean. In time also their distribution is
not less vague: so much so that it has been asserted of them that
“from the Cambrian age downwards, the families and even genera appear
identical with those now living.” Lastly, except perhaps in the case
of a few large “colonial forms,” we seldom if ever find, as is usual
{458} in most animals, a local predominance of one particular species.
On the contrary, in a little pinch of deep-sea mud or of some fossil
“Radiolarian earth,” we shall probably find scores, and it may be even
hundreds, of different forms. Moreover, the radiolarian skeletons
are of quite extraordinary delicacy and complexity, in spite of
their minuteness and the comparative simplicity of the “unicellular”
organisms within which they grow; and these complex conformations have
a wonderful and unusual appearance of geometric regularity. All these
_general_ considerations seem such as to prepare us for the special
need of some physical hypothesis of causation. The little skeletal
fabrics remind us of such objects as snow-crystals (themselves almost
endless in their diversity), rather than of a collection of distinct
animals, constructed in apparent accordance with functional needs, and
distributed in accordance with their fitness for particular situations.
Nevertheless great efforts have been made of recent years to attach “a
biological meaning” to these elaborate structures; and “to justify the
hope that in time the utilitarian character [of the skeleton] will be
more completely recognised[469].”

In the majority of cases, the skeleton of the Radiolaria is composed,
like that of so many sponges, of silica; in one large family, the
Acantharia (and perhaps in some others), it is composed, in great
part at least, of a very unusual constituent, namely strontium
sulphate[470]. There is no fundamental or important morphological
character in which the shells formed of these two constituents
differ from one another; and in no case can the chemical properties
of these inorganic materials be said to influence the form of the
complex skeleton or shell, save only in this general way that, by
their rigidity and toughness, they may give rise to a fabric far more
delicate and slender than we find developed among calcareous organisms.

A slight exception to this rule is found in the presence of true
crystals, which occur within the central capsules of certain {459}
Radiolaria, for instance the genus Collosphaera[471]. Johannes Müller
(whose knowledge and insight never fail to astonish us) remarked that
these were identical in form with crystals of celestine, a sulphate
of strontium and barium; and Bütschli’s discovery of sulphates of
strontium and of barium in kindred forms render it all but certain that
they are actually true crystals of celestine[472].

In its typical form, the Radiolarian body consists of a spherical
mass of protoplasm, around which, and separated from it by some sort
of porous “capsule,” lies a frothy mass, composed of protoplasm
honeycombed into a multitude of alveoli or vacuoles, filled with a
fluid which can scarcely differ much from sea-water[473]. According
to their surface-tension conditions, these vacuoles may appear more
or less isolated and spherical, or joining together in a “froth” of
polygonal cells; and in the latter, which is the commoner condition,
the cells tend to be of equal size, and the resulting polygonal
meshwork beautifully regular. In many cases, a large number of such
simple individual organisms are associated together, forming a floating
colony, and it is highly probable that many other forms, with whose
scattered skeletons we are alone acquainted, had in life formed part
likewise of a colonial organism.

In contradistinction to the sponges, in which the skeleton always
begins as a loose mass of isolated spicules, which only in a few
exceptional cases (such as Euplectella and Farrea) fuse into a
continuous network, the characteristic feature of the Radiolarians lies
in the possession of a continuous skeleton, in the form of a netted
mesh or perforated lacework, sometimes however replaced by and often
associated with minute independent spicules. Before we proceed to treat
of the more complex skeletons, we may begin, then, by dealing with
these comparatively simple cases where either the entire skeleton or
a considerable part of it is represented, not by a continuous fabric,
but by a quantity of loose, separate spicules, or aciculae, which seem,
like the spicules of Alcyonium, {460} to be developed as free and
isolated formations or deposits, precipitated in the colloid matrix,
with no relation of form to the cellular or vesicular boundaries.
These simple acicular spicules occupy a definite position in the
organism. Sometimes, as for instance among the fresh-water Heliozoa
(e.g. Raphidiophrys), they lie on the outer surface of the organism,
and not infrequently (when the spicules are few in number) they tend
to collect round the bases of the pseudopodia, or around the large
radiating spicules, or axial rays, in the cases where these latter are
present. When the spicules are thus localised around some prominent
centre, they tend to take up a position of symmetry in regard to it;
instead of forming a tangled or felted layer, they come to lie side by
side, in a radiating cluster round the focus. In other cases (as for
instance in the well-known Radiolarian _Aulacantha scolymantha_) the
felted layer of aciculae lies at some depth below the surface, forming
a sphere concentric with the entire spherical organism. In either case,
whether the layer of spicules be deep or be superficial, it tends to
mark a “surface of discontinuity,” a meeting place between two distinct
layers of protoplasm or between the protoplasm and the water around;
and it is obvious that, in either case, there are manifestations
of surface-energy at the boundary, which cause the spicules to be
retained there, and to take up their position in its plane. The case
is somewhat, though not directly, analogous to that of a cirrus cloud,
which marks the place of a surface of discontinuity in a stratified
atmosphere.

[Illustration: Fig. 218.]

We have, then, to enquire what are the conditions which shall, apart
from gravity, confine an extraneous body to a surface-film; and we may
do this very simply, by considering the surface-energy of the entire
system. In Fig. 218 we have two fluids in contact with one another
(let us call them water and protoplasm), and a body (_b_) which may
be immersed in either, or may be restricted to the boundary {461}
between. We have here three possible “interfacial contacts” each with
its own specific surface-energy, per unit of surface area: namely, that
between our particle and the water (let us call it α), that between the
particle and the protoplasm (β), and that between water and protoplasm
(γ). When the body lies in the boundary of the two fluids, let us say
half in one and half in the other, the surface-energies concerned are
equivalent to (_S_/2)α + (_S_/2)β; but we must also remember that, by
the presence of the particle, a small portion (equal to its sectional
area _s_) of the original contact-surface between water and protoplasm
has been obliterated, and with it a proportionate quantity of energy,
equivalent to _s_γ, has been set free. When, on the other hand, the
body lies entirely within one or other fluid, the surface-energies of
the system (so far as we are concerned) are equivalent to _S_α + _s_γ,
or _S_β + _s_γ, as the case may be. According as α be less or greater
than β, the particle will have a tendency to remain immersed in the
water or in the protoplasm; but if (_S_/2)(α + β) − _s_γ be less than
either _S_α or _S_β, then the condition of minimal potential will be
found when the particle lies, as we have said, in the boundary zone,
half in one fluid and half in the other; and, if we were to attempt
a more general solution of the problem, we should evidently have to
deal with possible conditions of equilibrium under which the necessary
balance of energies would be attained by the particle rising or sinking
in the boundary zone, so as to adjust the relative magnitudes of the
surface-areas concerned. It is obvious that this principle may, in
certain cases, help us to explain the position even of a _radial_
spicule, which is just a case where the surface of the solid spicule is
distributed between the fluids with a minimal disturbance, or minimal
replacement, of the original surface of contact between the one fluid
and the other.

In like manner we may provide for the case (a common and an important
one) where the protoplasm “creeps up” the spicule, covering it with
a delicate film. In Acanthocystis we have yet another special case,
where the radial spicules plunge only a certain distance into the
protoplasm of the cell, being arrested at a boundary-surface between
an inner and an outer layer of cytoplasm; here we have only to assume
that there is a tension {462} at this surface, between the two layers
of protoplasm, sufficient to balance the tensions which act directly on
the spicule[474].

In various Acanthometridae, besides such typical characters as the
radial symmetry, the concentric layers of protoplasm, and the capillary
surfaces in which the outer, vacuolated protoplasm is festooned
upon the projecting radii, we have another curious feature. On the
surface of the protoplasm where it creeps up the sides of the long
radial spicules, we find a number of elongated bodies, forming in
each case one or several little groups, and lying neatly arranged in
parallel bundles. A Russian naturalist, Schewiakoff, whose views have
been accepted in the text-books, tells us that these are muscular
structures, serving to raise or lower the conical masses of protoplasm
about the radial spicules, which latter serve as so many “tent-poles”
or masts, on which the protoplasmic membranes are hoisted up; and the
little elongated bodies are dignified with various names, such as
“myonemes” or “myophriscs,” in allusion to their supposed muscular
nature[475]. This explanation is by no means convincing. To begin
with, we have precisely similar festoons of protoplasm in a multitude
of other cases where the “myonemes” are lacking; from their minute
size (·006–·012 mm.) and the amount of contraction they are said to be
capable of, the myonemes can hardly be very efficient instruments of
traction; and further, for them to act (as is alleged) for a specific
purpose, namely the “hydrostatic regulation” of the organism giving it
power to sink or to swim, would seem to imply a mechanism of action
and of coordination which is difficult to conceive in these minute
and simple organisms. The fact is (as it seems to me), that the whole
method of explanation is unnecessary. Just as the supposed “hauling
up” of the protoplasmic festoons is at once explained by capillary
phenomena, so also, in all probability, is the position and arrangement
of the little elongated bodies. Whatever the actual nature of these
bodies may be, whether they are truly portions of differentiated
protoplasm, or whether they are foreign bodies or spicular structures
(as bodies occupying a similar position in other cases undoubtedly
are), we can explain their situation on the surface {463} of the
protoplasm, and their arrangement around the radial spicules, all on
the principles of surface-tension[476].

This last case is not of the simplest; and I do not forget that my
explanation of it, which is wholly theoretical, implies a doubt
of Schewiakoff’s statements, which are founded on direct personal
observation. This I am none too willing to do; but whether it be justly
done in this case or not, I hold that it is in principle justifiable
to look with great suspicion upon a number of kindred statements where
it is obvious that the observer has left out of account the purely
physical aspect of the phenomenon, and all the opportunities of simple
explanation which the consideration of that aspect might afford.

――――――――――

Whether it be wholly applicable to this particular and complex case or
no, our general theorem of the localisation and arrestment of solid
particles in a surface-film is of very great biological importance; for
on it depends the power displayed by many little naked protoplasmic
organisms of covering themselves with an “agglutinated” shell.
Sometimes, as in _Difflugia_, _Astrorhiza_ (Fig. 219) and others,
this covering consists of sand-grains picked up from the surrounding
medium, and sometimes, on the other hand, as in _Quadrula_, it consists
of solid particles which are said to arise, as inorganic deposits or
concretions, within the protoplasm itself, and which find their way
outwards to a position of equilibrium in the surface-layer; and in
both cases, the mutual capillary attractions between the particles,
confined to the boundary-layer but enjoying a certain measure of
freedom therein, tends to the orderly arrangement of the particles one
with another, and even to the appearance of a regular “pattern” as the
result of this arrangement.

[Illustration: Fig. 219. Arenaceous Foraminifera; _Astrorhiza limicola_
and _arenaria_. (From Brady’s _Challenger Monograph_.)]

The “picking up” by the protoplasmic organism of a solid particle with
which “to build its house” (for it is hard to avoid this customary use
of anthropomorphic figures of speech, misleading though they be), is
a physical phenomenon kindred to that by which an Amoeba “swallows” a
particle of food. This latter process has been reproduced or imitated
in various pretty experimental {465} ways. For instance, Rhumbler has
shewn that if a thread of glass be covered with shellac and brought
near a drop of chloroform suspended in water, the drop takes in the
spicule, robs it of its shellac covering, and then passes it out
again[477]. It is all a question of relative surface-energies, leading
to different degrees of “adhesion” between the chloroform and the
glass or its covering. Thus it is that the Amoeba takes in the diatom,
dissolves off its proteid covering, and casts out the shell.

Furthermore, as the whole phenomenon depends on a distribution of
surface-energy, the amount of which is specific to certain particular
substances in contact with one another, we have no difficulty
in understanding the _selective action_, which is very often a
conspicuous feature in the phenomenon[478]. Just as some caddis-worms
make their houses of twigs, and others of shells and again others
of stones, so some Rhizopods construct their agglutinated “test”
out of stray sponge-spicules, or frustules of diatoms, or again
of tiny mud particles or of larger grains of sand. In all these
cases, we have apparently to deal with differences in specific {466}
surface-energies, and also doubtless with differences in the total
available amount of surface-energy in relation to gravity or other
extraneous forces. In my early student days, Wyville Thomson used
to tell us that certain deep-sea “Difflugias,” after constructing a
shell out of particles of the black volcanic sand common in parts of
the North Atlantic, finished it off with “a clean white collar” of
little grains of quartz. Even this phenomenon may be accounted for on
surface-tension principles, if we assume that the surface-energy ratios
have tended to change, either with the growth of the protoplasm or by
reason of external variation of temperature or the like; and we are
by no means obliged to attribute the phenomenon to a manifestation of
volition, or taste, or aesthetic skill, on the part of the microscopic
organism. Nor, when certain Radiolaria tend more than others to attract
into their own substance diatoms and such-like foreign bodies, is it
scientifically correct to speak, as some text-books do, of species “in
which diatom selection has become _a regular habit_.” To do so is an
exaggerated misuse of anthropomorphic phraseology.

The formation of an “agglutinated” shell is thus seen to be a purely
physical phenomenon, and indeed a special case of a more general
physical phenomenon which has many other important consequences in
biology. For the shell to assume the solid and permanent character
which it acquires, for instance, in Difflugia, we have only to make
the additional assumption that some small quantities of a cementing
substance are secreted by the animal, and that this substance flows
or creeps by capillary attraction between all the interstices of the
little quartz grains, and ends by binding them all firmly together.
Rhumbler[479] has shewn us how these agglutinated tests, of spicules
or of sand-grains, can be precisely imitated, and how they are formed
with greater or less ease, and greater or less rapidity, according to
the nature of the materials employed, that is to say, according to the
specific surface-tensions which are involved. For instance if we mix up
a little powdered glass with chloroform, and set a drop of the mixture
in water, the glass particles gather neatly round the surface of the
drop so quickly that the eye cannot follow the {467} operation. If we
perform the same experiment with oil and fine sand, dropped into 70
per cent. alcohol, a still more beautiful artificial Rhizopod shell is
formed, but it takes some three hours to do.

It is curious that, just at the very time when Rhumbler was thus
demonstrating the purely physical nature of the Difflugian shell,
Verworn was studying the same and kindred organisms from the older
standpoint of an incipient psychology[480]. But, as Rhumbler himself
admits, Verworn was very careful not to overestimate the apparent signs
of volition, or selective choice, in the little organism’s use of the
material of its dwelling.

――――――――――

This long parenthesis has led us away, for the time being, from the
subject of the Radiolarian skeleton, and to that subject we must now
return. Leaving aside, then, the loose and scattered spicules, which
we have sufficiently discussed, the more perfect Radiolarian skeletons
consist of a continuous and regular structure; and the siliceous (or
other inorganic) material of which this framework is composed tends
to be deposited in one or other of two ways or in both combined: (1)
in the form of long spicular axes, usually conjoined at, or emanating
from, the centre of the protoplasmic body, and forming a symmetric
radial system; (2) in the form of a crust, developed in various ways,
either on the outer surface of the organism or in relation to the
various internal surfaces which separate its concentric layers or its
component vesicles. Not unfrequently, this superficial skeleton comes
to constitute a spherical shell, or a system of concentric or otherwise
associated spheres.

[Illustration: Fig. 220. “Reticulum plasmatique.” (After Carnoy.)]

We have already learned that a great part of the body of the
Radiolarian, and especially that outer portion to which Haeckel has
given the name of the “calymma,” is built up of a great mass of
“vesicles,” forming a sort of stiff froth, and equivalent in the
physical sense (though not necessarily in the biological sense) to
“cells,” inasmuch as the little vesicles have their own well-defined
boundaries, and their own surface phenomena. In short, all that we have
said of cell-surfaces, and cell conformations, in our discussion of
cells and of tissues, will apply in like manner, and under appropriate
conditions, to these. In certain cases, even in {468} so common and
simple a one as the vacuolated substance of an Actinosphaerium, we
may see a very close resemblance, or formal analogy, to an ordinary
cellular or “parenchymatous” tissue, in the close-packed arrangement
and consequent configuration of these vesicles, and even at times in
a slight membranous hardening of their walls. Leidy has figured[481]
some curious little bodies, like small masses of consolidated froth,
which seem to be nothing else than the dead and empty husks, or filmy
skeletons, of Actinosphaerium. And Carnoy[482] has demonstrated in
certain cell-nuclei an all but precisely similar framework, of extreme
delicacy and minuteness, as the result of partial solidification of
interstitial matter in a close-packed system of alveoli (Fig. 220).

[Illustration: Fig. 221. _Aulonia hexagona_, Hkl.]

[Illustration: Fig. 222. _Actinomma arcadophorum_, Hkl.]

Let us now suppose that, in our Radiolarian, the outer surface of the
animal is covered by a layer of froth-like vesicles, uniform or nearly
so in size. We know that their tensions will tend to conform them into
a “honeycomb,” or regular meshwork of hexagons, and that the free end
of each hexagonal prism will be a little spherical cap. Suppose now
that it be at the outer surface of the protoplasm (that namely which
is in contact with the surrounding sea-water), that the siliceous
particles have a tendency to be secreted or adsorbed; it will at once
follow that they will show a tendency to aggregate in the grooves which
separate the vesicles, and the result will be the development of a most
delicate sphere composed of tiny rods arranged in a regular hexagonal
network (e.g. _Aulonia_). Such a conformation is {469} extremely
common, and among its many variants may be found cases in which (e.g.
_Actinomma_), the vesicles have been less regular in size, and some
in which the hexagonal meshwork has been developed not only on one
outer surface, but at successive {470} surfaces, producing a system
of concentric spheres. If the siliceous material be not limited to
the linear junctions of the cells, but spread over a portion of the
outer spherical surfaces or caps, then we shall have the condition
represented in Fig. 223 (_Ethmosphaera_), where the shell appears
perforated by circular instead of hexagonal apertures, and the circular
pores are set on slight spheroidal eminences; and, interconnected with
such types as this, we have others in which the accumulating pellicles
of skeletal matter have extended from the edges into the substance of
the boundary walls and have so produced a system of films, normal to
the surface of the sphere, constituting a very perfect honeycomb, as in
_Cenosphaera favosa_ and _vesparia_[483].

[Illustration: Fig. 223. _Ethmosphaera conosiphonia_, Hkl.]

[Illustration: Fig. 224. Portions of shells of two “species” of
_Cenosphaera_: upper figure, _C. favosa_, lower, _C. vesparia_, Hkl.]

In one or two very simple forms, such as the fresh-water _Clathrulina_,
just such a spherical perforated shell is produced out of some organic,
acanthin-like substance; and in some examples of _Clathrulina_ the
chitinous lattice-work of the shell is just as {471} regular and
delicate, with the meshes just as beautifully hexagonal, as in the
siliceous shells of the oceanic Radiolaria. This is only another proof
(if proof be needed) that the peculiar conformation of these little
skeletons is not due to the material of which they are composed, but to
the moulding of that material upon an underlying vesicular structure.

[Illustration: Fig. 225. _Aulastrum triceros_, Hkl.]

Let us next suppose that, upon some such lattice-work as has just
been described, another and external layer of cells or vesicles is
developed, and that instead of (or perhaps only in addition to) a
second hexagonal lattice-work, which might develop concentrically to
the first in the boundary-furrows of this new layer of cells, the
siliceous matter now tends to be deposited radially, or normally to
the surface of the sphere, just in the lines where the external layer
of vesicles meet one another, three by three. The result will be
that, when the vesicles themselves are removed, a series of radiating
spicules will be revealed, directed outwards from each of the angles of
the original hexagon; as is seen in Fig. 225. And it may further happen
that these radiating skeletal rods are continued at their distal ends
into divergent rays, forming a triple fork, and corresponding (after a
fashion {472} which we have already described as occurring in certain
sponge-spicules) to the three superficial furrows between the adjacent
cells. This last is, as it were, an intermediate stage between the
simple rods and the complete formation of another concentric sphere of
latticed hexagons. Another possible case is when the large and uniform
vesicles of the outer protoplasm are mixed with, or replaced by, much
smaller vesicles, piled on one another in more or less concentric
layers; in this case the radiating rods will no longer be straight,
but will be bent into a zig-zag pattern, with angles in three vertical
planes, corresponding to the successive contacts of the groups of cells
around the axis (Fig. 226).

[Illustration: Fig. 226.]

[Illustration: Fig. 227. A Nassellarian skeleton, _Callimitra
carolotae_, Hkl.]

――――――――――

Among a certain group called the Nassellaria, we find geometrical
forms of peculiar simplicity and beauty,—such for instance as that
which I have represented in Fig. 227. It is obvious at a glance that
this is such a skeleton as may have been formed {473} (I think we
may go so far as to say _must_ have been formed) at the interfaces
of a little tetrahedral group of cells, the four equal cells of the
tetrahedron being in this particular case supplemented by a little
one in the centre of the system. We see, precisely as in the internal
boundary-system of an artificial group of four soap-bubbles, the plane
surfaces of contact, six in number; the relation to one another of
each triple set of interfacial planes, meeting one another at equal
angles of 120°; and finally the relation of the four lines or edges
of triple contact, which tend (but for the little central vesicle)
to meet at co-equal solid angles in the centre of the system, all as
we have described on p. 318. In short, each triple-walled re-entrant
angle of the little shell has essentially the configuration (or a part
thereof) of what we have called a “Maraldi pyramid” in our account of
the architecture of the honeycomb, on p. 329[484].

There are still two or three remarkable or peculiar features in this
all but mathematically perfect shell, and they are in part easy and in
part they seem more difficult of interpretation.

[Illustration: Fig. 228. An isolated portion of the skeleton of
_Dictyocha_.]

[Illustration: Fig. 229. _Dictyocha stapedia_, Hkl.]

We notice that the amount of solid matter deposited in the plane
interfacial boundaries is greatly increased at the outer margin of
each boundary wall, where it merges or coincides with the superficial
furrow which separates the free, spherical surfaces of the bubbles
from one another; and we may sometimes find that, along these edges,
the skeleton remains complete and strong, while it shows signs of
imperfect development or of breaking away over great part of the rest
of the interfacial surfaces. In this there is nothing anomalous, for
we have already recognised that it is at the edges or margins of the
interfacial partition-walls that the manifestation of surface-energy
will tend to reach its maximum. And just as we have seen that, in
certain of our “multicellular” spherical Radiolarians, it is at the
superficial {474} edges or borders of the partitions, and here only,
that skeletal formation occurs (giving rise to the netted shell with
its hexagonal meshes of Fig. 221), so also at times, in the case of
such little aggregates of cells or vesicles as the four-celled system
of Callimitra, it may happen that about the external boundary-_lines_,
and not in the interior boundary-_planes_, the whole of the skeletal
matter is aggregated. In Fig. 228 we see a curious little skeletal
structure or complex spicule, whose conformation is easily accounted
for after this fashion. Little spicules such as this form isolated
portions of the skeleton in the genus _Dictyocha_, and occur scattered
over the spherical surface of the organism (Fig. 229). The more or
less basket-shaped spicule has evidently been developed about a little
cluster of four cells or vesicles, lying in or on the plane of the
surface of the organism, and therefore arranged, not in the tetrahedral
form of Callimitra, but in the manner in which four contiguous cells
lying side by side normally set themselves, like the four cells of a
segmenting egg: that is to say with an intervening “polar furrow,”
whose ends mark the meeting place, at equal angles, of the cells in
groups of three.

The little projecting spokes, or spikes, which are set normally to the
main basket-work, seem to be incompleted portions of a larger basket,
or in other words imperfectly formed elements corresponding to the
interfacial contacts in the surrounding parts {475} of the system.
Similar but more complex formations, all explicable as basket-like
frameworks developed around a cluster of cells, are known in great
variety.

In our Nassellarian itself, and in many other cases where the plane
interfacial boundary-walls are skeletonised, we see that the siliceous
matter is not deposited in an even and continuous layer, like the waxen
walls of a bee’s cell, but constitutes a meshwork of fine curvilinear
threads; and the curves seem to run, on the whole, isogonally, and to
form three main series, one approximately parallel to, or concentric
with, the outer or free edge of the partition, and the other two
related severally to its two edges of attachment. Sometimes (as may
also be seen in our figure), the system is still further complicated
by a fourth series of linear elements, which tend to run radially from
the centre of the system to the free edge of each partition. As regards
the former, their arrangement is such as would result if deposition or
solidification had proceeded in waves, starting independently from each
of the three boundaries of the little partition-wall; and something
of this kind is doubtless what has happened. We are reminded at once
of the wave-like periodicity of the Liesegang phenomenon. But apart
from this we might conceive of other explanations. For instance, the
liquid film which originally constitutes the partition must easily be
thrown into _vibrations_, and (like the dust upon a Chladni’s plate)
minute particles of matter in contact with the film would tend to take
up their position in a symmetrical arrangement, in direct relation to
the nodal points or lines of the vibrating surface[485]. Some such
explanation as this (to my thinking) must be invoked to account for
the minute and varied and very beautiful patterns upon many diatoms,
the resemblance of which patterns (in certain of their simpler cases)
to the Chladni figures is sometimes striking and obvious. But the
many special problems which the diatom skeleton suggests I have not
attempted to consider.

[Illustration: Fig. 230.]

The last peculiarity of our Nassellarian lies in an apparent departure
from what we should at first expect in the way of its {476} external
symmetry. Were the system actually composed of four spherical vesicles
in mutual contact, the outer margin of each of the six interfacial
planes would obviously be a circular arc; and accordingly, at each
angle of the tetrahedron, we should expect to have a depressed, or
re-entrant angle, instead of a prominent cusp. This is all doubtless
due to some simple balance of tensions, whose precise nature and
distribution is meanwhile a matter of conjecture. But it seems as
though an extremely simple explanation would go a long way, and
possibly the whole way, to meet this particular case. In our ordinary
plane diagram of three cells, or soap-bubbles, in contact, we know
(and we have just said) that the tensions of the three partitions draw
inwards the outer walls of the system, till at each point of triple
contact (_P_) we tend to get a triradiate, equiangular junction. But
if we introduce another bubble into the centre of the system (Fig.
230), then, as Plateau shewed, the tensions of its walls and those of
the three partitions by which it is now suspended, again balance one
another, and the central bubble appears (in plane projection) as a
curvilinear, equilateral triangle. We have only got to convert this
plane diagram into that of a tetrahedral solid to obtain _almost_
precisely the configuration which we are seeking to explain. Now we
observe that, so far as our figure of Callimitra informs us, this
is just the shape of the little bubble which occupies the centre of
the tetrahedral system in that Radiolarian skeleton. And I conceive,
accordingly, that the entire organism was not limited to the four cells
or vesicles (together with the little central {477} fifth) which
we have hitherto been imagining, but there must have been an outer
tetrahedral system, enclosing the cells which fabricated the skeleton,
just as these latter enclosed, and deformed, the little bubble in
the centre of all. We have only to suppose that this hypothetical
tetrahedral series, forming the outer layer or surface of the whole
system, was for some chemico-physical reason incapable of secreting at
its interfacial contacts a skeletal fabric[486].

In this hypothetical case, the edges of the skeletal system would be
circular arcs, meeting one another at an angle of 120°, or, in the
solid pyramid, of 109°: and this latter is _very nearly_ the condition
which our little skeleton actually displays. But we observe in Fig.
227 that, in the immediate neighbourhood of the tetrahedral angle, the
circular arcs are slightly drawn out into projecting cusps (cf. Fig.
230, _B_). There is no S-shaped curvature of the tetrahedral edges as a
whole, but a very slight one, a very slight change of curvature; close
to the apex. This, I conceive, is nothing more than what, in a material
system, we are bound to have, to represent a “surface of continuity.”
It is a phenomenon precisely analogous to Plateau’s “bourrelet,”
which we have already seen to be a constant feature of all cellular
systems, rounding off the sharp angular contacts by which (in our more
elementary treatment) we expect one film to make its junction with
another[487].

――――――――――

In the foregoing examples of Radiolaria, the symmetry which the
organism displays would seem to be identical with that symmetry of
forces which is due to the assemblage of surface-tensions in the
whole system; this symmetry being displayed, in one class of cases,
in a complex spherical mass of froth, and in {478} another class
in a simpler aggregate of a few, otherwise isolated, vesicles. But
among the vast number of other known Radiolaria, there are certain
forms (especially among the Phaeodaria and Acantharia) which display
a still more remarkable symmetry, the origin of which is by no means
clear, though surface-tension doubtless plays a part in its causation.
These are cases in which (as in some of those already described) the
skeleton consists (1) of radiating spicular rods, definite in number
and position, and (2) of interconnecting rods or plates, tangential to
the more or less spherical body of the organism, whose form becomes,
accordingly, that of a geometric, polyhedral solid. It may be that
there is no mathematical difference, save one of degree, between such a
hexagonal polyhedron as we have seen in _Aulacantha_, and those which
we are about to describe; but the greater regularity, the numerical
symmetry, and the apparent simplicity of these latter, makes of them a
class apart, and suggests problems which have not been solved nor even
investigated.

[Illustration: Fig. 231. Skeletons of various Radiolarians, after
Haeckel. 1. _Circoporus sexfurcus_; 2. _C. octahedrus_; 3. _Circogonia
icosahedra_; 4. _Circospathis novena_; 5. _Circorrhegma dodecahedra_.]

The matter is sufficiently illustrated by the accompanying figures,
all drawn from Haeckel’s Monograph of the Challenger Radiolaria[488].
In one of these we see a regular octahedron, in another a regular,
or pentagonal dodecahedron, in a third a regular icosahedron. In all
cases the figure appears to be perfectly symmetrical, though neither
the triangular facets of the octahedron and icosahedron, nor the
pentagonal facets of the dodecahedron, are necessarily plane surfaces.
In all of these cases, the radial spicules correspond to the solid
angles of the figure; and they are, accordingly, six in number in the
octahedron, twenty in the dodecahedron, and twelve in the icosahedron.
If we add to these three figures the regular tetrahedron, which we have
had frequent occasion to study, and the cube (which is represented,
at least in outline, in the skeleton of the hexactinellid sponges),
we have completed the series of the five regular polyhedra known to
geometers, the _Platonic bodies_[489] of the older mathematicians. It
is at first sight all the more remarkable that we should here meet
{480} with the whole five regular polyhedra, when we remember that,
among the vast variety of crystalline forms known among minerals, the
regular dodecahedron and icosahedron, simple as they are from the
mathematical point of view, never occur. Not only do these latter never
occur in Crystallography, but (as is explained in text-books of that
science) it has been shewn that they cannot occur, owing to the fact
that their indices (or numbers expressing the relation of the faces
to the three primary axes) involve an irrational quantity: whereas it
is a fundamental law of crystallography, involved in the whole theory
of space-partitioning, that “the indices of any and every face of a
crystal are small whole numbers[490].” At the same time, an imperfect
pentagonal dodecahedron, whose pentagonal sides are non-equilateral, is
common among crystals. If we may safely judge from Haeckel’s figures,
the pentagonal dodecahedron of the Radiolarian is perfectly regular,
and we must presume, accordingly, that it is not brought about by
principles of space-partitioning similar to those which manifest
themselves in the phenomenon of crystallisation. It will be observed
that in all these radiolarian polyhedral shells, the surface of each
external facet is formed of a minute hexagonal network, whose probable
origin, in relation to a vesicular structure, is such as we have
already discussed.

[Illustration: Fig. 232. _Dorataspis_ sp.; diagrammatic.]

In certain allied Radiolaria (Fig. 232), which, like the dodecahedral
form figured in Fig. 231, 5, have twenty radial spines, these latter
are commonly described as being arranged in a certain very singular
way. It is stated that their arrangement may be referred {481} to a
series of five parallel circles on the sphere, corresponding to the
equator (_c_), the tropics (_b_, _d_) and the polar circles (_a_, _e_);
and that beginning with four equidistant spines in the equator, we
have alternating whorls of four, radiating outwards from the sphere
in each of the other parallel zones. This rule was laid down by the
celebrated Johannes Müller, and has ever since been used and quoted
as Müller’s law. The chief point in this alleged arrangement which
strikes us at first sight as very curious, is that there is said to
be no spine at either pole; and when we come to examine carefully the
figure of the organism, we find that the received description does not
do justice to the facts. We see, in the first place, from such figures
as Figs. 232, 234, that here, unlike our former cases, the radial
spines issue through the facets (and through _all_ the facets) of the
polyhedron, instead of through its solid angles; and accordingly,
that our twenty spines correspond (not, as before, to a dodecahedron)
but to some sort of an icosahedron. We see in the next place, that
this icosahedron is composed of faces, or plates, of two different
kinds, some hexagonal and some pentagonal; and when we look closer,
we discover that the whole figure is that of a hexagonal prism, whose
twelve solid angles are replaced by pentagonal facets. Both hexagons
and pentagons {482} appear to be perfectly equilateral, but if we
try to construct a plane-sided polyhedron of this kind, we soon find
that it is impossible; for into the angles between the six equatorial
hexagons those of the six united pentagons will not fit. The figure
however can be easily constructed if we replace the straight edges
(or some of them) by curves, and the plane facets by corresponding,
slightly curved, surfaces. The true symmetry of this figure, then, is
hexagonal, with a polar axis, produced into two polar spicules; with
six equatorial spicules, or rays; and with two sets of six spicular
rays, interposed between the polar axis and the equatorial rays, and
alternating in position with the latter.

 Müller’s description was emended by Brandt, and what is now known as
 “Brandt’s law,” viz. that the symmetry consists of two polar rays, and
 three whorls of six each, coincides with the above description so far
 as the spicular axes go: save only that Brandt specifically states
 that the intermediate whorls stand equidistant between the equator
 and the poles, i.e. in latitude 45°. While not far from the truth,
 this statement is not exact; for according to the geometry of the
 figure, the intermediate cycles obviously stand in a slightly higher
 latitude, but this latitude I have not attempted to determine; for the
 calculation seems to be a little troublesome owing to the curvature of
 the sides of the figure, and the enquiring mathematician will perform
 it more easily than I. Brandt, if I understand him rightly, did not
 propose his “law” as a substitute for Müller’s law, but as a second
 law applicable to a few particular cases. I on the other hand can find
 no case to which Müller’s law properly applies.

If we construct such a polyhedron, and set it in the position of
Fig. 232, _B_, we shall easily see that it is capable of explanation
(though improperly) in accordance with Müller’s law; for the four
equatorial rays of Müller (_c_) now correspond to the two polar and
to two opposite equatorial facets of our polyhedron: the four “polar”
rays of Müller (_a_ or _e_) correspond to two adjacent hexagons and
two intermediate pentagons of the figure: and Müller’s “tropical”
rays (_b_ or _d_) are those which emanate from the remaining four
pentagonal facets, in each half of the figure. In some cases, such as
Haeckel’s _Phatnaspis cristata_ (Fig. 233), we have an ellipsoidal
body, from which the spines emerge in the order described, but which
is not obviously divided by facets. In Fig. 234 I have indicated the
facets corresponding to the rays, and dividing the surface in the usual
symmetrical way. {483}

[Illustration: Fig. 233. _Phatnaspis cristata_, Hkl.]

[Illustration: Fig. 234. The same, diagrammatic.]

{484}

Within any polyhedron we may always inscribe another polyhedron,
whose corners correspond in number to the sides or facets of the
original figure, or (in alternative cases) to a certain number of
these sides; and a similar result is obtained by bevelling off the
corners of the original polyhedron. We may obtain a precisely similar
symmetrical result if (in such a case as these Radiolarians which we
are describing), we imagine the radial spines to be interconnected by
tangential rods, instead of by the complete facets which we have just
been dealing with. In our complicated polyhedron with its twenty radial
spines arranged in the manner described there are various symmetrical
ways in which we may imagine these interconnecting bars to be arranged.
The most symmetrical of these is one in which the whole surface is
divided into eighteen rhomboidal areas, obtained by systematically
connecting each group of four adjacent radii. This figure has eighteen
faces (_F_), twenty corners (_C_), and therefore thirty-six edges
(_E_), in conformity with Euler’s theorem, _F_ + _C_ = _E_ + 2.

[Illustration: Fig. 235. _Phractaspis prototypus_, Hkl.]

Another symmetrical arrangement will divide the surface into fourteen
rhombs and eight triangles. This latter arrangement is obtained by
linking up the radial rods as follows: _aaaa_, _aba_, _abcb_, _bcdc_,
etc. Here we have again twenty corners, but we have twenty-two faces;
the number of edges, or tangential spicular bars, will be found,
therefore, by the above formula, to be forty. In Haeckel’s figure of
_Phractaspis prototypus_ we have a spicular skeleton which appears to
be constructed precisely upon this plan, and to be derivable from the
faceted polyhedron precisely after this manner.

In all these latter cases it is the arrangement of the axial rods, or
in other words the “polar symmetry” of the entire organism, which lies
at the root of the matter, and which, if only {485} we could account
for it, would make it comparatively easy to explain the superficial
configuration. But there are no obvious mechanical forces by which we
can so explain this peculiar polarity. This at least is evident, that
it arises in the central mass of protoplasm, which is the essential
living portion of the organism as distinguished from that frothy
peripheral mass whose structure has helped us to explain so many
phenomena of the superficial or external skeleton. To say that the
arrangement depends upon a specific polarisation of the cell is merely
to refer the problem to other terms, and to set it aside for future
solution. But it is possible that we may learn something about the
lines in which _to seek for_ such a solution by considering the case
of Lehmann’s “fluid crystals,” and the light which they throw upon the
phenomena of molecular aggregation.

The phenomenon of “fluid crystallisation” is found in a number of
chemical bodies; it is exhibited at a specific temperature for each
substance; and it would seem to be limited to bodies in which there
is a more or less elongated, or “chain-like” arrangement of the atoms
in the molecule. Such bodies, at the appropriate temperature, tend
to aggregate themselves into masses, which are sometimes spherical
drops or globules (the so-called “spherulites”), and sometimes have
the definite form of needle-like or prismatic crystals. In either case
they remain liquid, and are also doubly refractive, polarising light
in brilliant colours. Together with them are formed ordinary solid
crystals, also with characteristic polarisation, and into such solid
crystals all the fluid material ultimately turns. It is evident that
in these liquid crystals, though the molecules are freely mobile,
just as are those of water, they are yet subject to, or endowed with,
a “directive force,” a force which confers upon them a definite
configuration or “polarity,” the _Gestaltungskraft_ of Lehmann.

Such an hypothesis as this had been gradually extruded from the
theories of mathematical crystallography[491]; and it had come to be
believed that the symmetrical conformation of a homogeneous crystalline
structure was sufficiently explained by the mere mechanical fitting
together of appropriate structural units along the easiest and simplest
lines of “close packing”: just as {486} a pile of oranges becomes
definite, both in outward form and inward structural arrangement,
without the play of any _specific_ directive force. But while our
conceptions of the tactical arrangement of crystalline molecules
remain the same as before, and our hypotheses of “modes of packing” or
of “space-lattices” remain as useful as ever for the definition and
explanation of the molecular arrangements, an entirely new theoretical
conception is introduced when we find such space-lattices maintained
in what has hitherto been considered the molecular freedom of a liquid
field; and we are constrained, accordingly, to postulate a specific
molecular force, or “Gestaltungskraft” (not unlike Kepler’s “facultas
formatrix”), to account for the phenomenon.

Now just as some sort of specific “Gestaltungskraft” had been of old
the _deus ex machina_ accounting for all crystalline phenomena (_gnara
totius geometriæ, et in ea exercita_, as Kepler said), and as such an
hypothesis, after being dethroned and repudiated, has now fought its
way back and has made good its right to be heard, so it may be also
in biology. We begin by an easy and general assumption of _specific
properties_, by which each organism assumes its own specific form; we
learn later (as it is the purpose of this book to shew) that throughout
the whole range of organic morphology there are innumerable phenomena
of form which are not peculiar to living things, but which are more
or less simple manifestations of ordinary physical law. But every now
and then we come to certain deep-seated signs of protoplasmic symmetry
or polarisation, which seem to lie beyond the reach of the ordinary
physical forces. It by no means follows that the forces in question
are not essentially physical forces, more obscure and less familiar
to us than the rest; and this would seem to be the crucial lesson for
us to draw from Lehmann’s surprising and most beautiful discovery.
For Lehmann seems actually to have demonstrated, in non-living,
chemical bodies, the existence of just such a determinant, just such
a “Gestaltungskraft,” as would be of infinite help to us if we might
postulate it for the explanation (for instance) of our Radiolarian’s
axial symmetry. But further than this we cannot go; for such analogy as
we seem to see in the Lehmann phenomenon soon evades us, and refuses
to be pressed home. Not only is it the case, as we have already {487}
seen, that certain of the geometric forms assumed by the symmetrical
Radiolarian shells are just such as the “space-lattice” theory would
seem to be inapplicable to, but it is in other ways obvious that
symmetry of _crystallisation_, whether liquid or solid, has no close
parallel, but only a series of analogies, in the protoplasmic symmetry
of the living cell.

{488}




CHAPTER X

A PARENTHETIC NOTE ON GEODETICS


We have made use in the last chapter of the mathematical principle of
Geodetics (or Geodesics) in order to explain the conformation of a
certain class of sponge-spicules; but the principle is of much wider
application in morphology, and would seem to deserve attention which it
has not yet received.

[Illustration: Fig. 236. Annular and spiral thickenings in the walls of
plant-cells.]

Defining, meanwhile, our geodetic line (as we have already done) as
the shortest distance between two points on the surface of a solid of
revolution, we find that the geodetics of the cylinder give us one of
the simplest of cases. Here it is plain that the geodetics are of three
kinds: (1) a series of annuli around the cylinder, that is to say,
a system of circles, in planes parallel to one another and at right
angles to the axis of the cylinder (Fig. 236, _a_); (2) a series of
straight lines parallel to the axis; and (3) a series of spiral curves
winding round the wall of the cylinder (_b_, _c_). These three systems
are all of frequent occurrence, and are all illustrated in the local
thickenings of the wall of the cylindrical cells or vessels of plants.

The spiral, or rather helicoid, geodetic is particularly common in
cylindrical structures, and is beautifully shewn for instance in the
spiral coil which stiffens the tracheal tubes of an insect, or the
so-called “tracheides” of a woody stem. A similar {489} phenomenon is
often witnessed in the splitting of a glass tube. If a crack appear in
a thin tube, such as a test-tube, it has a tendency to be prolonged in
its own direction, and the more perfectly homogeneous and isotropic be
the glass the more evenly will the split tend to follow the straight
course in which it began. As a result, the crack in our test-tube is
often seen to continue till the tube is split into a continuous spiral
ribbon.

In a right cone, the spiral geodetic falls into closer and closer coils
as the diameter of the cone narrows; and a very beautiful geodetic
of this kind is exemplified in the sutural line of a spiral shell,
such as Turritella, or in the striations which run parallel with the
spiral suture. Similarly, in an ellipsoidal surface, we have a spiral
geodetic, whose coils get closer together as we approach the ends of
the long axis of the ellipse; in the splitting of the integument of an
Equisetum-spore, by which are formed the spiral “elaters” of the spore,
we have a case of this kind, though the spiral is not sufficiently
prolonged to shew all its features in detail.

We have seen in these various cases, that our original definition of a
geodetic requires to be modified; for it is only subject to conditions
that it is “the shortest distance between two points on the surface of
the solid,” and one of the commonest of these restricting conditions is
that our geodetic may be constrained to go twice, or many times, round
the surface on its way. In short, we must redefine our geodetic, as a
curve drawn upon a surface, such that, if we take any two _adjacent_
points on the curve, the curve gives the shortest distance between
them. Again, in the geodetic systems which we meet with in morphology,
it sometimes happens that we have two opposite systems of geodetic
spirals separate and distinct from one another, as in Fig. 236, _c_;
and it is also common to find the two systems interfering with one
another, and forming a criss-cross, or reticulated arrangement. This is
a very common source of reticulated patterns.

Among the ciliated Infusoria, we have in the spiral lines along which
their cilia are arranged a great variety of beautiful geodetic curves;
though it is probable enough that in some complicated cases these are
not simple geodetics, but projections of curves other than a straight
line upon the surface of the solid. {490}

Lastly, a very instructive case is furnished by the arrangement of
the muscular fibres on the surface of a hollow organ, such as the
heart or the stomach. Here we may consider the phenomenon from the
point of view of mechanical efficiency, as well as from that of purely
descriptive or objective anatomy. In fact we have an _a priori_ right
to expect that the muscular fibres covering such hollow or tubular
organs will coincide with geodetic lines, in the sense in which we are
now using the term. For if we imagine a contractile fibre, or elastic
band, to be fixed by its two ends upon a curved surface, it is obvious
that its first effort of contraction will tend to expend itself in
accommodating the band to the form of the surface, in “stretching it
tight,” or in other words in causing it to assume a direction which is
the shortest possible line _upon the surface_ between the two extremes:
and it is only then that further contraction will have the effect of
constricting the tube and so exercising pressure on its contents. Thus
the muscular fibres, as they wind over the curved surface of an organ,
arrange themselves automatically in geodesic curves: in precisely
the same manner as we also automatically construct complex systems
of geodesics whenever we wind a ball of wool or a spindle of tow, or
when the skilful surgeon bandages a limb. In these latter cases we see
the production of those “figures-of-eight,” to which, in the case for
instance of the heart-muscles, Pettigrew and other anatomists have
ascribed peculiar importance. In the case of both heart and stomach
we must look upon these organs as developed from a simple cylindrical
tube, after the fashion of the glass-blower, as is further discussed on
p. 737 of this book, the modification of the simple cylinder consisting
of various degrees of dilatation and of twisting. In the primitive
undistorted cylinder, as in an artery or in the intestine, the muscular
fibres run in geodetic lines, which as a rule are not spiral, but are
merely either annular or longitudinal; these are the ordinary “circular
and longitudinal coats,” which form the normal musculature of all
tubular organs, or of the body-wall of a cylindrical worm[492]. If we
consider each muscular fibre as an elastic strand, imbedded in the
elastic membrane which constitutes the wall of the organ, it {491} is
evident that, whatever be the distortion suffered by the entire organ,
the individual fibre will follow the same course, which will still, in
a sense, be a geodetic. But if the distortion be considerable, as for
instance if the tube become bent upon itself, or if at some point its
walls bulge outwards in a diverticulum or pouch, it is obvious that the
old system of geodetics will only mark the shortest distance between
two points more or less approximate to one another, and that new
systems of geodetics will tend to appear, peculiar to the new surface,
and linking up points more remote from one another. This is evidently
the case in the human stomach. We still have the systems, or their
unobliterated remains, of circular and longitudinal muscles; but we
also see two new systems of fibres, both obviously geodetic (or rather,
when we look more closely, both parts of one and the same geodetic
system), in the form of annuli encircling the pouch or diverticulum at
the cardiac end of the stomach, and of oblique fibres taking a spiral
course from the neighbourhood of the oesophagus over the sides of the
organ.

――――――――――

In the heart we have a similar, but more complicated phenomenon. Its
musculature consists, in great part, of the original simple system of
circular and longitudinal muscles which enveloped the original arterial
tubes, which tubes, after a process of local thickening, expansion, and
especially _twisting_, came together to constitute the composite, or
double, mammalian heart; and these systems of muscular fibres, geodetic
to begin with, remain geodetic (in the sense in which we are using the
word) after all the twisting to which the primitive cylindrical tube or
tubes have been subjected. That is to say, these fibres still run their
shortest possible course, from start to finish, over the complicated
curved surface of the organ; and it is only because they do so that
their contraction, or longitudinal shortening, is able to produce
its direct effect, as Borelli well understood, in the contraction or
systole of the heart[493]. {492}

As a parenthetic corollary to the case of the spiral pattern upon the
wall of a cylindrical cell, we may consider for a moment the spiral
line which many small organisms tend to follow in their path of
locomotion[494]. The helicoid spiral, traced around the wall of our
cylinder, may be explained as a composition of two velocities, one a
uniform velocity in the direction of the axis of the cylinder, the
other a uniform velocity in a circle perpendicular to the axis. In a
somewhat analogous fashion, the smaller ciliated organisms, such as
the ciliate and flagellate Infusoria, the Rotifers, the swarm-spores
of various Protists, and so forth, have a tendency to combine a
direct with a revolving path in their ordinary locomotion. The means
of locomotion which they possess in their cilia are at best somewhat
primitive and inefficient; they have no apparent means of steering,
or modifying their direction; and, if their course tended to swerve
ever so little to one side, the result would be to bring them round
and round again in an approximately circular path (such as a man
astray on the prairie is said to follow), with little or no progress
in a definite longitudinal direction. But as a matter of fact, either
through the direct action of their cilia or by reason of a more or
less unsymmetrical form of the body, all these creatures tend more or
less to _rotate_ about their long axis while they swim. And this axial
rotation, just as in the case of a rifle-bullet, causes their natural
swerve, which is always in the same direction as regards their own
bodies, to be in a continually changing direction as regards space: in
short, to make a spiral course around, and more or less approximate to,
a straight axial line.

{493}




CHAPTER XI

THE LOGARITHMIC SPIRAL


The very numerous examples of spiral conformation which we meet with
in our studies of organic form are peculiarly adapted to mathematical
methods of investigation. But ere we begin to study them, we must take
care to define our terms, and we had better also attempt some rough
preliminary classification of the objects with which we shall have to
deal.

In general terms, a Spiral Curve is a line which, starting from a point
of origin, continually diminishes in curvature as it recedes from that
point; or, in other words, whose _radius of curvature_ continually
increases. This definition is wide enough to include a number of
different curves, but on the other hand it excludes at least one which
in popular speech we are apt to confuse with a true spiral. This
latter curve is the simple Screw, or cylindrical Helix, which curve,
as is very evident, neither starts from a definite origin, nor varies
in its curvature as it proceeds. The “spiral” thickening of a woody
plant-cell, the “spiral” thread within an insect’s tracheal tube, or
the “spiral” twist and twine of a climbing stem are not, mathematically
speaking, _spirals_ at all, but _screws or helices_. They belong to
a distinct, though by no means very remote, family of curves. Some
of these helical forms we have just now treated of, briefly and
parenthetically, under the subject of Geodetics.

[Illustration: Fig. 237. The shell of _Nautilus pompilius_, from a
radiograph: to shew the logarithmic spiral of the shell, together with
the arrangement of the internal septa. (From Messrs Green and Gardiner,
in _Proc. Malacol. Soc._ II, 1897.)]

Of true organic spirals we have no lack[495]. We think at once of the
beautiful spiral curves of the horns of ruminants, and of the still
more varied, if not more beautiful, spirals of molluscan shells.
Closely related spirals may be traced in the arrangement {494} of the
florets in the sunflower; a true spiral, though not, by the way, so
easy of investigation, is presented to us by the outline of a cordate
leaf; and yet again, we can recognise typical though transitory spirals
in the coil of an elephant’s trunk, in the “circling {495} spires” of
a snake, in the coils of a cuttle-fish’s arm, or of a monkey’s or a
chameleon’s tail.

Among such forms as these, and the many others which we might easily
add to them, it is obvious that we have to do with things which,
though mathematically similar, are biologically speaking fundamentally
different. And not only are they biologically remote, but they are also
physically different, in regard to the nature of the forces to which
they are severally due. For in the first place, the spiral coil of
the elephant’s trunk or of the chameleon’s tail is, as we have said,
but a transitory configuration, and is plainly the result of certain
muscular forces acting upon a structure of a definite, and normally an
essentially different, form. It is rather a position, or an _attitude_,
than a _form_, in the sense in which we have been using this latter
term; and, unlike most of the forms which we have been studying, it has
little or no direct relation to the phenomenon of Growth.

[Illustration: Fig. 238. A Foraminiferal shell (Globigerina).]

Again, there is a manifest and not unimportant difference between such
a spiral conformation as is built up by the separate and successive
florets in the sunflower, and that which, in the snail or Nautilus
shell, is apparently a single and indivisible unit. And a similar, if
not identical difference is apparent between the Nautilus shell and
the minute shells of the Foraminifera, which so closely simulate it;
inasmuch as the spiral shells of these latter are essentially composite
structures, combined out of successive and separate chambers, while
the molluscan shell, though it may (as in Nautilus) become secondarily
subdivided, has grown as one continuous tube. It follows from all this
that there cannot {496} possibly be a physical or dynamical, though
there may well be a mathematical _Law of Growth_, which is common to,
and which defines, the spiral form in the Nautilus, in the Globigerina,
in the ram’s horn, and in the disc of the sunflower.

Of the spiral forms which we have now mentioned, every one (with the
single exception of the outline of the cordate leaf) is an example of
the remarkable curve known as the Logarithmic Spiral. But before we
enter upon the mathematics of the logarithmic spiral, let us carefully
observe that the whole of the organic forms in which it is clearly and
permanently exhibited, however different they may be from one another
in outward appearance, in nature and in origin, nevertheless all
belong, in a certain sense, to one particular class of conformations.
In the great majority of cases, when we consider an organism in part
or whole, when we look (for instance) at our own hand or foot, or
contemplate an insect or a worm, we have no reason (or very little)
to consider one part of the existing structure as _older_ than
another; through and through, the newer particles have been merged and
commingled, by intussusception, among the old; the whole outline, such
as it is, is due to forces which for the most part are still at work to
shape it, and which in shaping it have shaped it as a whole. But the
horn, or the snail-shell, is curiously different; for in each of these,
the presently existing structure is, so to speak, partly old and partly
new; it has been conformed by successive and continuous increments; and
each successive stage of growth, starting from the origin, remains as
an integral and unchanging portion of the still growing structure, and
so continues to represent what at some earlier epoch constituted for
the time being the structure in its entirety.

In a slightly different, but closely cognate way, the same is true of
the spirally arranged florets of the sunflower. For here again we are
regarding serially arranged portions of a composite structure, which
portions, similar to one another in form, _differ in age_; and they
differ also in magnitude in a strict ratio according to their age.
Somehow or other, in the logarithmic spiral the _time-element_ always
enters in; and to this important fact, full of curious biological as
well as mathematical significance, we shall afterwards return. {497}

It is, as we have so often seen, an essential part of our whole
problem, to try to understand what distribution of forces is capable
of producing this or that organic form,—to give, in short, a
dynamical expression to our descriptive morphology. Now the _general_
distribution of forces which lead to the formation of a spiral (whether
logarithmic or other) is very easily understood; and need not carry us
beyond the use of very elementary mathematics.

[Illustration: Fig. 239.]

If we imagine growth to act in a perpendicular direction, as for
example the upward force of growth in a growing stem (_OA_), then, in
the absence of other forces, elongation will as a matter of course
proceed in an unchanging direction, that is to say the stem will grow
straight upwards. Suppose now that there be some constant _external
force_, such as the wind, impinging on the growing stem; and suppose
(for simplicity’s sake) that this external force be in a constant
direction (_AB_) perpendicular to the intrinsic force of growth. The
direction of actual growth will be in the line of the resultant of the
two forces: and, since the external force is (by hypothesis) constant
in direction, while the internal force tends always to act in the line
of actual growth, it is obvious that our growing organism will tend to
be bent into a curve, to which, for the time being, {498} the actual
force of growth will be acting at a tangent. So long as the two forces
continue to act, the curve will approach, but will never attain, the
direction of _AB_, perpendicular to the original direction _OA_. If the
external force be constant in amount the curve will approximate to the
form of a hyperbola; and, at any rate, it is obvious that it will never
tend to assume a spiral form.

In like manner, if we consider a horizontal beam, fixed at one end, the
imposition of a weight at the other will bend the beam into a curve,
which, as the beam elongates or the weight increases, will bring the
weighted end nearer and nearer to the vertical. But such a force,
constant in direction, will obviously never curve the beam into a
spiral,—a fact so patent and obvious that it would be superfluous to
state it, were it not that some naturalists have been in the habit of
invoking gravity as the force to which may be attributed the spiral
flexure of the shell.

But if, on the other hand, the deflecting force be _inherent_ in the
growing body, or so connected with it in a system that its direction
(instead of being constant, as in the former case) changes with the
direction of growth, and is perpendicular (or inclined at some constant
angle) to this changing direction of the growing force, then it is
plain that there is no such limit to the deflection from the normal,
but the growing curve will tend to wind round and round its point of
origin. In the typical case of the snail-shell, such an intrinsic force
is manifestly present in the action of the columellar muscle.

Many other simple illustrations can be given of a spiral course being
impressed upon what is primarily rectilinear motion, by any steady
deflecting force which the moving body carries, so to speak, along with
it, and which continually gives a lop-sided tendency to its forward
movement. For instance, we have been told that a man or a horse,
travelling over a great prairie, is very apt to find himself, after a
long day’s journey, back again near to his starting point. Here some
small and imperceptible bias, such as might for instance be caused by
one leg being in a minute degree longer or stronger than the other, has
steadily deflected the forward movement to one side; and has gradually
brought the traveller back, perhaps in a circle to the very point from
which he set out, {499} or else by a spiral curve, somewhere within
reach and recognition of it.

[Illustration: Fig. 240.]

We come to a similar result when we consider, for instance, a
cylindrical body in which forces of growth are at work tending to its
elongation, but these forces are unsymmetrically distributed. Let the
tendency to elongation along _AB_ be of a magnitude proportional to
_BB′_, and that along _CD_ be of a magnitude proportional to _DD′_;
and in each element parallel to _AB_ and _CD_, let a parallel force
of growth, proportionately intermediate in magnitude, be at work: and
let _EFF′_ be the middle line. Then at any cross-section _BFD_, if
we deduct the mean force _FF′_, we have a certain positive force at
_B_, equal to _Bb_, and an equal and opposite force at _D_, equal to
_Dd_. But _AB_ and _CD_ are not separate structures, but are connected
together, either by a solid core, or by the walls of a tubular shell;
and the forces which tend to separate _B_ and _D_ are opposed,
accordingly, by a _tension_ in _BD_. It follows therefore, that there
will be a resultant force _BG_, acting in a direction intermediate
between _Bb_ and _BD_, and also a resultant, _DH_, acting at _D_ in an
opposite direction; and accordingly, after a small increment of growth,
the growing end of the cylinder will come to lie, not in the direction
_BD_, but in the direction _GH_. The problem is therefore analogous
to that of a beam to which we apply a bending moment; and it is plain
that the unequal force of growth is equivalent to a “_couple_” which
will impart to our structure a curved form. For, if we regard the part
_ABDC_ as practically rigid, and the part _BB′D′D_ as pliable, this
couple {500} will tend to turn strips such as _B′D′_ about an axis
perpendicular to the plane of the diagram, and passing through an
intermediate point _F′_. It is plain, also, since all the forces under
consideration are _intrinsic to the system_, that this tendency will be
continuous, and that as growth proceeds the curving body will assume
either a circular or a spiral form. But the tension which we have here
assumed to exist in the direction _BD_ will obviously disappear if we
suppose a sufficiently rapid rate of growth in that direction. For if
we may regard the mouth of our tubular shell as _perfectly extensible_
in its own plane, so that it exerts no traction whatsoever on the
sides, then it will be drawn out into more and more elongated ellipses,
forming the more and more oblique orifices of a _straight_ tube. In
other words, in such a structure as we have presupposed, the existence
or maintenance of a constant ratio between the rates of extension or
growth in the vertical and transverse directions will lead, in general,
to the development of a logarithmic spiral; the magnitude of that ratio
will determine the character (that is to say, the constant angle) of
the spiral; and the spirals so produced will include, as special or
limiting cases, the circle and the straight line.

[Illustration: Fig. 241.]

[Illustration: Fig. 242.]

We may dispense with the hypothesis of bending moments, if we simply
presuppose that the increments of growth take place at a constant angle
to the growing surface (as _AB_), but more rapidly at _A_ (which we
shall call the “outer edge”) than at _B_, and that this difference
of velocity maintains a constant ratio. Let us also assume that the
whole structure is rigid, the new accretions solidifying as soon
as they are laid on. For example, {501} let Fig. 242 represent in
section the early growth of a Nautilus-shell, and let the part _ARB_
represent the earliest stage of all, which in Nautilus is nearly
semicircular. We have to find a law governing the growth of the shell,
such that each edge shall develop into an equiangular spiral; and
this law, accordingly, must be the same for each edge, namely that
at each instant the direction of growth makes a constant angle with
a line drawn from a fixed point (called the pole of the spiral) to
the point at which growth is taking place. This growth, we now find,
may be considered as effected by the continuous addition of similar
quadrilaterals. Thus, in Fig. 241, _AEDB_ is a quadrilateral with
_AE_, _DB_ parallel, and with the angle _EAB_ of a certain definite
magnitude, = γ. Let _AB_ and _ED_ meet, when produced, in _C_; and
call the angle _ACE_ (or _xCy_) = β. Make the angle _yCz_ = angle
_xCy_, = β. Draw _EG_, so that the angle _yEG_ = γ, meeting _Cz_ in
_G_; and draw _DF_ parallel to _EG_. It is then easy to show that
_AEDB_ and _EGFD_ are similar quadrilaterals. And, when we consider the
quadrilateral _AEDB_ as having infinitesimal sides, _AE_ and _BD_, the
angle γ tends to α, the constant angle of an equiangular spiral which
passes through the points _AEG_, and of a similar spiral which passes
through the points _BDF_; and the point _C_ is the pole of both of
these spirals. In a particular limiting case, when our quadrilaterals
are all equal as well as similar,—which will be the case when the angle
γ (or the angles _EAC_, etc.) is a {502} right angle,—the “spiral”
curve will be a circular arc, _C_ being the centre of the circle.

 Another, and a very simple illustration may be drawn from the
 “cymose inflorescences” of the botanists, though the actual mode of
 development of some of these structures is open to dispute, and their
 nomenclature is involved in extraordinary historical confusion[496].

 [Illustration: Fig. 243. _A_, a helicoid, _B_, a scorpioid cyme.]

 In Fig. 243_B_ (which represents the _Cicinnus_ of Schimper, or _cyme
 unipare scorpioide_ of Bravais, as seen in the Borage), we begin
 with a primary shoot from which is given off, at a certain definite
 angle, a secondary shoot: and from that in turn, on the same side
 and at the same angle, another shoot, and so on. The deflection,
 or curvature, is continuous and progressive, for it is caused by
 no external force but only by causes intrinsic in the system. And
 the whole system is symmetrical: the angles at which the successive
 shoots are given off being all equal, and the lengths of the shoots
 diminishing _in constant ratio_. The result is that the successive
 shoots, or successive increments of growth, are tangents to a curve,
 and this curve is a true logarithmic spiral. But while, in this simple
 case, the successive shoots are depicted as lying _in a plane_, it may
 also happen that, in addition to their successive angular divergence
 from one another within that plane, they also tend to diverge by
 successive equal angles _from_ that plane of reference; and by this
 means, there will be superposed upon the logarithmic spiral a helicoid
 twist or screw. And, in the particular case where this latter angle of
 divergence is just equal to 180°, or two right angles, the successive
 shoots will once more come to lie in a plane, but they will appear to
 come off from one another on _alternate_ sides, as in Fig. 243 _A_.
 This is the _Schraubel_ or _Bostryx_ of Schimper, the _cyme unipare
 hélicoide_ of Bravais. The logarithmic spiral is still latent in
 it, as in the other; but is concealed from view by the deformation
 resulting from the helicoid. The confusion of nomenclature would seem
 to have arisen from the fact that many botanists did not recognise (as
 the brothers Bravais did) the mathematical significance of the latter
 case; but were led, by the snail-like spiral of the scorpioid cyme, to
 transfer the name “helicoid” to it.

In the study of such curves as these, then, we speak of the point of
origin as the pole (_O_); a straight line having its extremity in the
pole and revolving about it, is called the radius vector; {503} and a
point (_P_) which is conceived as travelling along the radius vector
under definite conditions of velocity, will then describe our spiral
curve.

Of several mathematical curves whose form and development may be so
conceived, the two most important (and the only two with which we need
deal), are those which are known as (1) the equable spiral, or spiral
of Archimedes, and (2) the logarithmic, or equiangular spiral.

[Illustration: Fig. 244.]

The former may be illustrated by the spiral coil in which a sailor
coils a rope upon the deck; as the rope is of uniform thickness, so in
the whole spiral coil is each whorl of the same breadth as that which
precedes and as that which follows it. Using its ancient definition,
we may define it by saying, that “If a straight line revolve uniformly
about its extremity, a point which likewise travels uniformly along it
will describe the equable spiral[497].” Or, putting the same thing into
our more modern words, “If, while the radius vector revolve uniformly
about the pole, a point (_P_) travel with uniform velocity along it,
the curve described will be that called the equable spiral, or spiral
of Archimedes.” {504}

It is plain that the spiral of Archimedes may be compared to a
_cylinder_ coiled up. And it is plain also that a radius (_r_
= _OP_), made up of the successive and equal whorls, will increase in
_arithmetical_ progression: and will equal a certain constant quantity
(_a_) multiplied by the whole number of whorls, or (more strictly
speaking) multiplied by the whole angle (θ) through which it has
revolved: so that _r_ = _a_θ.

But, in contrast to this, in the logarithmic spiral of the Nautilus or
the snail-shell, the whorls gradually increase in breadth, and do so
in a steady and unchanging ratio. Our definition is as follows: “If,
instead of travelling with a _uniform_ velocity, our point move along
the radius vector with _a velocity increasing as its distance from
the pole_, then the path described is called a logarithmic spiral.”
Each whorl which the radius vector intersects will be broader than its
predecessor in a definite ratio; the radius vector will increase in
length in _geometrical_ progression, as it sweeps through successive
equal angles; and the equation to the spiral will be _r_ = _a_^θ. As
the spiral of Archimedes, in our example of the coiled rope, might be
looked upon as a coiled cylinder, so may the logarithmic spiral, in the
case of the shell, be pictured as a _cone_ coiled upon itself.

Now it is obvious that if the whorls increase very slowly indeed, the
logarithmic spiral will come to look like a spiral of Archimedes, with
which however it never becomes identical; for it is incorrect to say,
as is sometimes done, that the Archimedean spiral is a “limiting case”
of the logarithmic spiral. The Nummulite is a case in point. Here we
have a large number of whorls, very narrow, very close together, and
apparently of equal breadth, which give rise to an appearance similar
to that of our coiled rope. And, in a case of this kind, we might
actually find that the whorls _were_ of equal breadth, being produced
(as is apparently the case in the Nummulite) not by any very slow and
gradual growth in thickness of a continuous tube, but by a succession
of similar cells or chambers laid on, round and round, determined as
to their size by constant surface-tension conditions and therefore
of unvarying dimensions. But even in this case we should have no
Archimedean spiral, but only a logarithmic spiral in which the constant
angle approximated to 90°. {505}

 For, in the logarithmic spiral, when α tends to 90°, the expression
 _r_ = _a_^{θ cot α} tends to _r_ = _a_(1 + θ cot α); while the
 equation to the Archimedean spiral is _r_ = _b_θ. The nummulite must
 always have a central core, or initial cell, around which the coil
 is not only wrapped, _but out of which it springs_; and this initial
 chamber corresponds to our _a′_ in the expression _r_ = _a′_ + _a_θ
 cot α. The outer whorls resemble those of an Archimedean spiral,
 because of the other term _a_θ cot α in the same expression. It
 follows from this that in all such cases the whorls must be of
 excessively small breadth.

There are many other specific properties of the logarithmic spiral,
so interrelated to one another that we may choose pretty well any
one of them as the basis of our definition, and deduce the others
from it either by analytical methods or by the methods of elementary
geometry. For instance, the equation _r_ = _a_^θ may be written in the
form log _r_ = θ log _a_, or θ = (log _r_)/(log _a_), or (since _a_
is a constant), θ = _k_ log _r_. Which is as much as to say that the
vector angles about the pole are proportional to the logarithms of the
successive radii; from which circumstance the name of the “logarithmic
spiral” is derived.

[Illustration: Fig. 245.]

Let us next regard our logarithmic spiral from the dynamical point
of view, as when we consider the forces concerned in the growth of
a material, concrete spiral. In a growing structure, let the forces
of growth exerted at any point _P_ be a force _F_ acting along the
line joining _P_ to a pole _O_ and a force _T_ acting in a direction
perpendicular to _OP_; and let the magnitude of these forces be in
the same constant ratio at all points. It follows that the resultant
of the forces _F_ and _T_ (as _PQ_) makes a constant angle with the
radius vector. But the constancy of the angle between tangent and
radius vector at any point is a fundamental property of the logarithmic
spiral, and may be shewn to follow from our definition of the curve:
it gives to the curve its alternative name of _equiangular spiral_.
Hence in a structure growing under the above conditions the form of the
boundary will be a logarithmic spiral. {506}

[Illustration: Fig. 246.]

In such a spiral, radial growth and growth in the direction of the
curve bear a constant ratio to one another. For, if we consider a
consecutive radius vector, _OP′_, whose increment as compared with _OP_
is _dr_, while _ds_ is the small arc _PP′_, then

 _dr_/_ds_ = cos α = constant.

[Illustration: Fig. 247.]

In the concrete case of the shell, the distribution of forces will be,
originally, a little more complicated than this, though by resolving
the forces in question, the system may be reduced to this simple form.
And furthermore, the actual distribution of forces will not always be
identical; for example, there is a distinct difference between the
cases (as in the snail) where a columellar muscle exerts a definite
traction in the direction of the pole, and those (such as Nautilus)
where there is no columellar muscle, and where some other force must
be discovered, or postulated, to account for the flexure. In the most
frequent case, we have, as in Fig. 247, three forces to deal with,
acting at a point, _p_: _L_, acting in the direction of the tangent
to the curve, and representing the force of longitudinal growth; _T_,
perpendicular to _L_, and representing the organism’s tendency to
grow in breadth; and _P_, the traction exercised, in the direction
of the pole, by the columellar muscle. Let us resolve _L_ and _T_
into components along _P_ (namely _A′_, _B′_), and perpendicular to
_P_ (namely _A_, _B_); we have now only two forces to consider, viz.
_P_ − _A′_ − _B′_, and _A_ − _B_. And these two latter we can again
resolve, if we please, so as to deal only with forces in the direction
of _P_ and _T_. Now, the ratio of these forces remaining constant, the
locus of the point _p_ is an equiangular spiral. {507}

Furthermore we see how any _slight_ change in any one of the forces
_P_, _T_, _L_ will tend to modify the angle α, and produce a slight
departure from the absolute regularity of the logarithmic spiral.
Such slight departures from the absolute simplicity and uniformity
of the theoretic law we shall not be surprised to find, more or less
frequently, in Nature, in the complex system of forces presented by the
living organism.

In the growth of a shell, we can conceive no simpler law than this,
namely, that it shall widen and lengthen in the same unvarying
proportions: and this simplest of laws is that which Nature tends to
follow. The shell, like the creature within it, grows in size _but does
not change its shape_; and the existence of this constant relativity of
growth, or constant similarity of form, is of the essence, and may be
made the basis of a definition, of the logarithmic spiral.

Such a definition, though not commonly used by mathematicians, has
been occasionally employed; and it is one from which the other
properties of the curve can be deduced with great ease and simplicity.
In mathematical language it would run as follows: “Any [plane] curve
proceeding from a fixed point (which is called the pole), and such
that the arc intercepted between this point and any other whatsoever
on the curve is always similar to itself, is called an equiangular, or
logarithmic, spiral[498].”

In this definition, we have what is probably the most fundamental and
“intrinsic” property of the curve, namely the property of continual
similarity: and this is indeed the very property by reason of which
it is peculiarly associated with organic growth in such structures
as the horn or the shell, or the scorpioid cyme which is described
on p. 502. For it is peculiarly characteristic of the spiral of a
shell, for instance, that (under all normal circumstances) it does
not alter its shape as it grows; each increment is geometrically
similar to its predecessor, and the whole, at any epoch, is similar to
what constituted the whole at another and an earlier epoch. We feel
no surprise when the animal which secretes the shell, or any other
animal whatsoever, grows by such {508} _symmetrical_ expansion as to
preserve its form unchanged; though even there, as we have already
seen, the unchanging form denotes a nice balance between the rates of
growth in various directions, which is but seldom accurately maintained
for long. But the shell retains its unchanging form in spite of its
_asymmetrical_ growth; it grows at one end only, and so does the horn.
And this remarkable property of increasing by _terminal_ growth, but
nevertheless retaining unchanged the form of the entire figure, is
characteristic of the logarithmic spiral, and of no other mathematical
curve.

[Illustration: Fig. 248.]

We may at once illustrate this curious phenomenon by drawing the
outline of a little Nautilus shell within a big one. We know, or we
may see at once, that they are of precisely the same shape; so that,
if we look at the little shell through a magnifying glass, it becomes
identical with the big one. But we know, on the other hand, that the
little Nautilus shell grows into the big one, not by uniform growth or
magnification in all directions, as is (though only approximately) the
case when the boy grows into the man, but by growing _at one end only_.

――――――――――

Though of all curves, this property of continued similarity is found
only in the logarithmic spiral, there are very many rectilinear figures
in which it may be observed. For instance, as we may easily see, it
holds good of any right cone; for evidently, in Fig. 248, the little
inner cone (represented in its triangular section) may become identical
with the larger one either by magnification all round (as in _a_), or
simply by an increment at one end (as in _b_); indeed, in the case
of the cone, we have yet a third possibility, for the same result is
attained when it increases all round, save only at the base, that is to
say when the triangular section increases {509} on two of its sides,
as in _c_. All this is closely associated with the fact, which we have
already noted, that the Nautilus shell is but a cone rolled up; in
other words, the cone is but a particular variety, or “limiting case,”
of the spiral shell.

This property, which we so easily recognise in the cone, would
seem to have engaged the particular attention of the most ancient
mathematicians even from the days of Pythagoras, and so, with little
doubt, from the more ancient days of that Egyptian school whence he
derived the foundations of his learning[499]; and its bearing on our
biological problem of the shell, though apparently indirect, is yet so
close that it deserves our further consideration.

[Illustration: Fig. 249.]

[Illustration: Fig. 250.]

If, as in Fig. 249, we add to two sides of a square a symmetrical
L-shaped portion, similar in shape to what we call a “carpenter’s
square,” the resulting figure is still a square; and the portion which
we have added is called, by Aristotle (_Phys._ III, 4), a “gnomon.”
Euclid extends the term to include the case of any parallelogram[500],
whether rectangular or not (Fig. 250); and Hero of Alexandria
specifically defines a “gnomon” (as indeed Aristotle implicitly defines
it), as any figure which, being added to any figure whatsoever,
leaves the resultant figure similar to the original. Included in this
important definition is the case of numbers, considered geometrically;
that is to say, the εἰδητικοὶ ἀριθμοί, which can be translated into
_form_, by means of rows of dots or other signs (cf. Arist. _Metaph._
1092 b 12), or in the pattern of a tiled floor: all according to “the
mystical way of {510} Pythagoras, and the secret magick of numbers.”
Thus for example, the odd numbers are “gnomonic numbers,” because

 0 + 1 = 1^2,

 1^2 + 3 = 2^2,

 2^2 + 5 = 3^2,

 3^2 + 7 = 4^2 _etc._,

which relation we may illustrate graphically σχηματογραφεῖν by the
successive numbers of dots which keep the annexed figure a perfect
square[501]: as follows:

 · · · · · ·
 · · · · · ·
 · · · · · ·
 · · · · · ·
 · · · · · ·
 · · · · · ·

[Illustration: Fig. 251.]

[Illustration: Fig. 252.]

There are other gnomonic figures more curious still. For instance, if
we make a rectangle (Fig. 251) such that the two sides are in the ratio
of 1 : √2, it is obvious that, on doubling it, we obtain a precisely
similar figure; for 1 : √2 :: √2 : 2; and {511} each half of the
figure, accordingly, is now a gnomon to the other. Another elegant
example is when we start with a rectangle (_A_) whose sides are in the
proportion of 1 : ½(√5 − 1), or, approximately, 1 : 0·618. The gnomon
to this figure is a square (_B_) erected on its longer side, and so on
successively (Fig. 252).

[Illustration: Fig. 253.]

[Illustration: Fig. 254.]

In any triangle, as Aristotle tells us, one part is always a gnomon to
the other part. For instance, in the triangle _ABC_ (Fig. 253), let us
draw _CD_, so as to make the angle _BCD_ equal to the angle _A_. Then
the part _BCD_ is a triangle similar to the whole triangle _ABC_, and
_ADC_ is a gnomon to _BCD_. A very elegant case is when the original
triangle _ABC_ is an isosceles triangle having one angle of 36°, and
the other two angles, therefore, each equal to 72° (Fig. 254). Then,
by bisecting one of the angles of the base, we subdivide the large
isosceles triangle into two isosceles triangles, of which one is
similar to the whole figure and the other is its gnomon[502]. There is
good reason to believe that this triangle was especially studied by the
Pythagoreans; for it lies at the root of many interesting geometrical
constructions, such as the regular pentagon, and the mystical
“pentalpha,” and a whole range of other curious figures beloved of the
ancient mathematicians[503]. {512}

[Illustration: Fig. 255.]

If we take any one of these figures, for instance the isosceles
triangle which we have just described, and add to it (or subtract from
it) in succession a series of gnomons, so converting it into larger and
larger (or smaller and smaller) triangles all similar to the first,
we find that the apices (or other corresponding points) of all these
triangles have their _locus_ upon a logarithmic spiral: a result which
follows directly from that alternative definition of the logarithmic
spiral which I have quoted from Whitworth (p. 507).

[Illustration: Fig. 256. Logarithmic spiral derived from corresponding
points in a system of squares.]

Again, we may build up a series of right-angled triangles, each
of which is a gnomon to the preceding figure; and here again, a
logarithmic spiral is the locus of corresponding points in these
successive triangles. And lastly, whensoever we fill up space with
a {513} collection of either equal or similar figures, similarly
situated, as in Figs. 256, 257, there we can always discover a series
of inscribed or escribed logarithmic spirals.

[Illustration: Fig. 257. The same in a system of hexagons.]

Once more, then, we may modify our definition, and say that: “Any
plane curve proceeding from a fixed point (or pole), and such that the
vectorial area of any sector is always a gnomon to the whole preceding
figure, is called an equiangular, or logarithmic, spiral.” And we may
now introduce this new concept and nomenclature into our description
of the Nautilus shell and other related organic forms, by saying that:
(1) if a growing structure be built up of successive parts, similar
and similarly situated, we can always trace through corresponding
points a series of logarithmic spirals (Figs. 258, 259, etc.); (2) it
is characteristic of the growth of the horn, of the shell, and of all
other organic forms in which a logarithmic spiral can be recognised,
that _each successive increment of growth is a gnomon to the entire
pre-existing structure_. And conversely (3) it follows obviously, that
in the logarithmic spiral outline of the shell or of the horn we can
always inscribe an endless variety of other gnomonic figures, having
no necessary relation, save as a {514} mathematical accident, to the
nature or mode of development of the actual structure[504]. {515}

[Illustration: Fig. 258. A shell of Haliotis, with two of the many
lines of growth, or generating curves, marked out in black: the areas
bounded by these lines of growth being in all cases “gnomons” to the
pre-existing shell.]

[Illustration: Fig. 259. A spiral foraminifer (_Pulvinulina_), to show
how each successive chamber continues the symmetry of, or constitutes a
_gnomon_ to, the rest of the structure.]

[Illustration: Fig. 260. Another spiral foraminifer, _Cristellaria_.]

Of these three propositions, the second is of very great use and
advantage for our easy understanding and simple description of the
molluscan shell, and of a great variety of other structures whose
mode of growth is analogous, and whose mathematical properties are
therefore identical. We see at once that the successive chambers of
a spiral Nautilus (Fig. 237) or of a straight Orthoceras (Fig. 300),
each whorl or part of a whorl of a periwinkle or other gastropod
(Fig. 258), each new increment of the operculum of a gastropod (Fig.
263), each additional increment of an elephant’s tusk, or each new
chamber of a spiral foraminifer (Figs. 259 and 260), has its leading
characteristic at once described and its form so far explained by
the simple statement that it constitutes a _gnomon_ to the whole
previously existing structure. And herein lies the explanation of that
“time-element” in the development of organic spirals of which we have
spoken already, in a preliminary and empirical way. For it follows as
a simple corollary to this theorem of gnomons that we must not expect
to find the logarithmic spiral manifested in a structure whose parts
are simultaneously produced, as for instance in the margin of a leaf,
or among the many curves that make the contour of a fish. But we must
rather look for it wherever the organism retains for us, and still
presents to us at a single view, the successive phases of preceding
growth, the successive magnitudes attained, the successive outlines
occupied, as the organism or a part thereof pursued the even tenour
of its growth, year by year and day by day. And it easily follows
from this, that it is in the hard parts of organisms, and not the
soft, fleshy, actively growing parts, that this spiral is commonly and
characteristically found; not in the fresh mobile tissues whose form is
constrained merely by the active forces of the moment; but in things
like shell and tusk, and horn and claw, where the object is visibly
composed of parts {516} successively, and permanently, laid down. In
the main, the logarithmic spiral is characteristic, not of the living
tissues, but of the dead. And for the same reason, it will always or
nearly always be accompanied, and adorned, by a pattern formed of
“lines of growth,” the lasting record of earlier and successive stages
of form and magnitude.

――――――――――

It is evident that the spiral curve of the shell is, in a sense, a
vector diagram of its own growth; for it shews at each instant of time,
the direction, radial and tangential, of growth, and the unchanging
ratio of velocities in these directions. Regarding the _actual_
velocity of growth in the shell, we know very little (or practically
nothing), by way of experimental measurement; but if we make a
certain simple assumption, then we may go a good deal further in our
description of the logarithmic spiral as it appears in this concrete
case.

Let us make the assumption that _similar_ increments are added to the
shell in _equal_ times; that is to say, that the amount of growth in
unit time is measured by the areas subtended by equal angles. Thus,
in the outer whorl of a spiral shell a definite area marked out by
ridges, tubercles, etc., has very different linear dimensions to
the corresponding areas of the inner whorl, but the symmetry of the
figure implies that it subtends an equal angle with these; and it is
reasonable to suppose that the successive regions, marked out in this
way by successive natural boundaries or patterns, are produced in equal
intervals of time.

If this be so, the radii measured from the pole to the boundary of the
shell will in each case be proportional to the velocity of growth at
this point upon the circumference, and at the time when it corresponded
with the outer lip, or region of active growth; and while the direction
of the radius vector corresponds with the direction of growth in
thickness of the animal, so does the tangent to the curve correspond
with the direction, for the time being, of the animal’s growth in
length. The successive radii are a measure of the acceleration of
growth, and the spiral curve of the shell itself is no other than the
_hodograph_ of the growth of the contained organism. {517}

So far as we have now gone, we have studied the elementary properties
of the logarithmic spiral, including its fundamental property of
_continued similarity_; and we have accordingly learned that the shell
or the horn tends _necessarily_ to assume the form of this mathematical
figure, because in these structures growth proceeds by successive
increments, which are always similar in form, similarly situated, and
of constant relative magnitude one to another. Our chief objects in
enquiring further into the mathematical properties of the logarithmic
spiral will be: (1) to find means of confirming and verifying the fact
that the shell (or other organic curve) is actually a logarithmic
spiral; (2) to learn how, by the properties of the curve, we may
further extend our knowledge or simplify our descriptions of the shell;
and (3) to understand the factors by which the characteristic form of
any particular logarithmic spiral is determined, and so to comprehend
the nature of the specific or generic characters by which one spiral
shell is found to differ from another.

Of the elementary properties of the logarithmic spiral, so far as we
have now enumerated them, the following are those which we may most
easily investigate in the concrete case, such as we have to do with in
the molluscan shell: (1) that the polar radii of points whose vectorial
angles are in arithmetical progression, are themselves in geometrical
progression; and (2) that the tangent at any point of a logarithmic
spiral makes a constant angle (called the _angle of the spiral_) with
the polar radius vector.

[Illustration: Fig. 261.]

The former of these two propositions may be written in what is,
perhaps, a simpler form, as follows: radii which form equal angles
about the pole of the logarithmic spiral, are themselves continued
proportionals. That is to say, in Fig. 261, when the angle _ROQ_ is
equal to the angle _QOP_, then _OR_ : _OQ_ :: _OQ_ : _OP_.

A particular case of this proposition is when the equal angles are each
angles of 360°: that is to say when in each case the radius vector
makes a complete revolution, and when, therefore _P_, _Q_ and _R_ all
lie upon the same radius. {518}

It was by observing, with the help of very careful measurement,
this continued proportionality, that Moseley was enabled to verify
his first assumption, based on the general appearance of the shell,
that the shell of Nautilus was actually a logarithmic spiral, and
this demonstration he was immediately afterwards in a position to
generalise by extending it to all the spiral Ammonitoid and Gastropod
mollusca[505].

For, taking a median transverse section of a _Nautilus pompilius_, and
carefully measuring the successive breadths of the whorls (from the
dark line which marks what was originally the outer surface, before
it was covered up by fresh deposits on the part of the growing and
advancing shell), Moseley found that “the distance of any two of its
whorls measured upon a radius vector is one-third that of the two next
whorls measured upon the same radius vector[506]. Thus (in Fig. 262),
_ab_ is one-third of _bc_, _de_ of _ef_, _gh_ of _hi_, and _kl_ of
_lm_. The curve is therefore a logarithmic spiral.”

The numerical ratio in the case of the Nautilus happens to be one
of unusual simplicity. Let us take, with Moseley, a somewhat more
complicated example.

From the apex of a large specimen of _Turbo duplicatus_[507] a {519}
line was drawn across its whorls, and their widths were measured upon
it in succession, beginning with the last but one. The measurements
were, as before, made with a fine pair of compasses and a diagonal
scale. The sight was assisted by a magnifying glass. In a parallel
column to the following admeasurements are the terms of a geometric
progression, whose first term is the width of the widest whorl
measured, and whose common ratio is 1·1804.

[Illustration: Fig. 262.]

   Widths of successive        Terms of a geometrical progression,
 whorls measured in inches      whose first term is the width of
   and parts of an inch          the widest whorl, and whose
                                   common ratio is 1·1804

          1·31                          1·31
          1·12                          1·1098
           ·94                           ·94018
           ·80                           ·79651
           ·67                           ·67476
           ·57                           ·57164
           ·48                           ·48427
           ·41                           ·41026

The close coincidence between the observed and the calculated figures
is very remarkable, and is amply sufficient to justify the conclusion
that we are here dealing with a true logarithmic spiral.

Nevertheless, in order to verify his conclusion still further, and
to get partially rid of the inaccuracies due to successive small
{520} measurements, Moseley proceeded to investigate the same shell,
measuring not single whorls, but groups of whorls, taken several
at a time: making use of the following property of a geometrical
progression, that “if µ represent the ratio of the sum of every even
number (_m_) of its terms to the sum of half that number of terms, then
the common ratio (_r_) of the series is represented by the formula

 _r_ = (µ − 1)^{2/_m_} .”

Accordingly, Moseley made the following measurements, beginning from
the second and third whorls respectively:

         Width of
 ────────────────────────
 Six whorls  Three whorls  Ratio µ

    5·37         2·03       2·645
    4·55         1·72       2·645

 Four whorls  Two whorls

    4·15         1·74       2·385
    3·52         1·47       2·394

“By the ratios of the two first admeasurements, the formula gives

 _r_ = (1·645)^{1/3} = 1·1804.

By the mean of the ratios deduced from the second two admeasurements,
it gives

 _r_ = (1·389)^½ = 1·1806.

“It is scarcely possible to imagine a more accurate verification than
is deduced from these larger admeasurements, and we may with safety
annex to the species _Turbo duplicatus_ the characteristic number 1·18.”

By similar and equally concordant observations, Moseley found for
_Turbo phasianus_ the characteristic ratio, 1·75; and for _Buccinum
subulatum_ that of 1·13.

From the table referring to _Turbo duplicatus_, on page 519, it is
perhaps worth while to illustrate the logarithmic statement of the same
facts: that is to say, the elementary corollary to the fact that the
successive radii are in geometric progression, that their logarithms
differ from one another by a constant amount. {521}


_Turbo duplicatus._

 Relative widths of  Logarithms of      Difference of
 successive whorls   successive whorls  successive logarithms
 131                 2·11727              —
 112                 2·04922            ·06805
  94                 1·97313            ·07609
  80                 1·90309            ·07004
  67                 1·82607            ·07702
  57                 1·75587            ·07020
  48                 1·68124            ·07463
  41                 1·161278           ·06846
                        Mean difference ·07207

And ·07207 is the logarithm of 1·1805.

[Illustration: Fig. 263. Operculum of Turbo.]

The logarithmic spiral is not only very beautifully manifested in
the molluscan shell, but also, in certain cases, in the little lid
or “operculum” by which the entrance to the tubular shell is closed
after the animal has withdrawn itself within. In the spiral shell of
_Turbo_, for instance, the operculum is a thick calcareous structure,
with a beautifully curved outline, which grows by successive increments
applied to one portion of its edge, and shews, accordingly, a spiral
line of growth upon its surface. The successive increments leave their
traces on the surface of the operculum {522} (Fig. 264, 1), which
traces have the form of curved lines in Turbo, and of straight lines
in (e.g.) Nerita (Fig. 264, 2); that is to say, apart from the side
constituting the outer edge of the operculum (which side is always and
of necessity curved) the successive increments constitute curvilinear
triangles in the one case, and rectilinear triangles in the other.
The sides of these triangles are tangents to the spiral line of the
operculum, and may be supposed to generate it by their consecutive
intersections.

[Illustration: Fig. 264. Opercula of (1) Turbo, (2) Nerita. (After
Moseley.)]

In a number of such opercula, Moseley measured the breadths of the
successive whorls along a radius vector[508], just in the same way as
he did with the entire shell in the foregoing cases; and here is one
example of his results.

 _Operculum of Turbo sp.; breadth (in inches) of successive whorls,
 measured from the pole._

 Distance  Ratio  Distance  Ratio  Distance  Ratio  Distance  Ratio
  ·24              ·16              ·2               ·18
           2·28             2·31             2·30             2·30
  ·55              ·37              ·6               ·42
           2·32             2·30             2·30             2·24
 1·28              ·85             1·38              ·94

{523}

The ratio is approximately constant, and this spiral also is,
therefore, a logarithmic spiral.

But here comes in a very beautiful illustration of that property
of the logarithmic spiral which causes its whole shape to remain
unchanged, in spite of its apparently unsymmetrical, or unilateral,
mode of growth. For the mouth of the tubular shell, into which the
operculum has to fit, is growing or widening on all sides: while the
operculum is increasing, not by additions made at the same time all
round its margin, but by additions made only on one side of it at each
successive stage. One edge of the operculum thus remains unaltered as
it is advanced into each new position, and as it is placed in a newly
formed section of the tube, similar to but greater than the last.
Nevertheless, the two apposed structures, the chamber and its plug, at
all times fit one another to perfection. The mechanical problem (by no
means an easy one), is thus solved: “How to shape a tube of a variable
section, so that a piston driven along it shall, by one side of its
margin, coincide continually with its surface as it advances, provided
only that the piston be made at the same time continually to revolve in
its own plane.”

As Moseley puts it: “That the same edge which fitted a portion of
the first less section should be capable of adjustment, so as to
fit a portion of the next similar but greater section, supposes a
geometrical provision in the curved form of the chamber of great
apparent complication and difficulty. But God hath bestowed upon this
humble architect the practical skill of a learned geometrician, and he
makes this provision with admirable precision in that curvature of the
logarithmic spiral which he gives to the section of the shell. This
curvature obtaining, he has only to turn his operculum slightly round
in its own plane as he advances it into each newly formed portion of
his chamber, to adapt one margin of it to a new and larger surface and
a different curvature, leaving the space to be filled up by increasing
the operculum wholly on the other margin.”

But in many, and indeed more numerous Gastropod mollusca, the operculum
does not grow in this remarkable spiral fashion, but by the apparently
much simpler process of accretion by concentric rings. This suggests to
us another mathematical {524} feature of the logarithmic spiral. We
have already seen that the logarithmic spiral has a number of “limiting
cases,” apparently very diverse from one another. Thus the right cone
is a logarithmic spiral in which the revolution of the radius vector is
infinitely slow; and, in the same sense, the straight line itself is
a limiting case of the logarithmic spiral. The spiral of Archimedes,
though not a limiting case of the logarithmic spiral, closely resembles
one in which the angle of the spiral is very near to 90°, and the
spiral is coiled around a central core. But if the angle of the spiral
were actually 90°, the radius vector would describe a circle, identical
with the “core” of which we have just spoken; and accordingly it may
be said that the circle is, in this sense, a true limiting case of
the logarithmic spiral. In this sense, then, the circular concentric
operculum, for instance of Turritella or Littorina, does not represent
a breach of continuity, but a “limiting case” of the spiral operculum
of _Turbo_; the successive “gnomons” are now not lateral or terminal
additions, but complete concentric rings.

――――――――――

Viewed in regard to its own fundamental properties and to those of
its limiting cases, the logarithmic spiral is the simplest of all
known curves; and the rigid uniformity of the simple laws, or forces,
by which it is developed sufficiently account for its frequent
manifestation in the structures built up by the slow and steady growth
of organisms.

In order to translate into precise terms the whole form and growth
of a spiral shell, we should have to employ a mathematical notation,
considerably more complicated than any that I have attempted to make
use of in this book. But, in the most elementary language, we may
now at least attempt to describe the general method, and some of the
variations, of the mathematical development of the shell.

Let us imagine a closed curve in space, whether circular or elliptical
or of some other and more complex specific form, not necessarily in a
plane: such a curve as we see before us when we consider the mouth, or
terminal orifice, of our tubular shell; and let us imagine some one
characteristic point within this closed curve, such as its centre of
gravity. Then, starting from a fixed {525} origin, let this centre of
gravity describe an equiangular spiral in space, about a fixed axis
(namely the axis of the shell), while at the same time the generating
curve grows, with each angular increment of rotation, in such a way
as to preserve the symmetry of the entire figure, with or without a
simultaneous movement of translation along the axis.

[Illustration: Fig. 265. _Melo ethiopicus_, L.]

It is plain that the entire resulting shell may now be looked upon in
either of two ways. It is, on the one hand, an _ensemble of similar
closed curves_ spirally arranged in space, gradually increasing in
dimensions, in proportion to the increase of their vectorial angle
from the pole. In other words, we can imagine our shell cut up into a
system of rings, following one another in continuous spiral succession
from that terminal and largest one, which constitutes the lip of the
orifice of the shell. Or, on the other hand, we may figure to ourselves
the whole shell as made up of an _ensemble of spiral lines_ in space,
each spiral having been {526} traced out by the gradual growth and
revolution of a radius vector from the pole to a given point of the
generating curve.

Both systems of lines, the _generating spirals_ (as these latter may be
called), and the closed _generating curves_ corresponding to successive
margins or lips of the shell, may be easily traced in a great variety
of cases. Thus, for example, in Dolium, Eburnea, and a host of others,
the generating spirals are beautifully marked out by ridges, tubercles
or bands of colour. In Trophon, Scalaria, and (among countless others)
in the Ammonites, it is the successive generating curves which more
conspicuously leave their impress on the shell. And in not a few cases,
as in Harpa, _Dolium perdix_, etc., both alike are conspicuous, ridges
and colour-bands intersecting one another in a beautiful isogonal
system. {527}

[Illustration: Fig. 266. 1, _Harpa_; 2, _Dolium_. The ridges on the
shell correspond in (1) to generating curves, in (2) to generating
spirals.]

In the complete mathematical formula (such as I have not ventured
to set forth[509]) for any given turbinate shell, we should have,
accordingly, to include factors for at least the following elements:
(1) for the specific form of the section of the tube, which we have
called the generating curve; (2) for the specific rate of growth of
this generating curve; (3) for its specific rate of angular rotation
about the pole, perpendicular to the axis; (4) in turbinate (as opposed
to nautiloid) shells, for its rate of shear, or screw-translation
parallel to the axis. There are also other factors of which we should
have to take account, and which would help to make our whole expression
a very complicated one. We should find, for instance, (5) that in very
many cases our generating curve was not a plane curve, but a sinuous
curve in three dimensions; and we should also have to take account (6)
of the inclination of the plane of this generating curve to the axis,
a factor which will have a very important influence on the form and
appearance of the shell. For instance in Haliotis it is obvious that
the generating curve lies in a plane very oblique to the axis of the
shell. Lastly, we at once perceive that the ratios which happen to
exist between these various factors, the ratio for instance between
the growth-factor and the rate of angular revolution, will give us
endless possibilities of permutation of form. For instance (7) with a
given velocity of vectorial rotation, a certain rate of growth in the
generating curve will give us a spiral shell of which each successive
whorl will just touch its predecessor and no more; with a slower
growth-factor, the whorls will stand asunder, as in a ram’s horn;
with a quicker growth-factor, each whorl will cut or intersect its
predecessor, as in an Ammonite or the majority of gastropods, and so on
(cf. p. 541).

In like manner (8) the ratio between the growth-factor and the rate
of screw-translation parallel to the axis will determine the apical
angle of the resulting conical structure: will give us the difference,
for example, between the sharp, pointed cone of Turritella, the less
acute one of Fusus or Buccinum, and the {528} obtuse one of Harpa or
Dolium. In short it is obvious that _all_ the differences of form which
we observe between one shell and another are referable to matters of
_degree_, depending, one and all, upon the relative magnitudes of the
various factors in the complex equation to the curve.

――――――――――

The paper in which, nearly eighty years ago, Canon Moseley thus gave a
simple mathematical expression to the spiral forms of univalve shells,
is one of the classics of Natural History. But other students before
him had come very near to recognising this mathematical simplicity of
form and structure. About the year 1818, Reinecke had suggested that
the relative breadths of the adjacent whorls in an Ammonite formed a
constant and diagnostic character; and Leopold von Buch accepted and
developed the idea[510]. But long before, Swammerdam, with a deeper
insight, had grasped the root of the whole matter: for, taking a few
diverse examples, such as Helix and Spirula, he shewed that they and
all other spiral shells whatsoever were referable to one common type,
namely to that of a simple tube, variously curved according to definite
mathematical laws; that all manner of ornamentation, in the way of
spines, tuberosities, colour-bands and so forth, might be superposed
upon them, but the type was one throughout, and specific differences
were of a geometrical kind. “Omnis enim quae inter eas animadvertitur
differentia ex sola nascitur diversitate gyrationum: quibus si insuper
externa quaedam adjunguntur ornamenta pinnarum, sinuum, anfractuum,
planitierum, eminentiarum, profunditatum, extensionum, impressionum,
circumvolutionum, colorumque: ... tunc deinceps facile est,
quarumcumque Cochlearum figuras geometricas, curvosque, obliquos atque
rectos angulos, ad unicam omnes speciem redigere: ad oblongum videlicet
tubulum, qui vario modo curvatus, crispatus, extrorsum et introrsum
flexus, ita concrevit[511].” {529}

For some years after the appearance of Moseley’s paper, a number of
writers followed in his footsteps, and attempted, in various ways, to
put his conclusions to practical use. For instance, D’Orbigny devised a
very simple protractor, which he called a Helicometer[512], and which
is represented in Fig. 267. By means of this little instrument, the
apical angle of the turbinate shell was immediately read off, and could
then be used as a specific and diagnostic character. By keeping one
limb of the protractor parallel to the side of the cone while the other
was brought into line with the suture between two adjacent whorls,
another specific angle, the “sutural angle,” could in like manner be
recorded. And, by the linear scale upon the instrument, the relative
breadths of the consecutive whorls, and that of the terminal chamber
to the rest of the shell, might also, though somewhat roughly, be
determined. For instance, in _Terebra dimidiata_, the apical angle was
found to be 13°, the sutural angle 109°, and so forth.

[Illustration: Fig. 267. D’Orbigny’s Helicometer.]

It was at once obvious that, in such a shell as is represented in Fig.
267 the entire outline of the shell (always excepting that of the
immediate neighbourhood of {530} the mouth) could be restored from a
broken fragment. For if we draw our tangents to the cone, it follows
from the symmetry of the figure that we can continue the projection
of the sutural line, and so mark off the successive whorls, by simply
drawing a series of consecutive parallels, and by then filling into the
quadrilaterals so marked off a series of curves similar to one another,
and to the whorls which are still intact in the broken shell.

But the use of the helicometer soon shewed that it was by no means
universally the case that one and the same right cone was tangent to
all the turbinate whorls; in other words, there was not always one
specific apical angle which held good for the entire system. In the
great majority of cases, it is true, the same tangent touches all
the whorls, and is a straight line. But in others, as in the large
_Cerithium nodosum_, such a line is slightly convex to the axis of the
shell; and in the short spire of Dolium, for instance, the convexity
is marked, and the apex of the spire is a distinct cusp. On the other
hand, in Pupa and Clausilia, the common tangent is concave to the axis
of the shell.

So also is it, as we shall presently see, among the Ammonites: where
there are some species in which the ratio of whorl to whorl remains,
to all appearance, perfectly constant; others in which it gradually,
though only slightly increases; and others again in which it slightly
and gradually falls away. It is obvious that, among the manifold
possibilities of growth, such conditions as these are very easily
conceivable. It is much more remarkable that, among these shells,
the relative velocities of growth in various dimensions should be as
constant as it is, than that there should be an occasional departure
from perfect regularity. In such cases as these latter, the logarithmic
law of growth is only approximately true. The shell is no longer to be
represented as a _right_ cone which has been rolled up, but as a cone
which had grown trumpet-shaped, or conversely whose mouth had narrowed
in, and which in section is a curvilinear instead of a rectilinear
triangle. But all that has happened is that a new factor, usually of
small or all but imperceptible magnitude, has been introduced into the
case; so that the ratio, log _r_ = θ log α, is no longer constant, but
varies slightly, and in accordance with some simple law. {531}

Some writers, such as Naumann and Grabau, maintained that the
molluscan spiral was no true logarithmic spiral, but differed from it
specifically, and they gave to it the name of _Conchospiral_. They
pointed out that the logarithmic spiral originates in a mathematical
point, while the molluscan shell starts with a little embryonic shell,
or central chamber (the “protoconch” of the conchologists), around
which the spiral is subsequently wrapped. It is plain that this
undoubted and obvious fact need not affect the logarithmic law of the
shell as a whole; we have only to add a small constant to our equation,
which becomes _r_ = _m_ + _a_^θ.

There would seem, by the way, to be considerable confusion in the
books with regard to the so-called “protoconch.” In many cases it is
a definite structure, of simple form, representing the more or less
globular embryonic shell before it began to elongate into its conical
or spiral form. But in many cases what is described as the “protoconch”
is merely an empty space in the middle of the spiral coil, resulting
from the fact that the actual spiral shell has a definite magnitude to
begin with, and that we cannot follow it down to its vanishing point in
infinity. For instance, in the accompanying figure, the large space _a_
is styled the protoconch, but it is the little bulbous or hemispherical
chamber within it, at the end of the spire, which is the real beginning
of the tubular shell. The form and magnitude of the space _a_ are
determined by the “angle of retardation,” or ratio of rate of growth
between the inner and outer curves of the spiral shell. They are
independent of the shape and size of the embryo, and depend only (as
we shall see better presently) on the direction and relative rate of
growth of the double contour of the shell.

[Illustration: Fig. 268.]

――――――――――

Now that we have dealt, in a very general way, with some of the more
obvious properties of the logarithmic spiral, let us consider certain
of them a little more particularly, keeping in {532} view as our chief
object the investigation (on elementary lines) of the possible manner
and range of variation of the molluscan shell.

[Illustration: Fig. 269.]

There is yet another equation to the logarithmic spiral, very commonly
employed, and without the help of which we shall find that we cannot
get far. It is as follows:

 _r_ = ε^{θ cot α}.

This follows directly from the fact that the angle α (the angle between
the radius vector and the tangent to the curve) is constant.

For, then,

 tan α (= tan ϕ) = _r_ _d_θ/_dr_,

 therefore _dr_/_r_ = _d_θ cot α,

 and, integrating,

 log _r_ = θ cot α,

 or _r_ = ε^{θ cot α}.

――――――――――

As we have seen throughout our preliminary discussion, the two most
important constants (or chief “specific characters,” as the naturalist
would say) in any given logarithmic spiral, are (1) the magnitude of
the angle of the spiral, or “constant angle,” α, and (2) the rate of
increase of the radius vector for any given angle of revolution, θ.
Of this latter, the simplest case is when θ = 2π, or 360°; that is to
say when we compare the breadths, along the same radius vector, of two
successive whorls. As our two magnitudes, that of the constant angle,
and that of the ratio of the radii or breadths of whorl, are related to
one another, we may determine either of them by actual measurement and
proceed to calculate the other.

In any complete spiral, such as that of Nautilus, it is (as we have
seen) easy to measure any two radii (_r_), or the breadths in {533} a
radial direction of any two whorls (_W_). We have then merely to apply
the formula

 _r__{_n_ + 1}/_r__{_n_} = _e_^{θ cot α}, or _W__{_n_ + 1}/_W__{_n_}
     = _e_^{θ cot α},

which we may simply write _r_ = _e_^{θ cot α}, etc.; since our first
radius or whorl is regarded, for the purpose of comparison, as being
equal to unity.

Thus, in the diagram, _OC_/_OE_, or _EF_/_BD_, or _DC_/_EF_, being in
each case radii, or diameters, at right angles to one another, are all
equal to _e_^{π/2 cot α}. While in like manner, _EO_/_OF_, _EG_/_FH_,
or _GO_/_HO_, all equal _e_^{π cot α}; and _BC_/_BA_, or _CO_/_OB_
= _e_^{2π cot α}.

[Illustration: Fig. 270.]

As soon, then, as we have prepared tables for these values, the
determination of the constant angle α in a particular shell becomes a
very simple matter.

A complete table would be cumbrous, and it will be sufficient to deal
with the simple case of the ratio between the breadths of adjacent, or
immediately succeeding, whorls.

Here we have _r_ = _e_^{2π cot α}, or log _r_ = log _e_ × 2π × cot α,
from which we obtain the following figures[513]: {534}

  Ratio of breadth of each
 whorl to the next preceding      Constant angle
             _r_/1                     α
             1·1                     89°  8′
             1·25                    87  58
             1·5                     86  18
             2·0                     83  42
             2·5                     81  42
             3·0                     80   5
             3·5                     78  43
             4·0                     77  34
             4·5                     76  32
             5·0                     75  38
            10·0                     69  53
            20·0                     64  31
            50·0                     58   5
           100·0                     53  46
         1,000·0                     42  17
        10,000                       34  19
       100,000                       28  37
     1,000,000                       24  28
    10,000,000                       21  18
   100,000,000                       18  50
 1,000,000,000                       16  52

[Illustration: Fig. 271.]

We learn several interesting things from this short table. We see,
in the first place, that where each whorl is about three times the
breadth of its neighbour and predecessor, as is the case in Nautilus,
the constant angle is in the neighbourhood of 80°; and hence also that,
in all the ordinary Ammonitoid shells, and in all the typically spiral
shells of the Gastropods[514], the constant angle is also a large one,
being very seldom less than 80°, and usually between 80° and 85°. In
the next place, we see that with smaller angles the apparent form of
the spiral is greatly altered, and the very fact of its being a spiral
soon ceases to be apparent (Figs. 271, 272). Suppose one whorl to be an
inch in breadth, then, if the angle of the spiral were 80°, the {535}
next whorl would (as we have just seen) be about three inches broad;
if it were 70°, the next whorl would be nearly ten inches, and if it
were 60°, the next whorl would be nearly four feet broad. If the angle
were 28°, the next whorl would be a mile and a half in breadth; and if
it were 17°, the next would be some 15,000 miles broad.

[Illustration: Fig. 272.]

In other words, the spiral shells of gentle curvature, or of small
constant angle, such as Dentalium or Nodosaria, are true logarithmic
spirals, just as are those of Nautilus or Rotalia: from which they
differ only in degree, in the magnitude of an angular constant. But
this diminished magnitude of the angle causes the spiral to dilate with
such immense rapidity that, so to speak, “it never comes round”; and
so, in such a shell as Dentalium, we never see but a small portion of
the initial whorl.

[Illustration: Fig. 273.]

 We might perhaps be inclined to suppose that, in such a shell as
 Dentalium, the lack of a visible spiral convolution was only due to
 our seeing but a small portion of the curve, at a distance from the
 pole, and when, therefore, its {536} curvature had already greatly
 diminished. That is to say we might suppose that, however small the
 angle a, and however rapidly the whorls accordingly increased, there
 would nevertheless be a manifest spiral convolution in the immediate
 neighbourhood of the pole, as the starting point of the curve. But it
 may be shewn that this is not so.

 For, taking the formula _r_ = _a_ε^{θ cot α},

 this, for any given spiral, is equivalent to _a_ε^{_k_θ}.

 Therefore log(_r_/_a_) = _k_θ,

 or, 1/_k_ = θ/log(_r_/_a_).

 Then, if θ increase by 2π, while _r_ increases to _r__{1},

 1/_k_ = (θ + 2π)/log(_r__{1}/_a_),

 which leads, by subtraction to

 1/_k_ ⋅ log(_r__{1}/_r_) = 2π.

 Now, as α tends to 0, _k_ (i.e. cot α) tends to ∞, and therefore, as
 _k_ → ∞, log(_r__{1}/_r_) → ∞ and also _r__{1}/_r_ → ∞.

 Therefore if one whorl exists, the radius vector of the other
 is infinite; in other words, there is nowhere, even in the near
 neighbourhood of the pole, a complete revolution of the spire.
 Our spiral shells of small constant angle, such as Dentalium, may
 accordingly be considered to represent sufficiently well the true
 commencement of their respective spirals.

Let us return to the problem of how to ascertain, by direct
measurement, the spiral angle of any particular shell. The method
already employed is only applicable to complete spirals, that is to say
to those in which the angle of the spiral is large, and furthermore
it is inapplicable to portions, or broken fragments, of a shell. In
the case of the broken fragment, it is plain that the determination
of the angle is not merely of theoretic interest, but may be of great
practical use to the conchologist as being the one and only way by
which he may restore the outline of the missing portions. We have a
considerable choice of methods, which have been summarised by, and are
partly due to, a very careful student of the Cephalopoda, the late Rev.
J. F. Blake[515]. {537}

(1) The following method is useful and easy when we have a portion of
a single whorl, such as to shew both its inner and its outer edge.
A broken whorl of an Ammonite, a curved shell such as Dentalium, or
a horn of similar form to the latter, will fall under this head.
We have merely to draw a tangent, _GEH_, to the outer whorl at any
point _E_; then draw to the inner whorl a tangent parallel to _GEH_,
touching the curve in some point _F_. The straight line joining the
points of contact, _EF_, must evidently pass through the pole: and,
accordingly, the angle _GEF_ is the angle required. In shells which
bear _longitudinal_ striae or other ornaments, any pair of these will
suffice for our purpose, instead of the actual boundaries of the whorl.
But it is obvious that this method will be apt to fail us when the
angle α is very small; and when, consequently, the points _E_ and _F_
are very remote.

[Illustration: Fig. 274.]

(2) In shells (or horns) shewing rings, or other _transverse_
ornamentation, we may take it that these ornaments are set at a
constant angle to the spire, and therefore to the radii. The angle (θ)
between two of them, as _AC_, _BD_, is therefore equal to the angle
θ between the polar radii from _A_ and _B_, or from _C_ and _D_; and
therefore _BD_/_AC_ = _e_^{θ cot α}, which gives us the angle α in
terms of known quantities. {538}

[Illustration: Fig. 275. An Ammonite, to shew corrugated
surface-pattern.]

[Illustration: Fig. 276.]

(3) If only the outer edge be available, we have the ordinary
geometrical problem,—given an arc of an equiangular spiral, to find
its pole and spiral angle. The methods we may employ depend (1) on
determining directly the position of the pole, and (2) on determining
the radius of curvature.

[Illustration: Fig. 277.]

The first method is theoretically simple, but difficult in practice;
for it requires great accuracy in determining the points. Let _AD_,
_DB_, be two tangents drawn to the curve. Then a circle drawn through
the points _ABD_ will pass through the pole _O_; since the angles
_OAD_, _OBE_ (the supplement of _OBD_), are equal. The point _O_ may be
determined by the intersection of two such circles; and the angle _DBO_
is then the angle, α, required.

Or we may determine, graphically, at two points, the radii of
curvature, ρ_{1} ρ_{2}. Then, if _s_ be the length of the arc between
them (which may be determined with fair accuracy by rolling the margin
of the shell along a ruler)

 cot α = (ρ_{1} − ρ_{2})/_s_.

 The following method[516], given by Blake, will save actual
 determination of the radii of curvature.

 Measure along a tangent to the curve, the distance, _AC_, at which a
 certain small offset, _CD_, is made by the curve; and from another
 point _B_, measure the distance at which the curve makes an equal
 offset. Then, calling the offset μ; the arc _AB_, _s_; and _AC_, _BE_,
 respectively _x__{1}, _x__{2}, we have

 ρ_{1} = (_x__{1}^2 + μ^2)/2μ, approximately,

 and cot α = (_x__{2}^2 − _x__{1}^2)/2μ_s_.

Of all these methods by which the mathematical constants, or specific
characters, of a given spiral shell may be determined, the only one
of which much use has been made is that which Moseley first employed,
namely, the simple method of determining {539} the relative breadths
of the whorl at distances separated by some convenient vectorial angle
(such as 90°, 180°, or 360°).

Very elaborate measurements of a number of Ammonites have been made by
Naumann[517], by Sandberger[518], and by Grabau[519], among which we
may choose a couple of cases for consideration. In the following table
I have taken a portion of Grabau’s determinations of the breadth of the
whorls in _Ammonites_ (_Arcestes_)

 _Ammonites intuslabiatus._

                   Ratio of breadth of
 Breadth of whorls  successive whorls   The angle (α)
    (180° apart)      (360° apart)      as calculated

       0·30 mm.              —               —   —
       0·30                1·333            87° 23′
       0·40                1·500            86  19
       0·45                1·500            86  19
       0·60                1·444            86  39
       0·65                1·417            86  49
       0·85                1·692            85  13
       1·10                1·588            85  47
       1·35                1·545            86   2
       1·70                1·630            85  33
       2·20                1·441            86  40
       2·45                1·432            86  43
       3·15                1·735            85   0
       4·25                1·683            85  16
       5·30                1·482            86  25
       6·30                1·519            86  12
       8·05                1·635            85  32
      10·30                1·416            86  50
      11·40                1·252            87  57
      12·90                  —               —   —
                                            ──────
                                       Mean 86° 15′

{540}

_intuslabiatus_; these measurements Grabau gives for every 45° of
arc, but I have only set forth one quarter of these measurements,
that is to say, the breadths of successive whorls measured along one
diameter on both sides of the pole. The ratio between _alternate_
measurements is therefore the same ratio as Moseley adopted, namely the
ratio of breadth between _contiguous whorls_ along a radius vector. I
have then added to these observed values the corresponding calculated
values of the angle α, as obtained from our usual formula.

There is considerable irregularity in the ratios derived from these
measurements, but it will be seen that this irregularity only implies
a variation of the angle of the spiral between about 85° and 87°; and
the values fluctuate pretty regularly about the mean, which is 86° 15′.
Considering the difficulty of measuring the whorls, especially towards
the centre, and in particular the difficulty of determining with
precise accuracy the position of the pole, it is clear that in such a
case as this we are scarcely justified in asserting that the law of the
logarithmic spiral is departed from.

In some cases, however, it is undoubtedly departed from. Here for
instance is another table from Grabau, shewing the corresponding ratios
in an Ammonite of the group of _Arcestes tornatus_. In this case we see
a distinct tendency of the ratios to

                  _Ammonites tornatus._

                   Ratio of breadth of
 Breadth of whorls  successive whorls   The spiral angle
    (180° apart)      (360° apart)      (α) as calculated
      0·25 mm.             —                  —   —
      0·30               1·400               86° 56′
      0·35               1·667               85  21
      0·50               2·000               83  42
      0·70               2·000               83  42
      1·00               2·000               83  42
      1·40               2·100               83  16
      2·10               2·179               82  56
      3·05               2·238               82  42
      4·70               2·492               81  44
      7·60               2·574               81  27
     12·10               2·546               81  33
     19·35                 —                  —   —
                                             ──────
                                        Mean 83° 22′

{541}

increase as we pass from the centre of the coil outwards, and
consequently for the values of the angle α to diminish. The case is
precisely comparable to that of a cone with slightly curving sides: in
which, that is to say, there is a slight acceleration of growth in a
transverse as compared with the longitudinal direction.

――――――――――

In a tubular spiral, whether plane or helicoid, the consecutive whorls
may either be (1) isolated and remote from one another; or (2) they
may precisely meet, so that the outer border of one and the inner
border of the next just coincide; or (3) they may overlap, the vector
plane of each outer whorl cutting that of its immediate predecessor or
predecessors.

Looking, as we have done, upon the spiral shell as being essentially
a cone rolled up, it is plain that, for a given spiral angle,
intersection or non-intersection of the successive whorls will depend
upon _the apical angle_ of the original cone. For the wider the cone,
the more rapidly will its inner border tend to encroach on the outer
border of the preceding whorl.

But it is also plain that the greater be the apical angle of the
cone, and the broader, consequently, the cone itself be, the greater
difference will there be between the total _lengths_ of its inner
and outer border, under given conditions of flexure. And, since the
inner and outer borders are describing precisely the same spiral about
the pole, it is plain that we may consider the inner border as being
_retarded_ in growth as compared with the outer, and as being always
identical with a smaller and earlier part of the latter.

If λ be the ratio of growth between the outer and the inner curve,
then, the outer curve being represented by

 _r_ = _a_ _e_^{θ cot α},

the equation to the inner one will be

 _r′_ = _a_λ_e_^{θ cot α},

 or    _r′_ = _a_ _e_^{(θ − β)cot α},

and β may then be called the angle of retardation, to which the inner
curve is subject by virtue of its slower rate of growth. {542}

Dispensing with mathematical formulae, the several conditions may be
illustrated as follows:

[Illustration: Fig. 278.]

In the diagrams (Fig. 278), _O_ _P__{1} _P__{2} _P__{3}, etc.
represents a radius, on which _P__{1}, _P__{2}, _P__{3}, are the points
attained by the outer border of the tubular shell after as many entire
consecutive revolutions. And _P__{1}′, _P__{2}′, _P__{3}′, are the
points similarly intersected by the inner border; _OP_/_OP′_ being
always = λ, which is the ratio of growth, or “cutting-down factor.”
Then, obviously, when _O_ _P__{1} is less than _O_ _P__{2}′ the
whorls will be separated by an interspace (_a_); (2) when _O_ _P__{1}
= _O_ _P__{2}′ they will be in contact (_b_), and (3) when _O_ _P__{1}
is greater than _O_ _P__{2}′ there will a greater or less extent
of overlapping, that is to say of concealment of the surfaces of
the earlier by the later whorls (_c_). And as a further case (4),
it is plain that if λ be very large, that is to say if _O_ _P__{1}
be greater, not only than _O_ _P__{2}′ but also than _O_ _P__{3}′,
_O_ _P__{4}′, etc., we shall have complete, or all but complete
concealment by the last formed whorl, of the whole of its predecessors.
This latter condition is completely attained in _Nautilus pompilius_,
and approached, though not quite attained, in _N. umbilicatus_; and
the difference between these two forms, or “species,” is constituted
accordingly by a difference in the value of λ. (5) There is also a
final case, not easily distinguishable externally from (4), where _P′_
lies on {543} the opposite side of the radius vector to _P_, and is
therefore imaginary. This final condition is exhibited in Argonauta.

The limiting values of λ are easily ascertained.

[Illustration: Fig 279]

In Fig. 279 we have portions of two successive whorls, whose
corresponding points on the same radius vector (as _R_ and _R′_) are,
therefore, at a distance apart corresponding to 2π. Let _r_ and _r′_
refer to the inner, and _R_, _R′_ to the outer sides of the two whorls.
Then, if we consider

 _R_ = _a_ _e_^{θ cot α},

 it follows that _R′_ = _a_ _e_^{(θ + 2π)cot α},

 _r_ = λ_a_ _e_^{θ cot α} = _a_ _e_^{(θ − β)cot α},

 and _r′_ = λ_a_ _e_^{(θ + 2π)cot α} = _a_ _e_^{(θ + 2π − β)cot α}.

Now in the three cases (_a_, _b_, _c_) represented in Fig. 278, it is
plain that _r′_ ⪌ _R_, respectively. That is to say,

 λ_a_ _e_^{(θ + 2π)cot α} ⪌ _a_ _e_^{θ cot α},

 and λ_e_^{2π cot α} ⪌ 1.

The case in which λ_e_^{2π cot α} = 1, or −log λ = 2π cot α log ε, is
the case represented in Fig. 278, _b_: that is to say, the particular
case, for each value of α, where the consecutive whorls just touch,
without interspace or overlap. For such cases, then, we may tabulate
the values of λ, as follows:

 Constant angle      Ratio (λ) of rate of growth of inner border of tube,
 α of spiral             as compared with that of the outer border

      89°                           ·896
      88                            ·803
      87                            ·720
      86                            ·645
      85                            ·577
      80                            ·330
      75                            ·234
      70                            ·1016
      65                            ·0534

{544}

We see, accordingly, that in plane spirals whose constant angle
lies, say, between 65° and 70°, we can only obtain contact between
consecutive whorls if the rate of growth of the inner border of the
tube be a small fraction,—a tenth or a twentieth—of that of the outer
border. In spirals whose constant angle is 80°, contact is attained
when the respective rates of growth are, approximately, as 3 to 1;
while in spirals of constant angle from about 85° to 89°, contact is
attained when the rates of growth are in the ratio of from about 3/5 to
9/10.

[Illustration: Fig. 280.]

If on the other hand we have, for any given value of α, a value of λ
greater or less than the value given in the above table, then we have,
respectively, the conditions of separation or of overlap which are
exemplified in Fig. 278, _a_ and _c_. And, just as we have constructed
this table of values of λ for the particular case of simple contact
between the whorls, so we could construct similar tables for various
degrees of separation, or degrees of overlap.

For instance, a case which admits of simple solution is that in which
the interspace between the whorls is everywhere a mean proportional
between the breadths of the whorls themselves (Fig. 280). {545}

In this case, let us call _OA_ = _R_, _OC_ = _R__{1} and _OB_ = _r_. We
then have

                  _R__{1} = _OA_ = _a_ _e_^{θ cot α},

                  _R__{2} = _OC_ = _a_ _e_^{(θ + 2π) cot α},

                  _R__{1} _R__{2} = _a_ _e_^{2(θ + π) cot α} = _r_^2 [520].

 And              _r_^2 = (1/λ)^2 ⋅ ε^{2θ cot α},

 whence, equating, 1/λ = _e_^{π cot α}.

The corresponding values of λ are as follows:

                      Ratio (λ) of rates of growth of outer and inner
                      border, such as to produce a spiral with interspaces
                      between the whorls, the breadth of which
                      interspaces is a mean proportional between the
 Constant angle (α)   breadths of the whorls themselves

        90°                            1·00 (imaginary)
        89                              ·95
        88                              ·89
        87                              ·85
        86                              ·81
        85                              ·76
        80                              ·57
        75                              ·43
        70                              ·32
        65                              ·23
        60                              ·18
        55                              ·13
        50                              ·090
        45                              ·063
        40                              ·042
        35                              ·026
        30                              ·016

As regards the angle of retardation, β, in the formula

 _r′_ = λ_e_^{θ cot α}, or _r′_ = _e_^{(θ − β)cot α},

and in the case

 _r′_ = _e_^{(2π − β)cot α}, or −log λ = (2π − β)cot α,

{546}

it is evident that when β = 2π, that will mean that λ = 1. In other
words, the outer and inner borders of the tube are identical, and the
tube is constituted by one continuous line.

When λ is a very small fraction, that is to say when the rates of
growth of the two borders of the tube are very diverse, then β will
tend towards infinity—tend that is to say towards a condition in which
the inner border of the tube never grows at all. This condition is not
infrequently approached in nature. The nearly parallel-sided cone of
Dentalium, or the widely separated whorls of Lituites, are evidently
cases where λ nearly approaches unity in the one case, and is still
large in the other, β being correspondingly small; while we can easily
find cases where β is very large, and λ is a small fraction, for
instance in Haliotis, or in Gryphaea.

For the purposes of the morphologist, then, the main result of this
last general investigation is to shew that all the various types of
“open” and “closed” spirals, all the various degrees of separation or
overlap of the successive whorls, are simply the outward expression of
a varying ratio in the _rate of growth_ of the outer as compared with
the inner border of the tubular shell.

――――――――――

The foregoing problem of contact, or intersection, of the successive
whorls, is a very simple one in the case of the discoid shell but a
more complex one in the turbinate. For in the discoid shell contact
will evidently take place when the retardation of the inner as compared
with the outer whorl is just 360°, and the shape of the whorls need not
be considered.

As the angle of retardation diminishes from 360°, the whorls will stand
further and further apart in an open coil; as it increases beyond 360°,
they will more and more overlap; and when the angle of retardation is
infinite, that is to say when the true inner edge of the whorl does not
grow at all, then the shell is said to be completely involute. Of this
latter condition we have a striking example in Argonauta, and one a
little more obscure in _Nautilus pompilius_.

In the turbinate shell, the problem of contact is twofold, for we have
to deal with the possibilities of contact on the _same_ side of the
axis (which is what we have dealt with in the discoid) and {547} also
with the new possibility of contact or intersection on the _opposite_
side; it is this latter case which will determine the presence or
absence of an _umbilicus_, and whether, if present, it will be an
open conical space or a twisted cone. It is further obvious that, in
the case of the turbinate, the question of contact or no contact will
depend on the shape of the generating curve; and if we take the simple
case where this generating curve may be considered as an ellipse, then
contact will be found to depend on the angle which the major axis of
this ellipse makes with the axis of the shell. The question becomes a
complicated one, and the student will find it treated in Blake’s paper
already referred to.

When one whorl overlaps another, so that the generating curve cuts
its predecessor (at a distance of 2π) on the same radius vector, the
locus of intersection will follow a spiral line upon the shell, which
is called the “suture” by conchologists. It is evidently one of that
_ensemble_ of spiral lines in space of which, as we have seen, the
whole shell may be conceived to be constituted; and we might call it
a “contact-spiral,” or “spiral of intersection.” In discoid shells,
such as an Ammonite or a Planorbis, or in _Nautilus umbilicatus_,
there are obviously two such contact-spirals, one on each side of the
shell, that is to say one on each side of a plane perpendicular to the
axis. In turbinate shells such a condition is also possible, but is
somewhat rare. We have it for instance, in _Solarium perspectivum_,
where the one contact-spiral is visible on the exterior of the cone,
and the other lies internally, winding round the open cone of the
umbilicus[521]; but this second contact-spiral is usually imaginary,
or concealed within the whorls of the turbinated shell. Again, in
Haliotis, one of the contact-spirals is non-existent, because of the
extreme obliquity of the plane of the generating curve. In _Scalaria
pretiosa_ and in Spirula there is no contact-spiral, because the growth
of the generating curve has been too slow, in comparison with the
vector rotation of its plane. In Argonauta and in Cypraea, there is no
contact-spiral, because the growth of the generating curve has been too
quick. Nor, of course, is there any contact-spiral in Patella or in
Dentalium, because the angle α is too small ever to give us a complete
revolution of the spire. {548}

The various forms of straight or spiral shells among the Cephalopods,
which we have seen to be capable of complete definition by the help of
elementary mathematics, have received a very complicated descriptive
nomenclature from the palaeontologists. For instance, the straight
cones are spoken of as _orthoceracones_ or _bactriticones_, the loosely
coiled forms as _gyroceracones_ or _mimoceracones_, the more closely
coiled shells, in which one whorl overlaps the other, as _nautilicones_
or _ammoniticones_, and so forth. In such a succession of forms the
biologist sees undoubted and unquestioned evidence of ancestral
descent. For instance we read in Zittel’s _Palaeontology_[522]: “The
bactriticone obviously represents the primitive or primary radical of
the Ammonoidea, and the mimoceracone the next or secondary radical
of this order”; while precisely the opposite conclusion was drawn by
Owen, who supposed that the straight chambered shells of such fossil
cephalopods as Orthoceras had been produced by the gradual unwinding
of a coiled nautiloid shell[523]. _To such phylogenetic hypotheses
the mathematical or dynamical study of the forms of shells lends no
valid support._ If we have two shells in which the constant angle of
the spire be respectively 80° and 60°, that fact in itself does not at
all justify an assertion that the one is more primitive, more ancient,
or more “ancestral” than the other. Nor, if we find a third in which
the angle happens to be 70°, does that fact entitle us to say that
this shell is intermediate between the other two, in time, or in blood
relationship, or in any other sense whatsoever save only the strictly
formal and mathematical one. For it is evident that, though these
particular arithmetical constants manifest themselves in visible and
recognisable differences of form, yet they are not necessarily more
deep-seated or significant than are those which manifest themselves
only in difference of magnitude; and the student of phylogeny scarcely
ventures to draw conclusions as to the relative antiquity of two allied
organisms on the ground that one happens to be bigger or less, or
longer or shorter, than the other. {549}

At the same time, while it is obviously unsafe to rest conclusions
upon such features as these, unless they be strongly supported
and corroborated in other ways,—for the simple reason that there
is unlimited room for _coincidence_, or separate and independent
attainment of this or that magnitude or numerical ratio,—yet on the
other hand it is certain that, in particular cases, the evolution of
a race has actually involved gradual increase or decrease in some one
or more numerical factors, magnitude itself included,—that is to say
increase or decrease in some one or more of the actual and relative
velocities of growth. When we do meet with a clear and unmistakable
series of such progressive magnitudes or ratios, manifesting themselves
in a progressive series of “allied” forms, then we have the phenomenon
of “_orthogenesis_.” For orthogenesis is simply that phenomenon
of continuous lines or series of form (and also of functional or
physiological capacity), which was the foundation of the Theory of
Evolution, alike to Lamarck and to Darwin and Wallace; and which we
see to exist whatever be our ideas of the “origin of species,” or of
the nature and origin of “functional adaptations.” And to my mind,
the mathematical (as distinguished from the purely physical) study
of morphology bids fair to help us to recognise this phenomenon of
orthogenesis in many cases where it is not at once patent to the eye;
and also, on the other hand, to warn us, in many other cases, that
even strong and apparently complex resemblances in form may be capable
of arising independently, and may sometimes signify no more than the
equally accidental numerical coincidences which are manifested in
identity of length or weight, or any other simple magnitudes.

――――――――――

[Illustration: Fig. 281. An ammonitoid shell (_Macroscaphites_) to shew
change of curvature.]

I have already referred to the fact that, while in general a very great
and remarkable regularity of form is characteristic of the molluscan
shell, that complete regularity is apt to be departed from. We have
clear cases of such a departure in Pupa, Clausilia, and various Bulimi,
where the enveloping cone of the spire is not a right cone but a cone
whose sides are curved. It is plain that this condition may arise
in two ways: either by a gradual change in the ratio of growth of
the whorls, that is to say in the logarithmic spiral itself, or by a
change in the velocity of {550} translation along the axis, that is
to say in the helicoid which, in all turbinate shells, is superposed
upon the spiral. Very careful measurements will be necessary to
determine to which of these factors, or in what proportions to each,
the phenomenon is due. But in many Ammonitoidea where the helicoid
factor does not enter into the case, we have a clear illustration of
gradual and marked changes in the spiral angle itself, that is to say
of the ratio of growth corresponding to increase of vectorial angle.
We have seen from some of Naumann’s and Grabau’s measurements that
such a tendency to vary, such an acceleration or retardation, may be
detected even in Ammonites which present nothing abnormal to the eye.
But let us suppose that the spiral angle increases somewhat rapidly;
we shall then get a spiral with gradually narrowing whorls, and this
condition is characteristic of Oekotraustes, a subgenus of Ammonites.
If on the other hand, the angle α gradually diminishes, and even falls
away to zero, we shall have the spiral curve opening out, as it does in
Scaphites, Ancyloceras and Lituites, until the spiral coil is replaced
by a spiral curve so gentle as to seem all but straight. Lastly, there
are a few cases, such as _Bellerophon expansus_ and some Goniatites,
where the outer spiral does not perceptibly change, but the whorls
become more “embracing” or the whole shell more involute. Here it is
the angle of retardation, the ratio of growth between the outer and
inner parts of the whorl, which undergoes a gradual change.

――――――――――

In order to understand the relation of a close-coiled shell to one of
its straighter congeners, to compare (for example) an {551} Ammonite
with an Orthoceras, it is necessary to estimate the length of the right
cone which has, so to speak, been coiled up into the spiral shell. Our
problem then is, To find the length of a plane logarithmic spiral, in
terms of the radius and the constant angle α. In the annexed diagram,
if _OP_ be a radius vector, _OQ_ a line of reference perpendicular to
_OP_, and _PQ_ a tangent to the curve, _PQ_, or sec α, is equal in
length to the spiral arc _OP_. And this is practically obvious: for
_PP′_/_PR′_ = _ds_/_dr_ = sec α, and therefore sec α = _s_/_r_, or the
ratio of arc to radius vector.

[Illustration: Fig. 282.]

Accordingly, the ratio of _l_, the total length, to _r_, the radius
vector up to which the total length is to be measured, is expressed by
a simple table of secants; as follows:

  α          _l_/_r_

  5°           1·004
 10            1·015
 20            1·064
 30            1·165
 40            1·305
 50            1·56
 60            2·0
 70            2·9
 75            3·9
 80            5·8
 85           11·5
 86           14·3
 87           19·1
 88           28·7
 89           57·3
 89° 10′      68·8
     20       85·9
     30      114·6
     40      171·9
     50      343·8
     55      687·5
     59     3437·7
 90        Infinite

Putting the same table inversely, so as to shew the total {552} length
in whole numbers, in terms of the radius, we have as follows:

 Total length (in terms
     of the radius)        Constant angle

           2                60°
           3                70  31′
           4                75  32
           5                78  28
          10                84  16
          20                87   8
          30                88   6
          40                88  34
          50                88  51
         100                89  26
        1000                89  56′ 36″
      10,000                89  59  30

Accordingly, we see that (1), when the constant angle of the spiral
is small, the spiral itself is scarcely distinguishable from a
straight line, and its length is but very little greater than that
of its own radius vector. This remains pretty much the case for a
considerable increase of angle, say from 0° to 20° or more; (2) for a
very considerably greater increase of the constant angle, say to 50°
or more, the shell would only have the appearance of a gentle curve;
(3) the characteristic close coils of the Nautilus or Ammonite would
be typically represented only when the constant angle lies within a
few degrees on either side of about 80°. The coiled up spiral of a
Nautilus, with a constant angle of about 80°, is about six times the
length of its radius vector, or rather more than three times its own
diameter; while that of an Ammonite, with a constant angle of, say,
from 85° to 88°, is from about six to fifteen times as long as its own
diameter. And (4) as we approach an angle of 90° (at which point the
spiral vanishes in a circle), the length of the coil increases with
enormous rapidity. Our spiral would soon assume the appearance of the
close coils of a Nummulite, and the successive increments of breadth
in the successive whorls would become inappreciable to the eye. The
logarithmic spiral of high constant angle would, as we have already
seen, tend to become indistinguishable, without the most careful
measurement, from an Archimedean spiral. And it is obvious, moreover,
that our ordinary methods of {553} determining the constant angle of
the spiral would not in these cases be accurate enough to enable us to
measure the length of the coil: we should have to devise a new method,
based on the measurement of radii or diameters over a large number of
whorls.

The geometrical form of the shell involves many other beautiful
properties, of great interest to the mathematician, but which it is
not possible to reduce to such simple expressions as we have been
content to use. For instance, we may obtain an equation which shall
express completely the surface of any shell, in terms of polar or of
rectangular coordinates (as has been done by Moseley and by Blake),
or in Hamiltonian vector notation. It is likewise possible (though
of little interest to the naturalist) to determine the area of a
conchoidal surface, or the volume of a conchoidal solid, and to find
the centre of gravity of either surface or solid[524]. And Blake has
further shewn, with considerable elaboration, how we may deal with
the symmetrical distortion, due to pressure, which fossil shells
are often found to have undergone, and how we may reconstitute by
calculation their original undistorted form,—a problem which, were the
available methods only a little easier, would be very helpful to the
palaeontologist; for, as Blake himself has shewn, it is easy to mistake
a symmetrically distorted specimen of (for instance) an Ammonite,
for a new and distinct species of the same genus. But it is evident
that to deal fully with the mathematical problems contained in, or
suggested by, the spiral shell, would require a whole treatise, rather
than a single chapter of this elementary book. Let us then, leaving
mathematics aside, attempt to summarise, and perhaps to extend, what
has been said about the general possibilities of form in this class of
organisms.


_The Univalve Shell: a summary._

The surface of any shell, whether discoid or turbinate, may be imagined
to be generated by the revolution about a fixed axis of a closed curve,
which, remaining always geometrically similar to itself, increases
continually its dimensions: and, since the rate of growth of the
generating curve and its velocity of rotation follow the same law, the
curve traced in space by corresponding points {554} in the generating
curve is, in all cases, a logarithmic spiral. In discoid shells, the
generating figure revolves in a plane perpendicular to the axis, as
in Nautilus, the Argonaut and the Ammonite. In turbinate shells, it
slides continually along the axis of revolution, and the curve in space
generated by any given point partakes, therefore, of the character of a
helix, as well as of a logarithmic spiral; it may be strictly entitled
a helico-spiral. Such turbinate or helico-spiral shells include the
snail, the periwinkle and all the common typical Gastropods.

[Illustration: Fig. 283. Section of a spiral, or turbinate, univalve,
_Triton corrugatus_, Lam. (From Woodward.)]

The generating figure, as represented by the mouth of the shell, is
sometimes a plane curve, of simple form; in other and more numerous
cases, it becomes more complicated in form and its boundaries do not
lie in one plane: but in such cases as these we may replace it by its
“trace,” on a plane at some definite angle to the direction of growth,
for instance by its form as it appears in a section through the axis
of the helicoid shell. The generating curve is of very various shapes.
It is circular in Scalaria or Cyclostoma, and in Spirula; it may be
considered as a segment of a circle in Natica or in Planorbis. It
is approximately triangular in Conus, and rhomboidal in Solarium or
Potamides. It is very commonly more or less elliptical: the long axis
of the ellipse being parallel to the axis of the shell in Oliva and
Cypraea; all but perpendicular to it in many Trochi; and oblique to it
in many well-marked cases, such as Stomatella, Lamellaria, _Sigaretus
haliotoides_ (Fig. 284) and Haliotis. In _Nautilus pompilius_ it is
approximately a semi-ellipse, and in _N. umbilicatus_ rather more
than a semi-ellipse, the long axis lying in both cases perpendicular
to the axis of the shell[525]. Its {555} form is seldom open to easy
mathematical expression, save when it is an actual circle or ellipse;
but an exception to this rule may be found in certain Ammonites,
forming the group “Cordati,” where (as Blake points out) the curve
is very nearly represented by a cardioid, whose equation is _r_
= _a_(1 + cos θ).

The generating curve may grow slowly or quickly; its growth-factor
is very slow in Dentalium or Turritella, very rapid in Nerita, or
Pileopsis, or Haliotis or the Limpet. It may contain the axis in its
plane, as in Nautilus; it may be parallel to the axis, as in the
majority of Gastropods; or it may be inclined to the axis, as it is in
a very marked degree in Haliotis. In fact, in Haliotis the generating
curve is so oblique to the axis of the shell that the latter appears
to grow by additions to one margin only (cf. Fig. 258), as in the case
of the opercula of Turbo and Nerita referred to on p. 522; and this is
what Moseley supposed it to do.

[Illustration: Fig. 284. _A, Lamellaria perspicua; B, Sigaretus
haliotoides._

(After Woodward.)]

The general appearance of the entire shell is determined (apart from
the form of its generating curve) by the magnitude of three angles; and
these in turn are determined, as has been sufficiently explained, by
the ratios of certain velocities of growth. These angles are (1) the
constant angle of the logarithmic spiral (α); (2) in turbinate shells,
the enveloping angle of the cone, or (taking half that angle) the angle
(θ) which a tangent to the whorls makes with the axis of the shell; and
(3) an angle called the “angle of retardation” (β), which expresses the
retardation in growth of {556} the inner as compared with the outer
part of each whorl, and therefore measures the extent to which one
whorl overlaps, or the extent to which it is separated from, another.

The spiral angle (α) is very small in a limpet, where it is usually
taken as = 0°; but it is evidently of a significant amount, though
obscured by the shortness of the tubular shell. In Dentalium it is
still small, but sufficient to give the appearance of a regular curve;
it amounts here probably to about 30° to 40°. In Haliotis it is from
about 70° to 75°; in Nautilus about 80°; and it lies between 80° and
85°, or even more, in the majority of Gastropods.

The case of Fissurella is curious. Here we have, apparently, a conical
shell with no trace of spiral curvature, or (in other words) with a
spiral angle which approximates to 0°; but in the minute embryonic
shell (as in that of the limpet) a spiral convolution is distinctly
to be seen. It would seem, then, that what we have to do with here
is an unusually large growth-factor in the generating curve, which
causes the shell to dilate into a cone of very wide angle, the apical
portion of which has become lost or absorbed, and the remaining part
of which is too short to show clearly its intrinsic curvature. In the
closely allied Emarginula, there is likewise a well-marked spiral in
the embryo, which however is still manifested in the curvature of the
adult, nearly conical, shell. In both cases we have to do with a very
wide-angled cone, and with a high retardation-factor for its inner, or
posterior, border. The series is continued, from the apparently simple
cone to the complete spiral, through such forms as Calyptraea.

The angle α, as we have seen, is not always, nor rigorously, a constant
angle. In some Ammonites it may increase with age, the whorls becoming
closer and closer; in others it may decrease rapidly, and even fall
to zero, the coiled shell then straightening out, as in Lituites and
similar forms. It diminishes somewhat, also, in many Orthocerata,
which are slightly curved in youth, but straight in age. It tends to
increase notably in some common land-shells, the Pupae and Bulimi; and
it decreases in Succinea.

Directly related to the angle α is the ratio which subsists between
the breadths of successive whorls. The following table gives a few
illustrations of this ratio in particular cases, in addition to those
which we have already studied. {557}

 _Ratio of breadth of consecutive whorls._

 Pointed Turbinates

  _Telescopium fuscum_            1·14
  _Acus subulatus_                1·16
 *_Turritella terebellata_        1·18
 *_Turritella imbricata_          1·20
  _Cerithium palustre_            1·22
  _Turritella duplicata_          1·23
  _Melanopsis terebralis_         1·23
  _Cerithium nodulosum_           1·24
 *_Turritella carinata_           1·25
  _Acus crenulatus_               1·25
  _Terebra maculata_ (Fig. 285)   1·25
 *_Cerithium lignitarum_          1·26
  _Acus dimidiatus_               1·28
  _Cerithium sulcatum_            1·32
  _Fusus longissimus_             1·34
 *_Pleurotomaria conoidea_        1·34
  _Trochus niloticus_ (Fig. 286)  1·41
  _Mitra episcopalis_             1·43
  _Fusus antiquus_                1·50
  _Scalaria pretiosa_             1·56
  _Fusus colosseus_               1·71
  _Phasianella bulloides_         1·80
  _Helicostyla polychroa_         2·00

 Obtuse Turbinates and Discoids

  _Conus virgo_                   1·25
  _Conus litteratus_              1·40
  _Conus betulina_                1·43
 *_Helix nemoralis_               1·50
 *_Solarium perspectivum_         1·50
  _Solarium trochleare_           1·62
  _Solarium magnificum_           1·75
 *_Natica aperta_                 2·00
  _Euomphalus pentangulatus_      2·00
  _Planorbis corneas_             2·00
  _Solaropsis pellis-serpentis_   2·00
  _Dolium zonatum_                2·10
 *_Natica glaucina_               3·00
  _Nautilus pompilius_            3·00
  _Haliotis excavatus_            4·20
  _Haliotis parvus_               6·00
  _Delphinula atrata_             6·00
  _Haliotis rugoso-plicata_       9·30
  _Haliotis viridis_             10·00

 Those marked * from Naumann; the rest from Macalister[526].

In the case of turbinate shells, we require to take into account the
angle θ, in order to determine the spiral angle α from the ratio of the
breadths of consecutive whorls; for the short table given on p. 534 is
only applicable to discoid shells, in which the angle θ is an angle of
90°. Our formula, as mentioned on p. 518 now becomes

 _R_ = ε^{2π sin θ cot α}.

For this formula I have worked out the following table. {558}

 _Table shewing values of the spiral angle α corresponding to certain
 ratios of breadth of successive whorls of the shell, for various
 values of the apical semi-angle θ._

  Ratio
  _R_/1   θ = 5°      10°        15°       20°       30°       40°      50°       60°        70°      80°       90°

   1·1    80°  8′   85°  0′   86° 44′   87° 28′   88° 16′   88° 39′   88° 52′   89°  0′   89°  4′   89°  7′   89°  8′
   1·25   67  51    78  27    82  11    84   5    85  56    86  50    87  21    87  39    87  50    87  56    87  58
   1·5    53  30    69  37    76   0    79  21    82  39    84  16    85  13    85  44    86   4    86  15    86  18
   2·0    38  20    57  35    66  55    73  11    77  34    80  16    81  52    82  45    83  18    83  37    83  42
   2·5    30  53    50   0    60  35    67   0    73  45    77  13    79  19    80  26    81  11    81  35    81  42
   3·0    26  32    44  50    56   0    63   0    70  45    74  45    77  17    78  35    79  28    79  56    80   5
   3·5    23  37    41   5    52  25    59  50    68  15    72  45    75  35    77   2    78   1    78  33    78  43
   4·0    21  35    38  10    49  35    57  15    66  10    71   3    74   9    75  42    76  47    77  22    77  34
   4·5    20   0    36   0    47  15    55   5    64  25    69  35    72  54    74  33    75  43    76  20    76  35
   5·0    18  45    34  10    45  20    53  15    62  55    68  15    71  48    73  31    74  45    75  25    75  38
  10·0    13  25    25  20    35  15    43   5    53  45    60  20    64  57    67   4    68  42    69  35    69  53
  20·0    10  25    20   0    28  30    35  45    46  25    53  25    58  52    61  10    63   6    64  10    64  31
  50·0     8   0    15  35    22  35    28  50    38  45    45  55    52   1    54  18    56  28    57  42    58   6
 100·0     6  50    13  20    19  30    25   5    34  20    41  15    47  35    49  45    52   3    53  20    53  46

{559}

From this table, by interpolation, we may easily fill in the
approximate values of α, as soon as we have determined the apical angle
θ and measured the ratio _R_; as follows:

                             _R_        θ      α

 _Turritella_ sp.             1·12      7°    81°
 _Cerithium nodulosum_        1·24     15     82
 _Conus virgo_                1·25     70     88
 _Mitra episcopalis_          1·43     16     78
 _Scalaria pretiosa_          1·56     26     81
 _Phasianella bulloides_      1·80     26     80
 _Solarium perspectivum_      1·50     53     85
 _Natica aperta_              2·00     70     83
 _Planorbis corneus_          2·00     90     84
 _Euomphalus pentangulatus_   2·00     90     84

We see from this that shells so different in appearance as Cerithium,
Solarium, Natica and Planorbis differ very little indeed in the
magnitude of the spiral angle α, that is to say in the relative
velocities of radial and tangential growth. It is upon the angle θ that
the difference in their form mainly depends: that is to say the amount
of longitudinal shearing, or displacement parallel to the axis of the
shell.

[Illustration: Fig. 285. _Terebra maculata_, L.]

The enveloping angle, or rather semi-angle (θ), of the cone may be
taken as 90° in the discoid shells, such as Nautilus and Planorbis. It
is still a large angle, of 70° or 75°, in Conus or in Cymba, somewhat
less in Cassis, Harpa, Dolium or Natica; it is about 50° to 55° in the
various species of Solarium, about 35° in the typical Trochi, such as
_T. niloticus_ or _T. zizyphinus_, and about 25° or 26° in _Scalaria
pretiosa_ and _Phasianella bulloides_; it becomes a very acute angle,
of 15°, 10°, or even less, in Eulima, Turritella or Cerithium. The
costly _Conus gloria-maris_, one of the {560} great treasures of the
conchologist, differs from its congeners in no important particular
save in the somewhat “produced” spire, that is to say in the
comparatively low value of the angle θ.

[Illustration: Fig. 286. _Trochus niloticus_, L.]

A variation with advancing age of θ is common, but (as Blake points
out) it is often not to be distinguished or disentangled from an
alteration of α. Whether alone, or combined with a change in α, we find
it in all those many Gastropods whose whorls cannot all be touched by
the same enveloping cone, and whose spire is accordingly described
as _concave_ or _convex_. The former condition, as we have it in
Cerithium, and in the cusp-like spire of Cassis, Dolium and some Cones,
is much the commoner of the two. And such tendency to decrease may
lead to θ becoming a negative angle; in which case we have a depressed
spire, as in the Cypraeae.

When we find a “reversed shell,” a whelk or a snail for instance whose
spire winds to the left instead of to the right, we may describe it
mathematically by the simple statement that the angle θ has _changed
sign_. In the genus Ampullaria, or Apple-snails, inhabiting tropical
or sub-tropical rivers, we have a remarkable condition; for in
certain “species” the spiral turns to the right, in others to the
left, and in others again we have a flattened {561} “discoid” shell;
and furthermore we have numerous intermediate stages, on either
side, shewing right and left-handed spirals of varying degrees of
acuteness[527]. In this case, the angle θ may be said to vary, within
the limits of a genus, from somewhere about 35° to somewhere about 125°.

The angle of retardation (β) is very small in Dentalium and Patella;
it is very large in Haliotis. It becomes infinite in Argonauta and
in Cypraea. Connected with the angle of retardation are the various
possibilities of contact or separation, in various degrees, between
adjacent whorls in the discoid, and between both adjacent and opposite
whorls in the turbinated shell. But with these phenomena we have
already dealt sufficiently.


_Of Bivalve Shells._

Hitherto we have dealt only with univalve shells, and it is in these
that all the mathematical problems connected with the spiral, or
helico-spiral, are best illustrated. But the case of the bivalve shell,
of Lamellibranchs or of Brachiopods, presents no essential difference,
save only that we have here to do with two conjugate spirals, whose
two axes have a definite relation to one another, and some freedom of
rotatory movement relatively to one another.

The generating curve is particularly well seen in the bivalve, where it
simply constitutes what we call “the outline of the shell.” It is for
the most part a plane curve, but not always; for there are forms, such
as Hippopus, Tridacna and many Cockles, or Rhynchonella and Spirifer
among the Brachiopods, in which the edges of the two valves interlock,
and others, such as Pholas, Mya, etc., where in part they fail to meet.
In such cases as these the generating curves are conjugate, having a
similar relation, but of opposite sign, to a median plane of reference.
A great variety of form is exhibited by these generating curves among
the bivalves. In a good many cases the curve is approximately circular,
as in Anomia, Cyclas, Artemis, Isocardia; it is nearly semi-circular
in Argiope. It is approximately elliptical in Orthis and in Anodon; it
may be called semi-elliptical in Spirifer. It is a nearly rectilinear
{562} triangle in Lithocardium, and a curvilinear triangle in Mactra.
Many apparently diverse but more or less related forms may be shewn
to be deformations of a common type, by a simple application of the
mathematical theory of “Transformations,” which we shall have to study
in a later chapter. In such a series as is furnished, for instance, by
Gervillea, Perna, Avicula, Modiola, Mytilus, etc., a “simple shear”
accounts for most, if not all, of the apparent differences.

Upon the surface of the bivalve shell we usually see with great
clearness the “lines of growth” which represent the successive
margins of the shell, or in other words the successive positions
assumed during growth by the growing generating curve; and we have a
good illustration, accordingly, of how it is characteristic of the
generating curve that it should constantly increase, while never
altering its geometric similarity.

Underlying these “lines of growth,” which are so characteristic of a
molluscan shell (and of not a few other organic formations), there
is, then, a “law of growth” which we may attempt to enquire into and
which may be illustrated in various ways. The simplest cases are those
in which we can study the lines of growth on a more or less flattened
shell, such as the one valve of an oyster, a Pecten or a Tellina,
or some such bivalve mollusc. Here around an origin, the so-called
“umbo” of the shell, we have a series of curves, sometimes nearly
circular, sometimes elliptical, and often asymmetrical; and such
curves are obviously not “concentric,” though we are often apt to call
them so, but are always “co-axial.” This manner of arrangement may be
illustrated by various analogies. We might for instance compare it to
a series of waves, radiating outwards from a point, through a medium
which offered a resistance increasing, with the angle of divergence,
according to some simple law. We may find another, and perhaps a
simpler illustration as follows:

In a very simple and beautiful theorem, Galileo shewed that, if we
imagine a number of inclined planes, or gutters, sloping downwards (in
a vertical plane) at various angles from a common starting-point, and
if we imagine a number of balls rolling each down its own gutter under
the influence of gravity (and without hindrance from friction), then,
at any given instant, the locus of {563} all these moving bodies is a
circle passing through the point of origin. For the acceleration along
any one of the sloping paths, for instance _AB_ (Fig. 287), is such that

    _AB_ = ½_g_ cos θ ⋅ t^2
         = ½_g_ ⋅ _AB_/_AC_ ⋅ t^2.

 Therefore

    _t_^2 = 2/_g_ ⋅ _AC_.

[Illustration: Fig. 287.]

That is to say, all the balls reach the circumference of the circle at
the same moment as the ball which drops vertically from _A_ to _C_.

Where, then, as often happens, the generating curve of the shell is
approximately a circle passing through the point of origin, we may
consider the acceleration of growth along various radiants to be
governed by a simple mathematical law, closely akin to that simple law
of acceleration which governs the movements of a falling body. And,
_mutatis mutandis_, a similar definite law underlies the cases where
the generating curve is continually elliptical, or where it assumes
some more complex, but still regular and constant form.

It is easy to extend the proposition to the particular case where the
lines of growth may be considered elliptical. In such a case we have
_x_^2/_a_^2 + _y_^2/_b_^2 = 1, where _a_ and _b_ are the major and
minor axes of the ellipse.

Or, changing the origin to the vertex of the figure

        _x_^2/_a_^2 − 2_x_/_a_ + _y_^2/_b_^2 = 0,

 giving (_x_ − _a_)^2/_a_^2 + _y_^2/_b_^2 = 1.

Then, transferring to polar coordinates, where _r_ ⋅ cos θ = _x_,
_r_ ⋅ sin θ = _y_, we have

 (_r_ ⋅ cos^2 θ)/_a_^2 − (2 cos θ)/_a_ + (_r_ ⋅ sin θ)/_b_^2 = 0,

{564}

 which is equivalent to

 _r_ = (2_a_ _b_^2 cos θ)/((_b_^2 cos^2 θ) + (_a_^2 sin^2 θ)),

 or, eliminating the sine-function,

 _r_ = (2_a_ _b_^2 cos θ)/((_b_^2 − _a_^2) cos^2 θ + _a_^2).

Obviously, in the case when _a_ = _b_, this gives us the circular
system which we have already considered. For other values, or ratios,
of _a_ and _b_, and for all values of θ, we can easily construct a
table, of which the following is a sample:

 _Chords of an ellipse, whose major and minor axes (a, b) are in
 certain given ratios._

  θ  _a_/_b_ = 1/3   1/2      2/3      1/1     3/2    2/1     3/1
  0°      1·0        1·0     1·0     1·0     1·0     1·0     1·0
 10       1·01       1·01    1·002    ·985    ·948    ·902    ·793
 20       1·05       1·03    1·005    ·940    ·820    ·695    ·485
 30       1·115      1·065   1·005    ·866    ·666    ·495    ·289
 40       1·21       1·11     ·995    ·766    ·505    ·342    ·178
 50       1·34       1·145    ·952    ·643    ·372    ·232    ·113
 60       1·50       1·142    ·857    ·500    ·258    ·152    ·071
 70       1·59       1·015    ·670    ·342    ·163    ·092    ·042
 80       1·235       ·635    ·375    ·174    ·078    ·045    ·020
 90       0·0        0·0     0·0     0·0     0·0     0·0     0·0

[Illustration: Fig. 288.]

The coaxial ellipses which we then draw, from the values given in
the table, are such as are shewn in Fig. 288 for the ratio _a_/_b_
= 3/1, and in Fig. 289 for the ratio _a_/_b_ = ½; these are fair
approximations to the actual outlines, and to the actual arrangement
of the lines of growth, in such forms as Solecurtus or Cultellus, and
in Tellina or Psammobia. It is not difficult to introduce a constant
into our equation to meet the case of a shell which is somewhat
unsymmetrical on either side of the median axis. It is a somewhat
more troublesome matter, however, to bring these configurations into
relation with a “law of growth,” as was so easily done in the case
of the circular figure: in other words, to {565} formulate a law of
acceleration according to which points starting from the origin _O_,
and moving along radial lines, would all lie, at any future epoch, on
an ellipse passing through _O_; and this calculation we need not enter
into.

[Illustration: Fig. 289.]

All that we are immediately concerned with is the simple fact
that where a velocity, such as our rate of growth, varies with
its direction,—varies that is to say as a function of the angular
divergence from a certain axis,—then, in a certain simple case, we
get lines of growth laid down as a system of coaxial circles, and,
when the function is a more complex one, as a system of ellipses or of
other more complicated coaxial figures, which figures may or may not
be symmetrical on either side of the axis. Among our bivalve mollusca
we shall find the lines of growth to be approximately circular in, for
instance, Anomia; in Lima (e.g. _L. subauriculata_) we have a system
of nearly symmetrical ellipses with the vertical axis about twice the
transverse; in _Solen pellucidus_, we have again a system of lines of
growth which are not far from being symmetrical ellipses, in which
however the transverse is between three and four times as great as
the vertical axis. In the great majority of cases, we have a similar
phenomenon with the further complication of slight, but occasionally
very considerable, lateral asymmetry.

In certain little Crustacea (of the genus Estheria) the carapace
takes the form of a bivalve shell, closely simulating that of a
{566} lamellibranchiate mollusc, and bearing lines of growth in all
respects analogous to or even identical with those of the latter. The
explanation is very curious and interesting. In ordinary Crustacea the
carapace, like the rest of the chitinised and calcified integument,
is shed off in successive moults, and is restored again as a whole.
But in Estheria (and one or two other small crustacea) the moult is
incomplete: the old carapace is retained, and the new, growing up
underneath it, adheres to it like a lining, and projects beyond its
edge: so that in course of time the margins of successive old carapaces
appear as “lines of growth” upon the surface of the shell. In this mode
of formation, then (but not in the usual one), we obtain a structure
which “is partly old and partly new,” and whose successive increments
are all similar, similarly situated, and enlarged in a continued
progression. We have, in short, all the conditions appropriate and
necessary for the development of a logarithmic spiral; and this
logarithmic spiral (though it is one of small angle) gives its own
character to the structure, and causes the little carapace to partake
of the characteristic conformation of the molluscan shell.

The essential simplicity, as well as the great regularity of the
“curves of growth” which result in the familiar configurations of
our bivalve shells, sufficiently explain, in a general way, the ease
with which they may be imitated, as for instance in the so-called
“artificial shells” which Kappers has produced from the conchoidal form
and lamination of lumps of melted and quickly cooled paraffin[528].

 In the above account of the mathematical form of the bivalve shell,
 we have supposed, for simplicity’s sake, that the pole or origin
 of the system is at a point where all the successive curves touch
 one another. But such an arrangement is neither theoretically
 probable, nor is it actually the case; for it would mean that in a
 certain direction growth fell, not merely to a minimum, but to zero.
 As a matter of fact, the centre of the system (the “umbo” of the
 conchologists) lies not at the edge of the system, but very near to
 it; in other words, there is a certain amount of growth all round.
 But to take account of this condition would involve more troublesome
 mathematics, and it is obvious that the foregoing illustrations are a
 sufficiently near approximation to the actual case. {567}

Among the bivalves the spiral angle (α) is very small in the flattened
shells, such as Orthis, Lingula or Anomia. It is larger, as a rule, in
the Lamellibranchs than in the Brachiopods, but in the latter it is of
considerable magnitude among the Pentameri. Among the Lamellibranchs
it is largest in such forms as Isocardia and Diceras, and in the
very curious genus Caprinella; in all of these last-named genera its
magnitude leads to the production of a spiral shell of several whorls,
precisely as in the univalves. The angle is usually equal, but of
opposite sign, in the two valves of the Lamellibranch, and usually
of opposite sign but unequal in the two valves of the Brachiopod. It
is very unequal in many Ostreidae, and especially in such forms as
Gryphaea, or in Caprinella, which is a kind of exaggerated Gryphaea.
Occasionally it is of the same sign in both valves (that is to say,
both valves curve the same way) as we see sometimes in Anomia, and much
better in Productus or Strophomena.

[Illustration: Fig. 290. _Caprinella adversa._ (After Woodward.)]

[Illustration: Fig. 291. Section of _Productus_ (_Strophomena_) sp.
(From Woods.)]

Owing to the large growth-factor of the generating curve, and the
comparatively small angle of the spiral, the whole shell seldom
assumes a spiral form so conspicuous as to manifest in a typical
way the helical twist or shear which is so conspicuous in the {568}
majority of univalves, or to let us measure or estimate the magnitude
of the apical angle (θ) of the enveloping cone. This however we can
do in forms like Isocardia and Diceras; while in Caprinella we see
that the whorls lie in a plane perpendicular to the axis, forming a
discoidal spire. As in the latter shell, so also universally among
the Brachiopods, there is no lateral asymmetry in the plane of the
generating curve such as to lead to the development of a helix; but in
the majority of the Lamellibranchiata it is obvious, from the obliquity
of the lines of growth, that the angle θ is significant in amount.

――――――――――

[Illustration: Fig. 292. Skeletal loop of _Terebratula_. (From Woods.)]

The so-called “spiral arms” of Spirifer and many other Brachiopods
are not difficult to explain. They begin as a single structure, in
the form of a loop of shelly substance, attached to the dorsal valve
of the shell, in the neighbourhood of the hinge. This loop has a
curvature of its own, similar to but not necessarily identical with
that of the valve to which it is attached; and this curvature will
tend to be developed, by continuous and symmetrical growth, into a
fully formed logarithmic spiral, so far as it is permitted to do so
under the constraint of the shell in which it is contained. In various
Terebratulae we see the spiral growth of the loop, more or less
flattened and distorted by the restraining pressure of the ventral
valve. In a number of cases the loop remains small, but gives off two
nearly parallel branches or offshoots, which continue to grow. And
these, starting with just such a slight curvature as the loop itself
possessed, grow on and on till they may form close-wound spirals,
always provided that the “spiral angle” of the curve is such that
the resulting spire can be freely contained within the cavity of the
shell. Owing to the bilateral symmetry of the whole system, the case
will be rare, and unlikely to occur, in which each separate arm will
coil strictly _in a plane_, so as to constitute a discoid spiral; for
the original {569} direction of each of the two branches, parallel
to the valve (or nearly so) and outwards from the middle line, will
tend to constitute a curve of double curvature, and so, on further
growth, to develop into a helicoid. This is what actually occurs,
in the great majority of cases. But the curvature may be such that
the helicoid grows outwards from the middle line, or inwards towards
the middle line, a _very_ slight difference in the initial curvature
being sufficient to direct the spire the one way or the other; the
middle course of an undeviating discoid spire will be rare, from the
usual lack of any obvious controlling force to prevent its deviation.
The cases in which the helicoid spires point towards, or point away
from, the middle line are ascribed, in zoological classification, to
particular “families” of Brachiopods, the former

[Illustration: Fig. 293. Spiral arms of _Spirifer_. (From Woods.)]

[Illustration: Fig. 294. Inwardly directed spiral arms of _Atrypa_.]

condition defining (or helping to define) the Atrypidae and the
latter the Spiriferidae and Athyridae. It is obvious that the incipient
curvature of the arms, and consequently the form and direction of the
spirals, will be influenced by the surrounding pressures, and these in
turn by the general shape of the shell. We shall expect, accordingly,
to find the long outwardly directed spirals associated with shells
which are transversely elongated, as Spirifer is; while the more
rounded Atrypas will tend to the opposite condition. In a few cases, as
in Cyrtina or Reticularia, where the shell is comparatively narrow but
long, and where the uncoiled basal support of the arms is long also,
the spiral coils into which the latter grow are turned backwards, in
the direction where there is room for them. And in the few cases where
the shell is very considerably flattened, the spirals (if they find
room {570} to grow at all) will be constrained to do so in a discoid
or nearly discoid fashion, and this is actually the case in such
flattened forms as Koninckina or Thecidium.


_The Shells of Pteropods._

While mathematically speaking we are entitled to look upon the bivalve
shell of the Lamellibranch as consisting of two distinct elements,
each comparable to the entire shell of the univalve, we have no
biological grounds for such a statement; for the shell arises from a
single embryonic origin, and afterwards becomes split into portions
which constitute the two separate valves. We can perhaps throw some
indirect light upon this phenomenon, and upon several other phenomena
connected with shell-growth, by a consideration of the simple conical
or tubular shells of the Pteropods. The shells of the latter are in few
cases suitable for simple mathematical investigation, but nevertheless
they are of very considerable interest in connection with our general
problem.

The morphology of the Pteropods is by no means well understood, and
in speaking of them I will assume that there are still grounds for
believing (in spite of Boas’ and Pelseneer’s arguments) that they are
directly related to, or may at least be directly compared with, the
Cephalopoda[529].

[Illustration: Fig. 295. Pteropod shells: (1) _Cuvierina columnella_;
(2) _Cleodora chierchiae_; (3) _C. pygmaea_. (After Boas.)]

The simplest shells among the Pteropods have the form of a tube, more
or less cylindrical (Cuvierina), more often conical (Creseis, Clio);
and this tubular shell (as we have already had occasion to remark,
on p. 258), frequently tends, when it is very small and delicate, to
assume the character of an unduloid. (In such a case it is more than
likely that the tiny shell, or that portion of it which constitutes
the unduloid, has not grown by successive {571} increments or “rings
of growth,” but has developed as a whole.) A thickened “rib” is often,
perhaps generally, present on the dorsal side of the little conical
shell. In a few cases (Limacina, Peraclis) the tube becomes spirally
coiled, in a normal logarithmic spiral or helico-spiral.

[Illustration: Fig. 296. Diagrammatic transverse sections, or outlines
of the mouth, in certain Pteropod shells: A, B, _Cleodora australis_;
C, _C. pyramidalis_; D, _C. balantium_; E, _C. cuspidata_. (After
Boas.)]

[Illustration: Fig. 297. Shells of thecosome Pteropods (after Boas).
(1) _Cleodora cuspidata_; (2) _Hyalaea trispinosa_; (3) _H. globulosa_;
(4) _H. uncinata_; (5) _H. inflexa_.]

In certain cases (e.g. Cleodora, Hyalaea) the tube or cone is
curiously modified. In the first place, its cross-section, originally
{572} circular or nearly so, becomes flattened or compressed
dorso-ventrally; and the angle, or rather edge, where dorsal and
ventral walls meet, becomes more and more drawn out into a ridge or
keel. Along the free margin, both of the dorsal and the ventral portion
of the shell, growth proceeds with a regularly varying velocity, so
that these margins, or lips, of the shell become regularly curved or
markedly sinuous. At the same time, growth in a transverse direction
proceeds with an acceleration which manifests itself in a curvature
of the sides, replacing the straight borders of the original cone.
In other words, the cross-section of the cone, or what we have been
calling the generating curve, increases its dimensions more rapidly
than its distance from the pole.

[Illustration: Fig. 298. _Cleodora cuspidata._]

In the above figures, for instance in that of _Cleodora cuspidata_,
the markings of the shell which represent the successive edges of the
lip at former stages of growth, furnish us at once with a “graph”
of the varying velocities of growth as measured, radially, from the
apex. We can reveal more clearly the nature of these variations in the
following way which is simply tantamount to converting our radial into
rectangular coordinates. Neglecting curvature (if any) of the sides and
treating the shell (for simplicity’s sake) as a right cone, we lay off
equal angles from the apex _O_, along the radii _Oa_, _Ob_, etc. If
we then plot, as vertical equidistant ordinates, the magnitudes _Oa_,
_Ob_ ... _OY_, and again on to _Oa′_, we obtain a diagram such as the
following (Fig. 299); by {573} help of which we not only see more
clearly the way in which the growth-rate varies from point to point,
but we also recognise much better than before, the similar nature of
the law which governs this variation in the different species.

[Illustration: Fig. 299. Curves obtained by transforming radial
ordinates, as in Fig. 298, into vertical equidistant ordinates. 1,
_Hyalaea trispinosa_; 2, _Cleodora cuspidata_.]

[Illustration: Fig. 300. Development of the shell of _Hyalaea_
(_Cavolinia_) _tridentata_, Forskal: the earlier stages being the
“_Pleuropus longifilis_” of Troschel. (After Tesch.)]

Furthermore, the young shell having become differentiated into a dorsal
and a ventral part, marked off from one another by a lateral edge or
keel, and the inequality of growth being such as to cause each portion
to increase most rapidly in the median line, it follows that the entire
shell will appear to have been split into a dorsal and a ventral
plate, both connected with, and projecting from, {574} what remains
of the original undivided cone. Putting the same thing in other words,
we may say that the generating figure, which lay at first in a plane
perpendicular to the axis of the cone, has now, by unequal growth,
been sharply bent or folded, so as to lie approximately in two planes,
parallel to the anterior and posterior faces of the cone. We have only
to imagine the apical connecting portion to be further reduced, and
finally to disappear or rupture, and we should have a _bivalve shell_
developed out of the original simple cone.

In its outer and growing portion, the shell of our Pteropod now
consists of two parts which, though still connected together at the
apex, may be treated as growing practically independently. The shell is
no longer a simple tube, or simple cone, in which regular inequalities
of growth will lead to the development of a spiral; and this for the
simple reason that we have now two opposite maxima of growth, instead
of a maximum on the one side and a minimum on the other side of our
tubular shell. As a matter of fact, the dorsal and the ventral plate
tend to curve in opposite directions, towards the middle line, the
dorsal curving ventrally and the ventral curving towards the dorsal
side.

In the case of the Lamellibranch or the Brachiopod, it is quite
possible for both valves to grow into more or less pronounced spirals,
for the simple reason that they are _hinged_ upon one another; and each
growing edge, instead of being brought to a standstill by the growth of
its opposite neighbour, is free to move out of the way, by the rotation
about the hinge of the plane in which it lies.

[Illustration: Fig. 301. Pteropod shells, from the side: (1) _Cleodora
cuspidata_; (2) _Hyalaea longirostris_; (3) _H. trispinosa_. (After
Boas.)]

But where, as in the Pteropod, there is no such hinge, the dorsal and
ventral halves of the shell (or dorsal and ventral valves, if we may
call them so), if they curved towards one another (as they do in a
cockle), would soon interfere with one another’s progress, and the
development of a pair of conjugate spirals would become impossible.
Nevertheless, there is obviously, in both dorsal and ventral valve, a
_tendency_ to the development of a spiral curve, that of the ventral
valve being more marked than that of the larger and overlapping dorsal
one, exactly as in the two unequal valves of Terebratula. In many
cases (e.g. _Cleodora cuspidata_), the dorsal valve or plate, {575}
strengthened and stiffened by its midrib, is nearly straight, while
the curvature of the other is well displayed. But the case will be
materially altered and simplified if growth be arrested or retarded
in either half of the shell. Suppose for instance that the dorsal
valve grew so slowly that after a while, in comparison with the other,
we might speak of it as being absent altogether: or suppose that it
merely became so reduced in relative size as to form no impediment to
the continued growth of the ventral one; the latter would continue
to grow in the direction of its natural curvature, and would end
by forming a complete and coiled logarithmic spiral. It would be
precisely analogous to the spiral shell of Nautilus, and, in regard to
its ventral position, concave towards the dorsal side, it would even
deserve to be called directly homologous with it. Suppose, on the other
hand, that the ventral valve were to be greatly reduced, and even to
disappear, the dorsal valve would then pursue its unopposed growth;
and, were it to be markedly curved, it would come to form a logarithmic
spiral, concave towards the ventral side, as is the case in the shell
of Spirula[530]. Were the dorsal valve to be destitute of any marked
curvature (or in other words, to have but a low spiral angle), it would
form a simple plate, as in the shells of Sepia or Loligo. Indeed,
in the shells of these latter, and especially in that of Sepia, we
seem to recognise a manifest resemblance to the dorsal plate of the
Pteropod shell, as we have it (e.g.) in Cleodora or Hyalaea; {576} the
little “rostrum” of Sepia is but the apex of the primitive cone, and
the rounded anterior extremity has grown according to a law precisely
such as that which has produced the curved margin of the dorsal valve
in the Pteropod. The ventral portion of the original cone is nearly,
but not wholly, wanting. It is represented by the so-called posterior
wall of the “siphuncular space.” In many decapod cuttle-fishes also
(e.g. Todarodes, Illex, etc.) we still see at the posterior end of the
“pen,” a vestige of the primitive cone, whose dorsal margin only has
continued to grow; and the same phenomenon, on an exaggerated scale, is
represented in the Belemnites.

It is not at all impossible that we may explain on the same lines
the development of the curious “operculum” of the Ammonites. This
consists of a single horny plate (_Anaptychus_), or of a thicker, more
calcified plate divided into two symmetrical halves (_Aptychi_), often
found inside the terminal chamber of the Ammonite, and occasionally
to be seen lying _in situ_, as an operculum which partially closes
the mouth of the shell; this structure is known to exist even in
connection with the early embryonic shell. In form the Anaptychus,
or the pair of conjoined Aptychi, shew an upper and a lower border,
the latter strongly convex, the former sometimes slightly concave,
sometimes slightly convex, and usually shewing a median projection
or slightly developed rostrum. From this “rostral” border the curves
of growth start, and course round parallel to, finally constituting,
the convex border. It is this convex border which fits into the free
margin of the mouth of the Ammonite’s shell, while the other is
applied to and overlaps the preceding whorl of the spire. Now this
relationship is precisely what we should expect, were we to imagine as
our starting-point a shell similar to that of Hyalaea, in which however
the dorsal part of the split cone had become separate from the ventral
half, had remained flat, and had grown comparatively slowly, while at
the same time it kept slipping forward over the growing and coiling
spire into which the ventral half of the original shell develops[531].
In short, I think there is reason to believe, or at least to suspect,
that we {577} have in the shell and Aptychus of the Ammonites, two
portions of a once united structure; of which other Cephalopods retain
not both parts but only one or other, one as the ventrally situated
shell of Nautilus, the other as the dorsally placed shell for example
of Sepia or of Spirula.

In the case of the bivalve shells of the Lamellibranchs or of the
Brachiopods, we have to deal with a phenomenon precisely analogous
to the split and flattened cone of our Pteropods, save only that the
primitive cone has been split into two portions, not incompletely as
in the Pteropod (Hyalaea), but completely, so as to form two separate
valves. Though somewhat greater freedom is given to growth now that the
two valves are separate and hinged, yet still the two valves oppose
and hamper one another, so that in the longitudinal direction each is
capable of only a moderate curvature. This curvature, as we have seen,
is recognisable as a logarithmic spiral, but only now and then does the
growth of the spiral continue so far as to develop successive coils:
as it does in a few symmetrical forms such as _Isocardia cor_; and as
it does still more conspicuously in a few others, such as Gryphaea and
Caprinella, where one of the two valves is stunted, and the growth of
the other is (relatively speaking) unopposed.


_Of Septa._

Before we leave the subject of the molluscan shell, we have still
another problem to deal with, in regard to the form and arrangement
of the septa which divide up the tubular shell into chambers, in the
Nautilus, the Ammonite and their allies (Fig. 304, etc.).

The existence of septa in a Nautiloid shell may probably be accounted
for as follows. We have seen that it is a property of a cone that,
while growing by increments at one end only, it conserves its original
shape: therefore the animal within, which (though growing by a
different law) also conserves its shape, will continue to fill the
shell if it actually fills it to begin with: as does a snail or other
Gastropod. But suppose that our mollusc fills a part only of a conical
shell (as it does in the case of Nautilus); then, unless it alter its
shape, it must move upward as it grows in the growing cone, until it
come to occupy a space similar in form {578} to that which it occupied
before: just, indeed, as a little ball drops far down into the cone,
but a big one must stay farther up. Then, when the animal after a
period of growth has moved farther up in the shell, the mantle-surface
continues its normal secretory activity, and that portion which had
been in contact with the former septum secretes a septum anew. In
short, at any given epoch, the creature is not secreting a tube and a
septum by separate operations, but is secreting a shelly case about its
rounded body, of which case one part appears to us as the continuation
of the tube, and the other part, merging with it by indistinguishable
boundaries, appears to us as the septum[532].

The various forms assumed by the septa in spiral shells[533] present us
with a number of problems of great beauty, simple in their essence, but
whose full investigation would soon lead us into mathematics of a very
high order.

We do not know in great detail how these septa are laid down; but
the essential facts are clear[534]. The septum begins as a very
thin cuticular membrane (composed apparently of a substance called
conchyolin), which is secreted by the skin, or mantle-surface, of the
animal; and upon this membrane nacreous matter is gradually laid down
on the mantle-side (that is to say between the animal’s body and the
cuticular membrane which has been thrown off from it), so that the
membrane remains as a thin pellicle over the _hinder_ surface of the
septum, and so that, to begin with, the membranous septum is moulded on
the flexible and elastic surface of the animal, within which the fluids
of the body must exercise a uniform, or nearly uniform pressure.

Let us think, then, of the septa as they would appear in their
uncalcified condition, formed of, or at least superposed upon, an {579}
elastic membrane. They must then follow the general law, applicable to
all elastic membranes under uniform pressure, that the tension varies
inversely as the radius of curvature; and we come back once more to our
old equation of Laplace, that

 _P_ = _T_(1/_r_ + 1/_r′_).

Moreover, since the cavity below the septum is practically closed, and
is filled either with air or with water, _P_ will be constant over
the whole area of the septum. And further, we must assume, at least
to begin with, that the membrane constituting the incipient septum is
homogeneous or isotropic.

Let us take first the case of a straight cone, of circular section,
more or less like an Orthoceras; and let us suppose that the septum is
attached to the shell in a plane perpendicular to its axis. The septum
itself must then obviously be spherical. Moreover the extent of the
spherical surface is constant, and easily determined. For obviously, in
Fig. 302, the angle _LCL′_ equals the supplement of the angle (_LOL′_)
of the cone; that is to say, the circle of contact subtends an angle at
the centre of the spherical surface, which is constant, and which is
equal to π − 2θ. The case is not excluded where, owing to an asymmetry
of tensions, the septum meets the side walls of the cone at other than
a right angle, as in Fig. 303; and here, while the septa still remain
portions of spheres, the geometrical construction for the position of
their centres is equally easy.

[Illustration: Fig. 302.]

[Illustration: Fig. 303.]

If, on the other hand, the attachment of the septum to the inner walls
of the cone be in a plane oblique to the axis, then it is evident that
the outline of the septum will be an ellipse, and its surface an {580}
ellipsoid. If the attachment of the septum be not in one plane, but
form a sinuous line of contact with the cone, then the septum will be
a saddle-shaped surface, of great complexity and beauty. In all cases,
provided only that the membrane be isotropic, the form assumed will be
precisely that of a soap-bubble under similar conditions of attachment:
that is to say, it will be (with the usual limitations or conditions) a
surface of minimal area.

If our cone be no longer straight, but curved, then the septa will be
symmetrically deformed in consequence. A beautiful and interesting case
is afforded us by Nautilus itself. Here the outline of the septum,
referred to a plane, is approximately bounded by two elliptic curves,
similar and similarly situated, whose areas are to one another in a
definite ratio, namely as

 _A__{1}/_A__{2} = (_r__{1} _r′__{1})/(_r__{2} _r′__{2}) = ε^{−4π cot α},

and a similar ratio exists in Ammonites and all other close-whorled
spirals, in which however we cannot always make the simple assumption
of elliptical form. In a median section of Nautilus, we see each septum
forming a tangent to the inner and to the outer wall, just as it did
in a section of the straight Orthoceras; but the curvatures in the
neighbourhood of these two points of contact are not identical, for
they now vary inversely as the radii, drawn from the pole of the spiral
shell. The contour of the septum in this median plane is a spiral curve
identical with the original logarithmic spiral. Of this it is the
“invert,” and the fact that the original curve and its invert are both
identical is one of the most beautiful properties of the logarithmic
spiral[535].

[Illustration: Fig. 304. Section of _Nautilus_, shewing the contour
of the septa in the median plane: the septa being (in this plane)
logarithmic spirals, of which the shell-spiral is the evolute.]

But while the outline of the septum in median section is simple and
easy to determine, the curved surface of the septum in its entirety
is a very complicated matter, even in Nautilus which is one of the
simplest of actual cases. For, in the first place, since the form
of the septum, as seen in median section, is that of a logarithmic
spiral, and as therefore its curvature is constantly altering, it
follows that, in successive _transverse_ sections, the {581} curvature
is also constantly altering. But in the case of Nautilus, there are
other aspects of the phenomenon, which we can illustrate, but only
in part, in the following simple manner. Let us imagine a pack of
cards, in which we have cut out of each card a similar concave arc of
a logarithmic spiral, such as we actually see in the median section
of the septum of a Nautilus. Then, while we hold the cards together,
foursquare, in the ordinary position of the {582} pack, we have a
simple “ruled” surface, which in any longitudinal section has the form
of a logarithmic spiral but in any transverse section is a straight
horizontal line. If we shear or slide the cards upon one another,
thrusting the middle cards of the pack forward in advance of the
others, till the one end of the pack is a convex, and the other a
concave, ellipse, the cut edges which combine to represent our septum
will now form a curved surface

[Illustration: Fig. 305. Cast of the interior of _Nautilus_: to shew
the contours of the septa at their junction with the shell-wall.]

of much greater complexity; and this is part, but not by any means
all, of the deformation produced as a direct consequence of the form
in Nautilus of the section of the tube within which the septum has to
lie. And the complex curvature of the surface will be manifested in a
sinuous outline of the edge, or line of attachment of the septum to
the tube, and will vary according to the configuration of the latter.
In the case of Nautilus, it is easy to shew empirically (though not
perhaps easy to demonstrate {583} mathematically) that the sinuous
or saddle-shaped form of the “suture” (or line of attachment of the
septum to the tube) is such as can be precisely accounted for in this
manner. It is also easy to see that, when the section of the tube (or
“generating curve”) is more complicated in form, when it is flattened,
grooved, or otherwise ornamented, the curvature of the septum and
the outline of its sutural attachment will become very complicated
indeed[536]; but it will be comparatively simple in the case of the
first few sutures of the young shell, laid down before any overlapping
of whorls has taken place, and this comparative simplicity of the
first-formed sutures is a marked feature among Ammonites[537].

We have other sources of complication, besides those which are at
once introduced by the sectional form of the tube. For instance, the
siphuncle, or little inner tube which perforates the septa, exercises a
certain amount of tension, sometimes evidently considerable, upon the
latter; so that we can no longer consider each septum as an isotropic
surface, under uniform pressure; and there may be other structural
modifications, or inequalities, in that portion of the animal’s body
with which the septum is in contact, and by which it is conformed. It
is hardly likely, for all these reasons, that we shall ever attain to
a full and particular explanation of the septal surfaces and their
sutural outlines throughout the whole range of Cephalopod shells; but
in general terms, the problem is probably not beyond the reach of
mathematical analysis. The problem might be approached experimentally,
after the manner of Plateau’s experiments, by bending {584} a wire into
the complicated form of the suture-line, and studying the form of the
liquid film which constitutes the corresponding surface _minimae areae_.

[Illustration: Fig. 306. _Ammonites_ (_Sonninia_) _Sowerbyi_. (From
Zittel, after Steinmann and Döderlein.)]

[Illustration: Fig. 307. Suture-line of a Triassic Ammonite
(_Pinacoceras_). (From Zittel, after Hauer.)]

In certain Ammonites the septal outline is further complicated in
another way. Superposed upon the usual sinuous outline, with its
“lobes” and “saddles,” we have here a minutely ramified, or arborescent
outline, in which all the branches terminate in wavy, more or less
circular arcs,—looking just like the ‘landscape marble’ from the
Bristol Rhaetic. We have no difficulty in recognising in this a
surface-tension phenomenon. The figures are precisely such as we can
imitate (for instance) by pouring a {585} few drops of milk upon a
greasy plate, or of oil upon an alkaline solution.

We have very far from exhausted, we have perhaps little more than
begun, the study of the logarithmic spiral and the associated curves
which find exemplification in the multitudinous diversities of
molluscan shells. But, with a closing word or two, we must now bring
this chapter to an end.

In the spiral shell we have a problem, or a phenomenon, of growth,
immensely simplified by the fact that each successive increment is
irrevocably fixed in regard to magnitude and position, instead, of
remaining in a state of flux and sharing in the further changes which
the organism undergoes. In such a structure, then, we have certain
primary phenomena of growth manifested in their original simplicity,
undisturbed by secondary and conflicting phenomena. What actually
_grows_ is merely the lip of an orifice, where there is produced a ring
of solid material, whose form we have treated of under the name of the
generating curve; and this generating curve grows in magnitude without
alteration of its form. Besides its increase in areal magnitude, the
growing curve has certain strictly limited degrees of freedom, which
define its motions in space: that is to say, it has a vector motion
at right angles to the axis of the shell; and it has a sliding motion
along that axis. And, though we may know nothing whatsoever about the
actual velocities of any of these motions, we do know that they are so
correlated together that their _relative_ velocities remain constant,
and accordingly the form and symmetry of the whole system remain in
general unchanged.

But there is a vast range of possibilities in regard to every one of
these factors: the generating curve may be of various forms, and even
when of simple form, such as an ellipse, its axes may be set at various
angles to the system; the plane also in which it lies may vary, almost
indefinitely, in its angle relatively to that of any plane of reference
in the system; and in the several velocities of growth, of rotation and
of translation, and therefore in the ratios between all these, we have
again a vast range of possibilities. We have then a certain definite
type, or group of forms, mathematically isomorphous, but presenting
infinite diversities of outward appearance: which diversities, as
Swammerdam {586} said, _ex sola nascuntur diversitate gyrationum_; and
which accordingly are seen to have their origin in differences of rate,
or of magnitude, and so to be, essentially, neither more nor less than
_differences of degree_.

In nature, we find these forms presenting themselves with but little
relation to the character of the creature by which they are produced.
Spiral forms of certain particular kinds are common to Gastropods and
to Cephalopods, and to diverse families of each; while outside the
class of molluscs altogether, among the Foraminifera and among the
worms (as in Spirorbis, Spirographis, and in the Dentalium-like shell
of Ditrupa), we again meet with similar and corresponding forms.

Again, we find the same forms, or forms which (save for external
ornament) are mathematically identical, repeating themselves in all
periods of the world’s geological history; and, irrespective of climate
or local conditions, we see them mixed up, one with another, in the
depths and on the shores of every sea. It is hard indeed (to my mind)
to see where Natural Selection necessarily enters in, or to admit that
it has had any share whatsoever in the production of these varied
conformations. Unless indeed we use the term Natural Selection in a
sense so wide as to deprive it of any purely biological significance;
and so recognise as a sort of natural selection whatsoever nexus of
causes suffices to differentiate between the likely and the unlikely,
the scarce and the frequent, the easy and the hard: and leads
accordingly, under the peculiar conditions, limitations and restraints
which we call “ordinary circumstances,” one type of crystal, one form
of cloud, one chemical compound, to be of frequent occurrence and
another to be rare.

{587}




CHAPTER XII

THE SPIRAL SHELLS OF THE FORAMINIFERA


We have already dealt in a few simple cases with the shells of the
Foraminifera[538]; and we have seen that wherever the shell is but a
single unit or single chamber, its form may be explained in general
by the laws of surface tension: the assumption being that the little
mass of protoplasm which makes the simple shell behaves as a _fluid
drop_, the form of which is perpetuated when the protoplasm acquires
its solid covering. Thus the spherical Orbulinae and the flask-shaped
Lagenae represent drops in equilibrium, under various conditions of
freedom or constraint; while the irregular, amoeboid body of Astrorhiza
is a manifestation not of equilibrium, but of a varying and fluctuating
distribution of surface energy. When the foraminiferal shell becomes
multilocular, the same general principles continue to hold; the growing
protoplasm increases drop by drop, and each successive drop has its
particular phenomena of surface energy, manifested at its fluid
surface, and tending to confer upon it a certain place in the system
and a certain shape of its own.

It is characteristic and even diagnostic of this particular group of
Protozoa (1) that development proceeds by a well-marked alternation of
rest and of activity—of activity during which the protoplasm increases,
and of rest during which the shell is formed; (2) that the shell is
formed at the outer surface of the protoplasmic organism, and tends
to constitute a continuous or all but continuous covering; and it
follows (3) from these two factors taken together that each successive
increment is added on outside of and distinct from its predecessors,
that the successive parts or chambers of {588} the shell are of
different and successive ages, that one part of the shell is always
relatively new, and the rest old in various grades of seniority.

The forms which we set together in the sister-group of Radiolaria are
very differently characterised. Here the cells or vesicles of which
each little composite organism is made up are but little separated, and
in no way walled off, from one another; the hard skeletal matter tends
to be deposited in the form of isolated spicules or of little connected
rods or plates, at the angles, the edges or the interfaces of the
vesicles; the cells or vesicles form a coordinated and cotemporaneous
rather than a successive series. In a word, the whole quasi-fluid
protoplasmic body may be likened to a little mass of froth or foam:
that is to say, to an aggregation of simultaneously formed drops or
bubbles, whose physical properties and geometrical relations are very
different from those of a system of drops or bubbles which are formed
one after another, each solidifying before the next is formed.

[Illustration: Fig. 308. _Hastigerina_ sp.; to shew the “mouth.”]

With the actual origin or mode of development of the foraminiferal
shell we are now but little concerned. The main factor is the
adsorption, and subsequent precipitation at the surface of the
organism, of calcium carbonate,—the shell so formed being interrupted
by pores or by some larger interspace or “mouth” (Fig. 308), which
interruptions we may doubtless interpret as being due to unequal
distributions of surface energy. In many {589} cases the fluid
protoplasm “picks up” sand-grains and other foreign particles, after
a fashion which we have already described (p. 463); and it cements
these together with more or less of calcareous material. The calcareous
shell is a crystalline structure, and the micro-crystals of calcium
carbonate are so set that their little prisms radiate outwards in each
chamber through the thickness of the wall:—which symmetry is subject to
corresponding modification when the spherical chambers are more or less
symmetrically deformed[539].

In various ways the rounded, drop-like shells of the Foraminifera, both
simple and compound, have been artificially imitated. Thus, if small
globules of mercury be immersed in water in which a little chromic
acid is allowed to dissolve, as the little beads of quicksilver become
slowly covered with a crystalline coat of mercuric chromate they
assume various forms reminiscent of the monothalamic Foraminifera. The
mercuric chromate has a higher atomic volume than the mercury which
it replaces, and therefore the fluid contents of the drop are under
pressure, which increases with the thickness of the pellicle; hence
at some weak spot in the latter the contents will presently burst
forth, so forming a mouth to the little shell. Sometimes a long thread
is formed, just as in _Rhabdammina linearis_; and sometimes unduloid
swellings make their appearance on such a thread, just as in _R.
discreta_. And again, by appropriate modifications of the experimental
conditions, it is possible (as Rhumbler has shewn) to build up a
chambered shell[540].

In a few forms, such as Globigerina and its close allies, the shell is
beset during life with excessively long and delicate calcareous spines
or needles. It is only in oceanic forms that these are present, because
only when poised in water can such {590} delicate structures endure;
in dead shells, such as we are much more familiar with, every trace
of them is broken and rubbed away. The growth of these long needles
is explained (as we have already briefly mentioned, on p. 440) by the
phenomenon which Lehmann calls _orientirte Adsorption_—the tendency
for a crystalline structure to grow by accretion, not necessarily
in the outward form of a “crystal,” but continuing in any direction
or orientation which has once been impressed upon it: in this case
the spicular growth is simply in direct continuation of the radial
symmetry of the micro-crystalline elements of the shell-wall. Over
the surface of the shell the radiating spicules tend to occur in
a hexagonal pattern, symmetrically grouped around the pores which
perforate the shell. Rhumbler has suggested that this arrangement
is due to diffusion-currents, forming little eddies about the base
of the pseudopodia issuing from the pores: the idea being borrowed
from Bénard, to whom is due the discovery of this type or order of
vortices[541]. In one of Bénard’s experiments a thin layer of paraffin
is strewn with particles of graphite, then warmed to melting, whereupon
each little solid granule becomes the centre of a vortex; by the
interaction of these vortices the particles tend to be repelled to
equal distances from one another, and in the end they are found to be
arranged in a hexagonal pattern[542]. The analogy is plain between this
experiment and those diffusion experiments by which Leduc produces his
beautiful hexagonal systems of artificial cells, with which we have
dealt in a previous chapter (p. 320).

But let us come back to the shell itself, and consider particularly
its spiral form. That the shell in the Foraminifera should tend towards
a spiral form need not surprise us; for we have learned that one of
the fundamental conditions of the production of a concrete spiral is
just precisely what we have here, namely the gradual development of a
structure by means of successive increments superadded to its exterior,
which then form part, successively, of a permanent and rigid structure.
This condition {591} is obviously forthcoming in the foraminiferal, but
not at all in the radiolarian, shell. Our second fundamental condition
of the production of a logarithmic spiral is that each successive
increment shall be so posited and so conformed that its addition to
the system leaves the form of the whole system unchanged. We have now
to enquire into this latter condition; and to determine whether the
successive increments, or successive chambers, of the foraminiferal
shell actually constitute _gnomons_ to the entire structure.

It is obvious enough that the spiral shells of the Foraminifera closely
resemble true logarithmic spirals. Indeed so precisely do the minute
shells of many Foraminifera repeat or simulate the spiral shells of
Nautilus and its allies that to the naturalists of the early nineteenth
century they were known as the _Céphalopodes microscopiques_[543],
until Dujardin shewed that their little bodies comprised no complex
anatomy of organs, but consisted merely of that slime-like organic
matter which he taught us to call “sarcode,” and which we learned
afterwards from Schwann to speak of as “protoplasm.”

[Illustration: Fig. 309. _Nummulina antiquior_, R. and V. (After V. von
Möller.)]

One striking difference, however, is apparent between the shell
of Nautilus and the little nautiloid or rotaline shells of the
Foraminifera: namely that the septa in these latter, and in all other
{592} chambered Foraminifera, are convex outwards (Fig. 308), whereas
they are concave outwards in Nautilus (Fig. 304) and in the rest of
the chambered molluscan shells. The reason is perfectly simple. In
both cases the curvature of the septum was determined before it became
rigid, and at a time when it had the properties either of a fluid
film or an elastic membrane. In both cases the actual curvature is
determined by the tensions of the membrane and the pressures to which
it was exposed. Now it is obvious that the extrinsic pressure which
the tension of the membrane has to withstand is on opposite sides
in the two cases. In Nautilus, the pressure to be resisted is that
produced by the growing body of the animal, lying to the _outer side_
of the septum, in the outer, wider portion of the tubular shell. In
the Foraminifer the septum at the time of its formation was no septum
at all; it was but a portion of the convex surface of a drop-that
portion namely which afterwards became overlapped and enclosed by the
succeeding drop; and the curvature of the septum is concave towards the
pressure to be resisted, which latter is _inside_ the septum, being
simply the hydrostatic pressure of the fluid contents of the drop.
The one septum is, speaking generally, the reverse of the other; the
organism, so to speak, is outside the one and inside the other; and in
both cases alike, the septum tends to assume the form of a surface of
minimal area, as permitted, or as defined, by all the circumstances of
the case.

The logarithmic spiral is easily recognisable in typical cases[544]
(and especially where the spire makes more than one visible revolution
about the pole), by its fundamental property of continued similarity:
that is to say, by reason of the fact that the big many-chambered
shell is of just the same shape as the smaller and younger shell—which
phenomenon is apparent and even obvious in the nautiloid Foraminifera,
as in Nautilus itself: but nevertheless the nature of the curve must be
verified by careful measurement, just as Moseley determined or verified
it in his {593} original study of nautilus (cf. p. 518). This has
accordingly been done, by various writers: and in the first instance
by Valerian von Möller, in an elaborate study of Fusulina—a palaeozoic
genus whose little shells have built up vast tracts of carboniferous
limestone over great part of European Russia[545].

In this genus a growing surface of protoplasm may be conceived as
wrapping round and round a small initial chamber, in such a way as to
produce a fusiform or ellipsoidal shell—a transverse section of which
reveals the close-wound spiral coil. The following are examples of
measurements of the successive whorls in a couple of species of this
genus.

      _F. cylindrica_, Fischer   _F. Böcki_, v. Möller
                   Breadth (in millimetres).
 Whorl   Observed   Calculated   Observed   Calculated
    I      ·132         —          ·079         —
   II      ·195       ·198         ·120       ·119
  III      ·300       ·297         ·180       ·179
   IV      ·449       ·445         ·264       ·267
    V        —          —          ·396       ·401

In both cases the successive whorls are very nearly in the ratio of
1 : 1·5; and on this ratio the calculated values are based.

Here is another of von Möller’s series of measurements of _F.
cylindrica_, the measurements being those of opposite whorls—that is to
say of whorls 180° apart:

 Breadth in mm.  ·096  ·117  ·144  ·176  ·216  ·264  ·323  ·395
 Log. of mm.     ·982  ·068  ·158  ·246  ·334  ·422  ·509  ·597
 Diff. of logs.    —   ·086  ·090  ·088  ·088  ·088  ·087  ·088

The mean logarithmic difference is here ·088, = log 1·225; or the mean
difference of alternate logs (corresponding to a vector angle of 2π,
i.e. to consecutive measurements along the _same_ radius) is ·176,
= log 1·5, the same value as before. And this ratio of 1·5 between the
breadths of successive whorls corresponds (as we see by our table on p.
534) to a constant angle of about {594} 86°, or just such a spiral as
we commonly meet with in the Ammonites[546] (cf. p. 539).

In Fusulina, and in some few other Foraminifera (cf. Fig. 310, A), the
spire seems to wind evenly on, with little or no external sign of the
successive periods of growth, or successive chambers of the shell.
The septa which mark off the chambers, and correspond to retardations
or cessations in the periodicity of growth, are still to be found in
sections of the shell of Fusulina; but they are somewhat irregular and
comparatively inconspicuous; the measurements we have just spoken of
are taken without reference to the segments or chambers, but only with
reference to the whorls, or in other words with direct reference to the
vectorial angle.

[Illustration: Fig. 310. A, _Cornuspira foliacea_, Phil.; B,
_Operculina complanata_, Defr.]

The linear dimensions of successive chambers have been {595} measured
in a number of cases. Van Iterson[547] has done so in various
Miliolinidae, with such results as the following:

 _Triloculina rotunda_, d’Orb.

 No. of chamber             1   2   3   4   5    6    7    8    9   10
 Breadth of chamber in _µ_  —  34  45  61  84  114  142  182  246  319
 Breadth of chamber in _µ_,
   calculated               —  34  45  60  79  105  140  187  243  319

Here the mean ratio of breadth of consecutive chambers may be taken as
1·323 (that is to say, the eighth root of 319/34); and the calculated
values, as given above, are based on this determination.

Again, Rhumbler has measured the linear dimensions of a number of
rotaline forms, for instance _Pulvinulina menardi_ (Fig. 259): in which
common species he finds the mean linear ratio of consecutive chambers
to be about 1·187. In both cases, and especially in the latter, the
ratio is not strictly constant from chamber to chamber, but is subject
to a small secondary fluctuation[548].

[Illustration: Fig. 311. 1, 2, _Miliolina pulchella_, d’Orb.; 3–5, _M.
linnaeana_, d’Orb. (After Brady.)]

[Illustration: Fig. 312. _Cyclammina cancellata_, Brady.]

When the linear dimensions of successive chambers are in continued
proportion, then, in order that the whole shell may constitute a
logarithmic spiral, it is necessary that the several chambers should
subtend equal angles of revolution at the pole. In the case of the
Miliolidae this is obviously the case (Fig. 311); for in this family
the chambers lie in two rows (Biloculina), or three rows (Triloculina),
or in some other small number of series: so that the angles subtended
by them are large, simple fractions of the circular arc, such as 180°
or 120°. In many of the nautiloid forms, such as Cyclammina (Fig. 312),
the angles subtended, though of less magnitude, are still remarkably
constant, as we {597} may see by Fig. 313; where the angle subtended
by each chamber is made equal to 20°, and this diagrammatic figure is
not perceptibly different from the other. In some cases the subtended
angle is less constant; and in these it would be necessary to equate
the several linear dimensions with the corresponding vector angles,
according to our equation _r_ = _e_^{θ cot α}. It is probable that,
by so taking account of variations of θ, such variations of _r_ as
(according to Rhumbler’s measurements) Pulvinulina and other genera
appear to shew, would be found to diminish or even to disappear.

[Illustration: Fig. 313. _Cyclammina_ sp. (Diagrammatic.)]

The law of increase by which each chamber bears a constant ratio of
magnitude to the next may be looked upon as a simple consequence of the
structural uniformity or homogeneity of the organism; we have merely
to suppose (as this uniformity would naturally lead us to do) that the
rate of increase is at each instant proportional to the whole existing
mass. For if _V__{0}, _V__{1} etc., be the volumes of the successive
chambers, let _V__{1} bear a constant proportion to _V__{0}, so that
_V__{1} = _q_ _V__{0}, and let _V__{2} bear the same proportion to the
whole pre-existing volume: then

 _V__{2} = _q_(_V__{0} + _V__{1}) = _q_(_V__{0} + _q_ _V__{0})
         = _q_ _V__{0}(1 + _q_) and _V__{2}/_V__{1} = 1 + _q_.

{598}

This ratio of 1/(1 + _q_) is easily shewn to be the constant ratio
running through the whole series, from chamber to chamber; and if this
ratio of volumes be constant, so also are the ratios of corresponding
surfaces, and of corresponding linear dimensions, provided always that
the successive increments, or successive chambers, are similar in form.

We have still to discuss the similarity of form and the symmetry
of position which characterise the successive chambers, and which,
together with the law of continued proportionality of size, are the
distinctive characters and the indispensable conditions of a series of
“gnomons.”

The minute size of the foraminiferal shell or at least of each
successive increment thereof, taken in connection with the fluid or
semi-fluid nature of the protoplasmic substance, is enough to suggest
that the molecular forces, and especially the force of surface-tension,
must exercise a controlling influence over the form of the whole
structure; and this suggestion, or belief, is already implied in
our statement that each successive increment of growing protoplasm
constitutes a separate _drop_. These “drops,” partially concealed by
their successors, but still shewing in part their rounded outlines,
are easily recognisable in the various foraminiferal shells which are
illustrated in this chapter.

[Illustration: Fig. 314. _Orbulina universa_, d’Orb.]

The accompanying figure represents, to begin with, the spherical shell
characteristic of the common, floating, oceanic Orbulina. In the
specimen illustrated, a second chamber, superadded to the {599} first,
has arisen as a drop of protoplasm which exuded through the pores of
the first chamber, accumulated on its surface, and spread over the
latter till it came to rest in a position of equilibrium. We may take
it that this position of equilibrium is determined, at least in the
first instance, by the “law of the constant angle,” which holds, or
tends to hold, in all cases where the free surface of a given liquid
is in contact with a given solid, in presence of another liquid or a
gas. The corresponding equations are precisely the same as those which
we have used in discussing the form of a drop (on p. 294); though
some slight modification must be made in our definitions, inasmuch as
the consideration of surface-_tension_ is no longer appropriate at
the solid surfaces, and the concept of surface-_energy_ must take its
place. Be that as it may, it is enough for us to observe that, in such
a case as ours, when a given fluid (namely protoplasm) is in surface
contact with a solid (viz. a calcareous shell), in presence of another
fluid (sea-water), then the angle of contact, or angle by which the
common surface (or interface) of the two liquids abuts against the
solid wall, tends to be constant: and that being so, the drop will have
a certain definite form, depending (_inter alia_) on the form of the
surface with which it is in contact. After a period of rest, during
which the surface of our second drop becomes rigid by calcification,
a new period of growth will recur and a new drop of protoplasm be
accumulated. Circumstances remaining the same, this new drop will meet
the solid surface of the shell at the same angle as did the former one;
and, the other forces at work on the system remaining the same, the
form of the whole drop, or chamber, will be the same as before.

According to Rhumbler, this “law of the constant angle” is the
fundamental principle in the mechanical conformation of the
foraminiferal shell, and provides for the symmetry of form as well
as of position in each succeeding drop of protoplasm: which form
and position, once acquired, become rigid and fixed with the onset
of calcification. But Rhumbler’s explanation brings with it its own
difficulties. It is by no means easy of verification, for on the very
complicated curved surfaces of the shell it seems to me extraordinarily
difficult to measure, or even to recognise, the actual angle of
contact: of which angle of contact, by the way, {600} but little is
known, save only in the particular case where one of the three bodies
is air, as when a surface of water is exposed to air and in contact
with glass. It is easy moreover to see that in many of our Foraminifera
the angle of contact, though it may be constant in homologous positions
from chamber to chamber, is by no means constant at all points along
the boundary of each chamber. In Cristellaria, for instance (Fig. 315),
it would seem to be (and Rhumbler asserts that it actually is) about
90° on the outer side and only about 50° on the inner side of each
septal partition; in Pulvinulina (Fig. 259), according to Rhumbler,
the angles adjacent to the mouth are of 90°, and the opposite angles
are of 60°, in each chamber. For these and other similar discrepancies
Rhumbler would account by simply invoking the heterogeneity of the
protoplasmic drop: that is to say, by assuming that the protoplasm
has a different composition and different properties (including a
very different distribution of surface-energy), at points near to and
remote from the mouth of the shell. Whether the differences in angle
of contact be as great as Rhumbler takes them to be, whether marked
heterogeneities of the protoplasm occur, and whether these be enough to
account for the differences of angle, I cannot tell. But it seems to
me that we had better rest content with a general statement, and that
Rhumbler has taken too precise and narrow a view.

[Illustration: Fig. 315. _Cristellaria reniformis_, d’Orb.]

{601}

In the molecular growth of a crystal, although we must of
necessity assume that each molecule settles down in a position of
minimum potential energy, we find it very hard indeed to explain
precisely, even in simple cases and after all the labours of modern
crystallographers, why this or that position is actually a place of
minimum potential. In the case of our little Foraminifer (just as in
the case of the crystal), let us then be content to assert that each
drop or bead of protoplasm takes up a position of minimum potential
energy, in relation to all the circumstances of the case; and let us
not attempt, in the present state of our knowledge, to define that
position of minimum potential by reference to angle of contact or any
other particular condition of equilibrium. In most cases the whole
exposed surface, on some portion of which the drop must come to rest,
is an extremely complicated one, and the forces involved constitute
a system which, in its entirety, is more complicated still; but from
the symmetry of the case and the continuity of the whole phenomenon,
we are entitled to believe that the conditions are just the same, or
very nearly the same, time after time, from one chamber to another:
as the one chamber is conformed so will the next tend to be, and as
the one is situated relatively to the system so will its successor
tend to be situated in turn. The physical law of minimum potential
(including also the law of minimal area) is all that we need in order
to explain, _in general terms_, the continued similarity of one chamber
to another; and the physiological law of growth, by which a continued
proportionality of size tends to run through the series of successive
chambers, impresses upon this series of similar increments the form of
a logarithmic spiral.

In each particular case the nature of the logarithmic spiral, as
defined by its constant angle, will be chiefly determined by the rate
of growth; that is to say by the particular ratio in which each new
chamber exceeds its predecessor in magnitude. But shells having the
same constant angle (α) may still differ from one another in many
ways—in the general form and relative position of the chambers, in
their extent of overlap, and hence in the actual contour and appearance
of the shell; and these variations must correspond to particular
distributions of energy within the system, which is governed as a whole
by the law of minimum potential. {602}

Our problem, then, becomes reduced to that of investigating the
possible configurations which may be derived from the successive
symmetrical apposition of similar bodies whose magnitudes are in
continued proportion; and it is obvious, mathematically speaking,
that the various possible arrangements all come under the head of the
logarithmic spiral, together with the limiting cases which it includes.
Since the difference between one such form and another depends upon
the numerical value of certain coefficients of magnitude, it is plain
that any one must tend to pass into any other by small and continuous
gradations; in other words, that a _classification_ of these forms must
(like any classification whatsoever of logarithmic spirals or of any
other mathematical curves), be theoretic or “artificial.” But we may
easily make such an artificial classification, and shall probably find
it to agree, more or less, with the usual methods of classification
recognised by biological students of the Foraminifera.

Firstly we have the typically spiral shells, which occur in great
variety, and which (for our present purpose) we need hardly describe
further. We may merely notice how in certain cases, for instance
Globigerina, the individual chambers are little removed from spheres;
in other words, the area of contact between the adjacent chambers is
small. In such forms as Cyclammina and Pulvinulina, on the other hand,
each chamber is greatly overlapped by its successor, and the spherical
form of each is lost in a marked asymmetry. Furthermore, in Globigerina
and some others we have a tendency to the development of a helicoid
spiral in space, as in so many of our univalve molluscan shells. The
mathematical problem of how a shell should grow, under the assumptions
which we have made, would probably find its most general statement in
such a case as that of Globigerina, where the whole organism lives and
grows freely poised in a medium whose density is little different from
its own.

The majority of spiral forms, on the other hand, are plane or discoid
spirals, and we may take it that in these cases some force has
exercised a controlling influence, so as to keep all the chambers in a
plane. This is especially the case in forms like Rotalia or Discorbina
(Fig. 316), where the organism lives attached to a rock or a frond of
sea-weed; for here (just as in the case of {603} the coiled tubes
which little worms such as Serpula and Spirorbis make, under similar
conditions) the spiral disc is itself asymmetrical, its whorls being
markedly flattened on their attached surfaces.

[Illustration: Fig. 316. _Discorbina bertheloti_, d’Orb.]

We may also conceive, among other conditions, the very curious case in
which the protoplasm may entirely overspread the surface of the shell
without reaching a position of equilibrium; in which case a new shell
will be formed _enclosing_ the old one, {604} whether the old one be
in the form of a single, solitary chamber, or have already attained to
the form of a chambered or spiral shell. This is precisely what often
happens in the case of Orbulina, when within the spherical shell we
find a small, but perfectly formed, spiral “Globigerina[549].”

The various Miliolidae (Fig. 311), only differ from the typical spiral,
or rotaline forms, in the large angle subtended by each chamber, and
the consequent abruptness of their inclination to each other. In these
cases the _outward_ appearance of a spiral tends to be lost; and it
behoves us to recollect, all the more, that our spiral curve is not
necessarily identical with the _outline_ of the shell, but is always a
line drawn through corresponding points in the successive chambers of
the latter.

[Illustration: Fig. 317. A, _Tertularia trochus_, d’Orb. B, _T.
concava_, Karrer.]

We reach a limiting case of the logarithmic spiral when the chambers
are arranged in a straight line; and the eye will tend to associate
with this limiting case the much more numerous forms in which the
spiral angle is small, and the shell only exhibits a gentle curve with
no succession of enveloping whorls. This constitutes the Nodosarian
type (Fig. 87, p. 262); and here again, we must postulate some force
which has tended to keep the chambers in a rectilinear series: such for
instance as gravity, acting on a system of “hanging drops.” {605}

In Textularia and its allies (Fig. 317), we have a precise parallel
to the helicoid cyme of the botanists (cf. p. 502): that is to say we
have a screw translation, perpendicular to the plane of the underlying
logarithmic spiral. In other words, in tracing a genetic spiral through
the whole succession of chambers, we do so by a continuous vector
rotation, through successive angles of 180° (or 120° in some cases),
while the pole moves along an axis perpendicular to the original plane
of the spiral.

Another type is furnished by the “cyclic” shells of the Orbitolitidae,
where small and numerous chambers tend to be added on round and round
the system, so building up a circular flattened disc. This again we
perceive to be, mathematically, a limiting case of the logarithmic
spiral, where the spiral has become a circle and the constant angle is
now an angle of 90°.

Lastly there are a certain number of Foraminifera in which, without
more ado, we may simply say that the arrangement of the chambers is
irregular, neither the law of constant ratio of magnitude nor that of
constant form being obeyed. The chambers are heaped pell-mell upon one
another, and such forms are known to naturalists as the Acervularidae.

While in these last we have an extreme lack of regularity, we must not
exaggerate the regularity or constancy which the more ordinary forms
display. We may think it hard to believe that the simple causes, or
simple laws, which we have described should operate, and operate again
and again, in millions of individuals to produce the same delicate
and complex conformations. But we are taking a good deal for granted
if we assert that they do so, and in particular we are assuming,
with very little proof, the “constancy of species” in this group of
animals. Just as Verworn has shewn that the typical _Amoeba proteus_,
when a trace of alkali is added to the water in which it lives, tends,
by alteration of surface tensions, to protrude the more delicate
pseudopodia characteristic of _A. radiosa_,—and again when the water is
rendered a little more alkaline, to turn apparently into the so-called
_A. limax_,—so it is evident that a very slight modification in the
surface-energies concerned, might tend to turn one so-called species
into another among the Foraminifera. To what extent this process
actually occurs, we do not know. {606}

But that this, or something of the kind, does actually occur we can
scarcely doubt. For example in the genus Peneroplis, the first portion
of the shell consists of a series of chambers arranged in a spiral or
nautiloid series; but as age advances the spiral is apt to be modified
in various ways[550]. Sometimes the successive chambers grow rapidly
broader, the whole shell becoming fan-shaped. Sometimes the chambers
become narrower, till they no longer enfold the earlier chambers but
only come in contact each with its immediate predecessor: the result
being that the shell straightens out, and (taking into account the
earlier spiral portion) may be described as crozier-shaped. Between
these extremes of shape, and in regard to other variations of thickness
or thinness, roughness or smoothness, and so on, there are innumerable
gradations passing one into another and intermixed without regard to
geographical distribution:—“wherever Peneroplides abound this wide
variation exists, and nothing can be more easy than to pick out a
number of striking specimens and give to each a distinctive name,
but _in no other way can they be divided into_ ‘_species._’[551]”
Some writers have wondered at the peculiar variability of this
particular shell[552]; but for all we know of the life-history of the
Foraminifera, it may well be that a great number of the other forms
which we distinguish as separate species and even genera are no more
than temporary manifestations of the same variability[553]. {607}


_Conclusion._

If we can comprehend and interpret on some such lines as these the
form and mode of growth of the foraminiferal shell, we may also begin
to understand two striking features of the group, namely, on the one
hand the large number of diverse types or families which exist and the
large number of species and varieties within each, and on the other
the persistence of forms which in many cases seem to have undergone
little change or none at all from the Cretaceous or even from earlier
periods to the present day. In few other groups, perhaps only among the
Radiolaria, do we seem to possess so nearly complete a picture of all
possible transitions between form and form, and of the whole branching
system of the evolutionary tree: as though little or nothing of it had
ever perished, and the whole web of life, past and present, were as
complete as ever. It leads one to imagine that these shells have grown
according to laws so simple, so much in harmony with their material,
with their environment, and with all the forces internal and external
to which they are exposed, that none is better than another and none
fitter or less fit to survive. It invites one also to contemplate the
possibility of the lines of possible variation being here so narrow and
determinate that identical forms may have come independently into being
again and again.

While we can trace in the most complete and beautiful manner the
passage of one form into another among these little shells, and ascribe
them all at last (if we please) to a series which starts with the
simple sphere of Orbulina or with the amoeboid body of Astrorhiza, the
question stares us in the face whether this be an “evolution” which we
have any right to correlate with historic _time_. The mathematician
can trace one conic section into another, and “evolve” for example,
through innumerable graded ellipses, the circle from the straight
line: which tracing of continuous steps is a true “evolution,” though
time has no part therein. It was after this fashion that Hegel,
and for that matter Aristotle himself, was an evolutionist—to whom
evolution was {608} a mental concept, involving order and continuity
in thought, but not an actual sequence of events in time. Such a
conception of evolution is not easy for the modern biologist to grasp,
and harder still to appreciate. And so it is that even those who,
like Dreyer[554] and like Rhumbler, study the foraminiferal shell
as a physical system, who recognise that its whole plan and mode of
growth is closely akin to the phenomena exhibited by fluid drops under
particular conditions, and who explain the conformation of the shell
by help of the same physical principles and mathematical laws—yet all
the while abate no jot or tittle of the ordinary postulates of modern
biology, nor doubt the validity and universal applicability of the
concepts of Darwinian evolution. For these writers the _biogenetisches
Grundgesetz_ remains impregnable. The Foraminifera remain for them a
great family tree, whose actual pedigree is traceable to the remotest
ages; in which historical evolution has coincided with progressive
change; and in which structural fitness for a particular function
(or functions) has exercised its selective action and ensured “the
survival of the fittest.” By successive stages of historic evolution
we are supposed to pass from the irregular Astrorhiza to a Rhabdammina
with its more concentrated disc; to the forms of the same genus which
consist of but a single tube with central chamber; to those where
this chamber is more and more distinctly segmented; so to the typical
many-chambered Nodosariae; and from these, by another definite advance
and later evolution to the spiral Trochamminae. After this fashion,
throughout the whole varied series of the Foraminifera, Dreyer
and Rhumbler (following Neumayr) recognise so many successions of
related forms, one passing into another, and standing towards it in a
definite relationship of ancestry or descent. Each evolution of form,
from simpler to more complex, is deemed to have been attended by an
advantage to the organism, an enhancement of its chances of survival
or perpetuation; hence the historically older forms are, on the whole,
structurally the simpler; or conversely the simpler forms, such as the
simple sphere, were the first to come into being in primeval seas; and
finally, the gradual development and increasing {609} complication of
the individual within its own lifetime is held to be at least a partial
recapitulation of the unknown history of its race and dynasty[555].

We encounter many difficulties when we try to extend such concepts
as these to the Foraminifera. We are led for instance to assert, as
Rhumbler does, that the increasing complexity of the shell, and of the
manner in which one chamber is fitted on another, makes for advantage;
and the particular advantage on which Rhumbler rests his argument is
_strength_. Increase of strength, _die Festigkeitssteigerung_, is
according to him the guiding principle in foraminiferal evolution, and
marks the historic stages of their development in geologic time. But in
days gone by I used to see the beach of a little Connemara bay bestrewn
with millions upon millions of foraminiferal shells, simple Lagenae,
less simple Nodosariae, more complex Rotaliae: all drifted by wave and
gentle current from their sea-cradle to their sandy grave: all lying
bleached and dead: one more delicate than another, but all (or vast
multitudes of them) perfect and unbroken. And so I am not inclined to
believe that niceties of form affect the case very much: nor in general
that foraminiferal life involves a struggle for existence wherein
breakage is a constant danger to be averted, and increased strength an
advantage to be ensured[556].

In the course of the same argument Rhumbler remarks that Foraminifera
are absent from the coarse sands and gravels[557], as Williamson indeed
had observed many years ago: so averting, or {610} at least escaping,
the dangers of concussion. But this is after all a very simple matter
of mechanical analysis. The coarseness or fineness of the sediment on
the sea-bottom is a measure of the current: where the current is strong
the larger stones are washed clean, where there is perfect stillness
the finest mud settles down; and the light, fragile shells of the
Foraminifera find their appropriate place, like every other graded
sediment, in this spontaneous order of lixiviation.

The theorem of Organic Evolution is one thing; the problem of
deciphering the lines of evolution, the order of phylogeny, the degrees
of relationship and consanguinity, is quite another. Among the higher
organisms we arrive at conclusions regarding these things by weighing
much circumstantial evidence, by dealing with the resultant of many
variations, and by considering the probability or improbability of
many coincidences of cause and effect; but even then our conclusions
are at best uncertain, our judgments are continually open to revision
and subject to appeal, and all the proof and confirmation we can ever
have is that which comes from the direct, but fragmentary evidence of
palaeontology[558].

But in so far as forms can be shewn to depend on the play of physical
forces, and the variations of form to be directly due to simple
quantitative variations in these, just so far are we thrown back on our
guard before the biological conception of consanguinity, and compelled
to revise the vague canons which connect classification with phylogeny.

The physicist explains in terms of the properties of matter, and
classifies according to a mathematical analysis, all the drops and
forms of drops and associations of drops, all the kinds of froth and
foam, which he may discover among inanimate things; and his task ends
there. But when such forms, such conformations and configurations,
occur among _living_ things, then at once the biologist introduces
his concepts of heredity, of historical evolution, of succession in
time, of recapitulation of remote ancestry in individual growth, of
common origin (unless contradicted by direct evidence) of similar forms
remotely separated by geographic space or geologic time, of fitness
for a function, of {611} adaptation to an environment, of higher and
lower, of “better” and “worse.” This is the fundamental difference
between the “explanations” of the physicist and those of the biologist.

In the order of physical and mathematical complexity there is no
question of the sequence of historic time. The forces that bring about
the sphere, the cylinder or the ellipsoid are the same yesterday
and to-morrow. A snow-crystal is the same to-day as when the first
snows fell. The physical forces which mould the forms of Orbulina, of
Astrorhiza, of Lagena or of Nodosaria to-day were still the same, and
for aught we have reason to believe the physical conditions under which
they worked were not appreciably different, in that yesterday which we
call the Cretaceous epoch; or, for aught we know, throughout all that
duration of time which is marked, but not measured, by the geological
record.

In a word, the minuteness of our organism brings its conformation as a
whole within the range of the molecular forces; the laws of its growth
and form appear to lie on simple lines; what Bergson calls[559] the
“ideal kinship” is plain and certain, but the “material affiliation”
is problematic and obscure; and, in the end and upshot, it seems to me
by no means certain that the biologist’s usual mode of reasoning is
appropriate to the case, or that the concept of continuous historical
evolution must necessarily, or may safely and legitimately, be employed.

{612}




CHAPTER XIII

THE SHAPES OF HORNS, AND OF TEETH OR TUSKS: WITH A NOTE ON TORSION


We have had so much to say on the subject of shell-spirals that we must
deal briefly with the analogous problems which are presented by the
horns of sheep, goats, antelopes and other horned quadrupeds; and all
the more, because these horn-spirals are on the whole less symmetrical,
less easy of measurement than those of the shell, and in other ways
also are less easy of investigation. Let us dispense altogether in this
case with mathematics; and be content with a very simple account of the
configuration of a horn.

There are three types of horn which deserve separate consideration:
firstly, the horn of the rhinoceros; secondly the horns of the sheep,
the goat, the ox or the antelope, that is to say, of the so-called
hollow-horned ruminants; and thirdly, the solid bony horns, or
“antlers,” which are characteristic of the deer.

The horn of the rhinoceros presents no difficulty. It is
physiologically equivalent to a mass of consolidated hairs, and,
like ordinary hair, it consists of non-living or “formed” material,
continually added to by the living tissues at its base. In section,
that is to say in the form of its “generating curve,” the horn is
approximately elliptical, with the long axis fore-and-aft, or, in some
species, nearly circular. Its longitudinal growth proceeds with a
maximum velocity anteriorly, and a minimum posteriorly; and the ratio
of these velocities being constant, the horn curves into the form of
a logarithmic spiral in the manner that we have already studied. The
spiral is of small angle, but in the longer-horned species, such as the
great white rhinoceros (Ceratorhinus), the spiral form is distinctly
to be recognised. As the horn {613} occupies a median position on the
head,—a position, that is to say, of symmetry in respect to the field
of force on either side,—there is no tendency towards a lateral twist,
and the horn accordingly develops as a _plane_ logarithmic spiral. When
two horns coexist, the hinder one is much the smaller of the two: which
is as much as to say that the force, or rate, of growth diminishes as
we pass backwards, just as it does within the limits of the single
horn. And accordingly, while both horns have _essentially_ the same
shape, the spiral curvature is less manifest in the second one, simply
by reason of its comparative shortness.

The paired horns of the ordinary hollow-horned ruminants, such as the
sheep or the goat, grow under conditions which are in some respects
similar, but which differ in other and important respects from the
conditions under which the horn grows in the rhinoceros. As regards
its structure, the entire horn now consists of a bony core with
a covering of skin; the inner, or dermal, layer of the latter is
richly supplied with nutrient blood-vessels, while the outer layer,
or epidermis, develops the fibrous or chitinous material, chemically
and morphologically akin to a mass of cemented or consolidated hairs,
which constitutes the “sheath” of the horn. A zone of active growth at
the base of the horn keeps adding to this sheath, ring by ring, and
the specific form of this annular zone is, accordingly, that of the
“generating curve” of the horn. Each horn no longer lies, as it does
in the rhinoceros, in the plane of symmetry of the animal of which it
forms a part; and the limited field of force concerned in the genesis
and growth of the horn is bound, accordingly, to be more or less
laterally asymmetrical. But the two horns are in symmetry one with
another; they form “conjugate” spirals, one being the “mirror-image”
of the other. Just as in the hairy coat of the animal each hair, on
either side of the median “parting,” tends to have a certain definite
direction of its own axis, inclined away from the median axial plane of
the whole system, so is it both with the bony core of the horn and with
the consolidated mass of hairs or hair-like substance which constitutes
its sheath; the primary axis of the horn is more or less inclined to,
and may even be nearly perpendicular to, the axial plane of the animal.

The growth of the horny sheath is not continuous, but more or {614}
less definitely periodic: sometimes, as in the sheep, this periodicity
is particularly well-marked, and causes the horny sheath to be composed
of a series of all but separate rings, which are supposed to be formed
year by year, and so to record the age of the animal[560].

Just as we sought for the true generating curve in the orifice, or
“lip,” of the molluscan shell, so we might be apt to assume that in the
spiral horn the generating curve corresponded to the lip or margin of
one of the horny rings or annuli. This annular margin, or boundary of
the ring, is usually a sinuous curve, not lying in a plane, but such as
would form the boundary of an anticlastic surface of great complexity:
to the meaning and origin of which phenomenon we shall return
presently. But, as we have already seen in the case of the molluscan
shell, the complexities of the lip itself, or of the corresponding
lines of growth upon the shell, need not concern us in our study of
the development of the spiral: inasmuch as we may substitute for
these actual boundary lines, their “trace,” or projection on a plane
perpendicular to the axis—in other words the simple outline of a
transverse section of the whorl. In the horn, this transverse section
is often circular or nearly so, as in the oxen and many antelopes: it
now and then becomes of somewhat complicated polygonal outline, as
in a highland ram; but in many antelopes, and in most of the sheep,
the outline is that of an isosceles, or sometimes nearly equilateral
triangle, a form which is typically displayed, for instance, in _Ovis
Ammon_. The horn in this latter case is a trihedral prism, whose three
faces are, (1) an upper, or frontal face, in continuation of the plane
of the frontal bone; (2) an outer, or orbital, starting from the upper
margin of the orbit; and (3) an inner, or “nuchal,” abutting on the
parietal bone[561]. Along these three faces, and their corresponding
angles or edges, we can trace in the fibrous substance of the horn
a series of homologous spirals, such as we {615} have called in
a preceding chapter the “_ensemble_ of generating spirals” which
constitute the surface.

[Illustration: Fig. 318. Diagram of Ram’s horns. (After Sir Vincent
Brooke, from _P.Z.S._) _a_, frontal; _b_, orbital; _c_, nuchal surface.]

In some few cases, of which the male musk ox is one of the most
notable, the horn is not developed in a continuous spiral curve. It
changes its shape as growth proceeds; and this, as we have seen, is
enough to show that it does not constitute a logarithmic spiral. The
reason is that the bony exostoses, or horn-cores, about which the
horny sheath is shaped and moulded, neither grow continuously nor even
remain of constant size after attaining their full growth. But as the
horns grow heavy the bony core is bent downwards by their weight, and
so guides the growth of the horn in a new direction. Moreover as age
advances, the horn-core is further weakened and to a great extent
absorbed: and the horny sheath or horn proper, deprived of its support,
continues to grow, but in a flattened curve very different from its
original spiral[562]. The chamois is a somewhat analogous case. Here
the terminal, or oldest, part of the horn is curved; it tends to assume
a spiral form, though from its comparative shortness it seems merely to
be bent into a hook. But later on, the bony core within, as it grows
and strengthens, stiffens the horn, and guides it into a straighter
course or form. The same phenomenon {616} of change of curvature,
manifesting itself at the time when, or the place where, the horn is
freed from the support of the internal core, is seen in a good many
other antelopes (such as the hartebeest) and in many buffaloes; and the
cases where it is most manifest appear to be those where the bony core
is relatively short, or relatively weak.

[Illustration: Fig. 319. Head of Arabian Wild Goat, _Capra sinaitica_.
(After Sclater, from _P.Z.S._)]

But in the great majority of horns, we have no difficulty in
recognising a continuous logarithmic spiral, nor in referring it, as
before, to an unequal rate of growth (parallel to the axis) on two
opposite sides of the horn, the inequality maintaining a constant
ratio as long as growth proceeds. In certain antelopes, such as the
gemsbok, the spiral angle is very small, or in other words the horn
is very nearly straight; in other species of the same genus Oryx,
such as the Beisa antelope and the Leucoryx, a gentle {617} curve
(not unlike though generally less than that of a Dentalium shell) is
evident; and the spiral angle, according to the few measurements I have
made, is found to measure from about 20° to nearly 40°. In some of the
large wild goats, such as the Scinde wild goat, we have a beautiful
logarithmic spiral, with a constant angle of rather less than 70°;
and we may easily arrange a series of forms, such for example as the
Siberian ibex, the moufflon, _Ovis Ammon_, etc., and ending with the
long-horned Highland ram: in which, as we pass from one to another,
we recognise precisely homologous spirals, with an increasing angular
constant, the spiral angle being, for instance, about 75° or rather
less in _Ovis Ammon_, and in the Highland ram a very little more. We
have already seen that in the neighbourhood of 70° or 80° a small
change of angle makes a marked difference in the appearance of the
spire; and we know also that the actual length of the horn makes a very
striking difference, for the spiral becomes especially conspicuous to
the eye when a horn or shell is long enough to shew several whorls, or
at least a considerable part of one entire whorl.

Even in the simplest cases, such as the wild goats, the spiral is never
(strictly speaking) a plane or discoid spiral: but in greater or less
degree there is always superposed upon the plane logarithmic spiral a
helical spiral in space. Sometimes the latter is scarcely apparent,
for the helical curvature is comparatively small, and the horn (though
long, as in the said wild goats) is not nearly long enough to shew a
complete convolution: at other times, as in the ram, and still better
in many antelopes, such as the koodoo, the helicoid or corkscrew curve
of the horn is its most characteristic feature.

Accordingly we may study, as in the molluscan shell, the helicoid
component of the spire—in other words the variation in what we have
called (on p. 555) the angle _θ_. This factor it is which, more than
the constant angle of the logarithmic spiral, imparts a characteristic
appearance to the various species of sheep, for instance to the various
closely allied species of Asiatic wild sheep, or Argali. In all of
these the constant angle of the logarithmic spiral is very much the
same, but the shearing component differs greatly. And thus the long
drawn out horns of {618} _Ovis Poli_, four feet or more from tip to
tip, differ conspicuously from those of _Ovis Ammon_ or _O. hodgsoni_,
in which a very similar logarithmic spiral is wound (as it were) round
a much blunter cone.

――――――――――

The ram’s horn then, like the snail’s shell, is a curve of double
curvature, in which one component has imposed upon the structure a
plane logarithmic spiral, and the other has produced a continuous
displacement, or “shear,” proportionate in magnitude to, and
perpendicular or otherwise inclined in direction to, the axis of the
former spiral curvature. The result is precisely analogous to that
which we have studied in the snail and other spiral univalves; but
while the form, and therefore the resultant forces, are similar, the
original distribution of force is not the same: for we have not here,
as we had in the snail-shell, a “columellar” muscle, to introduce the
component acting in the direction of the axis. We have, it is true, the
central bony core, which in part performs an analogous function; but
the main phenomenon here is apparently a complex distribution of rates
of growth, perpendicular to the plane of the generating curve.

Let us continue to dispense with mathematics, for the mathematical
treatment of a curve of double curvature is never very simple, and let
us deal with the matter by experiment. We have seen that the generating
curve, or transverse section, of a typical ram’s horn is triangular in
form. Measuring (along the curve of the horn) the length of the three
edges of the trihedral structure in a specimen of _Ovis Ammon_, and
calling them respectively the outer, inner, and hinder edges (from
their position at the base of the horn, relatively to the skull),
I find the outer edge to measure 80 cm., the inner 74 cm., and the
posterior 45 cm.; let us say that, roughly, they are in the ratio of
9 : 8 : 5. Then, if we make a number of little cardboard triangles,
equip each with three little legs (I make them of cork), whose relative
lengths are as 9 : 8 : 5, and pile them up and stick them all together,
we straightway build up a curve of double curvature precisely analogous
to the ram’s horn: except only that, in this first approximation,
we have not allowed for the gradual increment (or decrement) of the
triangular surfaces, that is to say, for the _tapering_ of the horn due
to the growth in its own plane of the generating curve. {619}

In this case then, and in most other trihedral or three-sided horns,
one of the three components, or three unequal velocities of growth, is
of relatively small magnitude, but the other two are nearly equal one
to the other. It would involve but little change for these latter to
become precisely equal; and again but little to turn the balance of
inequality the other way. But the immediate consequence of this altered
ratio of growth would be that the horn would appear to wind the other
way, as it does in the antelopes, and also in certain goats, e.g. the
markhor, _Capra falconeri_.

 For these two opposite directions of twist Dr Wherry has introduced a
 convenient nomenclature. When the horn winds so that we follow it from
 base to apex in the direction of the hands of a watch, it is customary
 to call it a “left-handed” spiral. Such a spiral we have in the horn
 on the left-hand side of a ram’s head. Accordingly, Dr Wherry calls
 the condition _homonymous_, where, as in the sheep, a right-handed
 spiral is on the right side of the head, and a left-handed spiral on
 the left side; while he calls the opposite condition _heteronymous_,
 as we have it in the antelopes, where the right-handed twist is on the
 left side of the head, and the left-handed twist on the right-hand
 side. Among the goats, we may have either condition. Thus the
 domestic and most of the wild goats agree with the sheep; but in the
 markhor the twisted horns are heteronymous, as in the antelopes. The
 difference, as we have seen, is easily explained; and (very much as in
 the case of our opposite spirals in the apple-snail, referred to on p.
 560), it has no very deep importance.

Summarised then, in a very few words, the argument by which we account
for the spiral conformation of the horn is as follows: The horn
elongates by dint of continual growth within a narrow zone, or annulus,
at its base. If the rate of growth be identical on all sides of this
zone, the horn will grow straight; if it be greater on one side than
on the other, the horn will become curved: and it probably _will_
be greater on one side than on the other, because each single horn
occupies an unsymmetrical field with reference to the plane of symmetry
of the animal. If the maximal and minimal velocities of growth be
precisely at opposite sides of the zone of growth, the resultant spiral
will be a plane spiral; but if they be not precisely or diametrically
opposite, then the spiral will be a spiral in space, with a winding
or helical component; and it is by no means likely that the maximum
and minimum _will_ occur at precisely opposite ends of a diameter, for
{620} no such plane of symmetry is manifested in the field of force to
which the growing annulus corresponds or appertains.

Now we must carefully remember that the rates of growth of which we are
here speaking are the net rates of longitudinal increment, in which
increment the activity of the living cells in the zone of growth at the
base of the horn is only one (though it is the fundamental) factor.
In other words, if the horny sheath were continually being added to
with equal rapidity all round its zone of active growth, but at the
same time had its elongation more retarded on one side than the other
(prior to its complete solidification) by varying degrees of adhesion
or membranous attachment to the bone core within, then the net result
would be a spiral curve precisely such as would have arisen from
initial inequalities in the rate of growth itself. It seems highly
probable that this is a very important factor, and sometimes even the
chief factor in the case. The same phenomenon of attachment to the bony
core, and the consequent friction or retardation with which the sheath
slides over its surface, will lead to various subsidiary phenomena:
among others to the presence of transverse folds or corrugations upon
the horn, and to their unequal distribution upon its several faces or
edges. And while it is perfectly true that nearly all the characters
of the horn can be accounted for by unequal velocities of longitudinal
growth upon its different sides, it is also plain that the actual field
of force is a very complicated one indeed. For example, we can easily
see that (at least in the great majority of cases) the direction of
growth of the horny fibres of the sheath is by no means parallel to
the axis of the core within; accordingly these fibres will tend to
wind in a system of helicoid curves around the core, and not only this
helicoid twist but any other tendency to spiral curvature on the part
of the sheath will tend to be opposed or modified by the resistance of
the core within. But on the other hand living bone is a very plastic
structure, and yields easily though slowly to any forces tending to its
deformation; and so, to a considerable extent, the bony core itself
will tend to be modelled by the curvature which the growing sheath
assumes, and the final result will be determined by an equilibrium
between these two systems of forces. {621}

While it is not very safe, perhaps, to lay down any general rule as to
what horns are more, and what are less spirally curved, I think it may
be said that, on the whole, the thicker the horn, the greater is its
spiral curvature. It is the slender horns, of such forms as the Beisa
antelope, which are gently curved, and it is the robust horns of goats
or of sheep in which the curvature is more pronounced. Other things
being the same, this is what we should expect to find; for it is where
the transverse section of the horn is large that we may expect to find
the more marked differences in the intensity of the field of force,
whether of active growth or of retardation, on opposite sides or in
different sectors thereof.

[Illustration: Fig. 320. Head of _Ovis Ammon_, shewing St Venant’s
curves.]

But there is yet another and a very remarkable phenomenon which we may
discern in the growth of a horn, when it takes the form of a curve
of double curvature, namely, an effect of torsional strain; and this
it is which gives rise to the sinuous “lines of growth,” or sinuous
boundaries of the separate horny rings, of which we have already
spoken. It is not at first sight obvious that a mechanical strain of
torsion is necessarily involved in the growth of the horn. In our
experimental illustration (p. 618), we built up a twisted coil of
separate elements, and no torsional strain attended the development
of the system. So would it be if the horny sheath grew by successive
annular increments, free save for their relation to one another, and
having no attachment to the solid core within. But as a matter of fact
there is {622} such an attachment, by subcutaneous connective tissue,
to the bony core; and accordingly a torsional strain will be set up
in the growing horny sheath, again provided that the forces of growth
therein be directed more or less obliquely to the axis of the core; for
a “couple” is thus introduced, giving rise to a strain which the sheath
would not experience were it free (so to speak) to slip along, impelled
only by the pressure of its own growth from below. And furthermore,
the successive small increments of the growing horn (that is to say,
of the horny sheath) are not instantaneously converted from living to
solid and rigid substance; but there is an intermediate stage, probably
long-continued, during which the new-formed horny substance in the
neighbourhood of the zone of active growth is still plastic and capable
of deformation.

Now we know, from the celebrated experiments of St Venant[563], that
in the torsion of an elastic body, other than a cylinder of circular
section, a very remarkable state of strain is introduced. If the body
be thus cylindrical (whether solid or hollow), then a twist leaves
each circular section unchanged, in dimensions and in figure. But in
all other cases, such as an elliptic rod or a prism of any particular
sectional form, forces are introduced which act parallel to the axis of
the structure, and which warp each section into a complex anticlastic
surface. Thus in the case of a triangular and equilateral prism, such
as is shewn in section in Fig. 321, if the part of the rod represented
in the section be twisted by a force acting in the direction of the
arrow, then the originally plane section will be warped as indicated in
the diagram:—where the full contour-lines represent elevation above,
and the dotted lines represent depression below, the original level.
On the external surface of the prism, then, contour-lines which were
originally parallel and horizontal, will be found warped into sinuous
curves, such that, on each of the three faces, the curve will be convex
upwards on one half, and concave upwards on the other half of the face.
The ram’s horn, and still better that of _Ovis Ammon_, is comparable
to such a prism, save that in section it is not quite equilateral,
and that its three faces are not plane. The warping is therefore not
precisely identical on the three faces {623} of the horn; but, in the
general distribution of the curves, it is in complete accordance with
theory. Similar anticlastic curves are well seen in many antelopes; but
they are conspicuous by their absence in the _cylindrical_ horns of
oxen.

[Illustration: Fig. 321.]

[Illustration: Fig. 322.]

The better to illustrate this phenomenon, the nature of which is indeed
obvious enough from a superficial examination of the horn, I made a
plaster cast of one of the horny rings in a horn of _Ovis Ammon_, so
as to get an accurate pattern of its sinuous edge: and then, filling
the mould up with wet clay, I modelled an anticlastic surface, such
as to correspond as nearly as possible with the sinuous outline[564].
Finally, after making a plaster cast of this sectional surface, I drew
its contour-lines (as shewn in Fig. 322), with the help of a simple
form of spherometer. It will be seen that in great part this diagram
is precisely similar to St Venant’s diagram of the cross-section of
a twisted triangular prism; and this is especially the case in the
neighbourhood of the sharp angle of our prismatic section. That in
parts the diagram is somewhat asymmetrical is not to be wondered
at: and (apart from inaccuracies due to the somewhat rough means
by which it was made) this asymmetry can be sufficiently accounted
for by anisotropy of the material, by inequalities in thickness of
different parts of the horny sheath, and especially (I think) by
unequal distributions of rigidity due to the presence of the smaller
corrugations of the {624} horn. It is apparently on account of these
minor corrugations that, in such horns as the Highland ram’s, where
they are strongly marked, the main St Venant effect is not nearly so
well shewn as in the smoother horns such as those of _O. Ammon_ and its
immediate congeners[565].


_A further Note upon Torsion._

The phenomenon of torsion, to which we have been thus introduced, opens
up many wide questions in connection with form. Some of the associated
phenomena are admirably illustrated in the case of climbing plants; but
we can only deal with these still more briefly and parenthetically.

The subject of climbing plants has been elaborately dealt with not
only in Darwin’s books[566], but also by a very large number of
earlier and later writers. In “twining” plants, which constitute the
greater number of “climbers,” the essential phenomenon is a tendency
of the growing shoot to revolve about a vertical axis—a tendency long
ago discussed and investigated by such writers as Palm, H. von Mohl
and Dutrochet[567]. This tendency to revolution—“circumvolution,”
as Darwin calls it, “revolving nutation,” as Sachs puts it—is very
closely comparable to the process by which an antelope’s horn (such as
the koodoo’s) grows into its spiral or rather helicoid form; and it
is simply due, in like manner, to inequalities in the rate of growth
on different sides of the growing stem. There is only this difference
between the two cases, that in the antelope’s horn the zone of active
growth is confined to the base of the horn, while in the climbing stem
the same phenomenon is at work throughout the whole length of the
growing structure. This growth is in the main due to “turgescence,”
that is to the extension, or elongation, of ready-formed cells through
the imbibition of water; it is a phenomenon due to osmotic pressure.
The particular stimuli to which these movements (that is to say, these
inequalities of growth) have been {625} ascribed, such as contact
(thigmotaxis), exposure to light (heliotropism), and so forth, need not
be discussed here[568].

A simple stem growing upright in the dark, or in uniformly diffused
light, would be in a position of equilibrium to a field of force
radially symmetrical about its vertical axis. But this complete
radial symmetry will not often occur; and the radial anomalies may
be such as arise intrinsically from structural peculiarities in the
stem itself, or externally to it by reason of unequal illumination or
through various other localised forces. The essential fact, so far
as we are concerned, is that in twining plants we have a very marked
tendency to inequalities in longitudinal growth on different aspects
of the stem—a tendency which is but an exaggerated manifestation of
one which is more or less present, under certain conditions, in all
plants whatsoever. Just as in the case of the ruminants’ horns so
we find here, that this inequality may be, so to speak, positive or
negative, the maximum lying to the one side or the other of the twining
stem; and so it comes to pass that some climbers twine to the one side
and some to the other: the hop and the honeysuckle following the sun,
and the field-convolvulus twining in the reverse direction; there are
also some, like the woody nightshade (_Solanum Dulcamara_) which twine
indifferently either way.

Together with this circumnutatory movement, there is very generally
to be seen an actual _torsion_ of the twining stem—a twist, that is
to say, about its own axis; and Mohl made the curious observation,
confirmed by Darwin, that when a stem twines around a smooth
cylindrical stick the torsion does not take place, save “only in
that degree which follows as a mechanical necessity from the spiral
winding”: but that stems which had climbed around a rough stick were
all more or less, and generally much, twisted. Here Darwin did not
refrain from introducing that teleological argument which pervades
his whole train of reasoning: “The stem,” he says, “probably gains
rigidity by being twisted (on the same principle that a much twisted
rope {626} is stiffer than a slackly twisted one), and is thus
indirectly benefited so as to be able to pass over inequalities in its
spiral ascent, and to carry its own weight when allowed to revolve
freely.” The mechanical explanation would appear to be very simple, and
such as to render the teleological hypothesis unnecessary. In the case
of the roughened support, there is a temporary adhesion or “clinging”
between it and the growing stem which twines around it; and a system of
forces is thus set up, producing a “couple,” just as it was in the case
of the ram’s or antelope’s horn through direct adhesion of the bony
core to the surrounding sheath. The twist is the direct result of this
couple, and it disappears when the support is so smooth that no such
force comes to be exerted.

Another important class of climbers includes the so-called
“leaf-climbers.” In these, some portion of the leaf, generally the
petiole, sometimes (as in the fumitory) the elongated midrib, curls
round a support; and a phenomenon of like nature occurs in many, though
not all, of the so-called “tendril-bearers.” Except that a different
part of the plant, leaf or tendril instead of stem, is concerned in
the twining process, the phenomenon here is strictly analogous to
our former case; but in the resulting helix there is, as a rule,
this obvious difference, that, while the twining stem, for instance
of the hop, makes a slow revolution about its support, the typical
leaf-climber makes a close, firm coil: the axis of the latter is nearly
perpendicular and parallel to the axis of its support, while in the
twining stem the angle between the two axes is comparatively small.
Mathematically speaking, the difference merely amounts to this, that
the component in the direction of the vertical axis is large in the
one case, and the corresponding component is small, if not absent, in
the other; in other words, we have in the climbing stem a considerable
vertical component, due to its own tendency to grow in height, while
this longitudinal or vertical extension of the whole system is not
apparent, or little apparent, in the other cases. But from the fact
that the twining stem tends to run obliquely to its support, and the
coiling petiole of the leaf-climber tends to run transversely to the
axis of its support, there immediately follows this marked difference,
that the phenomenon {627} of _torsion_, so manifest in the former
case, will be absent in the latter.

――――――――――

There is one other phenomenon which meets us in the twining and twisted
stem, and which is doubtless illustrated also, though not so well, in
the antelope’s horn; it is a phenomenon which forms the subject of a
second chapter of St Venant’s researches on the effects of torsional
strain in elastic bodies. We have already seen how one effect of
torsion, in for instance a prism, is to produce strains parallel to the
axis, elevating parts and depressing other parts of each transverse
section. But in addition to this, the same torsion has the effect of
materially altering the form of the section itself, as we may easily
see by twisting a square or oblong piece of india-rubber. If we start
with a cylinder, such as a round piece of catapult india-rubber, and
twist it on its own long axis, we have already seen that it suffers no
other distortion; it still remains a cylinder, that is to say, it is
still in section everywhere circular. But if it be of any other shape
than cylindrical the case is quite different, for now the sectional
shape tends to alter under the strain of torsion. Thus, if our rod be
elliptical in section to begin with, it will, under torsion, become a
more elongated ellipse; if it be square, its angles will become more
prominent, and its sides will curve inwards, till at length the square
assumes the appearance of a four-pointed star, with rounded angles.
Furthermore, looking at the results of this process of modification, we
find experimentally that the resultant figures are more easily twisted,
less resistant to torsion, than were those from which we evolved
them; and this is a very curious physical or mathematical fact. So a
cylinder, which is especially resistant to torsion, is very easily bent
or flexed; while projecting ribs or angles, such as an engineer makes
in a bar or pillar of iron for the purpose of greatly increasing its
strength in the way of resistance to _bending_, actually make it much
weaker than before (for the same amount of metal per unit length) in
the way of resistance to _torsion_.

In the hop itself, and in a very considerable number of other twining
and twisting stems, the ribbed or channelled form of the stem is a
conspicuous feature. We may safely take it, (1) that {628} such
stems are especially susceptible of torsion; and (2) that the effect
of torsion will be to intensify any such peculiarities of sectional
outline which they may possess, though not to initiate them in an
originally cylindrical structure. In the leaf-climbers the case does
not present itself, for there, as we have seen, torsion itself is not,
or is very slightly, manifested. There are very distinct traces of the
phenomenon in the horns of certain antelopes, but the reason why it is
not a more conspicuous feature of the antelope’s horn or of the ram’s
is apparently a very simple one: namely, that the presence of the bony
core within tends to check that deformation which is perpendicular,
while it permits that which is parallel, to the axis of the horn.


_Of Deer’s Antlers._

But let us return to our subject of the shapes of horns, and consider
briefly our last class of these structures, namely the bony antlers
of the various species of elk and deer[569]. The problems which these
present to us are very different from those which we have had to do
with in the antelope or the sheep.

[Illustration: Fig. 323. Antlers of Swedish Elk. (After Lönnberg, from
_P.Z.S._)]

With regard to its structure, it is plain that the bony antler
corresponds, upon the whole, to the bony core of the antelope’s horn;
while in place of the hard horny sheath of the latter, we have the
soft “velvet,” which every season covers the new growing antler, and
protects the large nutrient blood-vessels by help of which the antler
grows[570]. The main difference lies in the fact that, in the one case,
the bony core, imprisoned within its sheath, is rendered incapable
of branching and incapable also of lateral expansion, and the whole
horn is only permitted to grow in length, while retaining a sectional
contour that is identical with (or but little altered from) that which
it possesses at its growing base: {629} but in the antler, on the
other hand, no such restraint is imposed, and the living, growing
fabric of bone may expand into a broad flattened plate over which
the blood-vessels run. In the immediate neighbourhood of the main
blood-vessels growth will be most active; in the interspaces between,
it may wholly fail: with the result that we may have great notches
cut out of the flattened plate, or may at length find it reduced to
the form of a simple branching structure. The main point, as it seems
to me, is that the “horn” is essentially an _axial rod_, while the
“antler” is essentially an outspread _surface_[571]. In other words,
I believe that the whole configuration of an antler is more easily
understood by conceiving it as a plate or a surface, more and more
notched and scolloped till but a slender skeleton may remain, than to
look upon it the other way, namely as an axial stem (or beam) giving
{630} off branches (or tines), the interspaces between which latter
may sometimes be filled up to form a continuous plate.

[Illustration: Fig. 324. Head and antlers of a Stag (_Cervus
Duvauceli_). (After Lydekker, from _P.Z.S._)]

In a sense it matters very little whether we regard the broad
plate-like antlers of the elk or the slender branching antlers of the
stag as the more primitive type; for we are not concerned here with
the question of hypothetical phylogeny. And even from the mathematical
point of view it makes little or no difference whether we describe
the plate as constituted by the interconnection of the branches, or
the branches derived by a process of notching or incision from the
plate. The important point for us is to recognise that (save for
occasional slight irregularities) the branching system in the one
_conforms_ essentially to the curved plate or surface which we see
plainly in the other. In short the arrangement of the branches is more
or less comparable to that of the veins in a leaf, or to that of the
blood-vessels as they course over the curved surface of an organ. It is
a process of ramification, not, like that of a tree, in various planes,
but strictly limited {631} to a single surface. And just as the veins
within a leaf are not necessarily confined (as they happen to be in
most ordinary leaves) to a _plane_ surface, but, as in the petal of a
tulip or the capsule of a poppy, may have to run their course within a
curved surface, so does the analogy of the leaf lead us directly to the
mode of branching which is characteristic of the antler. The surface to
which the branches of the antler tend to be confined is a more or less
spheroidal, or occasionally an ellipsoidal one; and furthermore, when
we inspect any well-developed pair of antlers, such as those of a red
deer, a sambur or a wapiti, we have no difficulty in seeing that the
two antlers make up between them _a single surface_, and constitute a
symmetrical figure, each half being the mirror-image of the other.

To put the case in another way, a pair of antlers (apart from
occasional slight irregularities) tends to constitute a figure such
that we could conceive an elastic sheet stretched over or round the
entire system, so as to form one continuous and even surface; and not
only would the surface curvature be on the whole smooth and even, but
the boundary of the surface would also tend to be an even curve: that
is to say the tips of all the tines would approximately have their
locus in a continuous curve.

It follows from this that if we want to make a simple model of a set
of antlers, we shall be very greatly helped by taking some appropriate
spheroidal surface as our groundwork or scaffolding. The best form of
surface is a matter for trial and investigation in each particular
case; but even in a sphere, by selecting appropriate areas thereof,
we can obtain sufficient varieties of surface to meet all ordinary
cases. With merely a bit of sculptor’s clay or plasticine, we should
be put hard to it to model the horns of a wapiti or a reindeer: but
if we start with an orange (or a round florence flask) and lay our
little tapered rolls of plasticine upon it, in simple natural curves,
it is surprising to see how quickly and successfully we can imitate
one type of antler after another. In doing so, we shall be struck by
the fact that our model may vary in its mode of branching within very
considerable limits, and yet look perfectly natural. For the same wide
range of variation is characteristic of the natural antlers themselves.
As Sir V. Brooke says (_op. cit._ p. 892), “No two antlers are ever
exactly alike; and the {632} variation to which the antlers are subject
is so great that in the absence of a large series they would be held
to be indicative of several distinct species[572].” But all these many
variations lie within a limited range, for they are all subject to our
general rule that the entire structure is essentially confined to a
single curved surface.

It is plain that in the curvatures both of the beam and of its tines,
in the angles by which these latter meet the beam, and in the contours
of the entire system, there are involved many elegant mathematical
problems with which we cannot at present attempt to deal. Nor must
we attempt meanwhile to enquire into the physical meaning or origin
of these phenomena, for as yet the clue seems to be lacking and we
should only heap one hypothesis upon another. That there is a complete
contrast of mathematical properties between the horn and the antler is
the main lesson with which, in the meantime, we must rest content.


_Of Teeth, and of Beak and Claw._

In a fashion similar to that manifested in the shell or the horn,
we find the logarithmic spiral to be implicit in a great many other
organic structures where the phenomena of growth proceed in a similar
way: that is to say, where about an axis there is some asymmetry
leading to unequal rates of longitudinal growth, and where the
structure is of such a kind that each new increment is added on as a
permanent and unchanging part of the entire conformation. Nail and
claw, beak and tooth, all come under this category. The logarithmic
spiral _always_ tends to manifest itself in such structures as these,
though it usually only attracts our attention in elongated structures,
where (that is to say) the radius vector has described a considerable
angle. When the canary-bird’s claws grow long from lack of use, or
when the incisor tooth of a rabbit or a rat grows long by reason of an
injury to the opponent tooth against which it was wont to bite, we know
that the tooth or claw tends to grow into a spiral curve, and we speak
of it as a malformation. But there has been no fundamental change of
form, save only an abnormal increase in length; {633} the elongated
tooth or claw has the selfsame curvature that it had when it was short,
but the spiral curvature becomes more and more manifest the longer it
grows. A curious analogous case is that of the New Zealand huia bird,
in which the beak of the female is described as being comparatively
short and straight, while that of the male is long and curved; it is
easy to see that there is a slight curvature also in the beak of the
female, and that the beak of the male shows nothing but the same curve
produced. In the case of the more curved beaks, such as those of an
eagle or a parrot, we may, if we please, determine the constant angle
of the logarithmic spiral, just as we have done in the case of the
Nautilus shell; and here again, as the bird grows older or the beak
longer, the spiral nature of the curve becomes more and more apparent,
as in the hooked beak of an old eagle, or as in the great beak of some
large parrot such as a hyacinthine macaw.

Let us glance at one or two instances to illustrate the spiral
curvature of teeth.

A dentist knows that every tooth has a curvature of its own, and that
in pulling the tooth he must follow the direction of the curve; but in
an ordinary tooth this curvature is scarcely visible, and is least so
when the diameter of the tooth is large compared with its length.

In the simply formed, more or less conical teeth, such as are those
of the dolphin, and in the more or less similarly shaped canines
and incisors of mammals in general, the curvature of the tooth is
particularly well seen. We see it in the little teeth of a hedgehog,
and in the canines of a dog or a cat it is very obvious indeed. When
the great canine of the carnivore becomes still further enlarged
or elongated, as in Machairodus, it grows into the strongly curved
sabre-tooth of that great extinct tiger. In rodents, it is the incisors
which undergo a great elongation; their rate of growth differs, though
but slightly, on the two sides, anterior and posterior, of the axis,
and by summation of these slight differences in the rapid growth of
the tooth an unmistakeable logarithmic spiral is gradually built up.
We see it admirably in the beaver, or in the great ground-rat, Geomys.
The elephant is a similar case, save that the tooth, or tusk, remains,
owing to comparative lack of wear, in a more perfect condition. In
the rodent (save only in those abnormal cases mentioned on the last
page) the {634} anterior, first-formed, part of the tooth wears
away as fast as it is added to from behind; and in the grown animal,
all those portions of the tooth near to the pole of the logarithmic
spiral have long disappeared. In the elephant, on the other hand,
we see, practically speaking, the whole unworn tooth, from point to
root; and its actual tip nearly coincides with the pole of the spiral.
If we assume (as with no great inaccuracy we may do) that the tip
actually coincides with the pole, then we may very easily construct
the continuous spiral of which the existing tusk constitutes a part;
and by so doing, we see the short, gently curved tusk of our ordinary
elephant growing gradually into the spiral tusk of the mammoth. No
doubt, just as in the case of our molluscan shells, we have a tendency
to variation, both individual and specific, in the constant angle of
the spiral; some elephants, and some species of elephant, undoubtedly
have a higher spiral angle than others. But in most cases, the angle
would seem to be such that a spiral configuration would become very
manifest indeed if only the tusk pursued its steady growth, unchanged
otherwise in form, till it attained the dimensions which we meet with
in the mammoth. In a species such as _Mastodon angustidens_, or _M.
arvernensis_, the specific angle is low and the tusk comparatively
straight; but the American mastodons and the existing species of
elephant have tusks which do not differ appreciably, except in
size, from the great spiral tusks of the mammoth, though from their
comparative shortness the spiral is little developed and only appears
to the eye as a gentle curve. Wherever the tooth is very long indeed,
as in the mammoth or the beaver, the effect of some slight and all
but inevitable lateral asymmetry in the rate of growth begins to shew
itself: in other words, the spiral is seen to lie not absolutely in a
plane, but to be a curve of double curvature, like a twisted horn. We
see this condition very well in the huge canine tusks of the Babirussa;
it is a conspicuous feature in the mammoth, and it is more or less
perceptible in any large tusk of the ordinary elephants.

The form of a molar tooth, which is essentially a branching or budding
system, and in which such longitudinal growth as gives rise to a spiral
curve is but little manifest, constitutes an entirely different problem
with which I shall not at present attempt to deal.

{635}




CHAPTER XIV

ON LEAF-ARRANGEMENT, OR PHYLLOTAXIS


The beautiful configurations produced by the orderly arrangement of
leaves or florets on a stem have long been an object of admiration and
curiosity. Leonardo da Vinci would seem, as Sir Theodore Cook tells
us, to have been the first to record his thoughts upon this subject;
but the old Greek and Egyptian geometers are not likely to have
left unstudied or unobserved the spiral traces of the leaves upon a
palm-stem, or the spiral curves of the petals of a lotus or the florets
in a sunflower.

The spiral leaf-order has been regarded by many learned botanists
as involving a fundamental law of growth, of the deepest and most
far-reaching importance; while others, such as Sachs, have looked
upon the whole doctrine of “phyllotaxis” as “a sort of geometrical or
arithmetical playing with ideas,” and “the spiral theory as a mode
of view gratuitously introduced into the plant.” Sachs even goes so
far as to declare this doctrine “in direct opposition to scientific
investigation, and based upon the idealistic direction of the
Naturphilosophie,”—the mystical biology of Oken and his school.

The essential facts of the case are not difficult to understand;
but the theories built upon them are so varied, so conflicting, and
sometimes so obscure, that we must not attempt to submit them to
detailed analysis and criticism. There are two chief ways by which we
may approach the question, according to whether we regard, as the more
fundamental and typical, one or other of the two chief modes in which
the phenomenon presents itself. That is to say, we may hold that the
phenomenon is displayed in its essential simplicity by the corkscrew
spirals, or helices, which mark the position of the leaves upon a
cylindrical stem or on an {636} elongated fir-cone; or, on the other
hand, we may be more attracted by, and regard as of greater importance,
the logarithmic spirals which we trace in the curving rows of florets
in the discoidal inflorescence of a sunflower. Whether one way or the
other be the better, or even whether one be not positively correct and
the other radically wrong, has been vehemently debated. In my judgment
they are, both mathematically and biologically, to be regarded as
inseparable and correlative phenomena.

The helical arrangement (as in the fir-cone) was carefully studied in
the middle of the eighteenth century by the celebrated Bonnet, with
the help of Calandrini, the mathematician. Memoirs published about
1835, by Schimper and Braun, greatly amplified Bonnet’s investigations,
and introduced a nomenclature which still holds its own in botanical
textbooks. Naumann and the brothers Bravais are among those who
continued the investigation in the years immediately following, and
Hofmeister, in 1868, gave an admirable account and summary of the work
of these and many other writers[573].

Starting from some given level and proceeding upwards, let us mark
the position of some one leaf (_A_) upon a cylindrical stem. Another,
and a younger leaf (_B_) will be found standing at a certain distance
_around_ the stem, and a certain distance _along_ the stem, {637}
from the first. The former distance may be expressed as a fractional
“divergence” (such as two-fifths of the circumference of the stem)
as the botanists describe it, or by an “angle of azimuth” (such as
ϕ = 144°) as the mathematician would be more likely to state it. The
position of _B_ relatively to _A_ must be determined, not only by
this angle ϕ, in the horizontal plane, but also by an angle (θ) in
the vertical plane; for the height of _B_ above the level of _A_, in
comparison with the diameter of the cylinder, will obviously make a
great difference in the appearance of the whole system, in short the
position of each leaf must be expressed by _F_(ϕ ⋅ sin θ). But this
matter botanical students have not concerned themselves with; in other
words, their studies have been limited (or mainly limited) to the
relation of the leaves to one another in _azimuth_.

Whatever relation we have found between _A_ and _B_, let precisely the
same relation subsist between _B_ and _C_: and so on. Let the growth
of the system, that is to say, be continuous and uniform; it is then
evident that we have the elementary conditions for the development of a
simple cylindrical helix; and this “primary helix” or “genetic spiral”
we can now trace, winding round and round the stem, through _A_, _B_,
_C_, etc. But if we can trace such a helix through _A_, _B_, _C_, it
follows from the symmetry of the system, that we have only to join _A_
to some other leaf to trace another spiral helix, such, for instance,
as _A_, _C_, _E_, etc.; parallel to which will run another and similar
one, namely in this case _B_, _D_, _F_, etc. And these spirals will run
in the opposite direction to the spiral _ABC_.

In short, the existence of one helical arrangement of points implies
and involves the existence of another and then another helical pattern,
just as, in the pattern of a wall-paper, our eye travels from one
linear series to another.

A modification of the helical system will be introduced when, instead
of the leaves appearing, or standing, in singular succession, we get
two or more appearing simultaneously upon the same level. If there
be two such, then we shall have two generating spirals precisely
equivalent to one another; and we may call them _A_, _B_, _C_, etc.,
and _A′_, _B′_, _C′_, and so on. These are the cases which we call
“whorled” leaves, or in the simplest case, where {638} the whorl
consists of two opposite leaves only, we call them decussate.

――――――――――

Among the phenomena of phyllotaxis, two points in particular have been
found difficult of explanation, and have aroused discussion. These are
(1), the presence of the logarithmic spirals such as we have already
spoken of in the sunflower; and (2) the fact that, as regards the
number of the helical or spiral rows, certain numerical coincidences
are apt to recur again and again, to the exclusion of others, and so to
become characteristic features of the phenomenon.

The first of these appears to me to present no difficulty. It is a
mere matter of strictly mathematical “deformation.” The stem which
we have begun to speak of as a cylinder is not strictly so, inasmuch
as it tapers off towards its summit. The curve which winds evenly
around this stem is, accordingly, not a true helix, for that term is
confined to the curve which winds evenly around the _cylinder_: it
is a curve in space which (like the spiral curve we have studied in
our turbinate shells) partakes of the characters of a helix and of a
logarithmic spiral, and which is in fact a logarithmic spiral with
its pole drawn out of its original plane by a force acting in the
direction of the axis. If we imagine a tapering cylinder, or cone,
projected, by vertical projection, on a plane, it becomes a circular
disc; and a helix described about the cone necessarily becomes in the
disc a logarithmic spiral described about a focus which corresponds
to the apex of our cone. In like manner we may project an identical
spiral in space upon such surfaces as (for instance) a portion of a
sphere or of an ellipsoid; and in all these cases we preserve the
spiral configuration, which is the more clearly brought into view the
more we reduce the vertical component by which it was accompanied.
The converse is, of course, equally true, and equally obvious,
namely that any logarithmic spiral traced upon a circular disc or
spheroidal surface will be transformed into a corresponding spiral
helix when the plane or spheroidal disc is extended into an elongated
cone approximating to a cylinder. This mathematical conception is
translated, in botany, into actual fact. The fir-cone may be looked
upon as a cylindrical axis contracted at both ends, until {639} it
becomes approximately an ellipsoidal solid of revolution, generated
about the long axis of the ellipse; and the semi-ellipsoidal capitulum
of the teasel, the more or less hemispherical one of the thistle, and
the flattened but still convex one of the sunflower, are all beautiful
and successive deformations of what is typically a long, conical, and
all but cylindrical stem. On the other hand, every stem as it grows
out into its long cylindrical shape is but a deformation of the little
spheroidal or ellipsoidal surface, or cone, which was its forerunner in
the bud.

This identity of the helical spirals around the stem with spirals
projected on a plane was clearly recognised by Hofmeister, who was
accustomed to represent his diagrams of leaf-arrangement either in one
way or the other, though not in a strictly geometrical projection[574].

――――――――――

According to Mr A. H. Church[575], who has dealt very carefully and
elaborately with the whole question of phyllotaxis, the logarithmic
spirals such as we see in the disc of the sunflower have a far greater
importance and a far deeper meaning than this brief treatment of mine
would accord to them: and Sir Theodore Cook, in his book on the _Curves
of Life_, has adopted and has helped to expound and popularise Mr
Church’s investigations.

Mr Church, regarding the problem as one of “uniform growth,” easily
arrives at the conclusion that, _if_ this growth can be conceived as
taking place symmetrically about a central point or “pole,” the uniform
growth would then manifest itself in logarithmic spirals, including
of course the limiting cases of the circle and straight line. With
this statement I have little fault to find; it is in essence identical
with much that I have said in a previous chapter. But other statements
of Mr Church’s, and many theories woven about them by Sir T. Cook
and himself, I am less able to follow. Mr Church tells us that the
essential phenomenon in the sunflower disc is a series of orthogonally
intersecting logarithmic spirals. Unless I wholly misapprehend Mr
Church’s meaning, I should say that this is very far from essential.
The spirals {640} intersect isogonally, but orthogonal intersection
would be only one particular case, and in all probability a very
infrequent one, in the intersection of logarithmic spirals developed
about a common pole. Again on the analogy of the hydrodynamic lines
of force in certain vortex movements, and of similar lines of force
in certain magnetic phenomena, Mr Church proceeds to argue that the
energies of life follow lines comparable to those of electric energy,
and that the logarithmic spirals of the sunflower are, so to speak,
lines of equipotential[576]. And Sir T. Cook remarks that this “theory,
if correct, would be fundamental for all forms of growth, though it
would be more easily observed in plant construction than in animals.”
The parallel I am not able to follow.

Mr Church sees in phyllotaxis an organic mystery, a something for which
we are unable to suggest any precise cause: a phenomenon which is to
be referred, somehow, to waves of growth emanating from a centre, but
on the other hand not to be explained by the division of an apical
cell, or any other histological factor. As Sir T. Cook puts it, “at the
growing point of a plant where the new members are being formed, there
is simply _nothing to see_.”

But it is impossible to deal satisfactorily, in brief space, either
with Mr Church’s theories, or my own objections to them[577]. Let
it suffice to say that I, for my part, see no subtle mystery in the
matter, other than what lies in the steady production of similar
growing parts, similarly situated, at similar successive intervals
of time. If such be the case, then we are bound to have in {641}
consequence a series of symmetrical patterns, whose nature will
depend upon the form of the entire surface. If the surface be that of
a cylinder we shall have a system, or systems, of spiral helices: if
it be a plane, with an infinitely distant focus, such as we obtain by
“unwrapping” our cylindrical surface, we shall have straight lines;
if it be a plane containing the focus within itself, or if it be any
other symmetrical surface containing the focus, then we shall have
a system of logarithmic spirals. The appearance of these spirals is
sometimes spoken of as a “subjective” phenomenon, but the description
is inaccurate: it is a purely mathematical phenomenon, an inseparable
secondary result of other arrangements which we, for the time being,
regard as primary. When the bricklayer builds a factory chimney, he
lays his bricks in a certain steady, orderly way, with no thought of
the spiral patterns to which this orderly sequence inevitably leads,
and which spiral patterns are by no means “subjective.” The designer
of a wall-paper not only has no intention of producing a pattern of
criss-cross lines, but on the contrary he does his best to avoid
them; nevertheless, so long as his design is a symmetrical one, the
criss-cross intersections inevitably come.

Let us, however, leave this discussion, and return to the facts of the
case.

――――――――――

Our second question, which relates to the numerical coincidences so
familiar to all students of phyllotaxis, is not to be set and answered
in a word.

Let us, for simplicity’s sake, avoid consideration of simultaneous or
whorled leaf origins, and consider only the more frequent cases where a
single “genetic spiral” can be traced throughout the entire system.

It is seldom that this primary, genetic spiral catches the eye, for
the leaves which immediately succeed one another in this genetic
order are usually far apart on the circumference of the stem, and it
is only in close-packed arrangements that the eye readily apprehends
the continuous series. Accordingly in such a case as a fir-cone, for
instance, it is certain of the secondary spirals or “parastichies”
which catch the eye; and among fir-cones, we can easily count these,
and we find them to be {642} on the whole very constant in number,
according to the species.

Thus in many cones, such as those of the Norway spruce, we can trace
five rows of scales winding steeply up the cone in one direction,
and three rows winding less steeply the other way; in certain other
species, such as the common larch, the normal number is eight rows
in the one direction and five in the other; while in the American
larch we have again three in the one direction and five in the other.
It not seldom happens that two arrangements grade into one another
on different parts of one and the same cone. Among other cases in
which such spiral series are readily visible we have, for instance,
the crowded leaves of the stone-crops and mesembryanthemums, and (as
we have said) the crowded florets of the composites. Among these we
may find plenty of examples in which the numbers of the serial rows
are similar to those of the fir-cones; but in some cases, as in the
daisy and others of the smaller composites, we shall be able to trace
thirteen rows in one direction and twenty-one in the other, or perhaps
twenty-one and thirty-four; while in a great big sunflower we may find
(in one and the same species) thirty-four and fifty-five, fifty-five
and eighty-nine, or even as many as eighty-nine and one hundred and
forty-four. On the other hand, in an ordinary “pentamerous” flower,
such as a ranunculus, we may be able to trace, in the arrangement of
its sepals, petals and stamens, shorter spiral series, three in one
direction and two in the other. It will be at once observed that these
arrangements manifest themselves in connection with very different
things, in the orderly interspacing of single leaves and of entire
florets, and among all kinds of leaf-like structures, foliage-leaves,
bracts, cone-scales, and the various parts or members of the flower.
Again we must be careful to note that, while the above numerical
characters are by much the most common, so much so as to be deemed
“normal,” many other combinations are known to occur.

The arrangement, as we have seen, is apt to vary when the entire
structure varies greatly in size, as in the disc of the sunflower.
It is also subject to less regular variation within one and the same
species, as can always be discovered when we examine a sufficiently
large sample of fir-cones. For instance, out of 505 {643} cones of
the Norway spruce, Beal[578] found 92 per cent. in which the spirals
were in five and eight rows; in 6 per cent. the rows were four and
seven, and in 4 per cent. they were four and six. In each case they
were nearly equally divided as regards direction; for instance of the
467 cones shewing the five-eight arrangement, the five-series ran in
right-handed spirals in 224 cases, and in left-handed spirals in 243.

Omitting the “abnormal” cases, such as we have seen to occur in a small
percentage of our cones of the spruce, the arrangements which we have
just mentioned may be set forth as follows, (the fractional number
used being simply an abbreviated symbol for the number of associated
helices or parastichies which we can count running in the opposite
directions): 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89, 89/144.
Now these numbers form a very interesting series, which happens to have
a number of curious mathematical properties[579]. We see, for instance,
that the denominator of each fraction is the numerator of the next; and
further, that each successive numerator, or denominator, is the sum
of the preceding two. Our immediate problem, then, is to determine,
if possible, how these numerical coincidences come about, and why
these particular numbers should be so commonly met with {644} as to
be considered “normal” and characteristic features of the general
phenomenon of phyllotaxis. The following account is based on a short
paper by Professor P. G. Tait[580].

[Illustration: Fig. 325.]

Of the two following diagrams, Fig. 325 represents the general case,
and Fig. 326 a particular one, for the sake of possibly greater
simplicity. Both diagrams represent a portion of a branch, or fir-cone,
regarded as cylindrical, and unwrapped to form a plane surface. _A_,
_a_, at the two ends of the base-line, represent the same initial leaf
or scale: _O_ is a leaf which can be reached from _A_ by _m_ steps in a
right-hand spiral (developed into the straight line _AO_), and by _n_
steps from _a_ in a left-handed spiral _aO_. Now it is obvious in our
fir-cone, that we can include _all_ the scales upon the cone by taking
so many spirals in the one direction, and again include them all by
so many in the other. Accordingly, in our diagrammatic construction,
the spirals _AO_ and _aO_ _must_, and always _can_, be so taken that
_m_ spirals parallel to _aO_, and _n_ spirals parallel to _AO_, shall
separately include all the leaves upon the stem or cone.

If _m_ and _n_ have a common factor, _l_, it can easily be shewn that
the arrangement is composite, and that there are _l_ fundamental, or
genetic spirals, and _l_ leaves (including _A_) which are situated
exactly on the line _Aa_. That is to say, we have here a _whorled_
arrangement, which we have agreed to leave unconsidered in favour of
the simpler case. We restrict ourselves, accordingly, to the cases
where there is but one genetic spiral, and when _therefore_ _m_ and _n_
are prime to one another.

Our fundamental, or genetic, spiral, as we have seen, is that which
passes from _A_ (or _a_) to the leaf which is situated nearest to
the base-line _Aa_. The fundamental spiral will thus be right-handed
(_A_, _P_, etc.) if _P_, which is nearer to _A_ than to _a_, be this
leaf—left-handed if it be _p_. That is to say, we make it a convention
that we shall always, for our fundamental spiral, run {645} round the
system, from one leaf to the next, _by the shortest way_.

[Illustration: Fig. 326.]

Now it is obvious, from the symmetry of the figure (as further shewn
in Fig. 326), that, besides the spirals running along _AO_ and _aO_,
we have a series running _from the steps on_ _aO_ to the steps on
_AO_. In other words we can find a leaf (_S_) upon _AO_, which, like
the leaf _O_, is reached directly by a spiral series from _A_ and from
_a_, such that _aS_ includes _n_ steps, and _AS_ (being part of the
old spiral line _AO_) now includes _m_ − _n_ steps. And, since _m_ and
_n_ are prime to one another (for otherwise the system would have been
a composite or whorled one), it is evident that we can continue this
process of convergence until we come down to a 1, 1 arrangement, that
is to say to a leaf which is reached by a single step, in opposite
directions from _A_ and from _a_, which leaf is therefore the first
leaf, next to _A_, of the fundamental or generating spiral. {646}

If our original lines along _AO_ and _aO_ contain, for instance, 13 and
8 steps respectively (i.e. _m_ = 13, _n_ = 8), then our next series,
observable in the same cone, will be 8 and (13 − 8) or 5; the next 5
and (8 − 5) or 3; the next 3, 2; and the next 2, 1; leading to the
ultimate condition of 1, 1. These are the very series which we have
found to be common, or normal; and so far as our investigation has yet
gone, it has proved to us that, if one of these exists, it entails,
_ipso facto_, the presence of the rest.

In following down our series, according to the above construction, we
have seen that at every step we have changed direction, the longer
and the shorter sides of our triangle changing places every time. Let
us stop for a moment, when we come to the 1, 2 series, or _AT_, _aT_
of Fig. 326. It is obvious that there is nothing to prevent us making
a new 1, 3 series if we please, by continuing the generating spiral
through three leaves, and connecting the leaf so reached directly with
our initial one. But in the case represented in Fig. 326, it is obvious
that these two series (_A_, 1, 2, 3, etc., and _a_, 3, 6, etc.) will be
running in the same direction; i.e. they will both be right-handed, or
both left-handed spirals. The simple meaning of this is that the third
leaf of the generating spiral was distant from our initial leaf by
_more than the circumference_ of the cylindrical stem; in other words,
that there were more than two, but _less than three_ leaves in a single
turn of the fundamental spiral.

Less than two there can obviously never be. When there are exactly
two, we have the simplest of all possible arrangements, namely that
in which the leaves are placed alternately on opposite sides of the
stem. When there are more than two, but less than three, we have the
elementary condition for the production of the series which we have
been considering, namely 1, 2; 2, 3; 3, 5, etc. To put the latter
part of this argument in more precise language, let us say that: If,
in our descending series, we come to steps 1 and _t_, where _t_ is
determined by the condition that 1 and _t_ + 1 would give spirals both
right-handed, or both left-handed; it follows that there are less
than _t_ + 1 leaves in a single turn of the fundamental spiral. And,
determined in this manner, it is found in the great majority of cases,
in fir-cones and a host of other examples of phyllotaxis, that _t_
= 2. In other words, in the {647} great majority of cases, we have
what corresponds to an arrangement next in order of simplicity to the
simplest case of all: next, that is to say, to the arrangement which
consists of opposite and alternate leaves.

“These simple considerations,” as Tait says, “explain completely the
so-called mysterious appearance of terms of the recurring series
1, 2, 3, 5, 8, 13, etc.[581] The other natural series, usually but
misleadingly represented by convergents to an infinitely extended
continuous fraction, are easily explained, as above, by taking _t_ = 3,
4, 5, etc., etc.” Many examples of these latter series have been given
by Dickson[582] and other writers.

――――――――――

We have now learned, among other elementary facts, that wherever any
one system of helical spirals is present, certain others invariably and
of necessity accompany it, and are definitely related to it. In any
diagram, such as Fig. 326, in which we represent our leaf-arrangement
by means of uniform and regularly interspaced dots, we can draw one
series of spirals after another, and one as easily as another. But
in our fir-cone, for instance, one particular series, or rather two
conjugate series, are always conspicuous, while the others are sought
and found with comparative difficulty.

[Illustration: Fig. 327.]

The phenomenon is illustrated by Fig. 327, _a_–_d_. The ground-plan of
all these diagrams is identically the same. The generating spiral in
each case represents a divergence of 3/8, or 135° of azimuth; and the
points succeed one another at the same successional distances parallel
to the axis. The rectangular outlines, which correspond to the exposed
surface of the leaves or cone-scales, are of equal area, and of equal
number. Nevertheless the appearances presented by these diagrams
are very different; for in one the eye catches a 5/8 arrangement,
in another a 3/5; and so on, down to an arrangement of 1/1. The
mathematical side of this very curious phenomenon I have not attempted
to investigate. But it is quite obvious that, in a system within {648}
which various spirals are implicitly contained, the conspicuousness
of one set or another does not depend upon angular divergence. It
depends on the relative proportions in length and breadth of the leaves
themselves; or, more strictly speaking, on the ratio of the diagonals
of the rhomboidal figure by which each leaf-area is circumscribed.
When, as in the fir-cone, the scales by mutual compression conform to
these rhomboidal outlines, their inclined edges at once guide the eye
in the direction of some one particular spiral; and we shall not fail
to notice that in such cases the usual {649} result is to give us
arrangements corresponding to the middle diagrams in Fig. 327, which
are the configurations in which the quadrilateral outlines approach
most nearly to a rectangular form, and give us accordingly the least
possible ratio (under the given conditions) of sectional boundary-wall
to surface area.

The manner in which one system of spirals may be caused to slide, so to
speak, into another, has been ingeniously demonstrated by Schwendener
on a mechanical model, consisting essentially of a framework which can
be opened or closed to correspond with one after another of the above
series of diagrams[583].

The determination of the precise angle of divergence of two consecutive
leaves of the generating spiral does not enter into the above general
investigation (though Tait gives, in the same paper, a method by which
it may be easily determined); and the very fact that it does not
so enter shews it to be essentially unimportant. The determination
of so-called “orthostichies,” or precisely vertical successions of
leaves, is also unimportant. We have no means, other than observation,
of determining that one leaf is vertically above another, and spiral
series such as we have been dealing with will appear, whether such
orthostichies exist, whether they be near or remote, or whether the
angle of divergence be such that no precise vertical superposition ever
occurs. And lastly, the fact that the successional numbers, expressed
as fractions, 1/2, 2/3, 3/5, represent a convergent series, whose final
term is equal to 0·61803..., the _sectio aurea_ or “golden mean” of
unity, is seen to be a mathematical coincidence, devoid of biological
significance; it is but a particular case of Lagrange’s theorem that
the roots of every numerical equation of the second degree can be
expressed by a periodic continued fraction. The same number has a
multitude of curious arithmetical properties. It is the final term of
all similar series to that with which we have been dealing, such for
instance as 1/3, 3/4, 4/7, etc., or 1/4, 4/5, 5/9, etc. It is a number
beloved of the circle-squarer, and of all those who seek to find, and
then to penetrate, the secrets of the Great Pyramid. It is deep-set in
Pythagorean as well as in Euclidean geometry. It enters (as the chord
of an angle of 36°), {650} into the thrice-isosceles triangle of
which we have spoken on p. 511; it is a number which becomes (by the
addition of unity) its own reciprocal; its properties never end. To
Kepler (as Naber tells us) it was a symbol of Creation, or Generation.
Its recent application to biology and art-criticism by Sir Theodore
Cook and others is not new. Naber’s book, already quoted, is full of
it. Zeising, in 1854, found in it the key to all morphology, and the
same writer, later on[584], declared it to dominate both architecture
and music. But indeed, to use Sir Thomas Browne’s words (though it
was of another number that he spoke): “To enlarge this contemplation
into all the mysteries and secrets accommodable unto this number, were
inexcusable Pythagorisme.”

If this number has any serious claim at all to enter into the
biological question of phyllotaxis, this must depend on the fact,
first emphasized by Chauncey Wright[585], that, if the successive
leaves of the fundamental spiral be placed at the particular azimuth
which divides the circle in this “sectio aurea,” then no two leaves
will ever be superposed; and thus we are said to have “the most
thorough and rapid distribution of the leaves round the stem, each
new or higher leaf falling over the angular space between the two
older ones which are nearest in direction, so as to divide it in
the same ratio (_K_), in which the first two or any two successive
ones divide the circumference. Now 5/8 and all successive fractions
differ inappreciably from _K_.” To this view there are many simple
objections. In the first place, even 5/8, or ·625, is but a moderately
close approximation to the “golden mean”; in the second place the
arrangements by which a better approximation is got, such as 8/13,
13/21, and the very close approximations such as 34/55, 55/89, 89/144,
etc., are comparatively rare, while the much less close approximations
of 3/5 or 2/3, or even 1/2, are extremely common. Again, the general
type of argument such as that which asserts that the plant is “aiming
at” something which we may call an “ideal angle” is one that cannot
commend itself to a plain student of physical science: nor is the
hypothesis rendered more acceptably when Sir T. Cook qualifies it by
telling us that “all that a plant can do {651} is to vary, to make
blind shots at constructions, or to ‘mutate’ as it is now termed;
and the most suitable of these constructions will in the long run be
isolated by the action of Natural Selection.” Finally, and this is the
most concrete objection of all, the supposed isolation of the leaves,
or their most complete “distribution to the action of the surrounding
atmosphere” is manifestly very little affected by any conditions which
are confined to the angle of azimuth. If we could imagine a case in
which all the leaves of the stem, or all the scales of a fir-cone, were
crushed down to one and the same level, into a simple ring or whorl of
leaves, then indeed they would have their most equable distribution
under the condition of the “ideal angle,” that is to say of the “golden
mean.” But if it be (so to speak) Nature’s object to set them further
apart than they actually are, to give them freer exposure to the air
than they actually have, then it is surely manifest that the simple way
to do so is to elongate the axis, and to set the leaves further apart,
lengthways on the stem. This has at once a far more potent effect than
any nice manipulation of the “angle of divergence.” For it is obvious
that in _F_(ϕ ⋅ sin θ) we have a greater range of variation by altering
θ than by altering ϕ. We come then, without more ado, to the conclusion
that the “Fibonacci series,” and its supposed usefulness, and the
hypothesis of its introduction into plant-structure through natural
selection, are all matters which deserve no place in the plain study of
botanical phenomena. As Sachs shrewdly recognised years ago, all such
speculations as these hark back to a school of mystical idealism.

{652}




CHAPTER XV

ON THE SHAPES OF EGGS, AND OF CERTAIN OTHER HOLLOW STRUCTURES


The eggs of birds and all other hard-shelled eggs, such as those of
the tortoise and the crocodile, are simple solids of revolution; but
they differ greatly in form, according to the configuration of the
plane curve by the revolution of which the egg is, in a mathematical
sense, generated. Some few eggs, such as those of the owl, the penguin,
or the tortoise, are spherical or very nearly so; a few more, such
as the grebe’s, the cormorant’s or the pelican’s, are approximately
ellipsoidal, with symmetrical or nearly symmetrical ends, and somewhat
similar are the so-called “cylindrical” eggs of the megapodes and the
sand-grouse; the great majority, like the hen’s egg, are ovoid, a
little blunter at one end than the other; and some, by an exaggeration
of this lack of antero-posterior symmetry, are blunt at one end but
characteristically pointed at the other, as is the case with the eggs
of the guillemot and puffin, the sandpiper, plover and curlew. It is
an obvious but by no means negligible fact that the egg, while often
pointed, is never flattened or discoidal; it is a prolate, but never an
oblate, spheroid.

The careful study and collection of birds’ eggs would seem to have
begun with the Count de Marsigli[586], the same celebrated naturalist
who first studied the “flowers” of the coral, and who wrote the
_Histoire physique de la mer_; and the specific form, as well as the
colour and other attributes of the egg have been again and again
discussed, and not least by the many dilettanti naturalists of the
eighteenth century who soon followed in Marsigli’s footsteps[587]. {653}

We need do no more than mention Aristotle’s belief, doubtless old
in his time, that the more pointed egg produces the male chicken,
and the blunter egg the hen; though this theory survived into modern
times[588] and perhaps still lingers on. Several naturalists, such as
Günther (1772) and Bühle (1818), have taken the trouble to disprove it
by experiment. A more modern and more generally accepted explanation
has been that the form of the egg is in direct relation to that of the
bird which has to be hatched within—a view that would seem to have
been first set forth by Naumann and Bühle, in their great treatise
on eggs[589], and adopted by Des Murs[590] and many other well-known
writers.

In a treatise by de Lafresnaye[591], an elaborate comparison is made
between the skeleton and the egg of the various birds, to shew, for
instance, how those birds with a deep-keeled sternum laid rounded
eggs, which alone could accommodate the form of the young. According
to this view, that “Nature had foreseen[592]” the form adapted to and
necessary for the growing embryo, it was easy to correlate the owl
with its spherical egg, the diver with its elliptical one, and in like
manner the round egg of the tortoise and the elongated one of the
crocodile with the shape of the creatures which had afterwards to be
hatched therein. A few writers, such as Thienemann[593], looked at the
same facts the other way, and asserted that the form of the egg was
determined by that of the bird by which it was laid, and in whose body
it had been conformed.

In more recent times, other theories, based upon the principles of
Natural Selection, have been current and very generally accepted, to
account for these diversities of form. The pointed, conical egg of the
guillemot is generally supposed to be an adaptation, {654} advantageous
to the species in the circumstances under which the egg is laid; the
pointed egg is less apt than a spherical one to roll off the narrow
ledge of rock on which this bird is said to lay its solitary egg,
and the more pointed the egg, so much the fitter and likelier is it
to survive. The fact that the plover or the sandpiper, breeding in
very different situations, lay eggs that are also conical, elicits
another explanation, to the effect that here the conical form permits
the many large eggs to be packed closely under the mother bird[594].
Whatever truth there be in these apparent adaptations to existing
circumstances, it is only by a very hasty logic that we can accept
them as a _vera causa_, or adequate explanation of the facts; and it
is obvious that, in the bird’s egg, we have an admirable case for the
direct investigation of the mechanical or physical significance of its
form[595].

Of all the many naturalists of the eighteenth and nineteenth centuries
who wrote on the subject of eggs, one alone (so far as I am aware)
ascribed the form of the egg to direct mechanical causes. Günther[596],
in 1772, declared that the more or less rounded or pointed form of
the egg is a mechanical consequence of the pressure of the oviduct
at a time when the shell is yet unformed or unsolidified; and that
accordingly, to explain the round egg of the owl or the kingfisher, we
have only to admit that the oviduct of these birds is somewhat larger
than that of most others, or less subject to violent contractions.
This statement contains, in essence, the whole story of the mechanical
conformation of the egg.

Let us consider, very briefly, the conditions to which the egg is
subject in its passage down the oviduct[597].

(1) The “egg,” as it enters the oviduct, consists of the yolk only,
enclosed in its vitelline membrane. As it passes down the first portion
of the oviduct, the white is gradually superadded, {655} and becomes in
turn surrounded by the “shell-membrane.” About this latter the shell is
secreted, rapidly and at a late period; the egg having meanwhile passed
on into a wider portion of the oviducal tube, called (by loose analogy,
as Owen says) the “uterus.” Here the egg assumes its permanent form,
here it becomes rigid, and it is to this portion of the “oviduct” that
our argument principally refers.

(2) Both the yolk and the entire egg tend to fill completely their
respective membranes, and, whether this be due to growth or imbibition
on the part of the contents or to contraction on the part of the
surrounding membranes, the resulting tendency is for both yolk and egg
to be, in the first instance, spherical, unless otherwise distorted by
external pressure.

(3) The egg is subject to pressure within the oviduct, which is an
elastic, muscular tube, along the walls of which pass peristaltic waves
of contraction. These muscular contractions may be described as the
contraction of successive annuli of muscle, giving annular (or radial)
pressure to successive portions of the egg; they drive the egg forward
against the frictional resistance of the tube, while tending at the
same time to distort its form. While nothing is known, so far as I am
aware, of the muscular physiology of the oviduct, it is well known in
the case of the intestine that the presence of an obstruction leads to
the development of violent contractions in its rear, which waves of
contraction die away, and are scarcely if at all propagated in advance
of the obstruction.

(4) It is known by observation that a hen’s egg is always laid blunt
end foremost.

(5) It can be shown, at least as a very common rule, that those eggs
which are most unsymmetrical, or most tapered off posteriorly, are
also eggs of a large size relatively to the parent bird. The guillemot
is a notable case in point, and so also are the curlews, sandpipers,
phaleropes and terns. We may accordingly presume that the more pointed
eggs are those that are large relatively to the tube or oviduct through
which they have to pass, or, in other words, are those which are
subject to the greatest pressure while being forced along. So general
is this relation that we may go still further, and presume with great
plausibility {656} in the few exceptional cases (of which the apteryx
is the most conspicuous) where the egg is relatively large though not
markedly unsymmetrical, that in these cases the oviduct itself is in
all probability large (as Günther had suggested) in proportion to the
size of the bird. In the case of the common fowl we can trace a direct
relation between the size and shape of the egg, for the first eggs laid
by a young pullet are usually smaller, and at the same time are much
more nearly spherical than the later ones; and, moreover, some breeds
of fowls lay proportionately smaller eggs than others, and on the whole
the former eggs tend to be rounder than the latter[598].

――――――――――

We may now proceed to inquire more particularly how the form of the egg
is controlled by the pressures to which it is subjected.

The egg, just prior to the formation of the shell, is, as we have seen,
a fluid body, tending to a spherical shape and _enclosed within a
membrane_.

Our problem, then, is: Given a practically incompressible fluid,
contained in a deformable capsule, which is either (_a_) entirely
inextensible, or (_b_) slightly extensible, and which is placed in
a long elastic tube the walls of which are radially contractile, to
determine the shape under pressure.

If the capsule be spherical, inextensible, and completely filled with
the fluid, absolutely no deformation can take place. The few eggs that
are actually or approximately spherical, such as those of the tortoise
or the owl, may thus be alternatively explained as cases where little
or no deforming pressure has been applied prior to the solidification
of the shell, or else as cases where the capsule was so little capable
of extension and so completely filled as to preclude the possibility of
deformation.

If the capsule be not spherical, but be inextensible, then deformation
can take place under the external radial compression, {657} only
provided that the pressure tends to make the shape more nearly
spherical, and then only on the further supposition that the capsule is
also not entirely filled as the deformation proceeds. In other words,
an incompressible fluid contained in an inextensible envelope cannot be
deformed without puckering of the envelope taking place.

Let us next assume, as the conditions by which this result may be
avoided, (_a_) that the envelope is to some extent extensible, or (_b_)
that the whole structure grows under relatively fixed conditions. The
two suppositions are practically identical with one another in effect.
It is obvious that, on the presumption that the envelope is only
moderately extensible, the whole structure can only be distorted to a
moderate degree away from the spherical or spheroidal form.

At all points the shape is determined by the law of the distribution of
_radial pressure within the given region of the tube_, surface friction
helping to maintain the egg in position. If the egg be under pressure
from the oviduct, but without any marked component either in a forward
or backward direction, the egg will be compressed in the middle, and
will tend more or less to the form of a cylinder with spherical ends.
The eggs of the grebe, cormorant, or crocodile may be supposed to
receive their shape in such circumstances.

When the egg is subject to the peristaltic contraction of the oviduct
during its formation, then from the nature and direction of motion of
the peristaltic wave the pressure will be greatest somewhere behind the
middle of the egg; in other words, the tube is converted for the time
being into a more conical form, and the simple result follows that the
anterior end of the egg becomes the broader and the posterior end the
narrower.

With a given shape and size of body, equilibrium in the tube may be
maintained under greater radial pressure towards one end than towards
the other. For example, a cylinder having conical ends, of semi-angles
θ and θ′ respectively, remains in equilibrium, apart from friction, if
_p_ cos^2 θ = _p′_ cos^2 θ′, so that at the more tapered end where θ
is small _p_ is small. Therefore the whole structure might assume such
a configuration, or grow under such conditions, finally becoming rigid
by solidification of the envelope. {658} According to the preceding
paragraph, we must assume some initial distribution of pressure, some
squeeze applied to the posterior part of the egg, in order to give it
its tapering form. But, that form once acquired, the egg may remain
in equilibrium both as regards form and position within the tube,
even after that excess of pressure on the posterior part is relieved.
Moreover, the above equation shews that a normal pressure no greater
and (within certain limits) actually less acting upon the posterior
part than on the anterior part of the egg after the shell is formed
will be sufficient to communicate to it a forward motion. This is an
important consideration, for it shews that the ordinary form of an egg,
and even the conical form of an extreme case such as the guillemot’s,
is directly favourable to the movement of the egg within the oviduct,
blunt end foremost.

The mathematical statement of the whole case is as follows: In our egg,
consisting of an extensible membrane filled with an incompressible
fluid and under external pressure, the equation of the envelope is
_p_{n}_ + _T_(1/_r_ + 1/_r′_) = _P_, where _p_{n}_ is the normal
component of external pressure at a point where _r_ and _r′_ are the
radii of curvature, _T_ is the tension of the envelope, and _P_ the
internal fluid pressure. This is simply the equation of an elastic
surface where _T_ represents the coefficient of elasticity; in other
words, a flexible elastic shell has the same mathematical properties
as our fluid, membrane-covered egg. And this is the identical equation
which we have already had so frequent occasion to employ in our
discussion of the forms of cells; save only that in these latter we
had chiefly to study the tension _T_ (i.e. the surface-tension of the
semi-fluid cell) and had little or nothing to do with the factor of
external pressure (_p_{n}_), which in the case of the egg becomes of
chief importance.

The above equation is the _equation of equilibrium_, so that it must be
assumed either that the whole body is at rest or that its motion while
under pressure is not such as to affect the result. Tangential forces,
which have been neglected, could modify the form by alteration of _T_.
In our case we must, and may very reasonably, assume that any movement
of the egg down the oviduct during the period when its form is being
impressed upon it is very slow, being possibly balanced by the advance
of the {659} peristaltic wave which causes the movement, as well as by
friction.

The quantity _T_ is the tension of the enclosing capsule—the
surrounding membrane. If _T_ be constant or symmetrical about the axis
of the body, the body is symmetrical. But the abnormal eggs that a
hen sometimes lays, cylindrical, annulated, or quite irregular, are
due to local weakening of the membrane, in other words, to asymmetry
of _T_. Not only asymmetry of _T_, but also asymmetry of _p_{n}_,
will render the body subject to deformation, and this factor, the
unknown but regularly varying, largely radial, pressure applied by
successive annuli of the oviduct, is the essential cause of the
form, and variations of form, of the egg. In fact, in so far as the
postulates correspond near enough to actualities, the above equation
is the equation of _all eggs_ in the universe. At least this is so if
we generalise it in the form _p_{n}_ + _T_/_r_ + _T′_/_r′_ = _P_ in
recognition of a possible difference between the principal tensions.

In the case of the spherical egg it is obvious that _p_{n}_ is
everywhere equal. The simplest case is where _p_{n}_ = 0, in other
words, where the egg is so small as practically to escape deforming
pressure from the tube. But we may also conceive the tube to be so
thin-walled and extensible as to press with practically equal force
upon all parts of the contained sphere. If while our egg be in process
of conformation the envelope be free at any part from external pressure
(that is to say, if _p_{n}_ = 0), then it is obvious that that part
(if of circular section) will be a portion of a sphere. This is not
unlikely to be the case actually or approximately at one or both poles
of the egg, and is evidently the case over a considerable portion of
the anterior end of the plover’s egg.

In the case of the conical egg with spherical ends, as is more or
less the case in the plover’s and the guillemot’s, then at either end
of the egg _r_ and _r′_ are identical, and they are greater at the
blunt anterior end than at the other. If we may assume that _p_{n}_
vanishes at the poles of the egg, then it is plain that _T_ varies in
the neighbourhood of these poles, and, further, that the tension _T_ is
greatest at and near the small end of the egg. It is here, in short,
that the egg is most likely to be irregularly distorted or {660} even
to burst, and it is here that we most commonly find irregularities of
shape in abnormal eggs.

If one portion of the envelope were to become practically stiff before
_p_ ceases to vary, that would be tantamount to a sudden variation of
_T_, and would introduce asymmetry by the imposition of a boundary
condition in addition to the above equation.

Within the egg lies the yolk, and the yolk is invariably spherical or
very nearly so, whatever be the form of the entire egg. The reason
is simple, and lies in the fact that the yolk is itself enclosed in
another membrane, between which and the outer membrane lies a fluid
the presence of which makes _p_{n}_ for the inner membrane practically
constant. The smallness of friction is indicated by the well-known fact
that the “germinal spot” on the surface of the yolk is always found
uppermost, however we may place and wherever we may open the egg; that
is to say, the yolk easily rotates within the egg, bringing its lighter
pole uppermost. So, owing to this lack of friction in the outer fluid,
or white, whatever shear is produced within the egg will not be easily
transmitted to the yolk, and, moreover, owing to the same fluidity, the
yolk will easily recover its normal sphericity after the egg-shell is
formed and the unequal pressure relieved.

These, then, are the general principles involved in, and illustrated
by, the configuration of an egg; and they take us as far as we
can safely go without actual quantitative determinations, in each
particular case, of the forces concerned.

――――――――――

In certain cases among the invertebrates, we again find instances of
hard-shelled eggs which have obviously been moulded by the oviduct,
or so-called “ootype,” in which they have lain: and not merely in
such a way as to shew the effects of peristaltic pressure upon a
uniform elastic envelope, but so as to impress upon the egg the more
or less irregular form of the cavity, within which it had been for a
time contained and compressed. After this fashion Dr Looss[599] of
Cairo has {661} explained the curious form of the egg in _Bilharzia_
(_Schistosoma_) _haematobium_, a formidable parasitic worm to which
is due a disease wide-spread in Africa and Arabia, and an especial
scourge of the Mecca pilgrims. The egg in this worm is provided at one
end with a little spine, which now and then is found to be placed not
terminally but laterally or ventrally, and which when so placed has
been looked upon as the mark of a supposed new species, _S. Mansoni_.
As Looss has now shewn, the little spine must be explained as having
been moulded within a little funnel-shaped expansion of the uterus,
just where it communicates with the common duct leading from the ovary
and yolk-gland; by the accumulation of eggs in the ootype, the one
last formed is crowded into a sideways position, and then, where the
side-wall of the egg bulges in the funnel-shaped orifice of the duct, a
little lateral “spine” is formed. In another species, _S. japonicum_,
the egg is described as bulging into a so-called “calotte,” or
bubble-like convexity at the end opposite to the spine. This, I think,
may, with very little doubt, be ascribed to hardening of the egg-shell
having taken place just at the period when partial relief from pressure
was being experienced by the egg in the neighbourhood of the dilated
orifice of the oviduct.

This case of Bilharzia is not, from our present point of view, a very
important one, but nevertheless it is interesting. It ascribes to a
mechanical cause a curious peculiarity of form; it shews, by reference
to this mechanical principle, that two conditions which were very
different to the systematic naturalist’s eye, were really only two
simple mechanical modifications of the same thing; and it destroys the
chief evidence for the existence of a supposed new species of worm,
a continued belief in which, among worms of such great pathogenic
importance, might lead to gravely erroneous pathological deductions.


_On the Form of Sea-urchins_

As a corollary to the problem of the bird’s egg, we may consider for a
moment the forms assumed by the shells of the sea-urchins. These latter
are commonly divided into two classes, the Regular and the Irregular
Echinids. The regular sea-urchins, save in {662} slight details
which do not affect our problem, have a complete radial symmetry.
The axis of the animal’s body is vertical, with mouth below and the
intestinal outlet above; and around this axis the shell is built as a
symmetrical system. It follows that in horizontal section the shell is
everywhere circular, and we shall have only to consider its form as
seen in vertical section or projection. The irregular urchins (very
inaccurately so-called) have the anal extremity of the body removed
from its central, dorsal situation; and it follows that they have now
a single plane of symmetry, about which the organism, shell and all,
is bilaterally symmetrical. We need not concern ourselves in detail
with the shapes of their shells, which may be very simply interpreted,
by the help of radial co-ordinates, as deformations of the circular or
“regular” type.

The sea-urchin shell consists of a membrane, stiffened into rigidity
by calcareous deposits, which constitute a beautiful skeleton of
separate, neatly fitting “ossicles.” The rigidity of the shell is more
apparent than real, for the entire structure is, in a sluggish way,
plastic; inasmuch as each little ossicle is capable of growth, and the
entire shell grows by increments to each and all of these multitudinous
elements, whose individual growth involves a certain amount of freedom
to move relatively to one another; in a few cases the ossicles are
so little developed that the whole shell appears soft and flexible.
The viscera of the animal occupy but a small part of the space within
the shell, the cavity being mainly filled by a large quantity of
watery fluid, whose density must be very near to that of the external
sea-water.

Apart from the fact that the sea-urchin continues to grow, it is plain
that we have here the same general conditions as in the egg-shell, and
that the form of the sea-urchin is subject to a similar equilibrium
of forces. But there is this important difference, that an external
muscular pressure (such as the oviduct administers during the
consolidation of egg-shell), is now lacking. In its place we have the
steady continuous influence of gravity, and there is yet another force
which in all probability we require to take into consideration.

While the sea-urchin is alive, an immense number of delicate
“tube-feet,” with suckers at their tips, pass through minute pores
{663} in the shell, and, like so many long cables, moor the animal to
the ground. They constitute a symmetrical system of forces, with one
resultant downwards, in the direction of gravity, and another outwards
in a radial direction; and if we look upon the shell as originally
spherical, both will tend to depress the sphere into a flattened cake.
We need not consider the radial component, but may treat the case as
that of a spherical shell symmetrically depressed under the influence
of gravity. This is precisely the condition which we have to deal with
in a drop of liquid lying on a plate; the form of which is determined
by its own uniform surface-tension, plus gravity, acting against the
uniform internal hydrostatic pressure. Simple as this system is, the
full mathematical investigation of the form of a drop is not easy, and
we can scarcely hope that the systematic study of the Echinodermata
will ever be conducted by methods based on Laplace’s differential
equation[600]; but we have no difficulty in seeing that the various
forms represented in a series of sea-urchin shells are no other than
those which we may easily and perfectly imitate in drops.

In the case of the drop of water (or of any other particular liquid)
the specific surface-tension is always constant, and the pressure
varies inversely as the radius of curvature; therefore the smaller the
drop the more nearly is it able to conserve the spherical form, and the
larger the drop the more does it become flattened under gravity. We can
represent the phenomenon by using india-rubber balls filled with water,
of different sizes; the little ones will remain very nearly spherical,
but the larger will fall down “of their own weight,” into the form of
more and more flattened cakes; and we see the same thing when we let
drops of heavy oil (such as the orthotoluidene spoken of on p. 219),
fall through a tall column of water, the little ones remaining round,
and the big ones getting more and more flattened as they sink. In the
case of the sea-urchin, the same series of forms may be assumed to
occur, irrespective of size, through variations in _T_, the specific
tension, or “strength,” of the enveloping shell. Accordingly we may
study, entirely from this point of view, such a series as the following
(Fig. 328). In a very few cases, such as the fossil Palaeechinus,
we have an approximately spherical {664} shell, that is to say a
shell so strong that the influence of gravity becomes negligible as
a cause of deformation. The ordinary species of Echinus begin to
display a pronounced depression, and this reaches its maximum in such
soft-shelled flexible forms as Phormosoma. On the general question
I took the opportunity of consulting Mr C. R. Darling, who is an
acknowledged expert in drops, and he at once agreed with me that such
forms as are represented in Fig. 328 are no other than diagrammatic
illustrations

[Illustration: Fig. 328. Diagrammatic vertical outlines of various
Sea-urchins: A, Palaeechinus; B, _Echinus acutus_; C, Cidaris; D, D′
Coelopleurus; E, E′ Genicopatagus; F, _Phormosoma luculenter_; G, P.
_tenuis_; H, Asthenosoma; I, Urechinus.]

of various kinds of drops, “most of which can easily be reproduced
in outline by the aid of liquids of approximately equal density to
water, although some of them are fugitive.” He found a difficulty in
the case of the outline which represents Asthenosoma, but the reason
for the anomaly is obvious; the flexible shell has flattened down
until it has come in contact with the hard skeleton of the jaws, or
“Aristotle’s lantern,” within, and the curvature of the outline is
accordingly disturbed. The elevated, conical shells such as those of
Urechinus and Coelopleurus evidently call for some further explanation;
for there is here some cause at work {665} to elevate, rather than to
depress the shell. Mr Darling tells me that these forms “are nearly
identical in shape with globules I have frequently obtained, in which,
on standing, bubbles of gas rose to the summit and pressed the skin
upwards, without being able to escape.” The same condition may be at
work in the sea-urchin; but a similar tendency would also be manifested
by the presence in the upper part of the shell of any accumulation of
substance lighter than water, such as is actually present in the masses
of fatty, oily eggs.


_On the Form and Branching of Blood-vessels_

Passing to what may seem a very different subject, we may investigate
a number of interesting points in connection with the form and
structure of the blood-vessels, on the same principle and by help of
the same equations as those we have used, for instance, in studying the
egg-shell.

We know that the fluid pressure (_P_) within the vessel is balanced by
(1) the tension (_T_) of the wall, divided by the radius of curvature,
and (2) the external pressure (_p_{n}_), normal to the wall: according
to our formula

 _P_ = _p_{n}_ + _T_(1/_r_ + 1/_r′_).

If we neglect the external pressure, that is to say any support which
may be given to the vessel by the surrounding tissues, and if we deal
only with a cylindrical vein or artery, this formula becomes simplified
to the form _P_ = _T_/_R_. That is to say, under constant pressure, the
tension varies as the radius. But the tension, per unit area of the
vessel, depends upon the thickness of the wall, that is to say on the
amount of membranous and especially of muscular tissue of which it is
composed.

Therefore, so long as the pressure is constant, the thickness of
the wall should vary as the radius, or as the diameter, of the
blood-vessel. But it is not the case that the pressure is constant,
for it gradually falls off, by loss through friction, as we pass from
the large arteries to the small; and accordingly we find that while,
for a time, the cross-sections of the larger and smaller vessels are
symmetrical figures, with the wall-thickness proportional to the
size of the tube, this proportion is gradually lost, and the walls
{666} of the small arteries, and still more of the capillaries,
become exceedingly thin, and more so than in strict proportion to the
narrowing of the tube.

――――――――――

In the case of the heart we have, within each of its cavities, a
pressure which, at any given moment, is constant over the whole
wall-area, but the thickness of the wall varies very considerably.
For instance, in the left ventricle, the apex is by much the thinnest
portion, as it is also that with the greatest curvature. We may assume,
therefore (or at least suspect), that the formula, _t_(1/_r_ + 1/_r′_)
= _C_, holds good; that is to say, that the thickness (_t_) of the
wall varies inversely as the mean curvature. This may be tested
experimentally, by dilating a heart with alcohol under a known
pressure, and then measuring the thickness of the walls in various
parts after the whole organ has become hardened. By this means it is
found that, for each of the cavities, the law holds good with great
accuracy[601]. Moreover, if we begin by dilating the right ventricle
and then dilate the left in like manner, until the whole heart is
equally and symmetrically dilated, we find (1) that we have had to use
a pressure in the left ventricle from six to seven times as great as in
the right ventricle, and (2) that the thickness of the walls is just in
the same proportion[602].

――――――――――

A great many other problems of a mechanical or hydrodynamical kind
arise in connection with the blood-vessels[603], and while these are
chiefly interesting to the physiologist they have also their interest
for the morphologist in so far as they bear upon structure and form.
As an example of such mechanical problems {667} we may take the
conditions which determine or help to determine the manner of branching
of an artery, or the angle at which its branches are given off; for,
as John Hunter said[604], “To keep up a circulation sufficient for
the part, and no more, Nature has varied the angle of the origin of
the arteries accordingly.” The general principle is that the form and
arrangement of the blood-vessels is such that the circulation proceeds
with a minimum of effort, and with a minimum of wall-surface, the
latter condition leading to a minimum of friction and being therefore
included in the first. What, then, should be the angle of branching,
such that there shall be the least possible loss of energy in the
course of the circulation? In order to solve this problem in any
particular case we should obviously require to know (1) how the loss
of energy depends upon the distance travelled, and (2) how the loss of
energy varies with the diameter of the vessel. The loss of energy is
evidently greater in a narrow tube than in a wide one, and greater,
obviously, in a long journey than a short. If the

[Illustration: Fig. 329.]

large artery, _AB_, give off a comparatively narrow branch leading
to _P_ (such as _CP_, or _DP_), the route _ACP_ is evidently shorter
than _ADP_, but on the other hand, by the latter path, the blood has
tarried longer in the wide vessel _AB_, and has had a shorter course
in the narrow branch. The relative advantage of the two paths will
depend on the loss of energy in the portion _CD_, as compared with that
in the alternative portion _CD′_, the latter being short and narrow,
the former long and wide. If we ask, then, which factor is the more
important, length or width, we may safely take it that the question is
one of degree: and that the factor of width will become much the more
important wherever the artery and its branch are markedly unequal in
size. In other words, it would seem that for small branches a large
angle of bifurcation, and for large branches a small one, is always the
better. Roux has laid down certain rules in regard to the branching of
arteries, which correspond with the general {668} conclusions which
we have just arrived at. The most important of these are as follows:
(1) If an artery bifurcate into two equal branches, these branches come
off at equal angles to the main stem. (2) If one of the two branches be
smaller than the other, then the main branch, or continuation of the
original artery, makes with the latter a smaller angle than does the
smaller or “lateral” branch. And (3) all branches which are so small
that they scarcely seem to weaken or diminish the main stem come off
from it at a large angle, from about 70° to 90°.

[Illustration: Fig. 330.]

We may follow Hess in a further investigation of this phenomenon. Let
_AB_ be an artery, from which a branch has to be given off so as to
reach _P_, and let _ACP_, _ADP_, etc., be alternative courses which
the branch may follow: _CD_, _DE_, etc., in the diagram, being equal
distances (= _l_) along _AB_. Let us call the angles _PCD_, _PCE_,
_x__{1}, _x__{2}, etc.: and the distances _CD′_, _DE′_, by which each
branch exceeds the next in length, we shall call _l__{1}, _l__{2}, etc.
Now it is evident that, of the courses shewn, _ACP_ is the shortest
which the blood can take, but it is also that by which its transit
through the narrow branch is the longest. We may reduce its transit
through the narrow branch more and more, till we come to _CGP_, or
rather to a point where the branch comes off at right angles to the
main stem; but in so doing we very considerably increase the whole
distance travelled. We may take it that there will be some intermediate
point which will strike the balance of advantage.

Now it is easy to shew that if, in Fig. 330, the route _ADP_ and _AEP_
(two contiguous routes) be equally favourable, then any other route on
either side of these, such as _ACP_ or _AFP_, must be less favourable
than either. Let _ADP_ and _AEP_, then, be equally favourable; that is
to say, let the loss of energy which the blood suffers in its passage
along these two routes be equal. {669} Then, if we make the distance
_DE_ very small, the angles _x__{2} and _x__{3} are nearly equal,
and may be so treated. And again, if _DE_ be very small, then _DE′E_
becomes a right angle, and _l__{2} (or _DE′_) = _l_ cos _x__{2}.

But if _L_ be the loss of energy per unit distance in the wide tube
_AB_, and _L′_ be the corresponding loss of energy in the narrow tube
_DP_, etc., then _lL_ = _l__{2} _L′_, because, as we have assumed, the
loss of energy on the route _DP_ is equal to that on the whole route
_DEP_. Therefore _lL_ = _lL′_ cos _x__{2}, and cos _x__{2} = _L_/_L′_.
That is to say, the most favourable angle of branching will be such
that the cosine of the angle is equal to the ratio of the loss of
energy which the blood undergoes, per unit of length, in the main
vessel, as compared with that which it undergoes in the branch.

While these statements are so far true, and while they undoubtedly
cover a great number of observed facts, yet it is plain that, as in
all such cases, we must regard them not as a complete explanation,
but as _factors_ in a complicated phenomenon: not forgetting that (as
the most learned of all students of the heart and arteries, Dr Thomas
Young, said in his Croonian lecture[605]) all such questions as these,
and all matters connected with the muscular and elastic powers of the
blood-vessels, “belong to the most refined departments of hydraulics.”
Some other explanation must be sought in order to account for a
phenomenon which particularly impressed John Hunter’s mind, namely the
gradually altering angle at which the successive intercostal arteries
are given off from the thoracic aorta: the special interest of this
case arising from the regularity and symmetry of the series, for “there
is not another set of arteries in the body whose origins are so much
the same, whose offices are so much the same, whose distances from
their origin to the place of use, and whose uses [? sizes][606] are so
much the same.”

{670}




CHAPTER XVI

ON FORM AND MECHANICAL EFFICIENCY


There is a certain large class of morphological problems of which we
have not yet spoken, and of which we shall be able to say but little.
Nevertheless they are so important, so full of deep theoretical
significance, and are so bound up with the general question of form and
of its determination as a result of growth, that an essay on growth
and form is bound to take account of them, however imperfectly and
briefly. The phenomena which I have in mind are just those many cases
where _adaptation_, in the strictest sense, is obviously present, in
the clearly demonstrable form of mechanical fitness for the exercise of
some particular function or action which has become inseparable from
the life and well-being of the organism.

When we discuss certain so-called “adaptations” to outward
circumstance, in the way of form, colour and so forth, we are often
apt to use illustrations convincing enough to certain minds but
unsatisfying to others—in other words, incapable of demonstration.
With regard to colouration, for instance, it is by colours “cryptic,”
“warning,” “signalling,” “mimetic,” and so on[607], that we prosaically
expound, and slavishly profess to justify, the vast Aristotelian
synthesis that Nature makes all things with a purpose and “does nothing
in vain.” Only for a moment let us glance at some few instances by
which the modern teleologist accounts for this or that manifestation
of colour, and is led on and on to beliefs and doctrines to which it
becomes more and more difficult to subscribe. {671}

Some dangerous and malignant animals are said (in sober earnest) to
wear a perpetual war-paint, in order to “remind their enemies that
they had better leave them alone[608].” The wasp and the hornet, in
gallant black and gold, are terrible as an army with banners; and the
Gila Monster (the poison-lizard of the Arizona desert) is splashed
with scarlet—its dread and black complexion stained with heraldry more
dismal. But the wasp-like livery of the noisy, idle hover-flies and
drone-flies is but stage armour, and in their tinsel suits the little
counterfeit cowardly knaves mimic the fighting crew.

The jewelled splendour of the peacock and the humming-bird, and the
less effulgent glory of the lyre-bird and the Argus pheasant, are
ascribed to the unquestioned prevalence of vanity in the one sex and
wantonness in the other[609].

The zebra is striped that it may graze unnoticed on the plain,
the tiger that it may lurk undiscovered in the jungle; the banded
Chaetodont and Pomacentrid fishes are further bedizened to the hues of
the coral-reefs in which they dwell[610]. The tawny lion is yellow as
the desert sand; but the leopard wears its dappled hide to blend, as it
crouches on the branch, with the sun-flecks peeping through the leaves.

The ptarmigan and the snowy owl, the arctic fox and the polar bear, are
white among the snows; but go he north or go he south, the raven (like
the jackdaw) is boldly and impudently black.

The rabbit has his white scut, and sundry antelopes their piebald
flanks, that one timorous fugitive may hie after another, spying the
warning signal. The primeval terrier or collie-dog {672} had brown
spots over his eyes that he might seem awake when he was sleeping[611]:
so that an enemy might let the sleeping dog lie, for the singular
reason that he imagined him to be awake. And a flock of flamingos,
wearing on rosy breast and crimson wings a garment of invisibility,
fades away into the sky at dawn or sunset like a cloud incarnadine[612].

To buttress the theory of natural selection the same instances of
“adaptation” (and many more) are used, which in an earlier but not
distant age testified to the wisdom of the Creator and revealed to
simple piety the high purpose of God. In the words of a certain learned
theologian[613], “The free use of final causes to explain what seems
obscure was temptingly easy .... Hence the finalist was often the man
who made a liberal use of the _ignava ratio_, or lazy argument: when
you failed to explain a thing by the ordinary process of causality,
you could “explain” it by reference to some purpose of nature or of
its Creator. This method lent itself with dangerous facility to the
well-meant endeavours of the older theologians to expound and emphasise
the beneficence of the divine purpose.” _Mutatis mutandis_, the passage
carries its plain message to the naturalist.

The fate of such arguments or illustrations is always the same. They
attract and captivate for awhile; they go to the building of a creed,
which contemporary orthodoxy defends under its severest penalties: but
the time comes when they lose their fascination, they somehow cease
to satisfy and to convince, their foundations are discovered to be
insecure, and in the end no man troubles to controvert them.

But of a very different order from all such “adaptations” as these,
are those very perfect adaptations of form which, for instance, fit a
fish for swimming or a bird for flight. Here we are {673} far above
the region of mere hypothesis, for we have to deal with questions of
mechanical efficiency where statical and dynamical considerations can
be applied and established in detail. The naval architect learns a
great part of his lesson from the investigation of the stream-lines
of a fish; and the mathematical study of the stream-lines of a bird,
and of the principles underlying the areas and curvatures of its wings
and tail, has helped to lay the very foundations of the modern science
of aeronautics. When, after attempting to comprehend the exquisite
adaptation of the swallow or the albatross to the navigation of the
air, we try to pass beyond the empirical study and contemplation of
such perfection of mechanical fitness, and to ask how such fitness came
to be, then indeed we may be excused if we stand wrapt in wonderment,
and if our minds be occupied and even satisfied with the conception of
a final cause. And yet all the while, with no loss of wonderment nor
lack of reverence, do we find ourselves constrained to believe that
somehow or other, in dynamical principles and natural law, there lie
hidden the steps and stages of physical causation by which the material
structure was so shapen to its ends[614].

But the problems associated with these phenomena are difficult at
every stage, even long before we approach to the unsolved secrets of
causation; and for my part I readily confess that I lack the requisite
knowledge for even an elementary discussion of the form of a fish or
of a bird. But in the form of a bone we have a problem of the same
kind and order, so far simplified and particularised that we may to
some extent deal with it, and may possibly even find, in our partial
comprehension of it, a partial clue to the principles of causation
underlying this whole class of problems.

――――――――――

Before we speak of the form of a bone, let us say a word about, the
mechanical properties of the material of which it is built[615], in
{674} relation to the strength it has to manifest or the forces it has
to resist: understanding always that we mean thereby the properties
of fresh or living bone, with all its organic as well as inorganic
constituents, for dead, dry bone is a very different thing. In all the
structures raised by the engineer, in beams, pillars and girders of
every kind, provision has to be made, somehow or other, for strength of
two kinds, strength to resist compression or crushing, and strength to
resist tension or pulling asunder. The evenly loaded column is designed
with a view to supporting a downward pressure, the wire-rope, like the
tendon of a muscle, is adapted only to resist a tensile stress; but in
many or most cases the two functions are very closely inter-related
and combined. The case of a loaded beam is a familiar one; though, by
the way, we are now told that it is by no means so simple as it looks,
and indeed that “the stresses and strains in this log of timber are
so complex that the problem has not yet been solved in a manner that
reasonably accords with the known strength of the beam as found by
actual experiment[616].” However, be that as it may, we know,

[Illustration: Fig. 331.]

roughly, that when the beam is loaded in the middle and supported
at both ends, it tends to be bent into an arc, in which condition its
lower fibres are being stretched, or are undergoing a tensile stress,
while its upper fibres are undergoing compression. It follows that in
some intermediate layer there is a “neutral zone,” where the fibres
of the wood are subject to no stress of either kind. In like manner,
a vertical pillar if unevenly loaded (as, for instance, the shaft
of our thigh-bone normally is) will tend to bend, and so to endure
compression on its concave, and tensile stress upon its convex side.
In many cases it is the business of the engineer to separate out, as
far as possible, the pressure-lines from the tension-lines, in order
to use separate modes of construction, or even different materials for
each. In a {675} suspension-bridge, for instance, a great part of the
fabric is subject to tensile strain only, and is built throughout of
ropes or wires; but the massive piers at either end of the bridge carry
the weight of the whole structure and of its load, and endure all the
“compression-strains” which are inherent in the system. Very much the
same is the case in that wonderful arrangement of struts and ties which
constitute, or complete, the skeleton of an animal. The “skeleton,”
as we see it in a Museum, is a poor and even a misleading picture of
mechanical efficiency[617]. From the engineer’s point of view, it is
a diagram showing all the compression-lines, but by no means all of
the tension-lines of the construction; it shews all the struts, but
few of the ties, and perhaps we might even say _none_ of the principal
ones; it falls all to pieces unless we clamp it together, as best we
can, in a more or less clumsy and immobilised way. But in life, that
fabric of struts is surrounded and interwoven with a complicated system
of ties: ligament and membrane, muscle and tendon, run between bone
and bone; and the beauty and strength of the mechanical construction
lie not in one part or in another, but in the complete fabric which
all the parts, soft and hard, rigid and flexible, tension-bearing and
pressure-bearing, make up together[618].

However much we may find a tendency, whether in nature or art, to
separate these two constituent factors of tension and compression,
we cannot do so completely; and accordingly the engineer seeks for a
material which shall, as nearly as possible, offer equal resistance
to both kinds of strain. In the following table—I borrow it from Sir
Donald MacAlister—we see approximately the relative breaking (or
tearing) limit and crushing limit in a few substances. {676}

 _Average Strength of Materials (in kg. per sq. mm.)._

                      Tensile      Crushing
                      strength     strength
 Steel                   100          145
 Wrought Iron             40           20
 Cast Iron                12           72
 Wood                      4            2
 Bone                      9–12        13–16

At first sight, bone seems weak indeed; but it has the great and
unusual advantage that it is very nearly as good for a tie as for a
strut, nearly as strong to withstand rupture, or tearing asunder, as
to resist crushing. We see that wrought-iron is only half as strong to
withstand the former as the latter; while in cast-iron there is a still
greater discrepancy the other way, for it makes a good strut but a very
bad tie indeed. Cast-steel is not only actually stronger than any of
these, but it also possesses, like bone, the two kinds of strength in
no very great relative disproportion.

When the engineer constructs an iron or steel girder, to take the place
of the primitive wooden beam, we know that he takes advantage of the
elementary principle we have spoken of, and saves weight and economises
material by leaving out as far as possible all the middle portion,
all the parts in the neighbourhood of the “neutral zone”; and in so
doing he reduces his girder to an upper and lower “flange,” connected
together by a “web,” the whole resembling, in cross-section, an I or an
⌶.

But it is obvious that, if the strains in the two flanges are to
be equal as well as opposite, and if the material be such as cast-iron
or wrought-iron, one or other flange must be made much thicker than
the other in order that it may be equally strong; and if at times the
two flanges have, as it were, to change places, or play each other’s
parts, then there must be introduced a margin of safety by making both
flanges thick enough to meet that kind of stress in regard to which the
material happens to be weakest. There is great economy, then, in any
material which is, as nearly as possible, equally strong in both ways;
and so we see that, from the engineer’s or contractor’s point of view,
bone is a very good and suitable material for purposes of construction.
{677}

The I or the H-girder or rail is designed to resist bending in one
particular direction, but if, as in a tall pillar, it be necessary to
resist bending in all directions alike, it is obvious that the tubular
or cylindrical construction best meets the case; for it is plain that
this hollow tubular pillar is but the I-girder turned round every
way, in a “solid of revolution,” so that on any two opposite sides
compression and tension are equally met and resisted, and there is now
no need for any substance at all in the way of web or “filling” within
the hollow core of the tube. And it is not only in the supporting
pillar that such a construction is useful; it is appropriate in every
case where _stiffness_ is required, where bending has to be resisted.
The long bone of a bird’s wing has little or no weight to carry, but
it has to withstand powerful bending moments; and in the arm-bone of a
long-winged bird, such as an albatross, we see the tubular construction
manifested in its perfection, the bony substance being reduced to a
thin, perfectly cylindrical, and almost empty shell. The quill of
the bird’s feather, the hollow shaft of a reed, the thin tube of the
wheat-straw bearing its heavy burden in the ear, are all illustrations
which Galileo used in his account of this mechanical principle[619].

Two points, both of considerable importance, present themselves here,
and we may deal with them before we go further. In the first place,
it is not difficult to see that, in our bending beam, the strain is
greatest at its middle; if we press our walking-stick hard against
the ground, it will tend to snap midway. Hence, if our cylindrical
column be exposed to strong bending stresses, it will be prudent and
economical to make its walls thickest in the middle and thinning off
gradually towards the ends; and if we look at a longitudinal section of
a thigh-bone, we shall see that this is just what nature has done. The
thickness of the walls is nothing less than a diagram, or “graph,” of
the “bending-moments” from one point to another along the length of the
bone.

[Illustration: Fig. 332.]

The second point requires a little more explanation. If we {678}
imagine our loaded beam to be supported at one end only (for instance,
by being built into a wall), so as to form what is called a “bracket”
or “cantilever,” then we can see, without much difficulty, that the
lines of stress in the beam run somewhat as in the accompanying
diagram. Immediately under the load, the “compression-lines” tend to
run vertically downward; but where the bracket is fastened to the
wall, there is pressure directed horizontally against the wall in the
lower part of the surface of attachment; and the vertical beginning
and the horizontal end of these pressure-lines must be continued into
one another in the form of some even mathematical curve—which, as
it happens, is part of a parabola. The tension-lines are identical
in form with the compression-lines, of which they constitute the
“mirror-image”; and where the two systems intercross, they do so
at right angles, or “orthogonally” to one another. Such systems of
stress-lines as these we shall deal with again; but let us take note
here of the important, though well-nigh obvious fact, that while in the
beam they both unite to carry the load, yet it is always possible to
weaken one set of lines at the expense of the other, and in some cases
to do altogether away with one set or the other. For example, when
we replace our end-supported beam by a curved bracket, bent upwards
or downwards as the case may be, we have evidently cut away in the
one case the greater part of the tension-lines, and in the other the
greater part of the compression-lines. And if instead of bridging a
stream with our beam of wood we bridge it with a rope, it is evident
that this new construction contains all the tension-lines, but none of
the compression-lines of the old. The biological interest connected
with this principle lies chiefly in the mechanical construction of
the rush or the straw, or any other typically cylindrical stem. The
material of which the stalk is constructed is very weak to withstand
compression, but parts of it have a very great tensile strength.
Schwendener, who was both botanist and engineer, has elaborately
investigated the factor of strength in the cylindrical stem, which
Galileo was the first to call attention to. {679} Schwendener[620]
shewed that the strength was concentrated in the little bundles of
“bast-tissue” but that these bast-fibres had a tensile strength per
square mm. of section, up to the limit of elasticity, not less than
that of steel-wire of such quality as was in use in his day.

For instance, we see in the following table the load which various
fibres, and various wires, were found capable of sustaining, not up
to the breaking-point, but up to the “elastic limit,” or point beyond
which complete recovery to the original length took place no longer
after release of the load.

                            Stress, or load in gms.   Strain, or amount
                                per sq. mm., at         of stretching,
                             Limit of Elasticity         per mille
 _Secale cereale_                   15–20                   4·4
 _Lilium auratum_                   19                      7·6
 _Phormium tenax_                   20                     13·0
 _Papyrus antiquorum_               20                     15·2
 _Molinia coerulea_                 22                     11·0
 _Pincenectia recurvata_            25                     14·5
 Copper wire                        12·1                    1·0
 Brass wire                         13·3                    1·35
 Iron wire                          21·9                    1·0
 Steel wire                         24·6                    1·2

In other respects, it is true, the plant-fibres were inferior to
the wires; for the former broke asunder very soon after the limit
of elasticity was passed, while the iron-wire could stand, before
snapping, three times the load which was measured by its limit of
elasticity: in the language of a modern engineer, the bast-fibres had
a low “yield-point,” little above the elastic limit. But nevertheless,
within certain limits, plant-fibre and wire were just as good and
strong one as the other. And then Schwendener proceeds to shew, in
many beautiful diagrams, the various ways in which these strands of
strong tensile tissue are arranged in various cases: sometimes, in the
simpler cases, forming numerous small bundles arranged in a peripheral
ring, not quite at the periphery, for a certain amount of space has
to be left for living and active tissue; sometimes in a sparser ring
of larger and {680} stronger bundles; sometimes with these bundles
further strengthened by radial balks or ridges; sometimes with all the
fibres set

[Illustration: Fig. 333.]

close together in a continuous hollow cylinder. In the case figured
in Fig. 333 Schwendener calculated that the resistance to bending
was at least twenty-five times as great as it would have been had
the six main bundles been brought close together in a solid core. In
many cases the centre of the stem is altogether empty; in all other
cases it is filled with soft tissue, suitable for the ascent of sap or
other functions, but never such as to confer mechanical rigidity. In
a tall conical stem, such as that of a palm-tree, we can see not only
these principles in the construction of the cylindrical trunk, but we
can observe, towards the apex, the bundles of fibre curving over and
intercrossing orthogonally with one another, exactly after the fashion
of our stress-lines in Fig. 332; but of course, in this case, we are
still dealing with tensile members, the opposite bundles taking on in
turn, as the tree sways, the alternate function of resisting tensile
strain[621].

――――――――――

Let us now come, at last, to the mechanical structure of bone, of which
we find a well-known and classical illustration in the various bones of
the human leg. In the case of the tibia, the bone is somewhat widened
out above, and its hollow shaft is capped by an almost flattened roof,
on which the weight of the body directly rest. It is obvious that,
under these circumstances, the engineer would find it necessary to
devise means for supporting this flat roof, and for distributing the
vertical pressures which impinge upon it to the cylindrical walls of
the shaft. {681}

[Illustration: Fig. 334. Head of the human femur in section. (After
Schäfer, from a photo by Prof. A. Robinson.)]

In the case of the bird’s wing-bone, the hollow of the bone is
practically empty, as we have already said, being filled only with
air save for a thin layer of living tissue immediately within the
cylinder of bone; but in our own bones, and all weight-carrying bones
in general, the hollow space is filled with marrow, blood-vessels and
other tissues; and among these living tissues lies a fine lattice-work
of little interlaced “trabeculae” of bone, forming the so-called
“cancellous tissue.” The older anatomists were content to describe
this cancellous tissue as a sort of “spongy network,” or irregular
honeycomb, until, some fifty years ago, a remarkable discovery was
made regarding it. It was found by Hermann Meyer (and afterwards shewn
in greater detail by Julius Wolff and others) that the trabeculae,
as seen in a longitudinal section of a long bone, were arranged in a
very definite and orderly way; in the femur, they spread in beautiful
curving {682} lines from the head to the tubular shaft of the bone,
and these bundles of lines were crossed by others, with so nice a
regularity of arrangement that each intercrossing was as nearly as
possible an orthogonal one: that is to say, the one set of fibres
crossed the other everywhere at right angles. A great engineer,
Professor Culmann of Zürich (to whom, by the way, we owe the whole
modern method of “graphic statics”), happened to see some of Meyer’s
drawings and preparations, and he recognised in a moment that in the
arrangement of the trabeculae we had

[Illustration: Fig. 335. Crane-head and femur. (After Culmann and H.
Meyer.)]

nothing more nor less than a diagram of the lines of stress, or
directions of compression and tension, in the loaded structure: in
short, that nature was strengthening the bone in precisely the manner
and direction in which strength was needed. In the accompanying diagram
of a crane-head, by Culmann, we recognise a slight modification (caused
entirely by the curved shape of the structure) of the still simpler
lines of tension and compression which we have already seen in our
end-supported beam as represented in Fig. 332. In the shaft of the
crane, the concave {683} or inner side, overhung by the loaded head,
is the “compression-member”; the outer side is the “tension-member”;
and the pressure-lines, starting from the loaded surface, gather
themselves together, always in the direction of the resultant pressure,
till they form a close bundle running down the compressed side of the
shaft: while the tension-lines, running upwards along the opposite side
of the shaft, spread out through the head, orthogonally to, and linking
together, the system of compression-lines. The head of the femur (Fig.
335) is a little more complicated in form and a little less symmetrical
than Culmann’s diagrammatic crane, from which it chiefly differs in
the fact that the load is divided into two parts, that namely which is
borne by the head of the bone, and that smaller portion which rests
upon the great trochanter; but this merely amounts to saying that a
_notch_ has been cut out of the curved upper surface of the structure,
and we have no difficulty in seeing that the anatomical arrangement
of the trabeculae follows precisely the mechanical distribution of
compressive and tensile stress or, in other words, accords perfectly
with the theoretical stress-diagram of the crane. The lines of stress
are bundled close together along the sides of the shaft, and lost or
concealed there in the substance of the solid wall of bone; but in and
near the head of the bone, a peripheral shell of bone does not suffice
to contain them, and they spread out through the central mass in the
actual concrete form of bony trabeculae[622]. {684}

_Mutatis mutandis_, the same phenomenon may be traced in any other bone
which carries weight and is liable to flexure; and in the _os calcis_
and the tibia, and more or less in all the bones of the lower limb, the
arrangement is found to be very simple and clear.

[Illustration: Fig. 336. Diagram of stress-lines in the human foot.
(From Sir D. MacAlister, after H. Meyer.)]

Thus, in the _os calcis_, the weight resting on the head of the bone
has to be transmitted partly through the backward-projecting heel to
the ground, and partly forwards through its articulation with the
cuboid bone, to the arch of the foot. We thus have, very much as
in a triangular roof-tree, two compression-members, sloping apart
from one another; and these have to be bound together by a “tie” or
tension-member, corresponding to the third, horizontal member of the
truss.

――――――――――

[Illustration: Fig. 337. Trabecular structure of the os calcis. (From
MacAlister.)]

So far, dealing wholly with the stresses and strains due to tension
and compression, we have altogether omitted to speak of a third very
important factor in the engineer’s calculations, namely what is known
as “shearing stress.” A shearing force is one which produces “angular
distortion” in a figure, or (what comes to the same thing) which tends
to cause its particles to {685} slide over one another. A shearing
stress is a somewhat complicated thing, and we must try to illustrate
it (however imperfectly) in the simplest possible way. If we build up
a pillar, for instance, of a pile of flat horizontal slates, or of a
pack of cards, a vertical load placed upon it will produce compression,
but will have no tendency to cause one card to slide, or shear, upon
another; and in like manner, if we make up a cable of parallel wires
and, letting it hang vertically, load it evenly with a weight, again
the tensile stress produced has no tendency to cause one wire to slip
or shear upon another. But the case would have been very different if
we had built up our pillar of cards or slates lying obliquely to the
lines of pressure, for then at once there would have been a tendency
for the elements of the pile to slip and slide asunder, and to produce
what the geologists call “a fault” in the structure.

 Somewhat more generally, if _AB_ be a bar, or pillar, of cross-section
 _a_ under a direct load _P_, giving a stress per unit area = _p_,
 then the whole pressure _P_ = _pa_. Let _CD_ be an oblique section,
 inclined at an angle θ to the cross-section; the pressure on _CD_
 will evidently be = _pa_ cos θ. But at any point _O_ in _CD_,
 the pressure _P_ may be resolved into the force _Q_ acting along
 _CD_, and _N_ perpendicular to it: where _N_ = _P_ cos θ, and _Q_
 = _P_ sin θ = _pa_ sin θ. The whole force _Q_ upon _CD_ = _q_ ⋅ area
 of _CD_, which is = _q_ ⋅ _a_/(cos θ). {686} Therefore _qa_/(cos θ)
 = _pa_ sin θ, therefore _q_ = _p_ sin θ cos θ, = ½_p_ sin 2θ.
 Therefore when sin 2θ = 1, that is, when θ = 45°, _q_ is a maximum,
 and = _p_/2; and when sin 2θ = 0, that is when θ = 0° or 90°, then _q_
 vanishes altogether.

[Illustration: Fig. 338.]

This is as much as to say, that a shearing stress vanishes altogether
along the lines of maximum compression or tension; it has a definite
value in all other positions, and a maximum value when it is inclined
at 45° to either, or half-way between the two. This may be further
illustrated in various simple ways. When we submit a cubical block of
iron to compression in the testing machine, it does not tend to give
way by crumbling all to pieces; but as a rule it disrupts by shearing,
and along some plane approximately at 45° to the axis of compression.
Again, in the beam which we have already considered under a bending
moment, we know that if we substitute for it a pack of cards, they
will be strongly sheared on one another; and the shearing stress is
greatest in the “neutral zone,” where neither tension nor compression
is manifested: that is to say in the line which cuts at equal angles of
45° the orthogonally intersecting lines of pressure and tension.

In short we see that, while shearing _stresses_ can by no means be got
rid of, the danger of rupture or breaking-down under shearing stress is
completely got rid of when we arrange the materials of our construction
wholly along the pressure-lines and tension-lines of the system; for
_along these lines_ there is no shear.

To apply these principles to the growth and development of our bone,
we have only to imagine a little trabecula (or group of trabeculae)
being secreted and laid down fortuitously in any direction within
the substance of the bone. If it lie in the direction of one of the
pressure-lines, for instance, it will be in a position of comparative
equilibrium, or minimal disturbance; but if it be inclined obliquely to
the pressure-lines, the shearing force will at once tend to act upon it
and move it away. This is neither more nor less than what happens when
we comb our {687} hair, or card a lock of wool: filaments lying in the
direction of the comb’s path remain where they were; but the others,
under the influence of an oblique component of pressure, are sheared
out of their places till they too come into coincidence with the lines
of force. So straws show how the wind blows—or rather how it has been
blowing. For every straw that lies askew to the wind’s path tends to be
sheared into it; but as soon as it has come to lie the way of the wind
it tends to be disturbed no more, save (of course) by a violence such
as to hurl it bodily away.

In the biological aspect of the case, we must always remember that our
bone is not only a living, but a highly plastic structure; the little
trabeculae are constantly being formed and deformed, demolished and
formed anew. Here, for once, it is safe to say that “heredity” need not
and cannot be invoked to account for the configuration and arrangement
of the trabeculae: for we can see them, at any time of life, in the
making, under the direct action and control of the forces to which
the system is exposed. If a bone be broken and so repaired that its
parts lie somewhat out of their former place, so that the pressure-and
tension-lines have now a new distribution, before many weeks are over
the trabecular system will be found to have been entirely remodelled,
so as to fall into line with the new system of forces. And as Wolff
pointed out, this process of reconstruction extends a long way off
from the seat of injury, and so cannot be looked upon as a mere
accident of the physiological process of healing and repair; for
instance, it may happen that, after a fracture of the _shaft_ of a
long bone, the trabecular meshwork is wholly altered and reconstructed
within the distant _extremities_ of the bone. Moreover, in cases of
transplantation of bone, for example when a diseased metacarpal is
repaired by means of a portion taken from the lower end of the ulna,
with astonishing quickness the plastic capabilities of the bony tissue
are so manifested that neither in outward form nor inward structure can
the old portion be distinguished from the new.

Herein then lies, so far as we can discern it, a great part at least of
the physical causation of what at first sight strikes us as a purely
functional adaptation: as a phenomenon, in other words, {688} whose
physical cause is as obscure as its final cause or end is, apparently,
manifest.

――――――――――

Partly associated with the same phenomenon, and partly to be looked
upon (meanwhile at least) as a fact apart, is the very important
physiological truth that a condition of _strain_, the result of a
_stress_, is a direct stimulus to growth itself. This indeed is no less
than one of the cardinal facts of theoretical biology. The soles of our
boots wear thin, but the soles of our feet grow thick, the more we walk
upon them: for it would seem that the living cells are “stimulated”
by pressure, or by what we call “exercise,” to increase and multiply.
The surgeon knows, when he bandages a broken limb, that his bandage is
doing something more than merely keeping the parts together: and that
the even, constant pressure which he skilfully applies is a direct
encouragement of growth and an active agent in the process of repair.
In the classical experiments of Sédillot[623], the greater part of the
shaft of the tibia was excised in some young puppies, leaving the whole
weight of the body to rest upon the fibula. The latter bone is normally
about one-fifth or sixth of the diameter of the tibia; but under the
new conditions, and under the “stimulus” of the increased load, it
grew till it was as thick or even thicker than the normal bulk of the
larger bone. Among plant tissues this phenomenon is very apparent,
and in a somewhat remarkable way; for a strain caused by a constant
or increasing weight (such as that in the stalk of a pear while the
pear is growing and ripening) produces a very marked increase of
_strength_ without any necessary increase of bulk, but rather by some
histological, or molecular, alteration of the tissues. Hegler, and also
Pfeffer, have investigated this subject, by loading the young shoot
of a plant nearly to its breaking point, and then redetermining the
breaking-strength after a few days. Some young shoots of the sunflower
were found to break with a strain of 160 gms.; but when loaded with 150
gms., and retested after two days, they were able to support 250 gms.;
and being again loaded with something short of this, by next day they
sustained 300 gms., and a few days later even 400 gms. {689}

Such experiments have been amply confirmed, but so far as I am aware,
we do not know much more about the matter: we do not know, for
instance, how far the change is accompanied by increase in number of
the bast-fibres, through transformation of other tissues; or how far
it is due to increase in size of these fibres; or whether it be not
simply due to strengthening of the original fibres by some molecular
change. But I should be much inclined to suspect that the latter had a
good deal to do with the phenomenon. We know nowadays that a railway
axle, or any other piece of steel, is weakened by a constant succession
of frequently interrupted strains; it is said to be “fatigued,” and
its strength is restored by a period of rest. The converse effect of
continued strain in a uniform direction may be illustrated by a homely
example. The confectioner takes a mass of boiled sugar or treacle (in a
particular molecular condition determined by the temperature to which
it has been exposed), and draws the soft sticky mass out into a rope;
and then, folding it up lengthways, he repeats the process again and
again. At first the rope is pulled out of the ductile mass without
difficulty; but as the work goes on it gets harder to do, until all the
man’s force is used to stretch the rope. Here we have the phenomenon
of increasing strength, following mechanically on a rearrangement of
molecules, as the original isotropic condition is transmuted more and
more into molecular asymmetry or anisotropy; and the rope apparently
“adapts itself” to the increased strain which it is called on to
bear, all after a fashion which at least suggests a parallel to the
increasing strength of the stretched and weighted fibre in the plant.
For increase of strength by rearrangement of the particles we have
already a rough illustration in our lock of wool or hank of tow. The
piece of tow will carry but little weight while its fibres are tangled
and awry: but as soon as we have carded it out, and brought all its
long fibres parallel and side by side, we may at once make of it a
strong and useful cord.

In some such ways as these, then, it would seem that we may
co-ordinate, or hope to co-ordinate, the phenomenon of growth with
certain of the beautiful structural phenomena which present themselves
to our eyes as “provisions,” or mechanical adaptations, for the display
of strength where strength is most required. {690} That is to say,
the origin, or causation, of the phenomenon would seem to lie, partly
in the tendency of growth to be accelerated under strain: and partly
in the automatic effect of shearing strain, by which it tends to
displace parts which grow obliquely to the direct lines of tension and
of pressure, while leaving those in place which happen to lie parallel
or perpendicular to those lines: an automatic effect which we can
probably trace as working on all scales of magnitude, and as accounting
therefore for the rearrangement of minute particles in the metal or the
fibre, as well as for the bringing into line of the fibres themselves
within the plant, or of the little trabeculae within the bone.

――――――――――

But we may now attempt to pass from the study of the individual bone
to the much wider and not less beautiful problems of mechanical
construction which are presented to us by the skeleton as a whole.
Certain problems of this class are by no means neglected by writers on
anatomy, and many have been handed down from Borelli, and even from
older writers. For instance, it is an old tradition of anatomical
teaching to point out in the human body examples of the three orders
of levers[624]; again, the principle that the limb-bones tend to be
shortened in order to support the weight of a very heavy animal is
well understood by comparative anatomists, in accordance with Euler’s
law, that the weight which a column liable to flexure is capable of
supporting varies inversely as the square of its length; and again, the
statical equilibrium of the body, in relation for instance to the erect
posture of man, has long been a favourite theme of the philosophical
anatomist. But the general method, based upon that of graphic statics,
to which we have been introduced in our study of a bone, has not, so
far as I know, been applied to the general fabric of the skeleton. Yet
it is plain that each bone plays {691} a part in relation to the whole
body, analogous to that which a little trabecula, or a little group
of trabeculae, plays within the bone itself: that is to say, in the
normal distribution of forces in the body, the bones tend to follow
the lines of stress, and especially the pressure-lines. To demonstrate
this in a comprehensive way would doubtless be difficult; for we should
be dealing with a framework of very great complexity, and should have
to take account of a great variety of conditions[625]. This framework
is complicated as we see it in the skeleton, where (as we have said)
it is only, or chiefly, the _struts_ of the whole fabric which are
represented; but to understand the mechanical structure in detail, we
should have to follow out the still more complex arrangement of the
_ties_, as represented by the muscles and ligaments, and we should also
require much detailed information as to the weights of the various
parts and as to the other forces concerned. Without these latter data
we can only treat the question in a preliminary and imperfect way. But,
to take once again a small and simplified part of a big problem, let us
think of a quadruped (for instance, a horse) in a standing posture, and
see whether the methods and terminology of the engineer may not help
us, as they did in regard to the minute structure of the single bone.

Standing four-square upon its forelegs and hindlegs, with the weight
of the body suspended between, the quadruped at once suggests to us
the analogy of a bridge, carried by its two piers. And if it occurs
to us, as naturalists, that we never look at a standing quadruped
without contemplating a bridge, so, conversely, a similar idea has
occurred to the engineer; for Professor Fidler, in this _Treatise on
Bridge-Construction_, deals with the chief descriptive part of his
subject under the heading of “The Comparative Anatomy of Bridges.” The
designation is most just, for in studying the various types of bridges
we are studying a series of well-planned _skeletons_[626]; and (at the
cost of a little pedantry) {692} we might go even further, and study
(after the fashion of the anatomist) the “osteology” and “desmology”
of the structure, that is to say the bones which are represented by
“struts,” and the ligaments, etc., which are represented by “ties.”
Furthermore after the methods of the comparative anatomist, we may
classify the families, genera and species of bridges according to their
distinctive mechanical features, which correspond to certain definite
conditions and functions.

In more ways than one, the quadrupedal bridge is a remarkable one;
and perhaps its most remarkable peculiarity is that it is a jointed
and flexible bridge, remaining in equilibrium under considerable
and sometimes great modifications of its curvature, such as we see,
for instance, when a cat humps or flattens her back. The fact that
_flexibility_ is an essential feature in the quadrupedal bridge, while
it is the last thing which an engineer desires and the first which
he seeks to provide against, will impose certain important limiting
conditions upon the design of the skeletal fabric; but to this matter
we shall afterwards return. Let us begin by considering the quadruped
at rest, when he stands upright and motionless upon his feet, and
when his legs exercise no function save only to carry the weight of
the whole body. So far as that function is concerned, we might now
perhaps compare the horse’s legs with the tall and slender piers of
some railway bridge; but it is obvious that these jointed legs are
ill-adapted to receive the _horizontal thrust_ of any _arch_ that may
be placed atop of them. Hence it follows that the curved backbone of
the horse, which appears to cross like an arch the span between his
shoulders and his flanks, cannot be regarded as an _arch_, in the
{693} engineer’s sense of the word. It resembles an arch in _form_,
but not in _function_, for it cannot act as an arch unless it be held
back at each end (as every arch is held back) by _abutments_ capable
of resisting the horizontal thrust; and these necessary abutments are
not present in the structure. But in various ways the engineer can
modify his superstructure so as to supply the place of these _external_
reactions, which in the simple arch are obviously indispensable.
Thus, for example, we may begin by inserting a straight steel tie,
_AB_ (Fig. 339), uniting the ends of the curved rib _AaB_; and this
tie will supply the place of the external reactions, converting the
structure into a “tied arch,” such as we may see in the roofs of many
railway-stations. Or we may go on to fill in the space between arch and
tie by a “web-system,” converting it into what the engineer describes
as a “parabolic bowstring girder” (Fig. 339_b_). In either case, the
structure becomes an

[Illustration: Fig. 339.]

independent “detached girder,” supported at each end but
not otherwise fixed, and consisting essentially of an upper
compression-member, _AaB_, and a lower tension-member, _AB_. But again,
in the skeleton of the quadruped, _the necessary tie_, _AB_, _is not
to be found_; and it follows that these comparatively simple types
of bridge do not correspond to, nor do they help us to understand,
the type of bridge which nature has designed in the skeleton of the
quadruped. Nevertheless if we try to look, as an engineer would look,
at the actual design of the animal skeleton and the actual distribution
of its load, we find that, the one is most admirably adapted to the
other, according to the strict principles of engineering construction.
The structure is not an arch, nor a tied arch, nor a bowstring girder:
but it is strictly and beautifully {694} comparable to the main girder
of a double-armed cantilever bridge.

Obviously, in our quadrupedal bridge, the superstructure does not
terminate (as it did in our former diagram) at the two points of
support, but it extends beyond them at each end, carrying the head at
one end and the tail at the other, upon a pair of projecting arms or
“cantilevers” (Fig. 346).

In a typical cantilever bridge, such as the Forth Bridge (Fig. 345), a
certain simplification is introduced. For each pier carries, in this
case, its own double-armed cantilever, linked by a short connecting
girder to the next, but so jointed to it that no weight is transmitted
from one cantilever to another. The bridge in short is _cut_ into
separate sections, practically independent of one another; at the
joints a certain amount of bending is not precluded, but shearing
strain is evaded; and each pier carries only its own load. By this
arrangement the engineer finds that design and construction are alike
simplified and facilitated. In the case of the horse, it is obvious
that the two piers of the bridge, that is to say the fore-legs and the
hind-legs, do not bear (as they do in the Forth Bridge) separate and
independent loads, but the whole system forms a continuous structure.
In this case, the calculation of the loads will be a little more
difficult and the corresponding design of the structure a little
more complicated. We shall accordingly simplify our problem very
considerably if, to begin with, we look upon the quadrupedal skeleton
as constituted of two separate systems, that is to say of two balanced
cantilevers, one supported on the fore-legs and the other on the hind;
and we may deal afterwards with the fact that these two cantilevers are
not independent, but are bound up in one common field of force and plan
of construction.

In the horse it is plain that the two cantilever systems into which we
may thus analyse the quadrupedal bridge are unequal in magnitude and
importance. The fore-part of the animal is much bulkier than its hind
quarters, and the fact that the fore-legs carry, as they so evidently
do, a greater weight than the hind-legs has long been known and is
easily proved; we have only to walk a horse onto a weigh-bridge, weigh
first his fore-legs and then his hind-legs, to discover that what we
may call his front half weighs {695} a good deal more than what is
carried on his hind feet, say about three-fifths of the whole weight of
the animal.

The great (or anterior) cantilever then, in the horse, is constituted
by the heavy head and still heavier neck on one side of the pier which
is represented by the fore-legs, and by the dorsal vertebrae carrying
a large part of the weight of the trunk upon the other side; and this
weight is so balanced over the fore-legs that the cantilever, while
“anchored” to the other parts of the structure, transmits but little of
its weight to the hind-legs, and the amount so transmitted will vary
with the position of the head and with the position of any artificial
load[627]. Under certain conditions, as when the head is thrust well
forward, it is evident that the hind-legs will be actually relieved of
a portion of the comparatively small load which is their normal share.

Our problem now is to discover, in a rough and approximate way, some
of the structural details which the balanced load upon the double
cantilever will impress upon the fabric.

――――――――――

Working by the methods of graphic statics, the engineer’s task is,
in theory, one of great simplicity. He begins by drawing in outline
the structure which he desires to erect; he calculates the stresses
and bending-moments necessitated by the dimensions and load on the
structure; he draws a new diagram representing these forces, and he
designs and builds his fabric on the lines of this statical diagram.
He does, in short, precisely what we have seen _nature_ doing in the
case of the bone. For if we had begun, as it were, by blocking out the
femur roughly, and considering its position and dimensions, its means
of support and the load which it has to bear, we could have proceeded
at once to draw the system of stress-lines which must occupy the field
of force: and to precisely these stress-lines has nature kept in the
building of the bone, down to the minute arrangement of its trabeculae.

The essential function of a bridge is to stretch across a certain span,
and carry a certain definite load; and this being so, the {696} chief
problem in the designing of a bridge is to provide due resistance to
the “bending-moments” which result from the load. These bending-moments
will vary from point to point along the girder, and taking the simplest
case of a uniform load supported at both ends, they will be represented
by points on a parabola. If the girder be of uniform depth, that is to
say if its two flanges,

[Illustration: Fig. 340. A, Span of proposed bridge. B, Stress diagram,
or diagram of bending-moments[628].]

respectively under tension and compression, be parallel to one
another, then the stress upon these flanges will vary as the
bending-moments, and will accordingly be very severe in the middle and
will dwindle towards the ends. But if we make the _depth_ of the girder
everywhere proportional to the bending-moments, that is

[Illustration: Fig. 341. The bridge constructed, as a parabolic girder.]

to say if we copy in the girder the outlines of the bending-moment
diagram, then our design will automatically meet the circumstances of
the case, for the horizontal stress in each flange will now be uniform
throughout the length of the girder. In short, in {697} Professor
Fidler’s words, “Every diagram of moments represents the outline of
a framed structure which will carry the given load with a uniform
horizontal stress in the principal members.”

[Illustration: Fig. 342.]

In the following diagrams (Fig. 342, _a_, _b_) (which are taken from
the original ones of Culmann), we see at once that the loaded beam or
bracket (_a_) has a “danger-point” close to its fixed base, that is to
say at the point remotest from its load. But in the parabolic bracket
(_b_) there is no danger-point at all, for the dimensions of the
structure are made to increase _pari passu_ with the bending-moments:
stress and resistance vary together. Again in Fig. 340, we have a
simple span (A), with its stress diagram (B); and in Fig. 341 we have
the corresponding parabolic girder, whose stresses are now uniform
throughout. In fact we see that, by a process of conversion, the stress
diagram in each case becomes the structural diagram in the other[629].
Now all this is but the modern rendering of one of Galileo’s most
famous propositions. In the Dialogue which we have already quoted
more than once[630], Sagredo says “It would be a fine thing if one
could discover the proper shape to give a solid in order to make it
equally resistant at every point, in which case a load placed at the
middle would not produce fracture more easily than if placed at any
other point.” And Galileo (in the person of Salviati) first puts the
problem into its more general form; and then shews us how, by giving
a parabolic outline to our beam, we have its simple and comprehensive
solution.

In the case of our cantilever bridge, we shew the primitive girder
{698} in Fig. 343, A, with its bending-moment diagram (B); and it
is evident that, if we turn this diagram upside down, it will still
be illustrative, just as before, of the bending-moments from point
to point: for as yet it is merely a diagram, or graph, of relative
magnitudes.

To either of these two stress diagrams, direct or inverted, we may fit
the design of the construction, as in Figs. 343, C and 344.

[Illustration: Fig. 343.]

[Illustration: Fig. 344.]

Now in different animals the amount and distribution of the load
differs so greatly that we can expect no single diagram, drawn from the
comparative anatomy of bridges, to apply equally well to all the cases
met with in the comparative anatomy of quadrupeds; but nevertheless
we have already gained an insight into the general principles of
“structural design” in the quadrupedal bridge.

In our last diagram the upper member of the cantilever is under {699}
tension; it is represented in the quadruped by the _ligamentum nuchae_
on the one side of the cantilever, and by the supraspinous ligaments of
the dorsal vertebrae on the other. The compression member is similarly
represented, on both sides of the cantilever, by the vertebral
column, or rather by the _bodies_ of the vertebrae; while the web,
or “filling,” of the girders, that is to say the upright or sloping
members which extend from one flange to the other, is represented on
the one hand by the spines of the vertebrae, and on the other hand, by
the oblique interspinous ligaments and muscles. The high spines over
the quadruped’s withers are no other than the high struts which rise
over the supporting piers in the parabolic girder, and correspond to
the position of the maximal bending-moments. The fact that these tall
vertebrae of the withers usually slope backwards, sometimes steeply,
in a quadruped, is easily and obviously explained[631]. For each
vertebra tends to act as a “hinged lever,” and its spine, acted on by
the tensions transmitted by the ligaments on either side, takes up its
position as the diagonal of the parallelogram of forces to which it is
exposed.

It happens that in these comparatively simple types of cantilever
bridge the whole of the parabolic curvature is transferred to one
or other of the principal members, either the tension-member or the
compression-member as the case may be. But it is of course equally
permissible to have both members curved, in opposite directions. This,
though not exactly the case in the Forth Bridge, is approximately
so; for here the main compression-member is curved or arched, and
the main tension-member slopes downwards on either side from its
maximal height above the piers. In short, the Forth Bridge is a nearer
approach than either of the other cantilever bridges which we have
{700} illustrated to the plan of the quadrupedal skeleton; for the main
compression-member almost exactly recalls the form of the vertebral
column, while the main tension-member, though not so closely similar to
the supraspinous and nuchal ligaments, corresponds to the plan of these
in a somewhat simplified form.

[Illustration: Fig. 345. A two-armed cantilever of the Forth Bridge.
Thick lines, compression-members (bones); thin lines, tension-members
(ligaments).]

We may now pass without difficulty from the two-armed cantilever
supported on a single pier, as it is in each separate section of the
Forth Bridge, or as we have imagined it to be in the forequarters of
a horse, to the condition which actually exists in that quadruped,
where a two-armed cantilever has its load distributed over two separate
piers. This is not precisely what an engineer calls a “continuous”
girder, for that term is applied to a girder which, as a continuous
structure, crosses two or more spans, while here there is only one. But
nevertheless, this girder

[Illustration: Fig. 346.]

is _effectively_ continuous from the head to the tip of the tail; and
at each point of support (_A_ and _B_) it is subjected to the negative
bending-moment due to the overhanging load on each of the projecting
cantilever arms _AH_ and _BT_. The diagram of bending-moments will
(according to the ordinary conventions) lie below {701} the base line
(because the moments are negative), and must take some such form as
that shown in the diagram: for the girder must suffer its greatest
bending stress not at the centre, but at the two points of support _A_
and _B_, where the moments are measured by the vertical ordinates. It
is plain that this figure only differs from a representation of _two_
independent two-armed cantilevers in the fact that there is no point
midway in the span where the bending-moment vanishes, but only a region
between the two piers in which its magnitude tends to diminish.

[Illustration: Fig. 347. Stress-diagram of horse’s backbone.]

The diagram effects a graphic summation of the positive and negative
moments, but its form may assume various modifications according to the
method of graphic summation which we may choose to adopt; and it is
obvious also that the form of the diagram may assume many modifications
of detail according to the actual distribution of the load. In all
cases the essential points to be observed are these: firstly that the
girder which is to resist the bending-moments induced by the load must
possess its two principal members—an upper tension-member or tie,
represented by ligament, and a lower compression-member represented by
bone: these members being united by a web represented by the vertebral
spines with their interspinous ligaments, and being placed one above
the other in the order named because the moments are negative;
secondly we observe that the depth of the web, or distance apart of
the principal members,—that is to say the height of the vertebral
spines,—must be proportional to the bending-moment at each point along
the length of the girder.

In the case of an animal carrying two-thirds of his weight upon his
fore-legs and only one-third upon his hind-legs, the bending-moment
diagram will be unsymmetrical, after the fashion of Fig. 347, the
vertical ordinate at _A_ being thrice the height of that at _B_. {702}

On the other hand the Dinosaur, with his light head and enormous tail
would give us a moment-diagram with the opposite kind of asymmetry,
the greatest bending stress being now found over the haunches, at _B_
(Fig. 348). A glance at the skeleton of _Diplodocus Carnegii_ will shew
us the high vertebral spines over the loins, in precise correspondence
with the requirements of this diagram: just as in the horse, under the
opposite conditions of load, the highest vertebral spines are those
of the withers, that is to say those of the posterior cervical and
anterior dorsal vertebrae.

[Illustration: Fig. 348. Stress-diagram of backbone of Dinosaur.]

We have now not only dealt with the general resemblance, both in
structure and in function, of the quadrupedal backbone with its
associated ligaments to a double-armed cantilever girder, but we have
begun to see how the characters of the vertebral system must differ
in different quadrupeds, according to the conditions imposed by the
varying distribution of the load: and in particular how the height of
the vertebral spines which constitute the web will be in a definite
relation, as regards magnitude and position, to the bending-moments
induced thereby. We should require much detailed information as to the
actual weights of the several parts of the body before we could follow
out quantitatively the mechanical efficiency of each type of skeleton;
but in an approximate way what we have already learnt will enable us to
trace many interesting correspondences between structure and function
in this particular part of comparative anatomy. We must, however, be
careful to note that the great cantilever system is not of necessity
constituted by the vertical column and its ligaments alone, but that
the pelvis, firmly united as it is to the sacral vertebrae, and
stretching backwards far beyond the acetabulum, becomes an intrinsic
part of the system; and helping (as it does) to carry the load of the
abdominal viscera, {703} constitutes a great portion of the posterior
cantilever arm, or even its chief portion in cases where the size and
weight of the tail are insignificant, as is the case in the majority of
terrestrial mammals.

We may also note here, that just as a bridge is often a “combined” or
composite structure, exhibiting a combination of principles in its
construction, so in the quadruped we have, as it were, another girder
supported by the same piers to carry the viscera; and consisting
of an inverted parabolic girder, whose compression-member is again
constituted by the backbone, its tension-member by the line of the
sternum and the abdominal muscles, while the ribs and intercostal
muscles play the part of the web or filling.

A very few instances must suffice to illustrate the chief variations in
the load, and therefore in the bending-moment diagram, and therefore
also in the plan of construction, of various quadrupeds. But let us
begin by setting forth, in a few cases, the actual weights which
are borne by the fore-limbs and the hind-limbs, in our quadrupedal
bridge[632].

                                         On     On
                                        Fore-  Hind-  % on   % on
                         Gross weight.  feet   feet   Fore-  Hind-
                          ton  cwts.    cwts.  cwts.  feet.  feet.
 Camel (Bactrian)          —   14·25     9·25   4·5    67·3   32·7
 Llama                     —    2·75     1·75    ·875  66·7   33·3
 Elephant (Indian)         1   15·75    20·5   14·75   58·2   41·8
 Horse                     —    8·25     4·75   3·5    57·6   42·4
 Horse (large Clydesdale)  —   15·5      8·5    7·0    54·8   45·2

It will be observed that in all these animals the load upon the
fore-feet preponderates considerably over that upon the hind, the
preponderance being rather greater in the elephant than in the horse,
and markedly greater in the camel and the llama than in the other two.
But while these weights are helpful and suggestive, it is obvious that
they do not go nearly far enough to give us a full insight into the
constructional diagram to which the animals are conformed. For such a
purpose we should {704} require to weigh the total load, not in two
portions, but in many; and we should also have to take close account
of the general form of the animal, of the relation between that form
and the distribution of the load, and of the actual directions of each
bone and ligament by which the forces of compression and tension were
transmitted. All this lies beyond us for the present; but nevertheless
we may consider, very briefly, the principal cases involved in our
enquiry, of which the above animals form only a partial and preliminary
illustration.

[Illustration: Fig. 349. Stress-diagram of Titanotherium.]

(1) Wherever we have a heavily loaded anterior cantilever arm, that is
to say whenever the head and neck represent a considerable fraction of
the whole weight of the body, we tend to have large bending-moments
over the fore-legs, and correspondingly high spines over the vertebrae
of the withers. This is the case in the great majority of four-footed,
terrestrial animals, the chief exceptions being found in animals
with comparatively small heads but large and heavy tails, such as
the anteaters or the Dinosaurian reptiles, and also (very naturally)
in animals such as the crocodile, where the “bridge” can scarcely be
said to be developed, for the long heavy body sags down to rest upon
the ground. The case is sufficiently exemplified by the horse, and
still more notably by the stag, the ox, or the pig. It is illustrated
in the accompanying diagram of the conditions in the great extinct
Titanotherium.

(2) In the elephant and the camel we have similar conditions, but
slightly modified. In both cases, and especially in the latter, the
weight on the fore-quarters is relatively large; and in both cases
the bending-moments are all the larger, by reason of the length and
forward extension of the camel’s neck, and the forward {705} position
of the heavy tusks of the elephant. In both cases the dorsal spines
are large, but they do not strike us as exceptionally so; but in both
cases, and especially in the elephant, they slope backwards in a marked
degree. Each spine, as already explained, must in all cases assume the
position of the diagonal in the parallelogram of forces defined by the
tensions acting on it at its extremity; for it constitutes a “hinged
lever,” by which the bending-moments on either side are automatically
balanced; and it is plain that the more the spine slopes backwards
the more it indicates a relatively large strain thrown upon the great
ligament of the neck, and a relief of strain upon the more directly
acting, but weaker, ligaments of the back and loins. In both cases,
the bending-moments would seem to be more evenly distributed over the
region of the back than, for instance, in the stag, with its light
hind-quarters and heavy load of antlers: and in both cases the high
“girder” is considerably prolonged, by an extension of the tall spines
backwards in the direction of the loins. When we come to such a case as
the mammoth, with its immensely heavy and immensely elongated tusks, we
perceive at once that the bending-moments over the fore-legs are now
very severe; and we see also that the dorsal spines in this region are
much more conspicuously elevated than in the ordinary elephant.

(3) In the case of the giraffe we have, without doubt, a very heavy
load upon the fore-legs, though no weighings are at hand to define the
ratio; but as far as possible this disproportionate load would seem to
be relieved, by help of a downward as well as backward thrust, through
the sloping back, to the unusually low hind-quarters. The dorsal spines
of the vertebrae are very high and strong, and the whole girder-system
very perfectly formed. The elevated, rather than protruding position of
the head lessens the anterior bending-moment as far as possible; but
it leads to a strong compressional stress transmitted almost directly
downwards through the neck: in correlation with which we observe that
the bodies of the cervical vertebrae are exceptionally large and strong
and steadily increase in size and strength from the head downwards.

(4) In the kangaroo, the fore-limbs are entirely relieved of their
load, and accordingly the tall spines over the withers, which {706}
were so conspicuous in all heavy-headed _quadrupeds_, have now
completely vanished. The creature has become bipedal, and body and tail
form the extremities of _a single_ balanced cantilever, whose maximal
bending-moments are marked by strong, high lumbar and sacral vertebrae,
and by iliac bones of peculiar form and exceptional strength.

Precisely the same condition is illustrated in the Iguanodon, and
better still by reason of the great bulk of the creature, and of the
heavy load which falls to be supported by the great cantilever and by
the hind-legs which form its piers. The long and heavy body and neck
require a balance-weight (as in the kangaroo) in the form of a long
heavy tail. And the double-armed cantilever, so constituted, shews a
beautiful parabolic curvature in the graded heights of the whole series
of vertebral spines, which rise to a maximum over the haunches and die
away slowly towards the neck and the tip of the tail.

(5) In the case of some of the great American fossil reptiles, such as
Diplodocus, it has always been a more or less disputed question whether
or not they assumed, like Iguanodon, an erect, bipedal attitude. In
all of these we see an elongated pelvis, and, in still more marked
degree, we see elevated spinous processes of the vertebrae over the
hind-limbs; in all of them we have a long heavy tail, and in most
of them we have a marked reduction in size and weight both of the
fore-limb and of the head itself. The great size of these animals is
not of itself a proof against the erect attitude; because it might well
have been accompanied by an aquatic or partially submerged habitat,
and the crushing stress of the creature’s huge bulk proportionately
relieved. But we must consider each such case in the whole light of
its own evidence; and it is easy to see that, just as the quadrupedal
mammal may carry the greater part but not all of its weight upon its
fore-limbs, so a heavy-tailed reptile may carry the greater part
upon its hind-limbs, without this process going so far as to relieve
its fore-limbs of all weight whatsoever. This would seem to be the
case in such a form as Diplodocus, and also in Stegosaurus, whose
restoration by Marsh is doubtless substantially correct[633]. The
fore-limbs, {707} though comparatively small, are obviously fashioned
for support, but the weight which they have to carry is far less than
that which the hind-limbs bear. The head is small and the neck short,
while on the other hand the hind-quarters and the tail are big and
massive. The backbone bends into a great, double-armed cantilever,
culminating over the pelvis and the hind-limbs, and here furnished with
its highest and strongest spines to separate the tension-member from
the compression-member of the girder. The fore-legs form a secondary
supporting pier to this great cantilever, the greater part of whose
weight is poised upon the hind-limbs alone.

[Illustration: Fig. 350. Diagram of Stegosaurus.]

(6) In a bird, such as an ostrich or a common fowl, the bipedal habit
necessitates the balancing of the load upon a single double-armed
cantilever-girder, just as in the Iguanodon and the kangaroo, but the
construction is effected in a somewhat different way. The great heavy
tail has entirely disappeared; but, though from the skeleton alone
it would seem that nearly all the bulk of the animal lay in front of
the hind-limbs, yet in the living bird we can easily perceive that
the great weight of the abdominal organs lies suspended _behind_ the
socket for the thigh-bone, and so hangs from the posterior lever-arm of
the cantilever, balancing the head and neck and thorax whose combined
weight hangs from {708} the anterior arm. The great cantilever
girder appears, accordingly, balanced over the hind-legs. It is now
constituted in part by the posterior dorsal or lumbar vertebrae, all
traces of special elevation having disappeared from the anterior
dorsals; but the greater part of the girder is made up of the great
iliac bones, placed side by side, and gripping firmly the sacral
vertebrae, often almost to the extinction of these latter. In the form
of these iliac bones, the arched curvature of their upper border, in
their elongation fore-and-aft to overhang both ways their supporting
pier, and in the coincidence of their greatest height with the median
line of support over the centre of gravity, we recognise all the
characteristic properties of the typical balanced cantilever[634].

(7) We find a highly important corollary in the case of aquatic
animals. For here the effect of gravity is neutralised; we have neither
piers nor cantilevers; and we find accordingly in all aquatic mammals
of whatsoever group—whales, seals or sea-cows—that the high arched
vertebral spines over the withers, or corresponding structures over the
hind-limbs, have both entirely disappeared.

Just as the cantilever girder tended to become obsolete in the aquatic
mammal so does it tend to weaken and disappear in the aquatic bird.
There is a very marked contrast between the high-arched strongly-built
pelvis in the ostrich or the hen, and the long, thin, comparatively
straight and weakly bone which represents it in a diver, a grebe or a
penguin.

But in the aquatic mammal, such as a whale or a dolphin (and not less
so in the aquatic bird), _stiffness_ must be ensured in order to enable
the muscles to act against the resistance of the water in the act of
swimming; and accordingly nature must provide against bending-moments
irrespective of gravity. In the dolphin, at any rate as regards its
tail end, the conditions will be not very different from those of a
column or beam with fixed ends, in which, under deflexion, there will
be two points of contrary flexure, as at _C_, _D_, in Fig. 351. {709}

[Illustration: Fig. 351.]

Here, between _C_ and _D_ we have a varying bending-moment, represented
by a continuous curve with its maximal elevation midway between the
points of inflexion. And correspondingly, in our dolphin, we have a
continuous series of high dorsal spines, rising to a maximum about
the middle of the animal’s body, and falling to nil at some distance
from the end of the tail. It is their business (as usual) to keep the
tension-member, represented by the strong supraspinous ligaments, wide
apart from the compression-member, which is as usual represented by
the backbone itself. But in our diagram we see that on the further
side of _C_ and _D_ we have a _negative_ curve of bending-moments,
or bending-moments in a contrary direction. Without inquiring how
these stresses are precisely met towards the dolphin’s head (where
the coalesced cervical vertebrae suggest themselves as a partial
explanation), we see at once that towards the tail they are met by the
strong series of chevron-bones, which in the caudal region, where tall
_dorsal_ spines are no longer needed, take their place _below_ the
vertebrae, in precise correspondence with the bending-moment diagram.
In many cases other than these aquatic ones, when we have to deal with
animals with long and heavy tails (like the Iguanodon and the kangaroo
of which we have already spoken), we are apt to meet with similar,
though usually shorter chevron-bones; and in all these cases we may
see without difficulty that a negative bending-moment is there to be
resisted.

In the dolphin we may find a good illustration of the fact that
not only is it necessary to provide for rigidity in the vertical
direction, but also in the horizontal, where a tendency to bending
must be resisted on either side. This function is effected in part
by the ribs with their associated muscles, but they extend but a
little way and their efficacy for this purpose can be but small. We
have, however, behind the region of the ribs and on either side of
the backbone a strong series of elongated and flattened transverse
processes, forming a web for the support of a tension-member in the
usual form of ligament, and so playing a part precisely analogous to
that performed by the dorsal spines in the same {710} animal. In an
ordinary fish, such as a cod or a haddock, we see precisely the same
thing: the backbone is stiffened by the indispensable help of its
_three series_ of ligament-connected processes, the dorsal and the
two transverse series. And here we see (as we see partly also among
the whales), that these three series of processes, or struts, tend
to be arranged well-nigh at equal angles, of 120°, with one another,
giving the greatest and most uniform strength of which such a system
is capable. On the other hand, in a flat fish, such as a plaice, where
from the natural mode of progression it is necessary that the backbone
should be flexible in one direction while stiffened in another, we find
the whole outline of the fish comparable to that of a double bowstring
girder, the compression-member being (as usual) the backbone, the
tension-member on either side being constituted by the interspinous
ligaments and muscles, while the web or filling is very beautifully
represented by the long and evenly graded spines, which spring
symmetrically from opposite sides of each individual vertebra.

――――――――――

The main result at which we have now arrived, in regard to the
construction of the vertebral column and its associated parts, is that
we may look upon it as a certain type of _girder_, whose depth, as
we cannot help seeing, is everywhere very nearly proportional to the
height of the corresponding ordinate in the diagram of moments: just
as it is in the girder of a cantilever bridge as designed by a modern
engineer. In short, after the nineteenth or twentieth century engineer
has done his best in framing the design of a big cantilever, he may
find that some of his best ideas bad, so to speak, been anticipated
ages ago in the fabric of the great saurians and the larger mammals.

But it is possible that the modern engineer might be disposed to
criticise the skeleton girder at two or three points; and in particular
he might think the girder, as we see it for instance in Diplodocus or
Stegosaurus, not deep enough for carrying the animal’s enormous weight
of some twenty tons. If we adopt a much greater depth (or ratio of
depth to length) as in the modern cantilever, we shall greatly increase
the _strength_ of the structure; but at the same time we should greatly
increase its _rigidity_, and {711} this is precisely what, in the
circumstances of the case, it would seem that nature is bound to avoid.
We need not suppose that the great saurian was by any means active and
limber; but a certain amount of activity and flexibility he was bound
to have, and in a thousand ways he would find the need of a backbone
that should be _flexible_ as well as _strong_. Now this opens up a new
aspect of the matter and is the beginning of a long, long story, for
in every direction this double requirement of strength and flexibility
imposes new conditions upon our design. To represent all the correlated
quantities we should have to construct not only a diagram of moments
but also a diagram of elastic deflexion and its so-called “curvature”;
and the engineer would want to know something more about the _material_
of the ligamentous tension-member—its modulus of elasticity in direct
tension, its elastic limit, and its safe working stress.

In various ways our structural problem is beset by “limiting
conditions.” Not only must rigidity be associated with flexibility, but
also stability must be ensured in various positions and attitudes; and
the primary function of support or weight-carrying must be combined
with the provision of _points d’appui_ for the muscles concerned in
locomotion. We cannot hope to arrive at a numerical or quantitative
solution of this complicate problem, but we have found it possible
to trace it out in part towards a qualitative solution. And speaking
broadly we may certainly say that in each case the problem has been
solved by nature herself, very much as she solves the difficult
problems of minimal areas in a system of soap-bubbles; so that each
animal is fitted with a backbone adapted to his own individual needs,
or (in other words) corresponding exactly to the mean resultant of the
stresses to which as a mechanical system it is exposed.

――――――――――

Throughout this short discussion of the principles of construction,
limited to one part of the skeleton, we see the same general principles
at work which we recognise in the plan and construction of an
individual bone. That is to say, we see a tendency for material to be
laid down just in the lines of _stress_, and so as to evade thereby
the distortions and disruptions due to _shear_. In these phenomena
there lies a definite law of growth, {712} whatever its ultimate
expression or explanation may come to be. Let us not press either
argument or hypothesis too far: but be content to see that skeletal
form, as brought about by growth, is to a very large extent determined
by mechanical considerations, and tends to manifest itself as a
diagram, or reflected image, of mechanical stress. If we fail, owing
to the immense complexity of the case, to unravel all the mathematical
principles involved in the construction of the skeleton, we yet gain
something, and not a little, by applying this method to the familiar
objects of our anatomical study: _obvia conspicimus, nubem pellente
mathesi_[635].

Before we leave this subject of mechanical adaptation, let us dwell
once more for a moment upon the considerations which arise from our
conception of a field of force, or field of stress, in which tension
and compression (for instance) are inevitably combined, and are met by
the materials naturally fitted to resist them. It has been remarked
over and over again how harmoniously the whole organism hangs together,
and how throughout its fabric one part is related and fitted to another
in strictly functional correlation. But this conception, though never
denied, is sometimes apt to be forgotten in the course of that process
of more and more minute analysis by which, for simplicity’s sake, we
seek to unravel the intricacies of a complex organism.

We tend, as we analyse a thing into its parts or into its properties,
to magnify these, to exaggerate their apparent independence, and to
hide from ourselves (at least for a time) the essential integrity and
individuality of the composite whole. We divide the body into its
organs, the skeleton into its bones, as in very much the same fashion
we make a subjective analysis of the mind, according to the teachings
of psychology, into component factors: but we know very well that
judgment and knowledge, courage or gentleness, love or fear, have no
separate existence, but are somehow mere manifestations, or imaginary
co-efficients, of a most complex integral. And likewise, as biologists,
we may go so far as to say that even the bones themselves are only in a
limited and even a deceptive sense, separate and individual things. The
skeleton begins as a _continuum_, and a _continuum_ it remains all life
long. The things that link bone with bone, {713} cartilage, ligaments,
membranes, are fashioned out of the same primordial tissue, and come
into being _pari passu_, with the bones themselves. The entire fabric
has its soft parts and its hard, its rigid and its flexible parts; but
until we disrupt and dismember its bony, gristly and fibrous parts, one
from another, it exists simply as a “skeleton,” as one integral and
individual whole.

A bridge was once upon a time a loose heap of pillars and rods and
rivets of steel. But the identity of these is lost, just as if they
were fused into a solid mass, when once the bridge is built; their
separate functions are only to be recognised and analysed in so far
as we can analyse the stresses, the tensions and the pressures, which
affect this part of the structure or that; and these forces are not
themselves separate entities, but are the resultants of an analysis of
the whole field of force. Moreover when the bridge is broken it is no
longer a bridge, and all its strength is gone. So is it precisely with
the skeleton. In it is reflected a field of force: and keeping pace, as
it were, in action and interaction with this field of force, the whole
skeleton and every part thereof, down to the minute intrinsic structure
of the bones themselves, is related in form and in position to the
lines of force, to the resistances it has to encounter; for by one
of the mysteries of biology, resistance begets resistance, and where
pressure falls there growth springs up in strength to meet it. And,
pursuing the same train of thought, we see that all this is true not
of the skeleton alone but of the whole fabric of the body. Muscle and
bone, for instance, are inseparably associated and connected; they are
moulded one with another; they come into being together, and act and
react together[636]. We may study them apart, but it is as a concession
to our weakness and to the narrow outlook of our minds. We see, dimly
perhaps, but yet with all the assurance of conviction, that between
muscle and bone there can be no change in the one but it is correlated
with changes in the other; that through and through they are linked in
indissoluble association; that they are only separate entities {714}
in this limited and subordinate sense, that they are _parts_ of a
whole which, when it loses its composite integrity, ceases to exist.

The biologist, as well as the philosopher, learns to recognise that
the whole is not merely the sum of its parts. It is this, and much
more than this. For it is not a bundle of parts but an organisation of
parts, of parts in their mutual arrangement, fitting one with another,
in what Aristotle calls “a single and indivisible principle of unity”;
and this is no merely metaphysical conception, but is in biology the
fundamental truth which lies at the basis of Geoffroy’s (or Goethe’s)
law of “compensation,” or “balancement of growth.”

Nevertheless Darwin found no difficulty in believing that “natural
selection will tend in the long run to reduce _any part_ of the
organisation, as soon as, through changed habits, it becomes
superfluous: without by any means causing some other part to be largely
developed in a corresponding degree. And conversely, that natural
selection may perfectly well succeed in largely developing an organ
without requiring as a necessary compensation the reduction of some
adjoining part[637].” This view has been developed into a doctrine
of the “independence of single characters” (not to be confused with
the germinal “unit characters” of Mendelism), especially by the
palaeontologists. Thus Osborn asserts a “principle of hereditary
correlation,” combined with a “principle of _hereditary separability_
whereby the body is a colony, a mosaic, of single individual and
separable characters[638].” I cannot think that there is more than a
small element of truth in this doctrine. As Kant said, “die Ursache der
Art der Existenz bei jedem Theile eines lebenden Körpers _ist im Ganzen
enthalten_.” And, according to the trend or aspect of our thought, we
may look upon the co-ordinated parts, now as related and fitted _to the
end or function_ of the whole, and now as related to or resulting _from
the physical causes_ inherent in the entire system of forces to which
the whole has been exposed, and under whose influence it has come into
being[639]. {715}

It would seem to me that the mechanical principles and phenomena which
we have dealt with in this chapter are of no small importance to the
morphologist, all the more when he is inclined to direct his study of
the skeleton exclusively to the problem of phylogeny; and especially
when, according to the methods of modern comparative morphology, he is
apt to take the skeleton to pieces, and to draw from the comparison of
a series of scapulae, humeri, or individual vertebrae, conclusions as
to the descent and relationship of the animals to which they belong.

It would, I dare say, be a gross exaggeration to see in every bone
nothing more than a resultant of immediate and direct physical or
mechanical conditions; for to do so would be to deny the existence, in
this connection, of a principle of heredity. And though I have tried
throughout this book to lay emphasis on the direct action of causes
other than heredity, in short to circumscribe the employment of the
latter as a working hypothesis in morphology, there can still be no
question whatsoever but that heredity is a vastly important as well as
a mysterious thing; it is one of the great factors in biology, however
we may attempt to figure to ourselves, or howsoever we may fail even to
imagine, its underlying physical explanation. But I maintain that it is
no less an exaggeration if we tend to neglect these direct physical and
mechanical modes of causation altogether, and to see in the characters
of a bone merely the results of variation and of heredity, and to
trust, in consequence, to those characters as a sure and certain and
unquestioned guide to affinity and phylogeny. Comparative anatomy has
its physiological side, which filled men’s minds in John Hunter’s day,
and in Owen’s day; it has its {716} classificatory and phylogenetic
aspect, which has all but filled men’s minds during the last couple of
generations; and we can lose sight of neither aspect without risk of
error and misconception.

It is certain that the question of phylogeny, always difficult, becomes
especially so in cases where a great change of physical or mechanical
conditions has come about, and where accordingly the physical and
physiological factors in connection with change of form are bound to
be large. To discuss these questions at length would be to enter on a
discussion of Lamarck’s philosophy of biology, and of many other things
besides. But let us take one single illustration.

The affinities of the whales constitute, as will be readily admitted,
a very hard problem in phylogenetic classification. We know now that
the extinct Zeuglodons are related to the old Creodont carnivores,
and thereby (though distantly) to the seals; and it is supposed, but
it is by no means so certain, that in turn they are to be considered
as representing, or as allied to, the ancestors of the modern toothed
whales[640]. The proof of any such a contention becomes, to my mind,
extraordinarily difficult and complicated; and the arguments commonly
used in such cases may be said (in Bacon’s phrase) to allure, rather
than to extort assent. Though the Zeuglodonts were aquatic animals, we
do not know, and we have no right to suppose or to assume, that they
swam after the fashion of a whale (any more than the seal does), that
they dived like a whale, and leaped like a whale. But the fact that
the whale does these things, and the way in which he does them, is
reflected in many parts of his skeleton—perhaps more or less in all:
so much so that the lines of stress which these actions impose are the
very plan and working-diagram of great part of his structure. That the
Zeuglodon has a scapula like that of a whale is to my mind no necessary
argument that he is akin by blood-relationship to a whale: that his
dorsal vertebrae are very different from a whale’s is no conclusive
argument that {717} such blood-relationship is lacking. The former
fact goes a long way to prove that he used his flippers very much as
a whale does; the latter goes still farther to prove that his general
movements and equilibrium in the water were totally different. The
whale may be descended from the Carnivora, or might for that matter,
as an older school of naturalists believed, be descended from the
Ungulates; but whether or no, we need not expect to find in him the
scapula, the pelvis or the vertebral column of the lion or of the cow,
for it would be physically impossible that he could live the life he
does with any one of them. In short, when we hope to find the missing
links between a whale and his terrestrial ancestors, it must be not
by means of conclusions drawn from a scapula, an axis, or even from a
tooth, but by the discovery of forms so intermediate in their general
structure as to indicate an organisation and, _ipso facto_, a mode of
life, intermediate between the terrestrial and the Cetacean form. There
is no valid syllogism to the effect that _A_ has a flat curved scapula
like a seal’s, and _B_ has a flat, curved scapula like a seal’s: and
therefore _A_ and _B_ are related to the seals and to each other; it
is merely a flagrant case of an “undistributed middle.” But there
is validity in an argument that _B_ shews in its general structure,
extending over this bone and that bone, resemblances both to _A_ and
to the seals: and that therefore he may be presumed to be related to
both, in his hereditary habits of life and in actual kinship by blood.
It is cognate to this argument that (as every palaeontologist knows) we
find clues to affinity more easily, that is to say with less confusion
and perplexity, in certain structures than in others. The deep-seated
rhythms of growth which, as I venture to think, are the chief basis of
morphological heredity, bring about similarities of form, which endure
in the absence of conflicting forces; but a new system of forces,
introduced by altered environment and habits, impinging on those
particular parts of the fabric which lie within this particular field
of force, will assuredly not be long of manifesting itself in notable
and inevitable modifications of form. And if this be really so, it
will further imply that modifications of form will tend to manifest
themselves, not so much in small and _isolated_ phenomena, in this
part of the fabric or in that, in a scapula for instance or a humerus:
but rather in {718} some slow, _general_, and more or less uniform
or graded modification, spread over a number of correlated parts, and
at times extending over the whole, or over great portions, of the
body. Whether any such general tendency to widespread and correlated
transformation exists, we shall attempt to discuss in the following
chapter.

{719}




CHAPTER XVII

ON THE THEORY OF TRANSFORMATIONS, OR THE COMPARISON OF RELATED
FORMS[641]


In the foregoing chapters of this book we have attempted to study the
inter-relations of growth and form, and the part which certain of the
physical forces play in this complex interaction; and, as part of
the same enquiry, we have tried in comparatively simple cases to use
mathematical methods and mathematical terminology in order to describe
and define the forms of organisms. We have learned in so doing that
our own study of organic form, which we call by Goethe’s name of
Morphology, is but a portion of that wider Science of Form which deals
with the forms assumed by matter under all aspects and conditions, and,
in a still wider sense, with forms which are theoretically imaginable.

The study of form may be descriptive merely, or it may become
analytical. We begin by describing the shape of an object in the simple
words of common speech: we end by defining it in the precise language
of mathematics; and the one method tends to follow the other in strict
scientific order and historical continuity. Thus, for instance, the
form of the earth, of a raindrop or a rainbow, the shape of the
hanging chain, or the path of a stone thrown up into the air, may all
be described, however inadequately, in common words; but when we have
learned to comprehend and to define the sphere, the catenary, or the
parabola, we have made a wonderful and perhaps a manifold advance. The
mathematical definition of a “form” has a quality of precision which
was quite lacking in our earlier stage of mere description; it is
expressed in few words, or in still briefer symbols, and these {720}
words or symbols are so pregnant with meaning that thought itself
is economised; we are brought by means of it in touch with Galileo’s
aphorism (as old as Plato, as old as Pythagoras, as old perhaps as
the wisdom of the Egyptians), that “the Book of Nature is written in
characters of Geometry.”

Next, we soon reach through mathematical analysis to mathematical
synthesis; we discover homologies or identities which were not obvious
before, and which our descriptions obscured rather than revealed: as
for instance, when we learn that, however we hold our chain, or however
we fire our bullet, the contour of the one or the path of the other is
always mathematically homologous. Lastly, and this is the greatest gain
of all, we pass quickly and easily from the mathematical conception
of form in its statical aspect to form in its dynamical relations: we
pass from the conception of form to an understanding of the forces
which gave rise to it; and in the representation of form and in the
comparison of kindred forms, we see in the one case a diagram of forces
in equilibrium, and in the other case we discern the magnitude and
the direction of the forces which have sufficed to convert the one
form into the other. Here, since a change of material form is only
effected by the movement of matter, we have once again the support of
the schoolman’s and the philosopher’s axiom, “_Ignorato motu, ignoratur
Natura_.”

――――――――――

In the morphology of living things the use of mathematical methods
and symbols has made slow progress; and there are various reasons for
this failure to employ a method whose advantages are so obvious in the
investigation of other physical forms. To begin with, there would seem
to be a psychological reason lying in the fact that the student of
living things is by nature and training an observer of concrete objects
and phenomena, and the habit of mind which he possesses and cultivates
is alien to that of the theoretical mathematician. But this is by no
means the only reason; for in the kindred subject of mineralogy, for
instance, crystals were still treated in the days of Linnaeus as wholly
within the province of the naturalist, and were described by him after
the simple methods in use for animals and plants: but as soon as Haüy
showed the application of mathematics to {721} the description and
classification of crystals, his methods were immediately adopted and a
new science came into being.

A large part of the neglect and suspicion of mathematical methods
in organic morphology is due (as we have partly seen in our opening
chapter) to an ingrained and deep-seated belief that even when we seem
to discern a regular mathematical figure in an organism, the sphere,
the hexagon, or the spiral which we so recognise merely resembles, but
is never entirely explained by, its mathematical analogue; in short,
that the details in which the figure differs from its mathematical
prototype are more important and more interesting than the features in
which it agrees, and even that the peculiar aesthetic pleasure with
which we regard a living thing is somehow bound up with the departure
from mathematical regularity which it manifests as a peculiar attribute
of life. This view seems to me to involve a misapprehension. There is
no such essential difference between these phenomena of organic form
and those which are manifested in portions of inanimate matter[642]. No
chain hangs in a perfect catenary and no raindrop is a perfect sphere:
and this for the simple reason that forces and resistances other than
the main one are inevitably at work. The same is true of organic form,
but it is for the mathematician to unravel the conflicting forces which
are at work together. And this process of investigation may lead us
on step by step to new phenomena, as it has done in physics, where
sometimes a knowledge of form leads us to the interpretation of forces,
and at other times a knowledge of the forces at work guides us towards
a better insight into form. I would illustrate this by the case of the
earth itself. After the fundamental advance had been made which taught
us that the world was round, Newton showed that the forces at work upon
it must lead to its being imperfectly spherical, and in the course of
time its oblate spheroidal shape was actually verified. But now, in
turn, it has been shown that its form is still more complicated, and
the next step will be to seek for the forces that have deformed the
oblate spheroid. {722}

The organic forms which we can define, more or less precisely, in
mathematical terms, and afterwards proceed to explain and to account
for in terms of force, are of many kinds, as we have seen; but
nevertheless they are few in number compared with Nature’s all but
infinite variety. The reason for this is not far to seek. The living
organism represents, or occupies, a field of force which is never
simple, and which as a rule is of immense complexity. And just as in
the very simplest of actual cases we meet with a departure from such
symmetry as could only exist under conditions of _ideal_ simplicity, so
do we pass quickly to cases where the interference of numerous, though
still perhaps very simple, causes leads to a resultant which lies far
beyond our powers of analysis. Nor must we forget that the biologist
is much more exacting in his requirements, as regards form, than the
physicist; for the latter is usually content with either an ideal or a
general description of form, while the student of living things must
needs be specific. The physicist or mathematician can give us perfectly
satisfying expressions for the form of a wave, or even of a heap of
sand; but we never ask him to define the form of any particular wave of
the sea, nor the actual form of any mountain-peak or hill[643]. {723}

For various reasons, then, there are a vast multitude of organic forms
which we are unable to account for, or to define, in mathematical
terms; and this is not seldom the case even in forms which are
apparently of great simplicity and regularity. The curved outline of a
leaf, for instance, is such a case; its ovate, lanceolate, or cordate
shape is apparently very simple, but the difficulty of finding for it a
mathematical expression is very great indeed. To define the complicated
outline of a fish, for instance, or of a vertebrate skull, we never
even seek for a mathematical formula.

But in a very large part of morphology, our essential task lies in the
comparison of related forms rather than in the precise definition of
each; and the _deformation_ of a complicated figure may be a phenomenon
easy of comprehension, though the figure itself have to be left
unanalysed and undefined. This process of comparison, of recognising
in one form a definite permutation or _deformation_ of another, apart
altogether from a precise and adequate understanding of the original
“type” or standard of comparison, lies within the immediate province of
mathematics, and finds its solution in the elementary use of a certain
method of the mathematician. This method is the Method of Co-ordinates,
on which is based the Theory of Transformations.

I imagine that when Descartes conceived the method of co-ordinates, as
a generalisation from the proportional diagrams of the artist and the
architect, and long before the immense possibilities of this analysis
could be foreseen, he had in mind a very simple purpose; it was perhaps
no more than to find a way of translating the _form_ of a curve into
_numbers_ and into _words_. This is precisely what we do, by the
method of co-ordinates, every time we study a statistical curve; and
conversely, we translate numbers into form whenever we “plot a curve”
to illustrate a table of mortality, a rate of growth, or the daily
variation of temperature or barometric pressure. In precisely the same
way it is possible to inscribe in a net of rectangular co-ordinates the
outline, for instance, of a fish, and so to translate {724} it into a
table of numbers, from which again we may at pleasure reconstruct the
curve.

But it is the next step in the employment of co-ordinates which is of
special interest and use to the morphologist; and this step consists in
the alteration, or “transformation,” of our system of co-ordinates and
in the study of the corresponding transformation of the curve or figure
inscribed in the co-ordinate network.

Let us inscribe in a system of Cartesian co-ordinates the outline
of an organism, however complicated, or a part thereof: such as a
fish, a crab, or a mammalian skull. We may now treat this complicated
figure, in general terms, as a function of _x_, _y_. If we submit
our rectangular system to “deformation,” on simple and recognised
lines, altering, for instance, the direction of the axes, the ratio
of _x_/_y_, or substituting for _x_ and _y_ some more complicated
expressions, then we shall obtain a new system of co-ordinates, whose
deformation from the original type the inscribed figure will precisely
follow. In other words, we obtain a new figure, which represents the
old figure _under strain_, and is a function of the new co-ordinates
in precisely the same way as the old figure was of the original
co-ordinates _x_ and _y_.

The problem is closely akin to that of the cartographer who transfers
identical data to one projection or another; and whose object is to
secure (if it be possible) a complete correspondence, _in each small
unit of area_, between the one representation and the other. The
morphologist will not seek to draw his organic forms in a new and
artificial projection; but, in the converse aspect of the problem, he
will inquire whether two different but more or less obviously related
forms can be so analysed and interpreted that each may be shown to be
a transformed representation of the other. This once demonstrated, it
will be a comparatively easy task (in all probability) to postulate the
direction and magnitude of the force capable of effecting the required
transformation. Again, if such a simple alteration of the system of
forces can be proved adequate to meet the case, we may find ourselves
able to dispense with many widely current and more complicated
hypotheses of biological causation. For it is a maxim in physics that
an effect ought not to be ascribed to {725} the joint operation of
many causes if few are adequate to the production of it: _Frustra fit
per plura, quod fieri potest per pauciora._

――――――――――

It is evident that by the combined action of appropriate forces any
material form can be transformed into any other: just as out of a
“shapeless” mass of clay the potter or the sculptor models his artistic
product; or just as we attribute to Nature herself the power to
effect the gradual and successive transformation of the simplest into
the most complex organism. In like manner it is possible, at least
theoretically, to cause the outline of any closed curve to appear
as a projection of any other whatsoever. But we need not let these
theoretical considerations deter us from our method of comparison of
_related_ forms. We shall strictly limit ourselves to cases where the
transformation necessary to effect a comparison shall be of a simple
kind, and where the transformed, as well as the original, co-ordinates
shall constitute an harmonious and more or less symmetrical system. We
should fall into deserved and inevitable confusion if, whether by the
mathematical or any other method, we attempted to compare organisms
separated far apart in Nature and in zoological classification. We are
limited, not by the nature of our method, but by the whole nature of
the case, to the comparison of organisms such as are manifestly related
to one another and belong to the same zoological class.

Our inquiry lies, in short, just within the limits which Aristotle
himself laid down when, in defining a “genus,” he showed that (apart
from those superficial characters, such as colour, which he called
“accidents”) the essential differences between one “species” and
another are merely differences of proportion, of relative magnitude, or
(as he phrased it) of “excess and defect.” “Save only for a difference
in the way of excess or defect, the parts are identical in the case of
such animals as are of one and the same genus; and by ‘genus’ I mean,
for instance, Bird or Fish.” And again: “Within the limits of the same
genus, as a general rule, most of the parts exhibit differences ... in
the way of multitude or fewness, magnitude or parvitude, in short,
in the way of excess or defect. For ‘the more’ and ‘the less’ may be
represented as {726} ‘excess’ and ‘defect[644].’ ” It is precisely
this difference of relative magnitudes, this Aristotelian “excess and
defect” in the case of form, which our co-ordinate method is especially
adapted to analyse, and to reveal and demonstrate as the main cause of
what (again in the Aristotelian sense) we term “specific” differences.

The applicability of our method to particular cases will depend
upon, or be further limited by, certain practical considerations or
qualifications. Of these the chief, and indeed the essential, condition
is, that the form of the entire structure under investigation should
be found to vary in a more or less uniform manner, after the fashion
of an approximately homogeneous and isotropic body. But an imperfect
isotropy, provided always that some “principle of continuity” run
through its variations, will not seriously interfere with our method;
it will only cause our transformed co-ordinates to be somewhat less
regular and harmonious than are those, for instance, by which the
physicist depicts the motions of a perfect fluid or a theoretic field
of force in a uniform medium.

Again, it is essential that our structure vary in its entirety, or
at least that “independent variants” should be relatively few. That
independent variations occur, that localised centres of diminished or
exaggerated growth will now and then be found, is not only probable but
manifest; and they may even be so pronounced as to appear to constitute
new formations altogether. Such independent variants as these Aristotle
himself clearly recognised: “It happens further that some have parts
that others have not; for instance, some [birds] have spurs and others
not, some have crests, or combs, and others not; but, as a general
rule, most parts and those that go to make up the bulk of the body are
either identical with one another, or differ from one another in the
way of contrast and of excess and defect. For ‘the more’ and ‘the less’
may be represented as ‘excess’ or ‘defect.’ ”

If, in the evolution of a fish, for instance, it be the case that its
several and constituent parts—head, body, and tail, or this fin and
that fin—represent so many independent variants, then our co-ordinate
system will at once become too complex to be intelligible; we shall
be making not one comparison but several {727} separate comparisons,
and our general method will be found inapplicable. Now precisely
this independent variability of parts and organs—here, there, and
everywhere within the organism—would appear to be implicit in our
ordinary accepted notions regarding variation; and, unless I am greatly
mistaken, it is precisely on such a conception of the easy, frequent,
and normal independent variability of parts that our conception of
the process of natural selection is fundamentally based. For the
morphologist, when comparing one organism with another, describes
the differences between them point by point, and “character” by
“character[645].” If he is from time to time constrained to admit the
existence of “correlation” between characters (as a hundred years ago
Cuvier first showed the way), yet all the while he recognises this fact
of correlation somewhat vaguely, as a phenomenon due to causes which,
except in rare instances, he can hardly hope to trace; and he falls
readily into the habit of thinking and talking of evolution as though
it had proceeded on the lines of his own descriptions, point by point,
and character by character[646].

But if, on the other hand, diverse and dissimilar fishes can
be referred as a whole to identical functions of very different
co-ordinate systems, this fact will of itself constitute a proof
that variation has proceeded on definite and orderly lines, that a
comprehensive “law of growth” has pervaded the whole structure in its
integrity, and that some more or less simple and recognisable system of
forces has been at work. It will not only show how real and deep-seated
is the phenomenon of “correlation,” in regard to form, but it will also
demonstrate the fact that a correlation which had seemed too complex
for analysis or {728} comprehension is, in many cases, capable of very
simple graphic expression. This, after many trials, I believe to be
in general the case, bearing always in mind that the occurrence of
independent or localised variations must often be considered.

 We are dealing in this chapter with the forms of related organisms,
 in order to shew that the differences between them are as a general
 rule simple and symmetrical, and just such as might have been brought
 about by a slight and simple change in the system of forces to which
 the living and growing organism was exposed. Mathematically speaking,
 the phenomenon is identical with one met with by the geologist, when
 he finds a bed of fossils squeezed flat or otherwise symmetrically
 deformed by the pressures to which they, and the strata which contain
 them, have been subjected. In the first step towards fossilisation,
 when the body of a fish or shellfish is silted over and buried, we
 may take it that the wet sand or mud exercises, approximately, a
 hydrostatic pressure—that is to say a pressure which is uniform in
 all directions, and by which the form of the buried object will not
 be appreciably changed. As the strata consolidate and accumulate, the
 fossil organisms which they contain will tend to be flattened by the
 vast superincumbent load, just as the stratum which contains them
 will also be compressed and will have its molecular arrangement more
 or less modified[647]. But the deformation due to direct vertical
 pressure in a horizontal stratum is not nearly so striking as are the
 deformations produced by the oblique or shearing stresses to which
 inclined and folded strata have been exposed, and by which their
 various “dislocations” have been brought about. And especially in
 mountain regions, where these dislocations are especially numerous and
 complicated, the contained fossils are apt to be so curiously and yet
 so symmetrically deformed (usually by a simple shear) that they may
 easily be interpreted as so many distinct and separate “species[648].”
 A great number of described species, and here and there a new genus
 (as the genus Ellipsolithes for an obliquely deformed Goniatite or
 Nautilus) are said to rest on no other foundation[649].

――――――――――

[Illustration: Fig. 352.]

[Illustration: Fig. 353.]

[Illustration: Fig. 354.]

[Illustration: Fig. 355.]

If we begin by drawing a net of rectangular equidistant co-ordinates
(about the axes _x_ and _y_), we may alter or _deform_ this {729}
network in various ways, several of which are very simple indeed.
Thus (1) we may alter the dimensions of our system, extending it
along one or other axis, and so converting each little square into a
corresponding and directly proportionate oblong (Fig. 353). It follows
that any figure which we may have inscribed in the original net, and
which we transfer to the new, will thereby be _deformed_ in strict
proportion to the deformation of the entire configuration, being still
defined by corresponding points in the network and being throughout
in conformity with the original figure. For instance, a circle
inscribed in the original “Cartesian” net will now, after extension
in the _y_-direction, be found elongated {730} into an ellipse. In
elementary mathematical language, for the original _x_ and _y_ we
have substituted _x__{1} and _c_ _y__{1}, and the equation to our
original circle, _x_^2 + _y_^2 = _a_^2, becomes that of the ellipse,
_x__{1}^2 + _c_^2 _y__{1}^2 = _a_^2.

If I draw the cannon-bone of an ox (Fig. 354, A), for instance, within
a system of rectangular co-ordinates, and then transfer the same
drawing, point for point, to a system in which for the _x_ of the
original diagram we substitute _x′_ = 2_x_/3, we obtain a drawing (B)
which is a very close approximation to the cannon-bone of the sheep. In
other words, the main (and perhaps the only) difference between the two
bones is simply that that of the sheep is elongated, along the vertical
axis, as compared with that of the ox in the relation of 3/2. And
similarly, the long slender cannon-bone of the giraffe (C) is referable
to the same identical type, subject to a reduction of breadth, or
increase of length, corresponding to _x″_ = _x_/3.

(2) The second type is that where extension is not equal or uniform
at all distances from the origin: but grows greater or less, as,
for instance, when we stretch a _tapering_ elastic band. In such
cases, as I have represented it in Fig. 355, the ordinate increases
logarithmically, and for _y_ we substitute ε^{_y_}. It is obvious that
this logarithmic extension may involve both abscissae and ordinates,
_x_ becoming ε^{_x_}, while _y_ becomes ε^{_y_}. The circle in our
original figure is now deformed into some such shape as that of Fig.
356. This method of deformation is a common one, and will often be of
use to us in our comparison of organic forms.

(3) Our third type is the “simple shear,” where the rectangular
co-ordinates become “oblique,” their axes being inclined to one another
at a certain angle ω. Our original rectangle now becomes such a figure
as that of Fig. 357. The system may now be described in terms of the
oblique axes _X_, _Y_; or may be directly referred to new rectangular
co-ordinates ξ, η by the simple transposition _x_ = ξ − η cot ω, _y_
= η cosec ω.

[Illustration: Fig. 356.]

[Illustration: Fig. 357.]

[Illustration: Fig. 358.]

(4) Yet another important class of deformations may be represented
by the use of radial co-ordinates, in which one set of lines are
represented as radiating from a point or “focus,” while the other set
are transformed into circular arcs cutting the radii orthogonally.
These radial co-ordinates are especially applicable {731} to cases
where there exists (either within or without the figure) some part
which is supposed to suffer no deformation; a simple illustration is
afforded by the diagrams which illustrate the flexure of a beam (Fig.
358). In biology these co-ordinates will be especially applicable in
cases where the growing structure includes a “node,” or point where
growth is absent or at a minimum; and about which node the rate of
growth may be assumed to increase symmetrically. Precisely such a
case is furnished us in a leaf of an ordinary dicotyledon. The leaf
of a {732} typical monocotyledon—such as a grass or a hyacinth, for
instance—grows continuously from its base, and exhibits no node or
“point of arrest.” Its sides taper off gradually from its broad base
to its slender tip, according to some law of decrement specific to the
plant; and any alteration in the relative velocities of longitudinal
and transverse growth will merely make the leaf a little broader or
narrower, and will effect no other conspicuous alteration in its
contour. But if there once come into existence a node, or “locus of no
growth,” about which we may assume the growth—which in the hyacinth
leaf was longitudinal and transverse—to take place radially and
transversely to the radii, then we shall

[Illustration: Fig. 359.]

at once see, in the first place, that the sloping and slightly
curved sides of the hyacinth leaf suffer a transformation into what we
consider a more typical and “leaf-like” shape, the sides of the figure
broadening out to a zone of maximum breadth and then drawing inwards
to the pointed apex. If we now alter the ratio between the radial and
tangential velocities of growth—in other words, if we increase the
angles between corresponding radii—we pass successively through the
various configurations which the botanist describes as the lanceolate,
the ovate, and finally the cordate leaf. These successive changes may
to some extent, and in appropriate cases, be traced as the individual
leaf grows {733} to maturity; but as a much more general rule, the
balance of forces, the ratio between radial and tangential velocities
of growth, remains so nicely and constantly balanced that the leaf
increases in size without conspicuous modification of form. It is
rather what we may call a long-period variation, a tendency for the
relative velocities to alter from one generation to another, whose
result is brought into view by this method of illustration.

[Illustration: Fig. 360. _Begonia daedalea._]

There are various corollaries to this method of describing the form
of a leaf which may be here alluded to, for we shall not return again
to the subject of radial co-ordinates. For instance, the so-called
unsymmetrical leaf[650] of a begonia, in which one side of the leaf
may be merely ovate while the other has a cordate outline, is seen to
be really a case of _unequal_, and not truly asymmetrical, growth on
either side of the midrib. There is nothing more mysterious in its
conformation than, for instance, in that of a forked twig in which
one limb of the fork has grown longer than the other. The case of the
begonia leaf is of sufficient interest to deserve illustration, and in
Fig. 360 I have outlined a leaf of the large _Begonia daedalea_. On the
smaller left-hand side of the leaf I have taken at random three points,
_a_, _b_, _c_, and have measured the angles, _AOa_, etc., which the
radii from the hilus of the leaf to these points make with the median
axis. On the other side of the leaf I have marked the points _a′_,
_b′_, _c′_, such that the radii drawn to this margin of the leaf are
equal to the former, _Oa′_ to _Oa_, etc. Now if the two sides of the
leaf are {734} mathematically similar to one another, it is obvious
that the respective angles should be in continued proportion, i.e.
as _AOa_ is to _AOa′_, so should _AOb_ be to _AOb′_. This proves to
be very nearly the case. For I have measured the three angles on one
side, and one on the other, and have then compared, as follows, the
calculated with the observed values of the other two:

                   _AOa_   _AOb_   _AOc_   _AOa′_   _AOb′_   _AOc′_
 Observed values     12°   28.5°    88°      —        —       157°
 Calculated values    —     —        —      21.5°    51.1°      —
 Observed values      —     —        —      20       52         —

The agreement is very close, and what discrepancy there is may be amply
accounted for, firstly, by the slight irregularity of the sinuous
margin of the leaf; and secondly, by the fact that the true axis or
midrib of the leaf is not straight but slightly curved, and therefore
that it is curvilinear and not rectilinear triangles which we ought
to have measured. When we understand these few points regarding
the peripheral curvature of the leaf, it is easy to see that its
principal veins approximate closely to a beautiful system of isogonal
co-ordinates. It is also obvious that we can easily pass, by a process
of shearing, from those cases where the principal veins start from the
base of the leaf to those, as in most dicotyledons, where they arise
successively from the midrib.

It may sometimes happen that the node, or “point of arrest,” is at the
upper instead of the lower end of the leaf-blade; and occasionally
there may be a node at both ends. In the former case, as we have it
in the daisy, the form of the leaf will be, as it were, inverted, the
broad, more or less heart-shaped, outline appearing at the upper end,
while below the leaf tapers gradually downwards to an ill-defined base.
In the latter case, as in _Dionaea_, we obtain a leaf equally expanded,
and similarly ovate or cordate, at both ends. We may notice, lastly,
that the shape of a solid fruit, such as an apple or a cherry, is a
solid of revolution, developed from similar curves and to be explained
on the same principle. In the cherry we have a “point of arrest” at the
base of the berry, where it joins its peduncle, and about this point
the fruit (in imaginary section) swells out into a cordate outline;
while in the {735} apple we have two such well-marked points of
arrest, above and below, and about both of them the same conformation
tends to arise. The bean and the human kidney owe their “reniform”
shape to precisely the same phenomenon, namely, to the existence of
a node or “hilus,” about which the forces of growth are radially and
symmetrically arranged.

――――――――――

Most of the transformations which we have hitherto considered (other
than that of the simple shear) are particular cases of a general
transformation, obtainable by the method of conjugate functions and
equivalent to the projection of the original figure on a new plane.
Appropriate transformations, on these general lines, provide for the
cases of a coaxial system where the Cartesian co-ordinates are replaced
by coaxial circles, or a confocal system in which they are replaced by
confocal ellipses and hyperbolas.

[Illustration: Fig. 361.]

Yet another curious and important transformation, belonging to the
same class, is that by which a system of straight lines becomes
transformed into a conformal system of logarithmic spirals: the
straight line _Y_ − _AX_ = _c_ corresponding to the logarithmic
spiral θ − _A_ log _r_ = _c_ (Fig. 361). This beautiful and simple
transformation lets us at once convert, for instance, the straight
conical shell of the Pteropod or the _Orthoceras_ into the logarithmic
spiral of the Nautiloid; it involves a mathematical symbolism which is
but a slight extension of that which we have employed in our elementary
treatment of the logarithmic spiral.

These various systems of coordinates, which we have now briefly
considered, are sometimes called “isothermal co-ordinates,” from
the fact that, when employed in this particular branch of physics,
they perfectly represent the phenomena of the conduction of heat,
the contour lines of equal temperature appearing, under appropriate
conditions, as the orthogonal lines of the co-ordinate system. And it
follows that {736} the “law of growth” which our biological analysis
by means of orthogonal co-ordinate systems presupposes, or at least
foreshadows, is one according to which the organism grows or develops
along _stream lines_, which may be defined by a suitable mathematical
transformation.

When the system becomes no longer orthogonal, as in many of the
following illustrations—for instance, that of _Orthagoriscus_ (Fig.
382),—then the transformation is no longer within the reach of
comparatively simple mathematical analysis. Such departure from the
typical symmetry of a “stream-line” system is, in the first instance,
sufficiently accounted for by the simple fact that the developing
organism is very far from being homogeneous and isotropic, or, in
other words, does not behave like a perfect fluid. But though under
such circumstances our co-ordinate systems may be no longer capable of
strict mathematical analysis, they will still indicate _graphically_
the relation of the new co-ordinate system to the old, and conversely
will furnish us with some guidance as to the “law of growth,” or play
of forces, by which the transformation has been effected.

――――――――――

Before we pass from this brief discussion of transformations in
general, let us glance at one or two cases in which the forces applied
are more or less intelligible, but the resulting transformations are,
from the mathematical point of view, exceedingly complicated.

The “marbled papers” of the bookbinder are a beautiful illustration of
visible “stream lines.” On a dishful of a sort of semi-liquid gum the
workman dusts a few simple lines or patches of colouring matter; and
then, by passing a comb through the liquid, he draws the colour-bands
into the streaks, waves, and spirals which constitute the marbled
pattern, and which he then transfers to sheets of paper laid down
upon the gum. By some such system of shears, by the effect of unequal
traction or unequal growth in various directions and superposed on
an originally simple pattern, we may account for the not dissimilar
marbled patterns which we recognise, for instance, on a large serpent’s
skin. But it must be remarked, in the case of the marbled paper, that
though the method of application of the forces is simple, yet in the
aggregate the system of forces set up by the many {737} teeth of the
comb is exceedingly complex, and its complexity is revealed in the
complicated “diagram of forces” which constitutes the pattern.

To take another and still more instructive illustration. To turn one
circle (or sphere) into two circles would be, from the point of view
of the mathematician, an extraordinarily difficult transformation;
but, physically speaking, its achievement may be extremely simple.
The little round gourd grows naturally, by its symmetrical forces of
expansive growth, into a big, round, or somewhat oval pumpkin or melon.
But the Moorish husbandman ties a rag round its middle, and the same
forces of growth, unaltered save for the presence of this trammel,
now expand the globular structure into two superposed and connected
globes. And again, by varying the position of the encircling band, or
by applying several such ligatures instead of one, a great variety of
artificial forms of “gourd” may be, and actually are, produced. It
is clear, I think, that we may account for many ordinary biological
processes of development or transformation of form by the existence of
trammels or lines of constraint, which limit and determine the action
of the expansive forces of growth that would otherwise be uniform and
symmetrical. This case has a close parallel in the operations of the
glassblower, to which we have already, more than once, referred in
passing[651]. The glassblower starts his operations with a _tube_,
which he first closes at one end so as to form a hollow vesicle, within
which his blast of air exercises a uniform pressure on all sides; but
the spherical conformation which this uniform expansive force would
naturally tend to produce is modified into all kinds of forms by the
trammels or resistances set up as the workman lets one part or another
of his bubble be unequally heated or cooled. It was Oliver Wendell
Holmes who first shewed this curious parallel between the operations
of the glassblower and those of Nature, when she starts, as she so
often does, with a simple tube[652]. The alimentary canal, {738} the
arterial system including the heart, the central nervous system of the
vertebrate, including the brain itself, all begin as simple tubular
structures. And with them Nature does just what the glassblower does,
and, we might even say, no more than he. For she can expand the tube
here and narrow it there; thicken its walls or thin them; blow off a
lateral offshoot or caecal diverticulum; bend the tube, or twist and
coil it; and infold or crimp its walls as, so to speak, she pleases.
Such a form as that of the human stomach is easily explained when it is
regarded from this point of view; it is simply an ill-blown bubble, a
bubble that has been rendered lopsided by a trammel or restraint along
one side, such as to prevent its symmetrical expansion—such a trammel
as is produced if the glassblower lets one side of his bubble get cold,
and such as is actually present in the stomach itself in the form of a
muscular band.

――――――――――

We may now proceed to consider and illustrate a few permutations or
transformations of organic form, out of the vast multitude which are
equally open to this method of inquiry.

[Illustration: Fig. 362.]

We have already compared in a preliminary fashion the metacarpal or
cannon-bone of the ox, the sheep, and the giraffe (Fig. 354); and we
have seen that the essential difference in form between these three
bones is a matter of relative length and breadth, such that, if we
reduce the figures to an identical standard of length (or identical
values of _y_), the breadth (or value of _x_) will be approximately
two-thirds that of the ox in the case of the sheep and one-third
that of the ox in the case of the giraffe. We may easily, for the
sake of closer comparison, determine these ratios more accurately,
for instance, if it be our purpose to compare the different racial
varieties within the limits of a single species. And in such cases, by
the way, as when we compare with one another various breeds or races of
cattle or of horses, the ratios {739} of length and breadth in this
particular bone are extremely significant[653].

If, instead of limiting ourselves to the cannon-bone, we inscribe the
entire foot of our several Ungulates in a co-ordinate system, the
same ratios of _x_ that served us for the cannon-bones still give us
a first approximation to the required comparison; but even in the
case of such closely allied forms as the ox and the sheep there is
evidently something wanting in the comparison. The reason is that the
relative elongation of the several parts, or individual bones, has not
proceeded equally or proportionately in all cases; in other words,
that the equations for _x_ will not suffice without some simultaneous
modification of the values of _y_ (Fig. 362). In such a case it may be
found possible to satisfy the varying values of _y_ by some logarithmic
or other formula; but, even if that be possible, it will probably be
somewhat difficult of discovery or verification in such a case as the
present, owing to the fact that we have too few well-marked points
of correspondence between the one object and the other, and that
especially along the shaft of such long bones as the cannon-bone of
the ox, the deer, the llama, or the giraffe there is a complete lack
of easily recognisable corresponding points. In such a case a brief
tabular statement of apparently corresponding values of _y_, or of
those obviously corresponding values which coincide with the boundaries
of the several bones of the foot, will, as in the following example,
enable us to dispense with a fresh equation.

                      _a_   _b_   _c_   _d_
 _y_ (Ox)         0   18    27    42    100
 _y′_ (Sheep)     0   10    19    36    100
 _y″_ (Giraffe)   0    5    10    24    100

This summary of values of _y′_, coupled with the equations for the
{740} value of _x_, will enable us, from any drawing of the ox’s foot,
to construct a figure of that of the sheep or of the giraffe with
remarkable accuracy.

[Illustration: Fig. 363.]

That underlying the varying amounts of extension to which the parts or
segments of the limb have been subject there is a law, or principle of
continuity, may be discerned from such a diagram as the above (Fig.
363), where the values of _y_ in the case of the ox are plotted as a
straight line, and the corresponding values for the sheep (extracted
from the above table) are seen to form a more or less regular and
even curve. This simple graphic result implies the existence of a
comparatively simple equation between _y_ and _y′_.

[Illustration: Fig. 364. (After Albert Dürer.)]

An elementary application of the principle of co-ordinates to the
study of proportion, as we have here used it to illustrate the
varying proportions of a bone, was in common use in the sixteenth and
seventeenth centuries by artists in their study of the human form. The
method is probably much more ancient, and may even be classical[654];
it is fully described and put in practice by Albert Dürer in his
_Geometry_, and especially in his _Treatise on Proportion_[655].
In this latter work, the manner in which the {741} human figure,
features, and facial expression are all transformed and modified by
slight variations in the relative magnitude of the parts is admirably
and copiously illustrated (Fig. 364).

[Illustration: Fig. 365.]

In a tapir’s foot there is a striking difference, and yet at the same
time there is an obvious underlying resemblance, between the middle
toe and either of its unsymmetrical lateral neighbours. Let us take
the median terminal phalanx and inscribe its outline in a net of
rectangular equidistant co-ordinates (Fig. 365, _a_). Let us then make
a similar network about axes which are no longer at right angles,
but inclined to one another at an angle of about 50° (_b_). If into
this new network we fill in, point for point, an outline precisely
corresponding to our original drawing of the middle toe, we shall find
that we have already represented the main features of the adjacent
lateral one. We shall, however, perceive that our new diagram looks a
little too bulky on one side, the inner side, of the lateral toe. If
now we substitute for our equidistant ordinates, ordinates which get
gradually closer and closer together as we pass towards the median
side of the toe, then we shall obtain a diagram which differs in no
essential respect from an actual outline copy of the lateral toe (_c_).
In short, the difference between the outline of the middle toe of the
tapir and the next lateral toe may be almost completely expressed
by saying that if the one be represented by rectangular equidistant
co-ordinates, the other will be represented by oblique co-ordinates,
whose axes make an angle of 50°, and in which the abscissal interspaces
decrease in a certain logarithmic ratio. We treated our original
complex curve or projection of the tapir’s toe as a function of the
form _F_ (_x_, _y_) = 0. The figure of the tapir’s lateral {742} toe
is a precisely identical function of the form _F_ (_e_^{_x_}, _y__{1})
= 0, where _x__{1}, _y__{1} are oblique co-ordinate axes inclined to
one another at an angle of 50°.

[Illustration: Fig. 366. (After Albert Dürer.)]

Dürer was acquainted with these oblique co-ordinates also, and I have
copied two illustrative figures from his book[656].

――――――――――

[Illustration: Fig. 367. _Oithona nana._]

[Illustration: Fig. 368. _Sapphirina._]

In Fig. 367 I have sketched the common Copepod _Oithona nana_,
{743} and have inscribed it in a rectangular net, with abscissae
three-fifths the length of the ordinates. Side by side (Fig. 368) is
drawn a very different Copepod, of the genus _Sapphirina_; and about
it is drawn a network such that each co-ordinate passes (as nearly as
possible) through points corresponding to those of the former figure.
It will be seen that two differences are apparent. (1) The values of
_y_ in Fig. 368 are large in the upper part of the figure, and diminish
rapidly towards its base. (2) The values of _x_ are very large in the
neighbourhood of the origin, but diminish rapidly as we pass towards
either side, away from the median vertical axis; and it is probable
that they do so according to a definite, but somewhat complicated,
ratio. If, instead of seeking for an actual equation, we simply
tabulate our values of _x_ and _y_ in the second figure as compared
with the first (just as we did in comparing the feet of the Ungulates),
we get the dimensions of a net in which, by simply projecting the
figure of _Oithona_, we obtain that of _Sapphirina_ without further
trouble, e.g.:

 _x_ (_Oithona_)       0   3    6    9   12   15    —
 _x′_ (_Sapphirina_)   0   8   10   12   13   14    —

 _y_ (_Oithona_)       0   5   10   15   20   25   30
 _y′_ (_Sapphirina_)   0   2    7    3   23   32   40

In this manner, with a single model or type to copy from, we may
record in very brief space the data requisite for the production of
approximate outlines of a great number of forms. For instance the
difference, at first sight immense, between the attenuated body of a
_Caprella_ and the thick-set body of a _Cyamus_ is obviously little,
and is probably nothing, more than a difference of relative magnitudes,
capable of tabulation by numbers and of complete expression by means of
rectilinear co-ordinates.

The Crustacea afford innumerable instances of more complex
deformations. Thus we may compare various higher Crustacea with one
another, even in the case of such dissimilar forms as a lobster and
a crab. It is obvious that the whole body of the former is elongated
as compared with the latter, and that the crab is relatively broad
in the region of the carapace, while it tapers off rapidly towards
its attenuated and abbreviated tail. In a general way, the elongated
rectangular system of co-ordinates {744} in which we may inscribe the
outline of the lobster becomes a shortened triangle in the case of
the crab. In a little more detail we may compare the outline of the
carapace in various crabs one with another: and the comparison will
be found easy and significant, even, in many cases, down to minute
details, such as the number and situation of the marginal spines,
though these are in other cases subject to independent variability.

[Illustration: Fig. 369. Carapaces of various crabs. 1, _Geryon_; 2,
_Corystes_; 3, _Scyramathia_; 4, _Paralomis_; 5, _Lupa_; 6, _Chorinus_.]

If we choose, to begin with, such a crab as _Geryon_ (Fig. 369, 1),
and inscribe it in our equidistant rectangular co-ordinates, we shall
see that we pass easily to forms more elongated in a transverse
{745} direction, such as _Matuta_ or _Lupa_ (5), and conversely, by
transverse compression, to such a form as _Corystes_ (2). In certain
other cases the carapace conforms to a triangular diagram, more or less
curvilinear, as in Fig. 4, which represents the genus _Paralomis_. Here
we can easily see that the posterior border is transversely elongated
as compared with that of _Geryon_, while at the same time the anterior
part is longitudinally extended as compared with the posterior. A
system of slightly curved and converging ordinates, with orthogonal and
logarithmically interspaced abscissal lines, as shown in the figure,
appears to satisfy the conditions.

In an interesting series of cases, such as the genus _Chorinus_,
or _Scyramathia_, and in the spider-crabs generally, we appear to
have just the converse of this. While the carapace of these crabs
presents a somewhat triangular form, which seems at first sight more
or less similar to those just described, we soon see that the actual
posterior border is now narrow instead of broad, the broadest part of
the carapace corresponding precisely, not to that which is broadest
in _Paralomis_, but to that which was broadest in _Geryon_; while the
most striking difference from the latter lies in an antero-posterior
lengthening of the forepart of the carapace, culminating in a great
elongation of the frontal region, with its two spines or “horns.” The
curved ordinates here converge posteriorly and diverge widely in front
(Figs. 3 and 6), while the decremental interspacing of the abscissae is
very marked indeed.

We put our method to a severer test when we attempt to sketch an entire
and complicated animal than when we simply compare corresponding
parts such as the carapaces of various Malacostraca, or related bones
as in the case of the tapir’s toes. Nevertheless, up to a certain
point, the method stands the test very well. In other words, one
particular mode and direction of variation is often (or even usually)
so prominent and so paramount throughout the entire organism, that one
comprehensive system of co-ordinates suffices to give a fair picture
of the actual phenomenon. To take another illustration from the
Crustacea, I have drawn roughly in Fig. 370, 1 a little amphipod of the
family Phoxocephalidae (_Harpinia_ sp.). Deforming the co-ordinates
of the figure into the {746} curved orthogonal system in Fig. 2,
we at once obtain a very fair representation of an allied genus,
belonging to a different family of amphipods, namely _Stegocephalus_.
As we proceed further from our type our co-ordinates will require
greater deformation, and the resultant figure will usually be somewhat
less accurate. In Fig. 3 I show a network, to which, if we transfer
our diagram of _Harpinia_ or of _Stegocephalus_, we shall obtain a
tolerable representation of the aberrant genus _Hyperia_, with its
narrow abdomen, its reduced pleural lappets, its great eyes, and its
inflated head.

[Illustration: Fig 370. 1. _Harpinia plumosa_ Kr. 2. _Stegocephalus
inflatus_ Kr. 3. _Hyperia galba_.]

――――――――――

The hydroid zoophytes constitute a “polymorphic” group, within which a
vast number of species have already been distinguished; and the labours
of the systematic naturalist are constantly adding to the number. The
specific distinctions are for the most part based, not upon characters
directly presented {747} by the living animal, but upon the form,
size and arrangement of the little cups, or “calycles,” secreted and
inhabited by the little individual polypes which compose the compound
organism. The variations, which are apparently infinite, of these
conformations are easily seen to be a question of relative magnitudes,
and are capable of complete expression, sometimes by very simple,
sometimes by somewhat more complex, co-ordinate networks.

[Illustration: Fig. 371. _a_, _Campanularia macroscyphus_, Allm.; _b_,
_Gonothyraea hyalina_, Hincks; _c_, _Clytia Johnstoni_, Alder.]

For instance, the varying shapes of the simple wineglass-shaped cups of
the Campanularidae are at once sufficiently represented and compared
by means of simple Cartesian co-ordinates (Fig. 371). In the two
allied families of Plumulariidae and Aglaopheniidae the calycles are
set unilaterally upon a jointed stem, and small cup-like structures
(holding rudimentary polypes) are associated with the large calycles
in definite number and position. These small calyculi are variable in
number, but in the great majority of cases they accompany the large
calycle in groups of three—two standing by its upper border, and one,
which is especially variable in form and magnitude, lying at its base.
The stem is liable to flexure and, in a high degree, to extension or
compression; and these variations extend, often on an exaggerated
scale, to the related calycles. As a result we find that we can draw
various systems of curved or sinuous co-ordinates, which express,
all but completely, the configuration of the various {748} hydroids
which we inscribe therein (Fig. 372). The comparative smoothness or
denticulation of the margin of the calycle, and the number of its
denticles, constitutes an independent variation, and requires separate
description; we have already seen (p. 236) that this denticulation is
in all probability due to a particular physical cause.

[Illustration: Fig. 372. _a_, _Cladocarpus crenatus_, F.; _b_,
_Aglaophenia pluma_, L.; _c_, _A. rhynchocarpa_, A.; _d_, _A cornuta_,
K.; _e_, _A. ramulosa_, K.]

――――――――――

[Illustration: Fig. 373. _Argyropelecus Olfersi._]

[Illustration: Fig. 374. _Sternoptyx diaphana._]

Among the fishes we discover a great variety of deformations, some of
them of a very simple kind, while others are more striking and more
unexpected. A comparatively simple case, involving a simple shear, is
illustrated by Figs. 373 and 374. Fig. 373 represents, within Cartesian
co-ordinates, a certain little oceanic fish known as _Argyropelecus
Olfersi_. Fig. 474 represents precisely the same outline, transferred
to a system of oblique co-ordinates whose {749} axes are inclined
at an angle of 70°; but this is now (as far as can be seen on the
scale of the drawing) a very good figure of an allied fish, assigned
to a different genus, under the name of _Sternoptyx diaphana_. The
deformation illustrated by this case of _Argyropelecus_ is precisely
analogous to the simplest and commonest kind of deformation to which
fossils are subject (as we have seen on p. 553) as the result of
shearing-stresses in the solid rock.

[Illustration: Fig. 375. _Scarus_ sp.]

[Illustration: Fig. 376. _Pomacanthus._]

Fig. 375 is an outline diagram of a typical Scaroid fish. Let us
deform its rectilinear co-ordinates into a system of (approximately)
coaxial circles, as in Fig. 376, and then filling into the new system,
space by space and point by point, our former diagram of _Scarus_,
we obtain a very good outline of an allied fish, belonging to a
neighbouring family, of the genus _Pomacanthus_. This case is all the
more interesting, because upon the body of our _Pomacanthus_ there are
striking colour bands, which correspond in direction very closely to
the lines of our new curved ordinates. In like manner, the still more
bizarre outlines of other fishes of the same family of Chaetodonts
will be found to correspond to very slight modifications of similar
co-ordinates; in other words, to small variations in the values of the
constants of the coaxial curves.

In Figs. 377–380 I have represented another series of Acanthopterygian
fishes, not very distantly related to the foregoing. If we start
this series with the figure of _Polyprion_, in Fig. 377, we see that
the outlines of _Pseudopriacanthus_ (Fig. 378) and of _Sebastes_ or
_Scorpaena_ (Fig. 379) are easily derived by substituting a system of
triangular, or radial, co-ordinates for the rectangular ones in {750}
which we had inscribed _Polyprion_. The very curious fish _Antigonia
capros_, an oceanic relative of our own “boar-fish,” conforms closely
to the peculiar deformation represented in Fig. 380.

[Illustration: Fig. 377. _Polyprion._]

[Illustration: Fig. 378. _Pseudopriacanthus altus._]

[Illustration: Fig. 379. _Scorpaena_ sp.]

[Illustration: Fig. 380. _Antigonia capros._]

[Illustration: Fig. 381. _Diodon._]

[Illustration: Fig. 382. _Orthagoriscus._]

Fig. 381 is a common, typical _Diodon_ or porcupine-fish, and in
Fig. 382 I have deformed its vertical co-ordinates into a system of
concentric circles, and its horizontal co-ordinates into a system of
curves which, approximately and provisionally, are made to resemble a
system of hyperbolas[657]. The old outline, transferred {751} in its
integrity to the new network, appears as a manifest representation of
the closely allied, but very different looking, sunfish, _Orthagoriscus
mola_. This is a particularly instructive case of deformation or
transformation. It is true that, in a mathematical sense, it is not a
perfectly satisfactory or perfectly regular deformation, for the system
is no longer isogonal; but nevertheless, it is symmetrical to the eye,
and obviously approaches to an isogonal system under certain conditions
of friction or constraint. And as such it accounts, by one single
integral transformation, for all the apparently separate and distinct
external differences between the two fishes. It leaves the parts
near to the origin of the system, the whole region of the head, the
opercular orifice and the pectoral fin, practically unchanged {752} in
form, size and position; and it shews a greater and greater apparent
modification of size and form as we pass from the origin towards the
periphery of the system.

In a word, it is sufficient to account for the new and striking contour
in all its essential details, of rounded body, exaggerated dorsal and
ventral fins, and truncated tail. In like manner, and using precisely
the same co-ordinate networks, it appears to me possible to shew the
relations, almost bone for bone, of the skeletons of the two fishes;
in other words, to reconstruct the skeleton of the one from our
knowledge of the skeleton of the other, under the guidance of the same
correspondence as is indicated in their external configuration.

――――――――――

The family of the crocodiles has had a special interest for the
evolutionist ever since Huxley pointed out that, in a degree only
second to the horse and its ancestors, it furnishes us with a close
and almost unbroken series of transitional forms, running down in
continuous succession from one geological formation to another. I
should be inclined to transpose this general statement into other
terms, and to say that the Crocodilia constitute a case in which,
with unusually little complication from the presence of independent
variants, the trend of one particular mode of transformation is
visibly manifested. If we exclude meanwhile from our comparison a few
of the oldest of the crocodiles, such as _Belodon_, which differ more
fundamentally from the rest, we shall find a long series of genera
in which we can refer not only the changing contours of the skull,
but even the shape and size of the many constituent bones and their
intervening spaces or “vacuities,” to one and the same simple system
of transformed co-ordinates. The manner in which the skulls of various
Crocodilians differ from one another may be sufficiently illustrated by
three or four examples.

[Illustration: Fig. 383. A, _Crocodilus porosus_. B, _C. americanus_.
C, _Notosuchus terrestris_.]

Let us take one of the typical modern crocodiles as our standard of
form, e.g. _C. porosus_, and inscribe it, as in Fig. 383, _a_, in the
usual Cartesian co-ordinates. By deforming the rectangular network
into a triangular system, with the apex of the triangle a little
way in front of the snout, as in _b_, we pass to such a form as _C.
americanus_. By an exaggeration of the same process we at once get
an approximation to the form of one of the sharp-snouted, {753} or
longirostrine, crocodiles, such as the genus _Tomistoma_; and, in
the species figured, the oblique position of the orbits, the arched
contour of the occipital border, and certain other characters suggest a
certain amount of curvature, such as I have represented in the diagram
(Fig. 383, _b_), on the part of the horizontal co-ordinates. In the
still more elongated skull of such a form as the Indian Gavial, the
whole skull has undergone a great longitudinal extension, or, in other
words, the ratio of _x_/_y_ is greatly diminished; and this extension
is not uniform, but is at a maximum in the region of the nasal and
maxillary bones. This especially elongated region is at the same time
narrowed in an exceptional degree, and its excessive narrowing is
represented by a curvature, convex towards the median axis, on the
part of the vertical ordinates. Let us take as a last illustration
one of the Mesozoic crocodiles, the little _Notosuchus_, from the
Cretaceous formation. This little crocodile is very different from
our type in the proportions of its skull. The region of the snout,
in front of and including the frontal bones, is greatly shortened;
from constituting fully two-thirds of the whole length of the
skull in _Crocodilus_, it now constitutes less than half, or, say,
three-sevenths of the whole; and the whole skull, and especially its
posterior part, is curiously compact, broad, and squat. The orbit is
unusually large. If in the diagram of this skull we select a number of
points obviously corresponding {754} to points where our rectangular
co-ordinates intersect particular bones or other recognisable features
in our typical crocodile, we shall easily discover that the lines
joining these points in _Notosuchus_ fall into such a co-ordinate
network as that which is represented in Fig. 383, _c_. To all intents
and purposes, then, this not very complex system, representing
one harmonious “deformation,” accounts for _all_ the differences
between the two figures, and is sufficient to enable one at any time
to reconstruct a detailed drawing, bone for bone, of the skull of
_Notosuchus_ from the model furnished by the common crocodile.

[Illustration: Fig. 384. Pelvis of (A) _Stegosaurus_; (B)
_Camptosaurus_.]

The many diverse forms of Dinosaurian reptiles, all of which manifest
a strong family likeness underlying much superficial diversity,
furnish us with plentiful material for comparison by the method of
transformations. As an instance, I have figured the pelvic bones of
_Stegosaurus_ and of _Camptosaurus_ (Fig. 384, _a_, _b_) to show that,
when the former is taken as our Cartesian type, a slight curvature
and an approximately logarithmic extension of the _x_-axis brings us
easily to the configuration of the other. In the original specimen of
_Camptosaurus_ described by Marsh[658], the anterior portion of the
iliac bone is missing; and in Marsh’s restoration this part of the
bone is drawn as though it came somewhat abruptly to a sharp point.
In my figure I {755} have completed this missing part of the bone
in harmony with the general co-ordinate network which is suggested
by our comparison of the two entire pelves; and I venture to think
that the result is more natural in appearance, and more likely to be
correct than was Marsh’s conjectural restoration. It would seem, in
fact, that there is an obvious field for the employment of the method
of co-ordinates in this task of reproducing missing portions of a
structure to the proper scale and in harmony with related types. To
this subject we shall presently return.

[Illustration: Fig. 385. Shoulder-girdle of _Cryptocleidus_. _a_,
young; _b_, adult.]

[Illustration: Fig. 386. Shoulder-girdle of _Ichthyosaurus_.]

In Fig. 385, _a_, _b_, I have drawn the shoulder-girdle of
_Cryptocleidus_, a Plesiosaurian reptile, half-grown in the one case
and full-grown in the other. The change of form during growth in
this region of the body is very considerable, and its nature is well
brought out by the two co-ordinate systems. In Fig. 386 I have drawn
the shoulder-girdle of an Ichthyosaur, referring it to _Cryptocleidus_
as a standard of comparison. The interclavicle, which is present in
_Ichthyosaurus_, is minute and hidden in _Cryptocleidus_; but the
numerous other differences between the two {756} forms, chief among
which is the great elongation in _Ichthyosaurus_ of the two clavicles,
are all seen by our diagrams to be part and parcel of one general and
systematic deformation.

[Illustration: Fig. 387. _a_, Skull of _Dimorphodon_. _b_, Skull of
_Pteranodon_.]

Before we leave the group of reptiles we may glance at the very
strangely modified skull of _Pteranodon_, one of the extinct flying
reptiles, or Pterosauria. In this very curious skull the region of
the jaws, or beak, is greatly elongated and pointed; the occipital
bone is drawn out into an enormous backwardly-directed crest; the
posterior part of the lower jaw is similarly produced backwards; the
orbit is small; and the quadrate bone is strongly inclined downwards
and forwards. The whole skull has a configuration which stands,
apparently, in the strongest possible contrast to that of a more normal
Ornithosaurian such as _Dimorphodon_. But if we inscribe the latter in
Cartesian coordinates (Fig. 387, _a_), and refer our _Pteranodon_ to
a system of oblique co-ordinates (_b_), in which the two co-ordinate
systems of parallel lines become each a pencil of diverging rays, we
make manifest a correspondence which extends uniformly throughout all
parts of these very different-looking skulls.

――――――――――

[Illustration: Fig. 388. Pelvis of _Archaeopteryx_.]

[Illustration: Fig. 389. Pelvis of _Apatornis_.]

We have dealt so far, and for the most part we shall continue to
deal, with our co-ordinate method as a means of comparing one known
structure with another. But it is obvious, as I have said, {757}
that it may also be employed for drawing hypothetical structures,
on the assumption that they have varied from a known form in some
definite way. And this process may be especially useful, and will be
most obviously legitimate, when we apply it to the particular case of
representing intermediate stages between two forms which are actually
known to exist, in other words, of reconstructing the transitional
stages through which the course of evolution must have successively
travelled if it has brought about the change from some ancestral type
to its presumed descendant. Some little time ago I sent to my friend,
Mr Gerhard Heilmann of Copenhagen, a few of my own rough co-ordinate
diagrams, including some in which the pelves of certain ancient and
primitive birds were compared one with another. Mr Heilmann, who is
both a skilled draughtsman and an able morphologist, returned me a set
of diagrams which are a vast improvement on my own, {758} and which
are reproduced in Figs. 388–393. Here we have, as extreme cases, the
pelvis of _Archaeopteryx_, the most ancient of known birds, and that of
_Apatornis_, one of the fossil “toothed”

[Illustration: Fig. 390. The co-ordinate systems of Figs. 388 and 389,
with three intermediate systems interpolated.]

[Illustration: Fig. 391. The first intermediate co-ordinate network,
with its corresponding inscribed pelvis.]

birds from the North American Cretaceous formations—a bird shewing
some resemblance to the modern terns. The pelvis of _Archaeopteryx_
is taken as our type, and referred accordingly to {759} Cartesian
co-ordinates (Fig. 388); while the corresponding coordinates of the
very different pelvis of _Apatornis_ are represented in Fig. 389. In
Fig. 390 the outlines of these two co-ordinate systems are superposed
upon one another, and those of three intermediate and equidistant
co-ordinate systems are interpolated between them. From each of these
latter systems, so determined by direct interpolation, a complete
co-ordinate diagram is drawn, and the corresponding outline of a pelvis
is found from each of these systems of co-ordinates, as in Figs. 391,
392. Finally, in Fig. 393 the complete series is represented, beginning
with the known pelvis of _Archaeopteryx_, and leading up by our three
intermediate hypothetical types to the known pelvis of _Apatornis_.

[Illustration: Fig. 392. The second and third intermediate co-ordinate
networks, with their corresponding inscribed pelves.]

――――――――――

Among mammalian skulls I will take two illustrations only, one drawn
from a comparison of the human skull with that of the higher apes, and
another from the group of Perissodactyle {760} Ungulates, the group
which includes the rhinoceros, the tapir, and the horse.

[Illustration: Fig. 393. The pelves of _Archaeopteryx_ and of
_Apatornis_, with three transitional types interpolated between them.]

Let us begin by choosing as our type the skull of _Hyrachyus agrarius_,
Cope, from the Middle Eocene of North America, as figured by Osborn in
his Monograph of the Extinct Rhinoceroses[659] (Fig. 394).

[Illustration: Fig. 394. Skull of _Hyrachyus agrarius_. (After
Osborn.)]

[Illustration: Fig. 395. Skull of _Aceratherium tridactylum_. (After
Osborn.)]

The many other forms of primitive rhinoceros described in the monograph
differ from _Hyrachyus_ in various details—in the characters of the
teeth, sometimes in the number of the toes, and so forth; and they also
differ very considerably in the general {761} appearance of the skull.
But these differences in the conformation of the skull, conspicuous
as they are at first sight, will be found easy to bring under the
conception of a simple and homogeneous transformation, such as would
result from the application of some not very complicated stress. For
instance, the corresponding co-ordinates of _Aceratherium tridactylum_,
as shown in Fig. 395, indicate that the essential difference between
this skull and the former one may be summed up by saying that the long
axis of the skull of _Aceratherium_ has undergone a slight double
curvature, while the upper parts of the skull have at the same time
been {762} subject to a vertical expansion, or to growth in somewhat
greater proportion than the lower parts. Precisely the same changes, on
a somewhat greater scale, give us the skull of an existing rhinoceros.

[Illustration: Fig. 396. Occipital view of the skulls of various
extinct rhinoceroses (_Aceratherium_ spp.). (After Osborn.)]

Among the species of _Aceratherium_, the posterior, or occipital,
view of the skull presents specific differences which are perhaps
more conspicuous than those furnished by the side view; and these
differences are very strikingly brought out by the series of conformal
transformations which I have represented in Fig. 396. In this case it
will perhaps be noticed that the correspondence is not always quite
accurate in small details. It could easily have been made much more
accurate by giving a slightly sinuous curvature to certain of the
co-ordinates. But as they stand, the correspondence indicated is very
close, and the simplicity of the figures illustrates all the better the
general character of the transformation.

By similar and not more violent changes we pass easily to such allied
forms as the Titanotheres (Fig. 397); and the well-known series of
species of _Titanotherium_, by which Professor Osborn has {763}
illustrated the evolution of this genus, constitutes a simple and
suitable case for the application of our method.

But our method enables us to pass over greater gaps than these, and
to discern the general, and to a very large extent even the detailed,
resemblances between the skull of the rhinoceros and those of the
tapir or the horse. From the Cartesian co-ordinates in which we have
begun by inscribing the skull of a primitive rhinoceros, we pass to
the tapir’s skull (Fig. 398), firstly, by converting the rectangular
into a triangular network, by which we represent the depression of the
anterior and the progressively increasing elevation of the posterior
part of the skull; and secondly, by giving to the vertical ordinates a
curvature such as to bring about a certain longitudinal compression, or
condensation, in the forepart of the skull, especially in the nasal and
orbital regions.

[Illustration: Fig. 397. _Titanotherium robustum_.]

[Illustration: Fig. 398. Tapir’s skull.]

The conformation of the horse’s skull departs from that of our
primitive Perissodactyle (that is to say our early type of rhinoceros,
_Hyrachyus_) in a direction that is nearly the opposite of that taken
by _Titanotherium_ and by the recent species of rhinoceros. For we
perceive, by Fig. 399, that the horizontal co-ordinates, which in these
latter cases became transformed into curves with the concavity upwards,
are curved, in the case of the horse, in the opposite direction. And
the vertical ordinates, which are also curved, somewhat in the same
fashion as in the tapir, are very nearly equidistant, instead of being,
as in that animal, crowded together anteriorly. Ordinates and abscissae
form an oblique {764} system, as is shown in the figure. In this case
I have attempted to produce the network beyond the region which is
actually required to include the diagram of the horse’s skull, in order
to show better the form of the general transformation, with a part only
of which we have actually to deal.

[Illustration: Fig. 399. Horse’s skull.]

[Illustration: Fig. 400. Rabbit’s skull.]

It is at first sight not a little surprising to find that we can pass,
by a cognate and even simpler transformation, from our Perissodactyle
skulls to that of the rabbit; but the fact that we can easily do so is
a simple illustration of the undoubted affinity which exists between
the Rodentia, especially the family of the Leporidae, and the more
primitive Ungulates. For my part, I would go further; for I think there
is strong reason to believe that the Perissodactyles are more closely
related to the Leporidae than the former are to the other Ungulates,
or than the Leporidae are to the rest of the Rodentia. Be that as it
may, it is obvious from Fig. 400 that the rabbit’s skull conforms
to a system of {765} co-ordinates corresponding to the Cartesian
co-ordinates in which we have inscribed the skull of _Hyrachyus_,
with the difference, firstly, that the horizontal ordinates of the
latter are transformed into equidistant curved lines, approximately
arcs of circles, with their concavity directed downwards; and
secondly, that the vertical ordinates are transformed into a pencil
of rays approximately orthogonal to the circular arcs. In short,
the configuration of the rabbit’s skull is derived from that of our
primitive rhinoceros by the unexpectedly simple process of submitting
the latter to a

[Illustration: Fig. 401. _A_, outline diagram of the Cartesian
co-ordinates of the skull of _Hyracotherium_ or _Eohippus_, as shewn
in Fig. 402, A. _H_, outline of the corresponding projection of the
horse’s skull. _B_–_G_, intermediate, or interpolated, outlines.]

strong and uniform flexure in the downward direction (cf. Fig. 358,
p. 731). In the case of the rabbit the configuration of the individual
bones does not conform quite so well to the general transformation as
it does when we are comparing the several Perissodactyles one with
another; and the chief departures from conformity will be found in the
size of the orbit and in the outline of the immediately surrounding
bones. The simple fact is that the relatively enormous eye of the
rabbit constitutes an independent variation, which cannot be brought
into the general and fundamental transformation, but must be dealt with
{768} separately. The enlargement of the eye, like the modification
in form and number of the teeth, is a separate phenomenon, which
supplements but in no way contradicts our general comparison of the
skulls taken in their entirety.

[Illustration: 〈two printed pages〉

Fig. 402. _A_, skull of _Hyracotherium_, from the Eocene, after W. B.
Scott; _H_, skull of horse, represented as a co-ordinate transformation
of that of _Hyracotherium_, and to the same scale of magnitude;
_B_–_G_, various artificial or imaginary types, reconstructed as
intermediate stages between _A_ and _H_; _M_, skull of _Mesohippus_,
from the Oligocene, after Scott, for comparison with _C_; _P_, skull of
_Protohippus_, from the Miocene, after Cope, for comparison with _E_;
_Pp_, lower jaw of _Protohippus placidus_ (after Matthew and Gidley),
for comparison with _F_; _Mi_, _Miohippus_ (after Osborn), _Pa_,
_Parahippus_ (after Peterson), shewing resemblance, but less perfect
agreement, with _C_ and _D_.]

――――――――――

Before we leave the Perissodactyla and their allies, let us look a
little more closely into the case of the horse and its immediate
relations or ancestors, doing so with the help of a set of diagrams
which I again owe to Mr Gerard Heilmann[660]. Here we start afresh,
with the skull (Fig. 402, _A_) of _Hyracotherium_ (or _Eohippus_),
inscribed in a simple Cartesian network. At the other end of the
series (_H_) is a skull of Equus, in its own corresponding network;
and the intermediate stages (_B_–_G_) are all drawn by direct and
simple interpolation, as in Mr Heilmann’s former series of drawings of
_Archaeopteryx_ and _Apatornis_. In this present case, the relative
magnitudes are shewn, as well as the forms, of the several skulls.
Alongside of these reconstructed diagrams, are set figures of certain
extinct “horses” (Equidae or Palaeotheriidae), and in two cases,
viz. _Mesohippus_ and _Protohippus_ (_M_, _P_), it will be seen that
the actual fossil skull coincides in the most perfect fashion with
one of the hypothetical forms or stages which our method shews to be
implicitly involved in the transition from _Hyracotherium_ to _Equus_.
In a third case, that of _Parahippus_ (_Pa_), the correspondence (as
Mr Heilmann points out) is by no means exact. The outline of this
skull comes nearest to that of the hypothetical transition stage
_D_, but the “fit” is now a bad one; for the skull of _Parahippus_
is evidently a longer, straighter and narrower skull, and differs in
other minor characters besides. In short, though some writers have
placed _Parahippus_ in the direct line of descent between _Equus_ and
_Eohippus_, we see at once that there is no place for it there, and
that it must, accordingly, represent a somewhat divergent branch or
offshoot of the Equidae[661]. It may be noticed, especially in the
case of _Protohippus_ {769} (_P_), that the configuration of the
angle of the jaw does not tally quite so accurately with that of our
hypothetical diagrams as do other parts of the skull. As a matter of
fact, this region is somewhat variable, in different species of a
genus, and even in different individuals of the same species; in the
small figure (_Pp_) of _Protohippus placidus_ the correspondence is
more exact.

[Illustration: Fig. 403. Human scapulae (after Dwight). _A_, Caucasian;
_B_, Negro; _C_, North American Indian (from Kentucky Mountains).]

In considering this series of figures we cannot but be struck, not
only with the regularity of the succession of “transformations,” but
also with the slight and inconsiderable differences which separate
the known and recorded stages, and even the two extremes of the whole
series. These differences are no greater (save in regard to actual
magnitude) than those between one human skull and another, at least
if we take into account the older or remoter races; and they are
again no greater, but if anything less, than the range of variation,
racial and individual, in certain other human bones, for instance the
scapula[662].

The variability of this latter bone is great, but it is neither {770}
surprising nor peculiar; for it is linked with all the considerations
of mechanical efficiency and functional modification which we dealt
with in our last chapter. The scapula occupies, as it were, a focus
in a very important field of force; and the lines of force converging
on it will be very greatly modified by the varying development of the
muscles over a large area of the body and of the uses to which they are
habitually put.

[Illustration: Fig. 404. Human skull.]

[Illustration: Fig. 405. Co-ordinates of chimpanzee’s skull, as a
projection of the Cartesian co-ordinates of Fig. 404.]

Let us now inscribe in our Cartesian co-ordinates the outline of a
human skull (Fig. 404), for the purpose of comparing it with the
skulls of some of the higher apes. We know beforehand that the main
differences between the human and the simian types depend upon the
enlargement or expansion of the brain and braincase in man, and the
relative diminution or enfeeblement of his jaws. Together with these
changes, the “facial angle” increases from an oblique angle to nearly a
right angle in man, {771} and the configuration of every constituent
bone of the face and skull undergoes an alteration. We do not know to
begin with, and we are not shewn by the ordinary methods of comparison,
how far these various changes form part of one harmonious and congruent
transformation, or whether we are to look, for instance, upon the
changes undergone by the frontal, the occipital, the maxillary, and
the mandibular regions as a congeries of separate modifications or
independent variants. But as soon as we have marked out a number of
points in the gorilla’s or chimpanzee’s skull, corresponding with
those which our co-ordinate network intersected in the human skull,
we find that these corresponding points may be at once linked up by
smoothly curved lines of intersection, which form a new system of
co-ordinates and constitute a simple “projection” of our human skull.
The network represented in Fig. 405 constitutes such a projection of
the human skull on what we may call, figuratively speaking, the “plane”
of the chimpanzee; and the full diagram in Fig. 406 demonstrates the
correspondence. In Fig. 407 I have shewn the similar deformation in
the case of a baboon, and it is obvious that the transformation is of
precisely the same order, and differs only in an increased intensity or
degree of deformation.

[Illustration: Fig. 406. Skull of chimpanzee.]

[Illustration: Fig. 407. Skull of baboon.]

In both dimensions, as we pass from above downwards and from behind
forwards, the corresponding areas of the network are seen to increase
in a gradual and approximately logarithmic order in the lower as
compared with the higher type of skull; and, in short, it becomes
at once manifest that the modifications of jaws, braincase, and the
regions between are all portions of one continuous and integral
process. It is of course easy to draw the {772} inverse diagrams,
by which the Cartesian co-ordinates of the ape are transformed into
curvilinear and non-equidistant co-ordinates in man.

From this comparison of the gorilla’s or chimpanzee’s with the
human skull we realise that an inherent weakness underlies the
anthropologist’s method of comparing skulls by reference to a small
number of axes. The most important of these are the “facial” and
“basicranial” axes, which include between them the “facial angle.”
But it is, in the first place, evident that these axes are merely the
principal axes of a system of co-ordinates, and that their restricted
and isolated use neglects all that can be learned from the filling in
of the rest of the co-ordinate network. And, in the second place, the
“facial axis,” for instance, as ordinarily used in the anthropological
comparison of one human skull with another, or of the human skull with
the gorilla’s, is in all cases treated as a straight line; but our
investigation has shewn that rectilinear axes only meet the case in
the simplest and most closely related transformations; and that, for
instance, in the anthropoid skull no rectilinear axis is homologous
with a rectilinear axis in a man’s skull, but what is a straight line
in the one has become a certain definite curve in the other.

Mr Heilmann tells me that he has tried, but without success, to obtain
a transitional series between the human skull and some prehuman,
anthropoid type, which series (as in the case of the Equidae) should
be found to contain other known types in direct linear sequence.
It appears impossible, however, to obtain such a series, or to
pass by successive and continuous gradations through such forms as
Mesopithecus, Pithecanthropus, _Homo neanderthalensis_, and the lower
or higher races of modern man. The failure is not the fault of our
method. It merely indicates that no one straight line of descent, or
of consecutive transformation, exists; but on the contrary, that among
human and anthropoid types, recent and extinct, we have to do with a
complex problem of divergent, rather than of continuous, variation.
And in like manner, easy as it is to correlate the baboon’s and
chimpanzee’s skulls severally with that of man, and easy as it is to
see that the chimpanzee’s skull is much nearer to the human type than
is the baboon’s, it is also not difficult to perceive that the series
is not, {773} strictly speaking, continuous, and that neither of
our two apes lies _precisely_ on the same direct line or sequence of
deformation by which we may hypothetically connect the other with man.

[Illustration: Fig. 408. Skull of dog, compared with the human skull of
Fig. 404.]

As a final illustration I have drawn the outline of a dog’s skull (Fig.
408), and inscribed it in a network comparable with the Cartesian
network of the human skull in Fig. 404. Here we attempt to bridge over
a wider gulf than we have crossed in any of our former comparisons.
But, nevertheless, it is obvious that our method still holds good, in
spite of the fact that there are various specific differences, such as
the open or closed orbit, etc., which have to be separately described
and accounted for. We see that the chief essential differences in plan
between the dog’s skull and the man’s lie in the fact that, relatively
speaking, the former tapers away in front, a triangular taking the
place of a rectangular conformation; secondly, that, coincident with
the tapering off, there is a progressive elongation, or pulling out,
of the whole forepart of the skull; and lastly, as a minor difference,
that the straight vertical ordinates of the human skull become
curved, with their convexity directed forwards, in the dog. While
the net result is that in the dog, just as in the chimpanzee, the
brain-pan is smaller and the jaws are larger than in man, it is now
conspicuously evident that the co-ordinate network of the ape is by no
means intermediate between those which fit the other two. The mode of
deformation is on different lines; and, while it may be correct to say
that the chimpanzee and the baboon are more brute-like, it would be
by no means accurate to assert that they are more dog-like, than man.
{774}

In this brief account of co-ordinate transformations and of their
morphological utility I have dealt with plane co-ordinates only, and
have made no mention of the less elementary subject of co-ordinates in
three-dimensional space. In theory there is no difficulty whatsoever
in such an extension of our method; it is just as easy to refer the
form of our fish or of our skull to the rectangular co-ordinates _x_,
_y_, _z_, or to the polar co-ordinates ξ, η, ζ, as it is to refer
their plane projections to the two axes to which our investigation has
been confined. And that it would be advantageous to do so goes without
saying; for it is the shape of the solid object, not that of the mere
drawing of the object, that we want to understand; and already we have
found some of our easy problems in solid geometry leading us (as in
the case of the form of the bivalve and even of the univalve shell)
quickly in the direction of co-ordinate analysis and the theory of
conformal transformations. But this extended theme I have not attempted
to pursue, and it must be left to other times, and to other hands.
Nevertheless, let us glance for a moment at the sort of simple cases,
the simplest possible cases, with which such an investigation might
begin; and we have found our plane co-ordinate systems so easily and
effectively applicable to certain fishes that we may seek among them
for our first and tentative introduction to the three-dimensional field.

It is obvious enough that the same method of description and analysis
which we have applied to one plane, we may apply to another: drawing
by observation, and by a process of trial and error, our various
cross-sections and the co-ordinate systems which seem best to
correspond. But the new and important problem which now emerges is to
_correlate_ the deformation or transformation which we discover in
one plane with that which we have observed in another: and at length,
perhaps, after grasping the general principles of such correlation, to
forecast approximately what is likely to take place in the other two
planes of reference when we are acquainted with one, that is to say, to
determine the values along one axis in terms of the other two.

Let us imagine a common “round” fish, and a common “flat” fish, such
as a haddock and a plaice. These two fishes are not as nicely adapted
for comparison by means of plane co-ordinates as {775} some which
we have studied, owing to the presence of essentially unimportant,
but yet conspicuous differences in the position of the eyes, or in
the number of the fins,—that is to say in the manner in which the
continuous dorsal fin of the plaice appears in the haddock to be cut
or scolloped into a number of separate fins. But speaking broadly, and
apart from such minor differences as these, it is manifest that the
chief factor in the case (so far as we at present see) is simply the
broadening out of the plaice’s body, as compared with the haddock’s,
in the dorso-ventral direction, that is to say, along the _y_ axis;
in other words, the ratio _x_/_y_ is much less, (and indeed little
more than half as great), in the haddock than in the plaice. But we
also recognise at once that while the plaice (as compared with the
haddock) is expanded in one direction, it is also flattened, or thinned
out, in the other: _y_ increases, but _z_ diminishes, relatively to
_x_. And furthermore, we soon see that this is a common or even a
general phenomenon. The high, expanded body in our Antigonia or in
our sun-fish is at the same time flattened or _compressed_ from side
to side, in comparison with the related fishes which we have chosen
as standards of reference or comparison; and conversely, such a fish
as the skate, while it is expanded from side to side in comparison
with a shark or dogfish, is at the same time flattened or _depressed_
in its vertical section. We proceed then, to enquire whether there
be any simple relation of _magnitude_ discernible between these twin
factors of expansion and compression; and the very fact that the two
dimensions tend to vary _inversely_ already assures us that, in the
general process of deformation, the _volume_ is less affected than
are the _linear dimensions_. Some years ago, when I was studying the
length-weight co-efficient in fishes (of which we have already spoken
in Chap. III, p. 98), that is to say the coefficient _k_ in the formula
_W_ = _k_ _L_^3, or _k_ = _W_/_L_^3, I was not a little surprised to
find that _k_ was all but identical in two such different looking
fishes as our haddock and our plaice: thus indicating that these two
fishes, little as they resemble one another externally (though they
belong to two closely related families), have approximately the same
_volume_ when they are equal in _length_; or, in other words, that the
extent to which the plaice’s body has become expanded or broadened is
_just about {776} compensated for_ by the extent to which it has also
got flattened or thinned. In short, if we could permit ourselves to
conceive of a haddock being directly transformed into a plaice, a very
large part of the change would be simply accounted for by supposing the
former fish to be “rolled out,” as a baker rolls a piece of dough. This
is, as it were, an extreme case of the _balancement des organes_, or
“compensation of parts.”

Simple Cartesian co-ordinates will not suffice very well to compare
the haddock with the plaice, for the deformation undergone by the
former in comparison with the latter is more on the lines of that
by which we have compared our Antigonia with our Polyprion; that is
to say, the expansion is greater towards the middle of the fish’s
length, and dwindles away towards either end. But again simplifying our
illustration to the utmost, and being content with a rough comparison,
we may assert that, when haddock and plaice are brought to the same
standard of length, we can inscribe them both (approximately) in
rectangular co-ordinate networks, such that _Y_ in the plaice is about
twice as great as _y_ in the haddock. But if the volumes of the two
fishes be equal, this is as much as to say that _xyz_ in the one case
(or rather the summation of all these values) is equal to _XYZ_ in the
other; and therefore (since _X_ = _x_, and _Y_ = 2_y_), it follows
that _Z_ = _z_/2. When we have drawn our vertical transverse section
of the haddock (or projected that fish in the _yz_ plane), we have
reason accordingly to anticipate that we can draw a similar projection
(or section) of the plaice by simply doubling the _y_’s and halving
the _z_’s: and, very approximately, this turns out to be the case. The
plaice is (in round numbers) just about twice as broad and also just
about half as thick as the haddock; and therefore the ratio of breadth
to thickness (or _y_ to _z_) is just about four times as great in the
one case as in the other.

It is true that this simple, or simplified, illustration carries us
but a very little way, and only half prepares us for much greater
complications. For instance, we have no right or reason to presume that
the equality of weights, or volumes, is a common, much less a general
rule. And again, in all cases of more complex deformation, such as
that by which we have compared Diodon with the sunfish, we must be
prepared for very much more {777} recondite methods of comparison and
analysis, leading doubtless to very much more complicated results.
In this last case, of Diodon and the sunfish, we have seen that the
vertical _expansion_ of the latter as compared with the former fish,
increases rapidly as we go backwards towards the tail; but we can by no
means say that the lateral _compression_ increases in like proportion.
If anything, it would seem that the said expansion and compression tend
to vary inversely; for the Diodon is very thick in front and greatly
thinned away behind, while the flattened sunfish is more nearly of the
same thickness all the way along. Interesting as the whole subject is
we must meanwhile leave it alone; recognising, however, that if the
difficulties of description and representation could be overcome, it is
by means of such co-ordinates in space that we should at last obtain an
adequate and satisfying picture of the processes of deformation and of
the directions of growth[663].

{778}




EPILOGUE.


In the beginning of this book I said that its scope and treatment were
of so prefatory a kind that of other preface it had no need; and now,
for the same reason, with no formal and elaborate conclusion do I bring
it to a close. The fact that I set little store by certain postulates
(often deemed to be fundamental) of our present-day biology the reader
will have discovered and I have not endeavoured to conceal. But it is
not for the sake of polemical argument that I have written, and the
doctrines which I do not subscribe to I have only spoken of by the
way. My task is finished if I have been able to shew that a certain
mathematical aspect of morphology, to which as yet the morphologist
gives little heed, is interwoven with his problems, complementary to
his descriptive task, and helpful, nay essential, to his proper study
and comprehension of Form. _Hic artem remumque repono._

And while I have sought to shew the naturalist how a few mathematical
concepts and dynamical principles may help and guide him, I have
tried to shew the mathematician a field for his labour,—a field which
few have entered and no man has explored. Here may be found homely
problems, such as often tax the highest skill of the mathematician, and
reward his ingenuity all the more for their trivial associations and
outward semblance of simplicity.

That I am no skilled mathematician I have had little need to confess,
but something of the use and beauty of mathematics I think I am able to
understand. I know that in the study of material things, number, order
and position are the threefold clue to exact knowledge; that these
three, in the mathematician’s hands, furnish the “first outlines for
a sketch of the Universe”; that by square and circle we are helped,
like Emile Verhaeren’s carpenter, to conceive “Les lois indubitables et
fécondes Qui sont la règle et la clarté du monde.”

For the harmony of the world is made manifest in Form and Number, and
the heart and soul and all the poetry of Natural {779} Philosophy
are embodied in the concept of mathematical beauty. A greater than
Verhaeren had this in mind when he told of “the golden compasses,
prepared In God’s eternal store.” A greater than Milton had magnified
the theme and glorified Him “who sitteth upon the circle of the earth,”
saying: He measureth the waters in the hollow of his hand, he meteth
out the heavens with his span, he comprehendeth the dust of the earth
in a measure.

Moreover the perfection of mathematical beauty is such (as Maclaurin
learned of the bee), that whatsoever is most beautiful and regular is
also found to be most useful and excellent.

The living and the dead, things animate and inanimate, we dwellers
in the world and this world wherein we dwell,—πάντα γα μὰν τὰ
γιγνωσκόμενα,—are bound alike by physical and mathematical law.
“Conterminous with space and coeval with time is the kingdom of
Mathematics; within this range her dominion is supreme; otherwise than
according to her order nothing can exist, and nothing takes place in
contradiction to her laws.” So said, some forty years ago, a certain
mathematician; and Philolaus the Pythagorean had said much the same.

But with no less love and insight has the science of Form and Number
been appraised in our own day and generation by a very great Naturalist
indeed:—by that old man eloquent, that wise student and pupil of
the ant and the bee, who died but yesterday, and who in his all but
saecular life tasted of the firstfruits of immortality; who curiously
conjoined the wisdom of antiquity with the learning of to-day; whose
Provençal verse seems set to Dorian music; in whose plainest words is a
sound as of bees’ industrious murmur; and who, being of the same blood
and marrow with Plato and Pythagoras, saw in Number “la clef de la
voûte,” and found in it “le comment et le pourquoi des choses.”




NOTES:

[1] These sayings of Kant and of Du Bois, and others like to them,
have been the text of many discourses: see, for instance, Stallo’s
_Concepts_, p. 21, 1882; Höber, _Biol. Centralbl._ XIX, p. 284, 1890,
etc. Cf. also Jellett, _Rep. Brit. Ass._ 1874, p. 1.

[2] “Quum enim mundi universi fabrica sit perfectissima, atque a
Creatore sapientissimo absoluta, nihil omnino in mundo contingit in quo
non maximi minimive ratio quaepiam eluceat; quamobrem dubium prorsus
est nullum quin omnes mundi effectus ex causis finalibus, ope methodi
maximorum et minimorum, aeque feliciter determinari queant atque ex
ipsis causis efficientibus.” _Methodus inveniendi_, etc. 1744 (_cit._
Mach, _Science of Mechanics_, 1902, p. 455).

[3] Cf. Opp. (ed. Erdmann), p. 106, “Bien loin d’exclure les causes
finales..., c’est de là qu’il faut tout déduire en Physique.”

[4] Cf. p. 162. “La force vitale dirige des phénomènes qu’elle ne
produit pas: les agents physiques produisent des phénomènes qu’ils ne
dirigent pas.”

[5] It is now and then conceded with reluctance. Thus Enriques, a
learned and philosophic naturalist, writing “della economia di sostanza
nelle osse cave” (_Arch. f. Entw. Mech._ XX, 1906), says “una certa
impronta di teleologismo quà e là è rimasta, mio malgrado, in questo
scritto.”

[6] Cf. Cleland, On Terminal Forms of Life, _J. Anat. and Phys._ XVIII,
1884.

[7] Conklin, Embryology of Crepidula, _Journ. of Morphol._ XIII, p.
203, 1897; Lillie, F. R., Adaptation in Cleavage, _Woods Holl Biol.
Lectures_, pp. 43–67, 1899.

[8] I am inclined to trace back Driesch’s teaching of Entelechy to no
less a person than Melanchthon. When Bacon (_de Augm._ IV, 3) states
with disapproval that the soul “has been regarded rather as a function
than as a substance,” R. L. Ellis points out that he is referring to
Melanchthon’s exposition of the Aristotelian doctrine. For Melanchthon,
whose view of the peripatetic philosophy had long great influence in
the Protestant Universities, affirmed that, according to the true view
of Aristotle’s opinion, the soul is not a substance, but an ἑντελέχεια,
or _function_. He defined it as δύναμις _quaedam ciens actiones_—a
description all but identical with that of Claude Bernard’s “_force
vitale_.”

[9] Ray Lankester, _Encycl. Brit._ (9th ed.), art. “Zoology,” p. 806,
1888.

[10] Alfred Russel Wallace, especially in his later years, relied
upon a direct but somewhat crude teleology. Cf. his _World of Life, a
Manifestation of Creative Power, Directive Mind and Ultimate Purpose_,
1910.

[11] Janet, _Les Causes Finales_, 1876, p. 350.

[12] The phrase is Leibniz’s, in his _Théodicée_.

[13] Cf. (_int. al._) Bosanquet, The Meaning of Teleology, _Proc.
Brit. Acad._ 1905–6, pp. 235–245. Cf. also Leibniz (_Discours de
Métaphysique; Lettres inédites, ed._ de Careil, 1857, p. 354; _cit._
Janet, p. 643), “L’un et l’autre est bon, l’un et l’autre peut être
utile ... et les auteurs qui suivent ces routes différentes ne
devraient point se maltraiter: _et seq._”

[14] The reader will understand that I speak, not of the “severe and
diligent inquiry” of variation or of “fortuity,” but merely of the
easy assumption that these phenomena are a sufficient basis on which
to rest, with the all-powerful help of natural selection, a theory of
definite and progressive evolution.

[15] _Revue Philosophique._ XXXIII, 1892.

[16] This general principle was clearly grasped by Dr George Rainey (a
learned physician of St Bartholomew’s) many years ago, and expressed in
such words as the following: “......it is illogical to suppose that in
the case of vital organisms a distinct force exists to produce results
perfectly within the reach of physical agencies, especially as in many
instances no end could be attained were that the case, but that of
opposing one force by another capable of effecting exactly the same
purpose.” (On Artificial Calculi, _Q.J.M.S._ (_Trans. Microsc. Soc._),
VI, p. 49, 1858.) Cf. also Helmholtz, _infra cit._, p. 9.

[17] Whereby he incurred the reproach of Socrates, in the _Phaedo_.

[18] In a famous lecture (Conservation of Forces applied to Organic
Nature, _Proc. Roy. Instit._, April 12, 1861), Helmholtz laid it down,
as “the fundamental principle of physiology,” that “There may be other
agents acting in the living body than those agents which act in the
inorganic world; but those forces, as far as they cause chemical and
mechanical influence in the body, must be _quite of the same character_
as inorganic forces: in this at least, that their effects must be
ruled by necessity, and must always be the same when acting in the
same conditions; and so there cannot exist any arbitrary choice in the
direction of their actions.” It would follow from this, that, like the
other “physical” forces, they must be subject to mathematical analysis
and deduction. Cf. also Dr T. Young’s Croonian Lecture On the Heart and
Arteries, _Phil. Trans._ 1809, p. 1; _Coll. Works_, I, 511.

[19] _Ektropismus, oder die physikalische Theorie des Lebens_, Leipzig,
1910.

[20] Wilde Lecture, _Nature_, March 12, 1908; _ibid._ Sept. 6, 1900, p.
485; _Aether and Matter_, p. 288. Cf. also Lord Kelvin, _Fortnightly
Review_, 1892, p. 313.

[21] Joly, The Abundance of Life, _Proc. Roy. Dublin Soc._ VII, 1890;
and in _Scientific Essays_, etc. 1915, p. 60 _et seq._

[22] Papillon, _Histoire de la philosophie moderne_, I, p. 300.

[23] With the special and important properties of _colloidal_ matter we
are, for the time being, not concerned.

[24] Cf. Hans Przibram, _Anwendung elementarer Mathematik auf
Biologische Probleme_ (in Roux’s _Vorträge_, Heft III), Leipzig, 1908,
p. 10.

[25] The subject is treated from an engineering point of view by Prof.
James Thomson, Comparisons of Similar Structures as to Elasticity,
Strength, and Stability, _Trans. Inst. Engineers, Scotland_, 1876
(_Collected Papers_, 1912, pp. 361–372), and by Prof. A. Barr, _ibid._
1899; see also Rayleigh, _Nature_, April 22, 1915.

[26] Cf. Spencer, The Form of the Earth, etc., _Phil. Mag._ XXX, pp.
194–6, 1847; also _Principles of Biology_, pt. II, ch. I, 1864 (p. 123,
etc.).

[27] George Louis Lesage (1724–1803), well known as the author of one
of the few attempts to explain gravitation. (Cf. Leray, _Constitution
de la Matière_, 1869; Kelvin, _Proc. R. S. E._ VII, p. 577, 1872, etc.;
Clerk Maxwell, _Phil. Trans._ vol. 157, p. 50, 1867; art. “Atom,”
_Encycl. Brit._ 1875, p. 46.)

[28] Cf. Pierre Prévost, _Notices de la vie et des écrits de Lesage_,
1805; quoted by Janet, _Causes Finales_, app. III.

[29] Discorsi e Dimostrazioni matematiche, intorno à due nuove
scienze, attenenti alla Mecanica, ed ai Movimenti Locali: appresso gli
Elzevirii, MDCXXXVIII. _Opere_, ed. Favaro, VIII, p. 169 seq. Transl.
by Henry Crew and A. de Salvio, 1914, p. 130, etc. See _Nature_, June
17, 1915.

[30] So Werner remarked that Michael Angelo and Bramanti could not have
built of gypsum at Paris on the scale they built of travertin in Rome.

[31] Sir G. Greenhill, Determination of the greatest height to which
a Tree of given proportions can grow, _Cambr. Phil. Soc. Pr._ IV, p.
65, 1881, and Chree, _ibid._ VII, 1892. Cf. Poynting and Thomson’s
_Properties of Matter_, 1907, p 99.

[32] In like manner the wheat-straw bends over under the weight of
the loaded ear, and the tip of the cat’s tail bends over when held
upright,—not because they “possess flexibility,” but because they
outstrip the dimensions within which stable equilibrium is possible in
a vertical position. The kitten’s tail, on the other hand, stands up
spiky and straight.

[33] _Modern Painters._

[34] The stem of the giant bamboo may attain a height of 60 metres,
while not more than about 40 cm. in diameter near its base, which
dimensions are not very far short of the theoretical limits (A. J.
Ewart, _Phil. Trans._ vol. 198, p. 71, 1906).

[35] _Trans. Zool. Soc._ IV, 1850, p. 27.

[36] It would seem to be a common if not a general rule that marine
organisms, zoophytes, molluscs, etc., tend to be larger than the
corresponding and closely related forms living in fresh water. While
the phenomenon may have various causes, it has been attributed
(among others) to the simple fact that the forces of growth are less
antagonised by gravity in the denser medium (cf. Houssay, _La Forme
et la Vie_, 1900, p. 815). The effect of gravity on outward _form_ is
illustrated, for instance, by the contrast between the uniformly upward
branching of a sea-weed and the drooping curves of a shrub or tree.

[37] The analogy is not a very strict one. We are not taking account,
for instance, of a proportionate increase in thickness of the
boiler-plates.

[38] Let _L_ be the length, _S_ the (wetted) surface, _T_ the tonnage,
_D_ the displacement (or volume) of a ship; and let it cross the
Atlantic at a speed _V_. Then, in comparing two ships, similarly
constructed but of different magnitudes, we know that _L_ = _V_^2,
_S_ = _L_^2 = _V_^4, _D_ = _T_ = _L_^3 = _V_^6; also _R_ (resistance)
= _S_ ⋅ _V_^2 = _V_^6; _H_ (horse-power) = _R_ ⋅ _V_ = _V_^7; and the
coal (_C_) necessary for the voyage = _H_/_V_ = _V_^6. That is to
say, in ordinary engineering language, to increase the speed across
the Atlantic by 1 per cent. the ship’s length must be increased 2 per
cent., her tonnage or displacement 6 per cent., her coal-consumpt also
6 per cent., her horse-power, and therefore her boiler-capacity, 7 per
cent. Her bunkers, accordingly, keep pace with the enlargement of the
ship, but her boilers tend to increase out of proportion to the space
available.

[39] This is the result arrived at by Helmholtz, Ueber ein Theorem
geometrisch ähnliche Bewegungen flüssiger Körper betreffend, nebst
Anwendung auf das Problem Luftballons zu lenken, _Monatsber. Akad.
Berlin_, 1873, pp. 501–14. It was criticised and challenged (somewhat
rashly) by K. Müllenhof, Die Grösse der Flugflächen, etc., _Pflüger’s
Archiv_, XXXV, p. 407, XXXVI, p. 548, 1885.

[40] Cf. also Chabrier, Vol des Insectes, _Mém. Mus. Hist. Nat. Paris_,
VI–VIII, 1820–22.

[41] _Aerial Flight_, vol. II (_Aerodonetics_), 1908, p. 150.

[42] By Lanchester, _op. cit._ p. 131.

[43] Cf. _L’empire de l’air; ornithologie appliquée à l’aviation_. 1881.

[44] _De Motu Animalium_, I, prop. cciv, ed. 1685, p. 243.

[45] Harlé, On Atmospheric Pressure in past Geological Ages, _Bull.
Geol. Soc. Fr._ XI, pp. 118–121; or _Cosmos_, p. 30, July 8, 1911.

[46] _Introduction to Entomology_, 1826, II, p. 190. K. and S., like
many less learned authors, are fond of popular illustrations of the
“wonders of Nature,” to the neglect of dynamical principles. They
suggest, for instance, that if the white ant were as big as a man, its
tunnels would be “magnificent cylinders of more than three hundred
feet in diameter”; and that if a certain noisy Brazilian insect were
as big as a man, its voice would be heard all the world over: “so that
Stentor becomes a mute when compared with these insects!” It is an easy
consequence of anthropomorphism, and hence a common characteristic of
fairy-tales, to neglect the principle of dynamical, while dwelling on
the aspect of geometrical, similarity.

[47] I.e. the available energy of muscle, in ft.-lbs. per lb. of
muscle, is the same for all animals: a postulate which requires
considerable qualification when we are comparing very different _kinds_
of muscle, such as the insect’s and the mammal’s.

[48] Prop. clxxvii. Animalia minora et minus ponderosa majores saltus
efficiunt respectu sui corporis, si caetera fuerint paria.

[49] See also (_int. al._), John Bernoulli, _de Motu Musculorum_,
Basil., 1694; Chabry, Mécanisme du Saut, _J. de l’Anat. et de
la Physiol._ XIX, 1883; Sur la longueur des membres des animaux
sauteurs, _ibid._ XXI, p. 356, 1885; Le Hello, De l’action des organes
locomoteurs, etc., _ibid._ XXIX, p. 65–93, 1893, etc.

[50] Recherches sur la force absolue des muscles des Invertébrés,
_Bull. Acad. E. de Belgique_ (3), VI, VII, 1883–84; see also _ibid._
(2), XX, 1865, XXII, 1866; _Ann. Mag. N. H._ XVII, p. 139, 1866, XIX,
p. 95, 1867. The subject was also well treated by Straus-Dürckheim,
in his _Considérations générales sur l’anatomie comparée des animaux
articulés_, 1828.

[51] The fact that the limb tends to swing in pendulum-time was first
observed by the brothers Weber (_Mechanik der menschl. Gehwerkzeuge_,
Göttingen, 1836). Some later writers have criticised the statement
(e.g. Fischer, Die Kinematik des Beinschwingens etc., _Abh. math. phys.
Kl. k. Sächs. Ges._ XXV–XXVIII, 1899–1903), but for all that, with
proper qualifications, it remains substantially true.

[52] Quoted in Mr John Bishop’s interesting article in Todd’s
_Cyclopaedia_, III, p. 443.

[53] There is probably also another factor involved here: for in
bending, and therefore shortening, the leg we bring its centre of
gravity nearer to the pivot, that is to say, to the joint, and so the
muscle tends to move it the more quickly.

[54] _Proc. Psychical Soc._ XII, pp. 338–355, 1897.

[55] For various calculations of the increase of surface due to
histological and anatomical subdivision, see E. Babak, Ueber die
Oberflächenentwickelung bei Organismen, _Biol. Centralbl._ XXX, pp.
225–239, 257–267, 1910. In connection with the physical theory of
surface-energy, Wolfgang Ostwald has introduced the conception of
_specific surface_, that is to say the ratio of surface to volume, or
_S_/_V_. In a cube, _V_ = _l_^3, and _S_ = 6_l_^2; therefore _S_/_V_
= 6/_l_. Therefore if the side _l_ measure 6 cm., the ratio _S_/_V_
= 1, and such a cube may be taken as our standard, or unit of specific
surface. A human blood-corpuscle has, accordingly, a specific surface
of somewhere about 14,000 or 15,000. It is found in physical chemistry
that surface energy becomes an important factor when the specific
surface reaches a value of 10,000 or thereby.

[56] Though the entire egg is not increasing in mass, this is not to
say that its living protoplasm is not increasing all the while at the
expense of the reserve material.

[57] Cf. Tait, _Proc. R.S.E._ V, 1866, and VI, 1868.

[58] _Physiolog. Notizen_ (9), p. 425, 1895. Cf. Strasbürger, Ueber die
Wirkungssphäre der Kerne und die Zellgrösse, _Histolog. Beitr._ (5),
pp. 95–129, 1893; J. J. Gerassimow, Ueber die Grösse des Zellkernes,
_Beih. Bot. Centralbl._ XVIII, 1905; also G. Levi and T. Terni, Le
variazioni dell’ indice plasmatico-nucleare durante l’intercinesi,
_Arch. Ital. di Anat._ X, p. 545, 1911.

[59] _Arch. f. Entw. Mech._ IV, 1898, pp. 75, 247.

[60] Conklin, E. G., Cell-size and nuclear-size, _J. Exp. Zool._ XII.
pp. 1–98, 1912.

[61] Thus the fibres of the crystalline lens are of the same size in
large and small dogs; Rabl, _Z. f. w. Z._ LXVII, 1899. Cf. (_int. al._)
Pearson, On the Size of the Blood-corpuscles in Rana, _Biometrika_,
VI, p. 403, 1909. Dr Thomas Young caught sight of the phenomenon,
early in last century: “The solid particles of the blood do not by
any means vary in magnitude in the same ratio with the bulk of the
animal,” _Natural Philosophy_, ed. 1845, p. 466; and Leeuwenhoek and
Stephen Hales were aware of it a hundred years before. But in this
case, though the blood-corpuscles show no relation of magnitude to the
size of the animal, they do seem to have some relation to its activity.
At least the corpuscles in the sluggish Amphibia are much the largest
known to us, while the smallest are found among the deer and other
agile and speedy mammals. (Cf. Gulliver, _P.Z.S._ 1875, p. 474, etc.)
This apparent correlation may have its bearing on modern views of the
surface-condensation or adsorption of oxygen in the blood-corpuscles,
a process which would be greatly facilitated and intensified by the
increase of surface due to their minuteness.

[62] Cf. P. Enriques, La forma come funzione della grandezza: Ricerche
sui gangli nervosi degli Invertebrati, _Arch. f. Entw. Mech._ XXV, p.
655, 1907–8.

[63] While the difference in cell-volume is vastly less than that
between the volumes, and very much less also than that between
the surfaces, of the respective animals, yet there _is_ a certain
difference; and this it has been attempted to correlate with the
need for each cell in the many-celled ganglion of the larger
animal to possess a more complex “exchange-system” of branches,
for intercommunication with its more numerous neighbours. Another
explanation is based on the fact that, while such cells as continue to
divide throughout life tend to uniformity of size in all mammals, those
which do not do so, and in particular the ganglion cells, continue to
grow, and their size becomes, therefore, a function of the duration of
life. Cf. G. Levi, Studii sulla grandezza delle cellule, _Arch. Ital.
di Anat. e di Embryolog._ V, p. 291, 1906.

[64] Boveri. _Zellen-studien, V. Ueber die Abhängigkeit der Kerngrösse
und Zellenzahl der Seeigellarven von der Chromosomenzahl der
Ausgangszellen._ Jena, 1905.

[65] Recent important researches suggest that such ultra-minute
“filter-passers” are the true cause of certain acute maladies commonly
ascribed to the presence of much larger organisms; cf. Hort, Lakin and
Benians, The true infective Agent in Cerebrospinal Fever, etc., _J.
Roy. Army Med. Corps_, Feb. 1910.

[66] _Zur Erkenntniss der Kolloide_, 1905, p. 122; where there will be
found an interesting discussion of various molecular and other minute
magnitudes.

[67] _Encyclopaedia Britannica_, 9th edit., vol. III, p. 42, 1875.

[68] Sur la limite de petitesse des organismes, _Bull. Soc. R. des
Sc. méd. et nat. de Bruxelles_, Jan. 1903; _Rec. d’œuvres_ (_Physiol.
générale_), p. 325.

[69] Cf. A. Fischer, _Vorlesungen über Bakterien_, 1897, p. 50.

[70] F. Hofmeister, quoted in Cohnheim’s _Chemie der Eiweisskörper_,
1900, p. 18.

[71] McKendrick arrived at a still lower estimate, of about 1250
proteid molecules in the minutest organisms. _Brit. Ass. Rep._ 1901, p.
808.

[72] Cf. Perrin, _Les Atomes_, 1914, p. 74.

[73] Cf. Tait, On Compression of Air in small Bubbles, _Proc. R. S. E._
V, 1865.

[74] _Phil. Mag._ XLVIII, 1899; _Collected Papers_, IV, p. 430.

[75] Carpenter, _The Microscope_, edit. 1862, p. 185.

[76] The modern literature on the Brownian Movement is very large,
owing to the value which the phenomenon is shewn to have in determining
the size of the atom. For a fuller, but still elementary account, see
J. Cox, _Beyond the Atom_, 1913, pp. 118–128; and see, further, Perrin,
_Les Atomes_, pp. 119–189.

[77] Cf. R. Gans, Wie fallen Stäbe und Scheiben in einer reibenden
Flüssigkeit? _Münchener Bericht_, 1911, p. 191; K. Przibram, Ueber die
Brown’sche Bewegung nicht kugelförmiger Teilchen, _Wiener Ber._ 1912,
p. 2339.

[78] Ueber die ungeordnete Bewegung niederer Thiere, _Pflüger’s
Archiv_, CLIII, p. 401, 1913.

[79] Sometimes we find one and the same diagram suffice, whether the
intervals of time be great or small; and we then invoke “Wolff’s
Law,” and assert that the life-history of the individual repeats, or
recapitulates, the history of the race.

[80] Our subject is one of Bacon’s “Instances of the Course,” or
studies wherein we “measure Nature by periods of Time.” In Bacon’s
_Catalogue of Particular Histories_, one of the odd hundred histories
or investigations which he foreshadowed is precisely that which we are
engaged on, viz. a “History of the Growth and Increase of the Body, in
the whole and in its parts.”

[81] Cf. Aristotle, _Phys._ vi, 5, 235 _a_ 11, ὲπεὶ γὰρ ἅπασα κίνησις
ἐν χρόνῳ, κτλ. Bacon emphasised, in like manner, the fact that “all
motion or natural action is performed in time: some more quickly, some
more slowly, but all in periods determined and fixed in the nature of
things. Even those actions which seem to be performed suddenly, and (as
we say) in the twinkling of an eye, are found to admit of degree in
respect of duration.” _Nov. Org._ XLVI.

[82] Cf. (e.g.) _Elem. Physiol._ ed. 1766, VIII, p. 114, “Ducimur
autem ad evolutionem potissimum, quando a perfecto animale retrorsum
progredimur, et incrementorum atque mutationum seriem relegimus. Ita
inveniemus perfectum illud animal fuisse imperfectius, alterius figurae
et fabricae, et denique rude et informe: et tamen idem semper animal
sub iis diversis phasibus fuisse, quae absque ullo saltu perpetuos
parvosque per gradus cohaereant.”

[83] _Beiträge zur Entwickelungsgeschichte des Hühnchens im Ei_, p. 40,
1817. Roux ascribes the same views also to Von Baer and to R. H. Lotze
(_Allg. Physiologie_, p. 353, 1851).

[84] Roux, _Die Entwickelungsmechanik_, p. 99, 1905.

[85] _Op. cit._ p. 302, “Magnum hoc naturae instrumentum, etiam in
corpore animato evolvendo potenter operatur; etc.”

[86] _Ibid._ p. 306. “Subtiliora ista, et aliquantum hypothesi mista,
tamen magnum mihi videntur speciem veri habere.”

[87] Cf. His, On the Principles of Animal Morphology, _Proc. R. S.
E._ XV, 1888, p. 294: “My own attempts to introduce some elementary
mechanical or physiological conceptions into embryology have not
generally been agreed to by morphologists. To one it seemed ridiculous
to speak of the elasticity of the germinal layers; another thought
that, by such considerations, we ‘put the cart before the horse’:
and one more recent author states, that we have better things to
do in embryology than to discuss tensions of germinal layers and
similar questions, since all explanations must of necessity be of
a phylogenetic nature. This opposition to the application of the
fundamental principles of science to embryological questions would
scarcely be intelligible had it not a dogmatic background. No other
explanation of living forms is allowed than heredity, and any which
is founded on another basis must be rejected ....... To think that
heredity will build organic beings without mechanical means is a piece
of unscientific mysticism.”

[88] Hertwig, O., _Zeit und Streitfragen der Biologie_, II. 1897.

[89] Cf. Roux, _Gesammelte Abhandlungen_, II, p. 31, 1895.

[90] _Treatise on Comparative Embryology_, I, p. 4, 1881.

[91] Cf. Fick, _Anal. Anzeiger_, XXV, p. 190, 1904.

[92] 1st ed. p. 444; 6th ed. p. 390. The student should not fail
to consult the passage in question; for there is always a risk of
misunderstanding or misinterpretation when one attempts to epitomise
Darwin’s carefully condensed arguments.

[93] “In omni rerum naturalium historia utile est _mensuras definiri
et numeros_,” Haller, _Elem. Physiol._ II, p. 258, 1760. Cf. Hales,
_Vegetable Staticks_, Introduction.

[94] Brussels, 1871. Cf. the same author’s _Physique sociale_, 1835,
and _Lettres sur la théorie des probabilités_, 1846. See also, for the
general subject, Boyd, R., Tables of weights of the Human Body, etc.
_Phil. Trans._ vol. CLI, 1861; Roberts, C., _Manual of Anthropometry_,
1878; Daffner, F., _Das Wachsthum des Menschen_ (2nd ed.), 1902, etc.

[95] Dr Johnson was not far wrong in saying that “life declines from
thirty-five”; though the Autocrat of the Breakfast-table, like Cicero,
declares that “the furnace is in full blast for ten years longer.”

[96] Joly, _The Abundance of Life_, 1915 (1890), p. 86.

[97] “_Lou pes, mèstre de tout_ [Le poids, maître de tout], _mèstre
sènso vergougno, Que te tirasso en bas de sa brutalo pougno_,” J. H.
Fabre, _Oubreto prouvençalo_, p. 61.

[98] The continuity of the phenomenon of growth, and the natural
passage from the phase of increase to that of decrease or decay, are
admirably discussed by Enriques, in “La morte,” _Riv. di Scienza_,
1907, and in “Wachsthum und seine analytische Darstellung,” _Biol.
Centralbl._ June, 1909. Haller (_Elem_. VII, p. 68) recognised
_decrementum_ as a phase of growth, not less important (theoretically)
than _incrementum_: “_tristis, sed copiosa, haec est materies_.”

[99] Cf. (_int. al._), Friedenthal, H., Das Wachstum des
Körpergewichtes ... in verschiedenen Lebensältern, _Zeit. f. allg.
Physiol._ IX, pp. 487–514, 1909.

[100] As Haller observed it to do in the chick (_Elem._ VIII, p. 294):
“Hoc iterum incrementum miro ordine ita distribuitur, ut in principio
incubationis maximum est: inde perpetuo minuatur.”

[101] There is a famous passage in Lucretius (v. 883) where he compares
the course of life, or rate of growth, in the horse and his boyish
master: _Principio circum tribus actis impiger annis Floret equus, puer
hautquaquam_, etc.

[102] Minot, C. S., Senescence and Rejuvenation, _Journ. of Physiol._
XII, pp. 97–153, 1891; The Problem of Age, Growth and Death, _Pop.
Science Monthly_ (June–Dec.), 1907.

[103] Quoted in Vierordt’s _Anatomische ... Daten und Tabellen_, 1906.
p. 13.

[104] _Unsere Körperform_, Leipzig, 1874.

[105] No such point of inflection appears in the curve of weight
according to C. M. Jackson’s data (On the Prenatal Growth of the Human
Body, etc., _Amer. Journ. of Anat._ IX, 1009, pp. 126, 156), nor in
those quoted by him from Ahlfeld, Fehling and others. But it is plain
that the very rapid increase of the monthly weights, approximately in
the ratio of the cubes of the corresponding lengths, would tend to
conceal any such breach of continuity, unless it happened to be very
marked indeed. Moreover in the case of Jackson’s data (and probably
also in the others) the actual age of the embryos was not determined,
but was estimated from their lengths. The following is Jackson’s
estimate of average weights at intervals of a lunar month:

 Months      0   1   2  3    4    5    6    7     8     9     10
 Wt in gms. ·0  ·04  3  36  120  330  600  1000  1500  2200  3200

[106] G. Kraus (after Wallich-Martius), _Ann. du Jardin bot. de
Buitenzorg_, XII, 1, 1894, p. 210. Cf. W. Ostwald, _Zeitliche
Eigenschaften_, etc. p. 56.

[107] Cf. Chodat, R., et Monnier, A., Sur la courbe de croissance des
végétaux, _Bull. Herb. Boissier_ (2), V, pp. 615, 616, 1905.

[108] Cf. Fr. Boas, Growth of Toronto Children, _Rep. of U.S. Comm.
of Education_, 1896–7, pp. 1541–1599, 1898; Boas and Clark Wissler,
Statistics of Growth, _Education Rep._ 1904, pp. 25–132, 1906; H. P.
Bowditch, _Rep. Mass. State Board of Health_, 1877; K. Pearson, On the
Magnitude of certain coefficients of Correlation in Man, _Pr. R. S._
LXVI, 1900.

[109] _l.c._ p. 42, and other papers there quoted.

[110] See, for an admirable résumé of facts, Wolfgang Ostwald, _Ueber
die Zeitliche Eigenschaften der Entwickelungsvorgänge_ (71 pp.),
Leipzig, 1908 (Roux’s _Vorträge_, Heft V): to which work I am much
indebted. A long list of observations on the growth-rate of various
animals is also given by H. Przibram, _Exp. Zoologie_, 1913, pt. IV
(_Vitalität_), pp. 85–87.

[111] Cf. St Loup, Vitesse de croissance chez les Souris, _Bull. Soc.
Zool. Fr._ XVIII, 242, 1893; Robertson, _Arch. f. Entwickelungsmech._
XXV, p. 587, 1908; Donaldson. _Boas Memorial Volume_, New York, 1906.

[112] Luciani e Lo Monaco, _Arch. Ital. de Biologie_, XXVII, p. 340,
1897.

[113] Schaper, _Arch. f. Entwickelungsmech._ XIV, p. 356, 1902. Cf.
Barfurth, Versuche über die Verwandlung der Froschlarven, _Arch. f.
mikr. Anat._ XXIX, 1887.

[114] Joh. Schmidt, Contributions to the Life-history of the Eel,
_Rapports du Conseil Intern. pour l’exploration de la Mer_, vol. V, pp.
137–274, Copenhague, 1906.

[115] That the metamorphoses of an insect are but phases in a process
of growth, was firstly clearly recognised by Swammerdam, _Biblia
Naturae_, 1737, pp. 6, 579 etc.

[116] From Bose, J. C., _Plant Response_, London, 1906, p. 417.

[117] This phenomenon, of _incrementum inequale_, as opposed to
_incrementum in universum_, was most carefully studied by Haller:
“Incrementum inequale multis modis fit, ut aliae partes corporis aliis
celerius increscant. Diximus hepar minus fieri, majorem pulmonem,
minimum thymum, etc.” (_Elem._ VIII (2), p. 34).

[118] See (_inter alia_) Fischel, A., Variabilität und Wachsthum des
embryonalen Körpers, _Morphol. Jahrb._ XXIV, pp. 369–404, 1896. Oppel,
_Vergleichung des Entwickelungsgrades der Organe zu verschiedenen
Entwickelungszeiten bei Wirbelthieren_, Jena, 1891. Faucon, A., _Pesées
et Mensurations fœtales à différents âges de la grossesse_. (Thèse.)
Paris, 1897. Loisel, G., Croissance comparée en poids et en longueur
des fœtus mâle et femelle dans l’espèce humaine, _C. R. Soc. de
Biologie_, Paris, 1903. Jackson, C. M., Pre-natal growth of the human
body and the relative growth of the various organs and parts, _Am. J.
of Anat._ IX, 1909; Post-natal growth and variability of the body and
of the various organs in the albino rat, _ibid._ XV, 1913.

[119] _l.c._ p. 1542.

[120] Variation and Correlation in Brain-weight, _Biometrika_, IV, pp.
13–104, 1905.

[121] _Die Säugethiere_, p. 117.

[122] _Amer. J. of Anatomy_, VIII, pp. 319–353, 1908. Donaldson
(_Journ. Comp. Neur. and Psychol._ XVIII, pp. 345–392, 1908) also
gives a logarithmic formula for brain-weight (_y_) as compared
with body-weight (_x_), which in the case of the white rat is _y_
= ·554 − ·569 log(_x_ − 8·7), and the agreement is very close. But the
formula is admittedly empirical and as Raymond Pearl says (_Amer. Nat._
1909, p. 303), “no ulterior biological significance is to be attached
to it.”

[123] _Biometrika_, IV, pp. 13–104, 1904.

[124] Donaldson, H. H., A Comparison of the White Rat with Man in
respect to the Growth of the entire Body, _Boas Memorial Vol._, New
York, 1906, pp. 5–26.

[125] Besides many papers quoted by Dubois on the growth and weight of
the brain, and numerous papers in _Biometrika_, see also the following:
Ziehen, Th., _Das Gehirn: Massverhältnisse_, in Bardeleben’s _Handb.
der Anat. des Menschen_, IV, pp. 353–386, 1899. Spitzka, E. A.,
Brain-weight of Animals with special reference to the Weight of the
Brain in the Macaque Monkey, _J. Comp. Neurol._ XIII, pp. 9–17, 1903.
Warneke, P., Mitteilung neuer Gehirn und Körpergewichtsbestimmungen
bei Säugern, nebst Zusammenstellung der gesammten bisher beobachteten
absoluten und relativen Gehirngewichte bei den verschiedenen Species,
_J. f. Psychol. u. Neurol._ XIII, pp. 355–403, 1909. Donaldson, H. H.,
On the regular seasonal Changes in the relative Weight of the Central
Nervous System of the Leopard Frog, _Journ. of Morph._ XXII, pp.
663–694, 1911.

[126] Cf. Jenkinson, Growth, Variability and Correlation in Young
Trout, _Biometrika_, VIII, pp. 444–455, 1912.

[127] Cf. chap. xvii, p. 739.

[128] “ ...I marked in the same manner as the Vine, young Honeysuckle
shoots, etc....; and I found in them all a gradual scale of unequal
extensions, those parts extending most which were tenderest,”
_Vegetable Staticks_, Exp. cxxiii.

[129] From Sachs, _Textbook of Botany_, 1882, p. 820.

[130] Variation and Differentiation in Ceratophyllum, _Carnegie Inst.
Publications_, No. 58, Washington, 1907.

[131] Cf. Lämmel, Ueber periodische Variationen in Organismen, _Biol.
Centralbl._ XXII, pp. 368–376, 1903.

[132] Herein lies the easy answer to a contention frequently raised
by Bergson, and to which he ascribes great importance, that “a mere
variation of size is one thing, and a change of form is another.” Thus
he considers “a change in the form of leaves” to constitute “a profound
morphological difference.” _Creative Evolution_, p. 71.

[133] I do not say that the assumption that these two groups of earwigs
were of different ages is altogether an easy one; for of course,
even in an insect whose metamorphosis is so simple as the earwig’s,
consisting only in the acquisition of wings or wing-cases, we usually
take it for granted that growth proceeds no more after the final stage,
or “adult form” is attained, and further that this adult form is
attained at an approximately constant age, and constant magnitude. But
even if we are not permitted to think that the earwig may have grown,
or moulted, after once the elytra were produced, it seems to me far
from impossible, and far from unlikely, that prior to the appearance
of the elytra one more stage of growth, or one more moult took place
in some cases than in others: for the number of moults is known to be
variable in many species of Orthoptera. Unfortunately Bateson tells
us nothing about the sizes or total lengths of his earwigs; but his
figures suggest that it was bigger earwigs that had the longer tails;
and that the rate of growth of the tails had had a certain definite
ratio to that of the bodies, but not necessarily a simple ratio of
equality.

[134] Jackson, C. M., _J. of Exp. Zool._ XIX, 1915, p. 99; cf. also
Hans Aron, Unters. über die Beeinflüssung der Wachstum durch die
Ernährung, _Berl. klin. Wochenbl._ LI, pp. 972–977, 1913, etc.

[135] The temperature limitations of life, and to some extent of
growth, are summarised for a large number of species by Davenport,
_Exper. Morphology_, cc. viii, xviii, and by Hans Przibram, _Exp.
Zoologie_, IV, c. v.

[136] Réaumur: _L’art de faire éclore et élever en toute saison des
oiseaux domestiques, foit par le moyen de la chaleur du fumier_, Paris,
1749.

[137] Cf. (_int. al._) de Vries, H., Matériaux pour la connaissance
de l’influence de la température sur les plantes, _Arch. Néerl._ V,
385–401, 1870. Köppen, Wärme und Pflanzenwachstum, _Bull. Soc. Imp.
Nat. Moscou._ XLIII, pp. 41–110, 1870.

[138] Blackman, F. F., _Ann. of Botany_, XIX, p. 281, 1905.

[139] For various instances of a “temperature coefficient” in
physiological processes, see Kanitz, _Zeitschr. f. Elektrochemie_,
1907, p. 707; _Biol. Centralbl._ XXVII, p. 11, 1907; Hertzog, R.
O., Temperatureinfluss auf die Entwicklungsgeschwindigkeit der
Organismen, _Zeitschr. f. Elektrochemie_, XI, p 820, 1905; Krogh,
Quantitative Relation between Temperature and Standard Metabolism,
_Int. Zeitschr. f. physik.-chem. Biologie_, I, p. 491, 1914; Pütter,
A., Ueber Temperaturkoefficienten, _Zeitschr. f. allgem. Physiol._
XVI, p. 574, 1914. Also Cohen, _Physical Chemistry for Physicians and
Biologists_ (English edition), 1903; Pike, F. H., and Scott. E. L., The
Regulation of the Physico-chemical Condition of the Organism, _American
Naturalist_, Jan. 1915, and various papers quoted therein.

[140] Cf. Errera, L., _L’Optimum_, 1896 (_Rec. d’Oeuvres, Physiol.
générale_, pp. 338–368, 1910); Sachs, _Physiologie d. Pflanzen_, 1882,
p. 233; Pfeffer, _Pflanzenphysiologie_, ii, p. 78, 1904; and cf. Jost,
Ueber die Reactionsgeschwindigkeit im Organismus, _Biol. Centralbl._
XXVI, pp. 225–244, 1906.

[141] After Köppen, _Bull. Soc. Nat. Moscou_, XLIII, pp. 41–110, 1871.

[142] _Botany_, p. 387.

[143] Leitch, I., Some Experiments on the Influence of Temperature on
the Rate of Growth in _Pisum sativum, Ann. of Botany_, XXX, pp. 25–46,
1916. (Cf. especially Table III, p. 45.)

[144] Blackman, F. F., Presidential Address in Botany, _Brit. Ass._
Dublin, 1908.

[145] _Rec. de l’Inst. Bot. de Bruxelles_, VI, 1906.

[146] Hertwig, O., Einfluss der Temperatur auf die Entwicklung von
_Rana fusca_ und _R. esculenta_, _Arch. f. mikrosk. Anat._ LI, p.
319, 1898. Cf. also Bialaszewicz, K., Beiträge z. Kenntniss d.
Wachsthumsvorgänge bei Amphibienembryonen, _Bull. Acad. Sci. de
Cracovie_, p. 783, 1908; Abstr. in _Arch. f. Entwicklungsmech._ XXVIII,
p. 160, 1909.

[147] Der Grad der Beschleunigung tierischer Entwickelung durch
erhöhte Temperatur, _A. f. Entw._ Mech. XX. p. 130, 1905. More
recently, Bialaszewicz has determined the coefficient for the rate of
segmentation in Rana as being 2·4 per 10° C.

[148] _Das Wachstum des Menschen_, p. 329, 1902.

[149] The _diurnal_ periodicity is beautifully shewn in the case of
the Hop by Joh. Schmidt (_C. R. du Laboratoire de Carlsberg_, X, pp.
235–248, Copenhague, 1913).

[150] _Trans. Botan. Soc. Edinburgh_, XVIII, 1891, p. 456.

[151] I had not received, when this was written, Mr Douglass’s paper,
On a method of estimating Rainfall by the Growth of Trees, _Bull.
Amer. Geograph. Soc._ XLVI, pp. 321–335, 1914. Mr Douglass does not
fail to notice the long period here described; but he lays more stress
on the occurrence of shorter cycles (of 11, 21 and 33 years), well
known to meteorologists. Mr Douglass is inclined (and I think rightly)
to correlate the variations in growth directly with fluctuations
in rainfall, that is to say with alternate periods of moisture and
aridity; but he points out that the temperature curves (and also the
sunspot curves) are markedly similar.

[152] It may well be that the effect is not due to light after all; but
to increased absorption of heat by the soil, as a result of the long
hours of exposure to the sun.

[153] On growth in relation to light, see Davenport, _Exp. Morphology_,
II, ch. xvii. In some cases (as in the roots of Peas), exposure to
light seems to have no effect on growth; in other cases, as in diatoms
(according to Whipple’s experiments, quoted by Davenport, II, p.
423), the effect of light on growth or multiplication is well-marked,
measurable, and apparently capable of expression by a logarithmic
formula. The discrepancy would seem to arise from the fact that, while
light-energy always tends to be absorbed by the chlorophyll of the
plant, converted into chemical energy, and stored in the shape of
starch or other reserve materials, the actual rate of growth depends
on the rate at which these reserves are drawn on: and this is another
matter, in which light-energy is no longer directly concerned.

[154] Cf. for instance, Nägeli’s classical account of the effect of
change of habitat on Alpine and other plants: _Sitzungsber. Baier.
Akad. Wiss._ 1865, pp. 228–284.

[155] Cf. Blackman, F. F., Presidential Address in Botany, _Brit. Ass._
Dublin, 1908. The fact was first enunciated by Baudrimont and St Ange,
Recherches sur le développement du fœtus, _Mém. Acad. Sci._ XI, p. 469,
1851.

[156] Cf. Loeb, _Untersuchungen zur physiol. Morphologie der Thiere_,
1892; also Experiments on Cleavage, _J. of Morph._ VII, p. 253, 1892;
Zusammenstellung der Ergebnisse einiger Arbeiten über die Dynamik des
thierischen Wachsthum, _Arch. f. Entw. Mech._ XV, 1902–3, p. 669;
Davenport, On the Rôle of Water in Growth, _Boston Soc. N. H._ 1897;
Ida H. Hyde, _Am. J. of Physiol._ XII, 1905, p. 241, etc.

[157] _Pflüger’s Archiv_, LV, 1893.

[158] Beiträge zur Kenntniss der Wachstumsvorgänge bei
Amphibienembryonen, _Bull. Acad. Sci. de Cracovie_, 1908, p. 783; cf.
_Arch. f. Entw. Mech._ XXVIII, p. 160, 1909; XXXIV, p. 489, 1912.

[159] Fehling, H., _Arch. für Gynaekologie_, XI, 1877; cf. Morgan,
_Experimental Zoology_, p. 240, 1907.

[160] Höber, R., Bedeutung der Theorie der Lösungen für Physiologie und
Medizin, _Biol. Centralbl._ XIX, 1899; cf. pp. 272–274.

[161] Schmankewitsch has made other interesting observations on change
of size and form, after some generations, in relation to change of
density; e.g. in the flagellate infusorian _Anisonema acinus_, Bütschli
(_Z. f. w. Z._ XXIX, p. 429, 1877).

[162] These “Fezzan-worms,” when first described, were supposed to be
“insects’ eggs”; cf. Humboldt, _Personal Narrative_, VI, i, 8, note;
Kirby and Spence, Letter X.

[163] Cf. _Introd. à l’étude de la médecine expérimentale_, 1885, p.
110.

[164] Cf. Abonyi, _Z. f. w. Z._ CXIV, p. 134, 1915. But Frédéricq has
shewn that the amount of NaCl in the blood of Crustacea (_Carcinus
moenas_) varies, and all but corresponds, with the density of the water
in which the creature has been kept (_Arch. de Zool. Exp. et Gén._ (2),
III, p. xxxv, 1885); and other results of Frédéricq’s, and various
data given or quoted by Bottazzi (Osmotischer Druck und elektrische
Leitungsfähigkeit der Flüssigkeiten der Organismen, in Asher-Spiro’s
_Ergebn. d. Physiologie_, VII, pp. 160–402, 1908) suggest that the case
of the brine-shrimps must be looked upon as an extreme or exceptional
one.

[165] Cf. Schmankewitsch, _Z. f. w. Zool._ XXV, 1875, XXIX, 1877, etc.;
transl. in appendix to Packard’s _Monogr. of N. American Phyllopoda_,
1883, pp. 466–514; Daday de Deés, _Ann. Sci. Nat._ (_Zool._), (9),
XI, 1910; Samter und Heymons, _Abh. d. K. pr. Akad. Wiss._ 1902;
Bateson, _Mat. for the Study of Variation_, 1894, pp. 96–101; Anikin,
_Mitth. Kais. Univ. Tomsk_, XIV: _Zool. Centralbl._ VI, pp. 756–760,
1908; Abonyi, _Z. f. w. Z._ CXIV, pp. 96–168, 1915 (with copious
bibliography), etc.

[166] According to the empirical canon of physiology, that (as
Frédéricq expresses it) “L’être vivant est agencé de telle manière que
chaque influence perturbatrice provoque d’elle-même la mise en activité
de l’appareil compensateur qui doit neutraliser et réparer le dommage.”

[167] Such phenomena come precisely under the head of what Bacon called
_Instances of Magic_: “By which I mean those wherein the material or
efficient cause is scanty and small as compared with the work or effect
produced; so that even when they are common, they seem like miracles,
some at first sight, others even after attentive consideration. These
magical effects are brought about in three ways ... [of which one is]
by excitation or invitation in another body, as in the magnet which
excites numberless needles without losing any of its virtue, _or in
yeast and such-like_.” _Nov. Org._, cap. li.

[168] Monnier, A., Les matières minérales, et la loi d’accroissement
des Végétaux, _Publ. de l’Inst. de Bot. de l’Univ. de Genève_
(7), III, 1905. Cf. Robertson, On the Normal Rate of Growth of an
Individual, and its Biochemical Significance, _Arch. f. Entw. Mech._
XXV, pp. 581–614, XXVI, pp. 108–118, 1908; Wolfgang Ostwald, _Die
zeitlichen Eigenschaften der Entwickelungsvorgänge_, 1908; Hatai, S.,
Interpretation of Growth-curves from a Dynamical Standpoint, _Anat.
Record_, V, p. 373, 1911.

[169] _Biochem. Zeitschr._ II, 1906, p. 34.

[170] Even a crystal may be said, in a sense, to display
“autocatalysis”: for the bigger its surface becomes, the more rapidly
does the mass go on increasing.

[171] Cf. Loeb, The Stimulation of Growth, _Science_, May 14, 1915.

[172] _B. coli-communis_, according to Buchner, tends to double in
22 minutes; in 24 hours, therefore, a single individual would be
multiplied by something like 10^{28}; _Sitzungsber. München. Ges.
Morphol. u. Physiol._ III, pp. 65–71, 1888. Cf. Marshall Ward, Biology
of _Bacillus ramosus_, etc. _Pr. R. S._ LVIII, 265–468, 1895. The
comparatively large infusorian Stylonichia, according to Maupas, would
multiply in a month by 10^{43}.

[173] Cf. Enriques, Wachsthum und seine analytisehe Darstellung, _Biol.
Centralbl._ 1909, p. 337.

[174] Cf. (_int. al._) Mellor, _Chemical Statics and Dynamics_, 1904,
p. 291.

[175] Cf. Robertson, _l.c._

[176] See, for a brief resumé of this subject, Morgan’s _Experimental
Zoology_, chap. xvi.

[177] _Amer. J. of Physiol._, X, 1904.

[178] _C.R._ CXXI, CXXII, 1895–96.

[179] Cf. Loeb, _Science_, May 14, 1915.

[180] Cf. Baumann u. Roos, Vorkommen von Iod im Thierkörper, _Zeitschr.
für Physiol. Chem._ XXI, XXII, 1895, 6.

[181] Le Néo-Vitalisme, _Rev. Scientifique_, Mars 1911, p. 22 (of
reprint).

[182] _La vie et la mort_, p. 43, 1902.

[183] Cf. Dendy, _Evolutionary Biology_, 1912, p. 408; _Brit. Ass.
Report_ (Portsmouth), 1911, p. 278.

[184] Lucret. v, 877. “Lucretius nowhere seems to recognise the
possibility of improvement or change of species by ‘natural selection’;
the animals remain as they were at the first, except that the weaker
and more useless kinds have been crushed out. Hence he stands in marked
contrast with modern evolutionists.” Kelsey’s note, _ad loc._

[185] Even after we have so narrowed the scope and sphere of natural
selection, it is still hard to understand; for the causes of
_extinction_ are often wellnigh as hard to comprehend as are those of
the _origin_ of species. If we assert (as has been lightly done) that
Smilodon perished owing to its gigantic tusks, that Teleosaurus was
handicapped by its exaggerated snout, or Stegosaurus weighed down by
its intolerable load of armour, we may be reminded of other kindred
forms to show that similar conditions did not necessarily lead to
extermination, or that rapid extinction ensued apart from any such
visible or apparent disadvantages. Cf. Lucas, F. A., On Momentum in
Variation, _Amer. Nat._ xli, p. 46, 1907.

[186] See Professor T. H. Morgan’s _Regeneration_ (316 pp.), 1901 for
a full account and copious bibliography. The early experiments on
regeneration, by Vallisneri, Réaumur, Bonnet, Trembley, Baster, and
others, are epitomised by Haller, _Elem. Physiologiae_, VIII, p. 156
_seq._

[187] _Journ. Experim. Zool._ VII, p. 397, 1909.

[188] _Op. cit._ p. 406, Exp. IV.

[189] The experiments of Loeb on the growth of Tubularia in various
saline solutions, referred to on p. 125, might as well or better have
been referred to under the heading of regeneration, as they were
performed on cut pieces of the zoophyte. (Cf. Morgan, _op. cit._ p. 35.)

[190] _Powers of the Creator_, I, p. 7, 1851. See also _Rare and
Remarkable Animals_, II, pp. 17–19, 90, 1847.

[191] Lillie, F. R., The smallest Parts of Stentor capable of
Regeneration, _Journ. of Morphology_, XII, p. 239, 1897.

[192] Boveri, Entwicklungsfähigkeit kernloser Seeigeleier, etc., _Arch.
f. Entw. Mech._ II, 1895. See also Morgan, Studies of the partial
larvae of Sphaerechinus, _ibid._ 1895; J. Loeb, On the Limits of
Divisibility of Living Matter, _Biol. Lectures_, 1894, etc.

[193] Cf. Przibram, H., Scheerenumkehr bei dekapoden Crustaceen, _Arch.
f. Entw. Mech._ XIX, 181–247, 1905; XXV, 266–344, 1907. Emmel, _ibid._
XXII, 542, 1906; Regeneration of lost parts in Lobster, _Rep. Comm.
Inland Fisheries, Rhode Island_, XXXV, XXXVI, 1905–6; _Science_ (n.s.),
XXVI, 83–87, 1907. Zeleny, Compensatory Regulation, _J. Exp. Zool._ II,
1–102, 347–369, 1905; etc.

[194] Lobsters are occasionally found with two symmetrical claws:
which are then usually serrated, sometimes (but very rarely) both
blunt-toothed. Cf. Calman, _P.Z.S._ 1906, pp. 633, 634, and _reff._

[195] Wilson, E. B., Reversal of Symmetry in _Alpheus heterochelis_,
_Biol. Bull._ IV, p. 197, 1903.

[196] _J. Exp. Zool._ VII, p. 457, 1909.

[197] _Biologica_, III, p. 161, June. 1913.

[198] _Anatomical and Pathological Observations_, p. 3, 1845;
_Anatomical Memoirs_, II, p. 392, 1868.

[199] Giard, A., L’œuf et les débuts de l’évolution, _Bull. Sci. du
Nord de la Fr._ VIII, pp. 252–258, 1876.

[200] _Entwickelungsvorgänge der Eizelle_, 1876; _Investigations on
Microscopic Foams and Protoplasm_, p. 1, 1894.

[201] _Journ. of Morphology_, I, p. 229, 1887.

[202] While it has been very common to look upon the phenomena of
mitosis as sufficiently explained by the results _towards which_ they
seem to lead, we may find here and there a strong protest against this
mode of interpretation. The following is a case in point: “On a tenté
d’établir dans la mitose dite primitive plusieurs catégories, plusieurs
types de mitose. On a choisi le plus souvent comme base de ces systèmes
des concepts abstraits et téléologiques: répartition plus ou moins
exacte de la chromatine entre les deux noyaux-fils suivant qu’il y
a ou non des chromosomes (_Dangeard_), distribution particulière et
signification dualiste des substances nucléaires (substance kinétique
et substance générative ou héréditaire, _Hartmann et ses élèves_), etc.
Pour moi tous ces essais sont à rejeter catégoriquement à cause de leur
caractère finaliste; de plus, ils sont construits sur des concepts non
démontrés, et qui parfois représentent des généralisations absolument
erronées.” A. Alexeieff, _Archiv für Protistenkunde_, XIX, p. 344, 1913.

[203] This is the old philosophic axiom writ large: _Ignorato motu,
ignoratur natura_; which again is but an adaptation of Aristotle’s
phrase, ἡ ἀρχὴ τῆς κινήσεως, as equivalent to the “Efficient Cause.”
FitzGerald holds that “all explanation consists in a description of
underlying motions”; _Scientific Writings_, 1902, p. 385.

[204] As when Nägeli concluded that the organism is, in a certain
sense, “vorgebildet”; _Beitr. zur wiss. Botanik_, II, 1860. Cf. E. B.
Wilson, _The Cell, etc._, p. 302.

[205] “La matière arrangée par une sagesse divine doit être
essentiellement organisée partout ... il y a machine dans les parties
de la machine Naturelle à l’infini.” _Sur le principe de la Vie_,
p. 431 (Erdmann). This is the very converse of the doctrine of the
Atomists, who could not conceive a condition “_ubi dimidiae partis pars
semper habebit Dimidiam partem, nec res praefiniet ulla_.”

[206] Cf. an interesting passage from the _Elements_ (I, p. 445,
Molesworth’s edit.), quoted by Owen, _Hunterian Lectures on the
Invertebrates_, 2nd ed. pp. 40, 41, 1855.

[207] “Wir müssen deshalb den lebenden Zellen, abgesehen von der
Molekularstructur der organischen Verbindungen welche sie enthält, noch
eine andere und in anderer Weise complicirte Structur zuschreiben,
und diese es ist welche wir mit dem Namen _Organisation_ bezeichnen,”
Brücke, Die Elementarorganismen, _Wiener Sitzungsber._ XLIV, 1861, p.
386; quoted by Wilson, _The Cell_, etc. p. 289. Cf. also Hardy, _Journ.
of Physiol._ XXIV, 1899, p. 159.

[208] Precisely as in the Lucretian _concursus_, _motus_, _ordo_,
_positura_, _figurae_, whereby bodies _mutato ordine mutant naturam_.

[209] Otto Warburg, Beiträge zur Physiologie der Zelle, insbesondere
über die Oxidationsgeschwindigkeit in Zellen; in Asher-Spiro’s
_Ergebnisse der Physiologie_, XIV, pp. 253–337, 1914 (see p. 315). (Cf.
Bayliss, _General Physiology_, 1915, p. 590).

[210] Hardy, W. B., On some Problems of Living Matter (Guthrie
Lecture), _Tr. Physical Soc. London_, xxviii, p. 99–118, 1916.

[211] As a matter of fact both phrases occur, side by side, in Graham’s
classical paper on “Liquid Diffusion applied to Analysis,” _Phil.
Trans._ CLI, p. 184, 1861; _Chem. and Phys. Researches_ (ed. Angus
Smith), 1876, p. 554.

[212] L. Rhumbler, Mechanische Erklärung der Aehnlichkeit zwischen
Magnetischen Kraftliniensystemen und Zelltheilungsfiguren, _Arch. f.
Entw. Mech._ XV, p. 482, 1903.

[213] Gallardo, A., Essai d’interpretation des figures caryocinétiques,
_Anales del Museo de Buenos-Aires_ (2), II, 1896; La division de la
cellule, phenomène bipolaire de caractère electro-colloidal, _Arch. f.
Entw. Mech._ XXVIII, 1909, etc.

[214] _Arch. f. Entw. Mech._ III, IV, 1896–97.

[215] On various theories of the mechanism of mitosis, see (e.g.)
Wilson, _The Cell in Development_, etc., pp. 100–114; Meves,
_Zelltheilung_, in Merkel u. Bonnet’s _Ergebnisse der Anatomie_, etc.,
VII, VIII, 1897–8; Ida H. Hyde, _Amer. Journ. of Physiol._ XII, pp.
241–275, 1905; and especially Prenant, A., Theories et interprétations
physiques de la mitose, _J. de l’Anat. et Physiol._ XLVI, pp. 511–578,
1910.

[216] Hartog, M., Une force nouvelle: le mitokinétisme, _C.R._ 11 Juli,
1910; Mitokinetism in the Mitotic Spindle and in the Polyasters, _Arch.
f. Entw. Mech._ XXVII, pp. 141–145, 1909; cf. _ibid._ XL, pp. 33–64,
1914. Cf. also Hartog’s papers in _Proc. R. S._ (B), LXXVI, 1905;
_Science Progress_ (n. s.), I, 1907; _Riv. di Scienza_, II, 1908; _C.
R. Assoc. fr. pour l’Avancem. des Sc._ 1914, etc.

[217] The configurations, as obtained by the usual experimental
methods, were of course known long before Faraday’s day, and
constituted the “convergent and divergent magnetic curves” of
eighteenth century mathematicians. As Leslie said, in 1821, they were
“regarded with wonder by a certain class of dreaming philosophers,
who did not hesitate to consider them as the actual traces of an
invisible fluid, perpetually circulating between the poles of the
magnet.” Faraday’s great advance was to interpret them as indications
of _stress in a medium_,—of tension or attraction along the lines, and
of repulsion transverse to the lines, of the diagram.

[218] Cf. also the curious phenomenon in a dividing egg described as
“spinning” by Mrs G. F. Andrews, _J. of Morph._ XII, pp. 367–389, 1897.

[219] Whitman, _J. of Morph._ II, p. 40, 1889.

[220] “Souvent il n’y a qu’une séparation _physique_ entre le
cytoplasme et le suc nucléaire, comme entre deux liquides immiscibles,
etc.;” Alexeieff, Sur la mitose dite “primitive,” _Arch. f.
Protistenk._ XXIX, p. 357, 1913.

[221] The appearance of “vacuolation” is a result of endosmosis or the
diffusion of a less dense fluid into the denser plasma of the cell.
_Caeteris paribus_, it is less apparent in marine organisms than in
those of freshwater, and in many or most marine Ciliates and even
Rhizopods a contractile vacuole has not been observed (Bütschli, in
Bronn’s _Protozoa_, p. 1414); it is also absent, and probably for the
same reason, in parasitic Protozoa, such as the Gregarines and the
Entamoebae. Rossbach shewed that the contractile vacuole of ordinary
freshwater Ciliates was very greatly diminished in a 5 per cent.
solution of NaCl, and all but disappeared in a 1 per cent. solution of
sugar (_Arb. z. z. Inst. Würzburg_, 1872, cf. Massart, _Arch. de Biol._
LX, p. 515, 1889). _Actinophrys sol_, when gradually acclimatised
to sea-water, loses its vacuoles, and _vice versa_ (Gruber, _Biol.
Centralbl._ IX, p. 22, 1889); and the same is true of Amoeba (Zuelzer,
_Arch. f. Entw. Mech._ 1910, p. 632). The gradual enlargement of the
contractile vacuole is precisely analogous to the change of size of a
bubble until the gases on either side of the film are equally diffused,
as described long ago by Draper (_Phil. Mag._ (n. s.), XI, p. 559,
1837). Rhumbler has shewn that contractile or pulsating vacuoles may
be well imitated in chloroform-drops, suspended in water in which
various substances are dissolved (_Arch. f. Entw. Mech._ VII, 1898,
p. 103). The pressure within the contractile vacuole, always greater
than without, diminishes with its size, being inversely proportional
to its radius; and when it lies near the surface of the cell, as in
a Heliozoon, it bursts as soon as it reaches a thinness which its
viscosity or molecular cohesion no longer permits it to maintain.

[222] Cf. p. 660.

[223] The elongated or curved “macronucleus” of an Infusorian is
to be looked upon as a single mass of chromatin, rather than as an
aggregation of particles in a fluid drop, as in the case described. It
has a shape of its own, in which ordinary surface-tension plays a very
subordinate part.

[224] _Théorie physico-chimique de la Vie_, p. 73, 1910; _Mechanism of
Life_, p. 56, 1911.

[225] Whence the name “mitosis” (Greek μίτος, a thread), applied first
by Flemming to the whole phenomenon. Kollmann (_Biol. Centralbl._
II, p. 107, 1882) called it _divisio per fila_, or _divisio laqueis
implicata_. Many of the earlier students, such as Van Beneden (Rech.
sur la maturation de l’œuf, _Arch. de Biol._ IV, 1883), and Hermann
(Zur Lehre v. d. Entstehung d. karyokinetischen Spindel, _Arch. f.
mikrosk. Anat._ XXXVII, 1891) thought they recognised actual muscular
threads, drawing the nuclear material asunder towards the respective
foci or poles; and some such view was long maintained by other writers,
Boveri, Heidenhain, Flemming, R. Hertwig, and many more. In fact, the
existence of contractile threads, or the ascription to the spindle
rather than to the poles or centrosomes of the active forces concerned
in nuclear division, formed the main tenet of all those who declined to
go beyond the “contractile properties of protoplasm” for an explanation
of the phenomenon. (Cf. also J. W. Jenkinson, _Q. J. M. S._ XLVIII, p.
471, 1904.)

[226] Cf. Bütschli, O., Ueber die künstliche Nachahmung der
karyokinetischen Figur, _Verh. Med. Nat. Ver. Heidelberg_, V, pp. 28–41
(1892), 1897.

[227] Arrhenius, in describing a typical colloid precipitate, does so
in terms that are very closely applicable to the ordinary microscopic
appearance of the protoplasm of the cell. The precipitate consists, he
says, “en un réseau d’une substance solide contenant peu d’eau, dans
les mailles duquel est inclus un fluide contenant un peu de colloide
dans beaucoup d’eau ... Evidemment cette structure se forme à cause
de la petite différence de poids spécifique des deux phases, et de la
consistance gluante des particules séparées, qui s’attachent en forme
de réseau.” _Rev. Scientifique_, Feb. 1911.

[228] F. Schwartz, in Cohn’s _Beitr. z. Biologie der Pflanzen_, V, p.
1, 1887.

[229] Fischer, _Anat. Anzeiger_, IX, p. 678, 1894, X, p. 769, 1895.

[230] See, in particular, W. B. Hardy, On the structure of Cell
Protoplasm, _Journ. of Physiol._ XXIV, pp. 158–207, 1889; also Höber,
_Physikalische Chemie der Zelle und der Gewebe_, 1902. Cf. (_int. al._)
Flemming, _Zellsubstanz, Kern und Zelltheilung_ 1882, p. 51, etc.

[231] My description and diagrams (Figs 42–51) are based on those of
Professor E. B. Wilson.

[232] If the word _permeability_ be deemed too directly suggestive of
the phenomena of _magnetism_ we may replace it by the more general
term of _specific inductive capacity_. This would cover the particular
case, which is by no means an improbable one, of our phenomena being
due to a “surface charge” borne by the nucleus itself and also by
the chromosomes: this surface charge being in turn the result of a
difference in inductive capacity between the body or particle and its
surrounding medium. (Cf. footnote, p. 187.)

[233] On the effect of electrical influences in altering the
surface-tensions of the colloid particles, see Bredig, _Anorganische
Fermente_, pp. 15, 16, 1901.

[234] _The Cell_, etc. p. 66.

[235] Lillie, R. S., _Amer. J. of Physiol._ VIII, p. 282, 1903.

[236] We have not taken account in the above paragraphs of the obvious
fact that the supposed symmetrical field of force is distorted by
the presence in it of the more or less permeable bodies; nor is it
necessary for us to do so, for to that distorted field the above
argument continues to apply, word for word.

[237] M. Foster, _Lectures on the History of Physiology_, 1901, p. 62.

[238] _Op. cit._ pp. 110 and 91.

[239] Lamb, A. B., A new Explanation of the Mechanism of Mitosis,
_Journ. Exp. Zool._ V, pp. 27–33, 1908.

[240] _Amer. J. of Physiol._ VIII, pp. 273–283, 1903 (_vide supra_, p.
181); cf. _ibid._ XV, pp. 46–84, 1905. Cf. also _Biological Bulletin_,
IV, p. 175. 1903.

[241] In like manner Hardy has shewn that colloid particles migrate
with the negative stream if the reaction of the surrounding fluid
be alkaline, and _vice versa_. The whole subject is much wider than
these brief allusions suggest, and is essentially part of Quincke’s
theory of Electrical Diffusion or Endosmosis: according to which the
particles and the fluid in which they float (or the fluid and the
capillary walls through which it flows) each carry a charge, there
being a discontinuity of potential at the surface of contact, and hence
a field of force leading to powerful tangential or shearing stresses,
communicating to the particles a velocity which varies with the
density per unit area of the surface charge. See W. B. Hardy’s paper
on Coagulation by Electricity, _Journ. of Physiol._ XXIV, p. 288–304,
1899, also Hardy and H. W. Harvey, Surface Electric Charges of Living
Cells, _Proc. R. S._ LXXXIV (B), pp. 217–226, 1911, and papers quoted
therein. Cf. also E. N. Harvey’s observations on the convection of
unicellular organisms in an electric field (Studies on the Permeability
of Cells, _Journ. of Exper. Zool._ X, pp. 508–556, 1911).

[242] On Differences in Electrical Potential in Developing Eggs, _Amer.
Journ. of Physiol._ XII, pp. 241–275, 1905. This paper contains an
excellent summary of various physical theories of the segmentation of
the cell.

[243] Gray has recently demonstrated a temporary increase of electrical
conductivity in sea-urchin eggs during the process of fertilisation
(The Electrical Conductivity of fertilised and unfertilised Eggs,
_Journ. Mar. Biol. Assoc._ X, pp. 50–59, 1913).

[244] Schewiakoff, Ueber die karyokinetische Kerntheilung der _Euglypha
alveolata, Morph. Jahrb._ XIII, pp. 193–258, 1888 (see p. 216).

[245] Coe, W. R., Maturation and Fertilization of the Egg of
Cerebratulus, _Zool. Jahrbücher_ (_Anat. Abth._), XII, pp. 425–476,
1899.

[246] Thus, for example, Farmer and Digby (On Dimensions of Chromosomes
considered in relation to Phylogeny, _Phil. Trans._ (B), CCV, pp. 1–23,
1914) have been at pains to shew, in confutation of Meek (_ibid._
CCIII, pp. 1–74, 1912), that the width of the chromosomes cannot be
correlated with the order of phylogeny.

[247] Cf. also _Arch. f. Entw. Mech._ X, p. 52, 1900.

[248] Cf. Loeb, _Am. J. of Physiol._ VI, p. 32, 1902; Erlanger, _Biol.
Centralbl._ XVII, pp. 152, 339, 1897; Conklin, _Biol. Lectures_, _Woods
Holl_, p. 69, etc. 1898–9.

[249] Robertson, T. B., Note on the Chemical Mechanics of Cell
Division, _Arch. f. Entw. Mech._ XXVII, p. 29, 1909, XXXV, p. 692.
1913. Cf. R. S. Lillie, _J. Exp. Zool._ XXI, pp. 369–402, 1916.

[250] Cf. D’Arsonval, _Arch. de Physiol._ p. 460, 1889; Ida H. Hyde,
_op. cit._ p. 242.

[251] Cf. Plateau’s remarks (_Statique des liquides_, II, p. 154) on
the _tendency_ towards equilibrium, rather than actual equilibrium, in
many of his systems of soap-films.

[252] But under artificial conditions, “polyspermy” may take place,
e.g. under the action of dilute poisons, or of an abnormally high
temperature, these being all, doubtless, conditions under which the
surface-tension is diminished.

[253] Fol, H., _Recherches sur la fécondation_, 1879. Roux, W.,
Beiträge zur Entwickelungsmechanik des Embryo, _Arch. f. Mikr. Anat._
XIX, 1887. Whitman, C. O., Oökinesis, _Journ. of Morph._ I, 1887.

[254] Wilson. _The Cell_, p. 77.

[255] Eight and twelve are by much the commonest numbers, six and
sixteen coming next in order. If we may judge by the list given by E.
B. Wilson (_The Cell_, p. 206), over 80 % of the observed cases lie
between 6 and 16, and nearly 60 % between 8 and 12.

[256] _Theory of Cells_, p. 191.

[257] _The Cell in Development_, etc. p. 59; cf. pp. 388, 413.

[258] E.g. Brücke, _Elementarorganismen_, p. 387: “Wir müssen in der
Zelle einen kleinen Thierleib sehen, und dürfen die Analogien, welche
zwischen ihr und den kleinsten Thierformen existiren, niemals aus den
Augen lassen.”

[259] Whitman, C. O., The Inadequacy of the Cell-theory, _Journ. of
Morphol._ VIII, pp. 639–658, 1893; Sedgwick, A., On the Inadequacy of
the Cellular Theory of Development, _Q.J.M.S._ XXXVII, pp. 87–101,
1895, XXXVIII, pp. 331–337, 1896. Cf. Bourne, G. C., A Criticism of the
Cell-theory; being an answer to Mr Sedgwick’s article, etc., _ibid._
XXXVIII, pp. 137–174, 1896.

[260] Cf. Hertwig, O., _Die Zelle und die Gewebe_, 1893, p. 1;
“Die Zellen, in welche der Anatom die pflanzlichen und thierischen
Organismen zerlegt, sind die Träger der Lebensfunktionen; sie sind,
wie Virchow sich ausgedrückt hat, die ‘Lebenseinheiten.’ Von diesem
Gesichtspunkt aus betrachtet, erscheint der Gesammtlebensprocess eines
zusammengesetzten Organismus nichts Anderes zu sein als das höchst
verwickelte Resultat der einzelnen Lebensprocesse seiner zahlreichen,
verschieden functionirenden Zellen.”

[261] _Journ. of Morph._ VIII, p. 653, 1893.

[262] Neue Grundlegungen zur Kenntniss der Zelle, _Morph. Jahrb._ VIII,
pp. 272, 313, 333, 1883.

[263] _Journ. of Morph._ II, p. 49, 1889.

[264] _Phil. Trans._ CLI, p. 183, 1861; _Researches_, ed. Angus Smith,
1877, p. 553.

[265] Cf. Kelvin, On the Molecular Tactics of a Crystal, _The Boyle
Lecture_, Oxford, 1893, _Baltimore Lectures_, 1904, pp. 612–642. Here
Kelvin was mainly following Bravais’s (and Frankenheim’s) theory
of “space-lattices,” but he had been largely anticipated by the
crystallographers. For an account of the development of the subject in
modern crystallography, by Sohncke, von Fedorow, Schönfliess, Barlow
and others, see Tutton’s _Crystallography_, chap. ix, pp. 118–134, 1911.

[266] In a homogeneous crystalline arrangement, _symmetry_ compels a
locus of one property to be a plane or set of planes; the locus in this
case being that of least surface potential energy.

[267] This is what Graham called the _water of gelatination_, on the
analogy of _water of crystallisation_; _Chem. and Phys. Researches_, p.
597.

[268] Here, in a non-crystalline or random arrangement of particles,
symmetry ensures that the potential energy shall be the same per unit
area of all surfaces; and it follows from geometrical considerations
that the total surface energy will be least if the surface be spherical.

[269] Lehmann, O., _Flüssige Krystalle, sowie Plasticität von
Krystallen im allgemeinen_, etc., 264 pp. 39 pll., Leipsig, 1904. For a
semi-popular, illustrated account, see Tutton’s _Crystals_ (Int. Sci.
Series), 1911.

[270] As Graham said of an allied phenomenon (the so-called
blood-crystals of Funke), it “illustrates the maxim that in nature
there are no abrupt transitions, and that distinctions of class are
never absolute.”

[271] Cf. Przibram, H., Kristall-analogien zur Entwickelungsmechanik
der Organismen, _Arch. f. Entw. Mech._ XXII, p. 207, 1906 (with copious
bibliography); Lehmann, Scheinbar lebende Kristalle und Myelinformen,
_ibid._ XXVI, p. 483, 1908.

[272] The idea of a “surface-tension” in liquids was first enunciated
by Segner, _De figuris superficierum fluidarum_, in _Comment. Soc. Roy.
Göttingen_, 1751, p. 301. Hooke, in the _Micrographia_ (1665, Obs.
VIII, etc.), had called attention to the globular or spherical form of
the little morsels of steel struck off by a flint, and had shewn how
to make a powder of such spherical grains, by heating fine filings to
melting point. “This Phaenomenon” he said “proceeds from a propriety
which belongs to all kinds of fluid Bodies more or less, and is caused
by the Incongruity of the Ambient and included Fluid, which so acts
and modulates each other, that they acquire, as neer as is possible, a
_spherical_ or _globular_ form....”

[273] _Science of Mechanics_, 1902, p. 395; see also Mach’s article
Ueber die physikalische Bedeutung der Gesetze der Symmetrie, _Lotos_,
XXI, pp. 139–147, 1871.

[274] Similarly, Sir David Brewster and others made powerful lenses
by simply dropping small drops of Canada balsam, castor oil, or other
strongly refractive liquids, on to a glass plate: _On New Philosophical
Instruments_ (Description of a new Fluid Microscope), Edinburgh, 1813,
p. 413.

[275] Beiträge z. Physiologie d. Protoplasma, _Pflüger’s Archiv_, II,
p. 307, 1869.

[276] _Poggend. Annalen_, XCIV, pp. 447–459, 1855. Cf. Strethill
Wright, _Phil. Mag._ Feb. 1860.

[277] Haycraft and Carlier pointed out (_Proc. R.S.E._ XV, pp.
220–224, 1888) that the amoeboid movements of a white blood-corpuscle
are only manifested when the corpuscle is in contact with some solid
substance: while floating freely in the plasma or serum of the blood,
these corpuscles are spherical, that is to say they are at rest and in
equilibrium. The same fact has recently been recorded anew by Ledingham
(On Phagocytosis from an adsorptive point of view, _Journ. of Hygiene_,
XII, p. 324, 1912). On the emission of pseudopodia as brought about
by changes in surface tension, see also (_int. al._) Jensen, Ueber
den Geotropismus niederer Organismen, _Pflüger’s Archiv_, LIII, 1893.
Jensen remarks that in Orbitolites, the pseudopodia issuing through
the pores of the shell first float freely, then as they grow longer
bend over till they touch the ground, whereupon they begin to display
amoeboid and streaming motions. Verworn indicates (_Allg. Physiol._
1895, p. 429), and Davenport says (_Experim. Morphology_, II, p. 376)
that “this persistent clinging to the substratum is a ‘thigmotropic’
reaction, and one which belongs clearly to the category of
‘response.’ ” (Cf. Pütter, Thigmotaxis bei Protisten, _A. f. Physiol._
1900, Suppl. p. 247.) But it is not clear to my mind that to account
for this simple phenomenon we need invoke other factors than gravity
and surface-action.

[278] Cf. Pauli, _Allgemeine physikalische Chemie d. Zellen u. Gewebe_,
in Asher-Spiro’s _Ergebnisse der Physiologie_, 1912; Przibram,
_Vitalität_, 1913, p. 6.

[279] The surface-tension theory of protoplasmic movement has been
denied by many. Cf. (e.g.), Jennings, H. S., Contributions to the Study
of the Behaviour of the Lower Organisms, _Carnegie Inst._ 1904, pp.
130–230; Dellinger, O. P., Locomotion of Amoebae, etc. _Journ. Exp.
Zool._ III, pp. 337–357, 1906; also various papers by Max Heidenhain,
in _Anatom. Hefte_ (Merkel und Bonnet), etc.

[280] These various movements of a liquid surface, and other still more
striking movements such as those of a piece of camphor floating on
water, were at one time ascribed by certain physicists to a peculiar
force, _sui generis_, the _force épipolique_ of Dutrochet: until
van der Mensbrugghe shewed that differences of surface tension were
enough to account for this whole series of phenomena (Sur la tension
superficielle des liquides considérée au point de vue de certains
mouvements observés à leur surface, _Mém. Cour. Acad. de Belgique_,
XXXIV, 1869; cf. Plateau, p. 283).

[281] Cf. _infra_, p. 306.

[282] Cf. p. 32.

[283] Or, more strictly speaking, unless its thickness be less than
twice the range of the molecular forces.

[284] It follows that the tension, depending only on the
surface-conditions, is independent of the thickness of the film.

[285] This simple but immensely important formula is due to Laplace
(_Mécanique Céleste_, Bk. x. suppl. _Théorie de l’action capillaire_,
1806).

[286] Sur la surface de révolution dont la courbure moyenne est
constante, _Journ. de M. Liouville_, VI, p. 309, 1841.

[287] See _Liquid Drops and Globules_, 1914, p. 11. Robert Boyle used
turpentine in much the same way. For other methods see Plateau, _op.
cit._ p. 154.

[288] Felix Plateau recommends the use of a weighted thread, or
plumb-line, drawn up out of a jar of water or oil; _Phil. Mag._ XXXIV,
p. 246, 1867.

[289] Cf. Boys, C. V., On Quartz Fibres, _Nature_, July 11, 1889;
Warburton, C., The Spinning Apparatus of Geometric Spiders, _Q.J.M.S._
XXXI, pp. 29–39, 1890.

[290] J. Blackwall, _Spiders of Great Britain_ (Ray Society), 1859, p.
10; _Trans. Linn. Soc._ XVI, p. 477, 1833.

[291] The intermediate spherules appear, with great regularity
and beauty, whenever a liquid jet breaks up into drops; see the
instantaneous photographs in Poynting and Thomson’s _Properties of
Matter_, pp. 151, 152, (ed. 1907).

[292] Kühne, _Untersuchungen über das Protoplasma_, 1864, p. 75, etc.

[293] _A Study of Splashes_, 1908, p. 38, etc.; Segmentation of a
Liquid Annulus, _Proc. Roy. Soc._ XXX, pp. 49–60, 1880.

[294] Cf. _ibid._ pp. 17, 77. The same phenomenon is beautifully and
continuously evident when a strong jet of water from a tap impinges on
a curved surface and then shoots off it.

[295] See a _Study of Splashes_, p. 54.

[296] A case which we have not specially considered, but which may be
found to deserve consideration in biology, is that of a cell or drop
suspended in a liquid of _varying_ density, for instance in the upper
layers of a fluid (e.g. sea-water) at whose surface condensation is
going on, so as to produce a steady density-gradient. In this case the
normally spherical drop will be flattened into an oval form, with its
maximum surface-curvature lying at the level where the densities of the
drop and the surrounding liquid are just equal. The sectional outline
of the drop has been shewn to be not a true oval or ellipse, but a
somewhat complicated quartic curve. (Rice, _Phil. Mag._ Jan. 1915.)

[297] Indeed any non-isotropic _stiffness_, even though _T_ remained
uniform, would simulate, and be indistinguishable from, a condition of
non-stiffness and non-isotropic _T_.

[298] A non-symmetry of _T_ and _T′_ might also be capable of
explanation as a result of “liquid crystallisation.” This hypothesis is
referred to, in connection with the blood-corpuscles, on p. 272.

[299] The case of the snow-crystals is a particularly interesting one;
for their “distribution” is in some ways analogous to what we find, for
instance, among our microscopic skeletons of Radiolarians. That is to
say, we may one day meet with myriads of some one particular form or
species only, and another day with myriads of another; while at another
time and place we may find species intermingled in inexhaustible
variety. (Cf. e.g. J. Glaisher, _Ill. London News_, Feb. 17, 1855;
_Q.J.M.S._ III, pp. 179–185, 1855).

[300] Cf. Bergson, _Creative Evolution_, p. 107: “Certain Foraminifera
have not varied since the Silurian epoch. Unmoved witnesses of the
innumerable revolutions that have upheaved our planet, the Lingulae are
today what they were at the remotest times of the palaeozoic era.”

[301] Ray Lankester, _A.M.N.H._ (4), XI, p. 321, 1873.

[302] Leidy, Parasites of the Termites, _J. Nat. Sci., Philadelphia_,
VIII, pp. 425–447, 1874–81; cf. Saville Kent’s _Infusoria_, II, p. 551.

[303] _Op. cit._ p. 79.

[304] Brady, _Challenger Monograph_, pl. XX, p. 233.

[305] That the Foraminifera not only can but do hang from the surface
of the water is confirmed by the following apt quotation which I owe
to Mr E. Heron-Allen: “Quand on place, comme il a été dit, le dépôt
provenant du lavage des fucus dans un flacon que l’on remplit de
nouvelle eau, on voit au bout d’une heure environ les animaux [_Gromia
dujardinii_] se mettre en mouvement et commencer à grimper. Six heures
après ils tapissent l’extérieur du flacon, de sorte que les plus élevés
sont à trente-six ou quarante-deux millimetres du fond; le lendemain
beaucoup d’entre eux, _après avoir atteint le niveau du liquide, ont
continué à ramper à sa surface, en se laissant pendre au-dessous_
comme certains mollusques gastéropodes.” (Dujardin, F., Observations
nouvelles sur les prétendus céphalopodes microscopiques, _Ann. des Sci.
Nat._ (2), III, p. 312, 1835.)

[306] Cf. Boas, _Spolia Atlantica_, 1886, pl. 6.

[307] This cellular pattern would seem to be related to the “cohesion
figures” described by Tomlinson in various surface-films (_Phil. Mag._
1861 to 1870); to the “tesselated structure” in liquids described by
Professor James Thomson in 1882 (_Collected Papers_, p. 136); and to
the _tourbillons cellulaires_ of Prof. H. Bénard (_Ann. de Chimie_
(7), XXIII, pp. 62–144, 1901, (8), XXIV, pp. 563–566, 1911), _Rev.
génér. des Sci._ XI, p. 1268, 1900; cf. also E. H. Weber. (_Poggend.
Ann._ XCIV, p. 452, 1855, etc.). The phenomenon is of great interest
and various appearances have been referred to it, in biology, geology,
metallurgy and even astronomy: for the flocculent clouds in the solar
photosphere shew an analogous configuration. (See letters by Kerr
Grant, Larmor, Wager and others, in _Nature_, April 16 to June 11,
1914.) In many instances, marked by strict symmetry or regularity, it
is very possible that the interference of waves or ripples may play
its part in the phenomenon. But in the majority of cases, it is fairly
certain that localised centres of action, or of diminished tension, are
present, such as might be provided by dust-particles in the case of
Darling’s experiment (cf. _infra_, p. 590).

[308] Ueber physikalischen Eigenschaften dünner, fester Lamellen, _S.B.
Berlin. Akad._ 1888, pp. 789, 790.

[309] Certain palaeontologists (e.g. Haeusler and Spandel) have
maintained that in each family or genus the plain smooth-shelled
forms are the primitive and ancient ones, and that the ribbed and
otherwise ornamented shells make their appearance at later dates in
the course of a definite evolution (cf. Rhumbler, _Foraminiferen der
Plankton-Expedition_, 1911, i, p. 21). If this were true it would be of
fundamental importance: but this book of mine would not deserve to be
written.

[310] _A Study of Splashes_, p. 116.

[311] See _Silliman’s Journal_, II, p. 179, 1820; and cf. Plateau, _op.
cit._ II, pp. 134, 461.

[312] The presence or absence of the contractile vacuole or vacuoles
is one of the chief distinctions, in systematic zoology, between the
Heliozoa and the Radiolaria. As we have seen on p. 165 (footnote),
it is probably no more than a physical consequence of the different
conditions of existence in fresh water and in salt.

[313] Cf. Doflein, _Lehrbuch der Protozoenkunde_, 1911, p. 422.

[314] Cf. Minchin, _Introduction to the Study of the Protozoa_, 1914 p.
293, Fig. 127.

[315] Cf. C. A. Kofoid and Olive Swezy, On Trichomonad Flagellates,
etc. _Pr. Amer. Acad. of Arts and Sci._ LI, pp. 289–378, 1915.

[316] D. L. Mackinnon, Herpetomonads from the Alimentary Tract of
certain Dungflies, _Parasitology_, III, p. 268, 1910.

[317] _Proc. Roy. Soc._ XII, pp. 251–257, 1862–3.

[318] Cf. (_int. al._) Lehmann, Ueber scheinbar lebende Kristalle und
Myelinformen, _Arch. f. Entw. Mech._ XXVI, p. 483, 1908; _Ann. d.
Physik_, XLIV, p. 969, 1914.

[319] Cf. B. Moore and H. C. Roaf, On the Osmotic Equilibrium of the
Red Blood Corpuscle, _Biochem. Journal_, III, p. 55, 1908.

[320] For an attempt to explain the form of a blood-corpuscle by
surface-tension alone, see Rice, _Phil. Mag._ Nov. 1914; but cf.
Shorter, _ibid._ Jan. 1915.

[321] Koltzoff, N. K., Studien über die Gestalt der Zelle, _Arch. f.
mikrosk. Anat._ LXVII, pp. 364–571, 1905; _Biol. Centralbl._ XXIII,
pp. 680–696, 1903, XXVI, pp. 854–863, 1906; _Arch. f. Zellforschung_,
II, pp. 1–65, 1908, VII, pp. 344–423, 1911; _Anat. Anzeiger_, XLI, pp.
183–206, 1912.

[322] Cf. _supra_, p. 129.

[323] As Bethe points out (Zellgestalt, Plateausche Flüssigkeitstigur
und Neurofibrille, _Anat. Anz._ XL. p. 209, 1911), the spiral fibres
of which Koltzoff speaks must lie _in the surface_, and not within the
substance, of the cell whose conformation is affected by them.

[324] See for a further but still elementary account, Michaelis,
_Dynamics of Surfaces_, 1914, p. 22 _seq._; Macallum,
_Oberflächenspannung und Lebenserscheinungen_, in Asher-Spiro’s
_Ergebnisse der Physiologie_, XI, pp. 598–658, 1911; see also W.
W. Taylor’s _Chemistry of Colloids_, 1915, p. 221 _seq._, Wolfgang
Ostwald, _Grundriss der Kolloidchemie_, 1909, and other text-books of
physical chemistry; and Bayliss’s _Principles of General Physiology_,
pp. 54–73, 1915.

[325] The first instance of what we now call an adsorptive phenomenon
was observed in soap-bubbles. Leidenfrost, in 1756, was aware that the
outer layer of the bubble was covered by an “oily” layer. A hundred
years later Dupré shewed that in a soap-solution the soap tends to
concentrate at the surface, so that the surface-tension of a very weak
solution is very little different from that of a strong one (_Théorie
mécanique de la chaleur_, 1869, p. 376; cf. Plateau, II, p. 100).

[326] This identical phenomenon was the basis of Quincke’s
theory of amoeboid movement (Ueber periodische Ausbreitung von
Flüssigkeitsoberflächen, etc., _SB. Berlin. Akad._ 1888, pp. 791–806;
cf. _Pflüger’s Archiv_, 1879, p. 136).

[327] J. Willard Gibbs, Equilibrium of Heterogeneous Substances, _Tr.
Conn. Acad._ III, pp. 380–400, 1876, also in _Collected Papers_, I,
pp. 185–218, London, 1906; J. J. Thomson, _Applications of Dynamics
to Physics and Chemistry_, 1888 (Surface tension of solutions), p.
190. See also (_int. al._) the various papers by C. M. Lewis, _Phil.
Mag._ (6), XV, p. 499, 1908, XVII, p. 466, 1909, _Zeitschr. f. physik.
Chemie_, LXX, p. 129, 1910; Milner, _Phil. Mag._ (6), XIII, p. 96,
1907, etc.

[328] G. F. FitzGerald, On the Theory of Muscular Contraction, _Brit.
Ass. Rep._ 1878; also in _Scientific Writings_, ed. Larmor, 1902, pp.
34, 75. A. d’Arsonval, Relations entre l’électricité animale et la
tension superficielle, _C. R._ CVI, p. 1740. 1888; cf. A. Imbert, Le
mécanisme de la contraction musculaire, déduit de la considération des
forces de tension superficielle, _Arch. de Phys._ (5), IX, pp. 289–301,
1897.

[329] Ueber die Natur der Bindung der Gase im Blut und in seinen
Bestandtheilen, _Kolloid. Zeitschr._ II, pp. 264–272, 294–301, 1908;
cf. Loewy, Dissociationsspannung des Oxyhaemoglobin im Blut, _Arch. f.
Anat. und Physiol._ 1904, p. 231.

[330] We may trace the first steps in the study of this phenomenon
to Melsens, who found that thin films of white of egg become firm
and insoluble (Sur les modifications apportées à l’albumine ... par
l’action purement mécanique, _C. R. Acad. Sci._ XXXIII, p. 247, 1851);
and Harting made similar observations about the same time. Ramsden has
investigated the same subject, and also the more general phenomenon
of the formation of albuminoid and fatty membranes by adsorption: cf.
Koagulierung der Eiweisskörper auf mechanischer Wege, _Arch. f. Anat.
u. Phys._ (_Phys. Abth._) 1894, p. 517; Abscheidung fester Körper in
Oberflächenschichten _Z. f. phys. Chem._ XLVII, p. 341, 1902; _Proc.
R. S._ LXXII, p. 156, 1904. For a general review of the whole subject
see H. Zangger, Ueber Membranen und Membranfunktionen, in Asher-Spiro’s
_Ergebnisse der Physiologie_, VII, pp. 99–160, 1908.

[331] Cf. Taylor, _Chemistry of Colloids_, p. 252.

[332] Strasbürger, Ueber Cytoplasmastrukturen, etc. _Jahrb. f. wiss.
Bot._ XXX, 1897; R. A. Harper, Kerntheilung und freie Zellbildung im
Ascus, _ibid._; cf. Wilson, _The Cell in Development, etc._ pp. 53–55.

[333] Cf. A. Gurwitsch, _Morphologie und Biologie der Zelle_, 1904,
pp. 169–185; Meves, Die Chondriosomen als Träger erblicher Anlagen,
_Arch. f. mikrosk. Anat._ 1908, p. 72; J. O. W. Barratt, Changes in
Chondriosomes, etc. _Q.J.M.S._ LVIII, pp. 553–566, 1913, etc.; A.
Mathews, Changes in Structure of the Pancreas Cell, etc., _J. of
Morph._ XV (Suppl.), pp. 171–222, 1899.

[334] The question whether chromosomes, chondriosomes or chromidia
be the true vehicles or transmitters of “heredity” is not without
its analogy to the older problem of whether the pineal gland or the
pituitary body were the actual seat and domicile of the soul.

[335] Cf. C. C. Dobell, Chromidia and the Binuclearity Hypotheses; a
review and a criticism, _Q.J.M.S._ LIII, 279–326, 1909; Prenant, A.,
Les Mitochondries et l’Ergastoplasme, _Journ. de l’Anat. et de la
Physiol._ XLVI, pp. 217–285, 1910 (both with copious bibliography).

[336] Traube in particular has maintained that in differences of
surface-tension we have the origin of the active force productive
of osmotic currents, and that herein we find an explanation, or an
approach to an explanation, of many phenomena which were formerly
deemed peculiarly “vital” in their character. “Die Differenz der
Oberflächenspannungen oder der Oberflächendruck eine Kraft darstellt,
welche als treibende Kraft der Osmose, an die Stelle des nicht mit
dem Oberflächendruck identischen osmotischen Druckes, zu setzen ist,
etc.” (Oberflächendruck und seine Bedeutung im Organismus, _Pflüger’s
Archiv_, CV, p. 559, 1904.) Cf. also Hardy (_Pr. Phys. Soc._ XXVIII, p.
116, 1916), “If the surface film of a colloid membrane separating two
masses of fluid were to change in such a way as to lower the potential
of the water in it, water would enter the region from both sides at
once. But if the change of state were to be propagated as a wave of
change, starting at one face and dying out at the other face, water
would be carried along from one side of the membrane to the other. A
succession of such waves would maintain a flow of fluid.”

[337] On the Distribution of Potassium in animal and vegetable Cells;
_Journ. of Physiol._ XXXII, p. 95, 1905.

[338] The reader will recognise that there is a fundamental difference,
and contrast, between such experiments as these of Professor Macallum’s
and the ordinary staining processes of the histologist. The latter
are (as a general rule) purely empirical, while the former endeavour
to reveal the true microchemistry of the cell. “On peut dire que la
microchimie n’est encore qu’à la période d’essai, et que l’avenir de
l’histologie et spécialement de la cytologie est tout entier dans la
microchimie” (Prenant, A., Méthodes et résultats de la Microchimie,
_Journ. de l’Anat. et de la Physiol._ XLVI, pp. 343–404, 1910).

[339] Cf. Macallum, Presidential Address, Section I, _Brit. Ass. Rep._
(Sheffield), 1910, p. 744.

[340] In accordance with a simple _corollary_ to the Gibbs-Thomson law.

[341] It can easily be proved (by equating the increase of energy
stored in an increased surface to the work done in increasing that
surface), that the tension measured per unit breadth, _T__{_ab_}, is
equal to the energy per unit area, _E__{_ab_}.

[342] The presence of this little liquid “bourrelet,” drawn from the
material of which the partition-walls themselves are composed, is
obviously tending to a reduction of the internal surface-area. And it
may be that it is as well, or better, accounted for on this ground than
on Plateau’s assumption that it represents a “surface of continuity.”

[343] A similar “bourrelet” is admirably seen at the line of junction
between a floating bubble and the liquid on which it floats; in which
case it constitutes a “masse annulaire,” whose mathematical properties
and relation to the form of the _nearly_ hemispherical bubble, have
been investigated by van der Mensbrugghe (cf. Plateau, _op. cit._,
p. 386). The form of the superficial vacuoles in Actinophrys or
Actinosphaerium involves an identical problem.

[344] In an actual calculation we must of course always take account of
the tensions on _both sides_ of each film or membrane.

[345] Hofmeister, _Pringsheim’s Jahrb._ III, p. 272, 1863; _Hdb. d.
physiol. Bot._ I, 1867, p. 129.

[346] Sachs, Ueber die Anordnung der Zellen in jüngsten
Pflanzentheilen, _Verh. phys. med. Ges. Würzburg_, XI, pp. 219–242,
1877; Ueber Zellenanordnung und Wachsthum, _ibid._ XII, 1878; Ueber
die durch Wachsthum bedingte Verschiebung kleinster Theilchen in
trajectorischen Curven, _Monatsber. k. Akad. Wiss. Berlin_, 1880;
_Physiology of Plants_, chap. xxvii, pp. 431–459, Oxford, 1887.

[347] Schwendener, Ueber den Bau und das Wachsthum des Flechtenthallus,
_Naturf. Ges. Zürich_, Febr. 1860, pp. 272–296.

[348] Reinke, _Lehrbuch der Botanik_, 1880, p. 519.

[349] Cf. Leitgeb, _Unters. über die Lebermoose_, II, p. 4, Graz, 1881.

[350] Rauber, Neue Grundlegungen zur Kenntniss der Zelle, _Morph.
Jahrb._ VIII, pp. 279, 334, 1882.

[351] _C. R. Acad. Sc._ XXXIII, p. 247, 1851; _Ann. de chimie et de
phys._ (3), XXXIII, p. 170, 1851; _Bull. R. Acad. Belg._ XXIV, p. 531,
1857.

[352] Klebs, _Biolog. Centralbl._ VII, pp. 193–201, 1887.

[353] L. Errera, Sur une condition fondamentale d’équilibre des
cellules vivantes, _C. R._, CIII, p. 822, 1886; _Bull. Soc. Belge
de Microscopie_, XIII, Oct. 1886; _Recueil d’œuvres_ (_Physiologie
générale_), 1910, pp. 201–205.

[354] L. Chabry, Embryologie des Ascidiens, _J. Anat. et Physiol._
XXIII, p. 266, 1887.

[355] Robert, Embryologie des Troques, _Arch. de Zool. exp. et gén._
(3), X, 1892.

[356] “Dass der Furchungsmodus etwas für das Zukünftige unwesentliches
ist,” _Z. f. w. Z._ LV, 1893, p. 37. With this statement compare,
or contrast, that of Conklin, quoted on p. 4; cf. also pp. 157, 348
(footnotes).

[357] de Wildeman, Etudes sur l’attache des cloisons cellulaires, _Mém.
Couronn. de l’Acad. R. de Belgique_, LIII, 84 pp., 1893–4.

[358] It was so termed by Conklin in 1897, in his paper on Crepidula
(_J. of Morph._ XIII, 1897). It is the _Querfurche_ of Rabl (_Morph.
Jahrb._ V, 1879); the _Polarfurche_ of O. Hertwig (_Jen. Zeitschr._
XIV, 1880); the _Brechungslinie_ of Rauber (Neue Grundlage zur K. der
Zelle, _M. Jb._ VIII, 1882). It is carefully discussed by Robert, Dév.
des Troques, _Arch. de Zool. Exp. et Gén._ (3), X, 1892, p. 307 seq.

[359] Thus Wilson (_J. of Morph._ VIII, 1895) declared that in
Amphioxus the polar furrow was occasionally absent, and Driesch took
occasion to criticise and to throw doubt upon the statement (_Arch. f.
Entw. Mech._ I, 1895, p. 418).

[360] Precisely the same remark was made long ago by Driesch: “Das so
oft sehematisch gezeichnete Vierzellenstadium mit zwei sich in zwei
Punkten scheidende Medianen kann man wohl getrost aus der Reihe des
Existierenden streichen,” _Entw. mech. Studien, Z. f. w. Z._ LIII, p.
166, 1892. Cf. also his _Math. mechanische Bedeutung morphologischer
Probleme der Biologie_, Jena, 59 pp. 1891.

[361] Compare, however, p. 299.

[362] _Ricreatione dell’ occhio e della mente, nell’ Osservatione delle
Chiocciole_, Roma, 1681.

[363] Cf. some of J. H. Vincent’s photographs of ripples, in _Phil.
Mag._ 1897–1899; or those of F. R. Watson, in _Phys. Review_, 1897,
1901, 1916. The appearance will depend on the rate of the wave, and in
turn on the surface-tension; with a low tension one would probably see
only a moving “jabble.” FitzGerald thought diatom-patterns might be due
to electromagnetic vibrations (_Works_, p. 503, 1902).

[364] Cushman, J. A. and Henderson, W. P., _Amer. Nat._ XL, pp.
797–802, 1906.

[365] This does not merely neglect the _broken_ ones but _all_ whose
centres lie between this circle and a hexagon inscribed in it.

[366] For more detailed calculations see a paper by “H.M.” [? H.
Munro], in _Q. J. M. S._ VI, p. 83, 1858.

[367] Cf. Hartog, The Dual Force of the Dividing Cell, _Science
Progress_ (n.s.), I, Oct. 1907, and other papers. Also Baltzer, _Ueber
mehrpolige Mitosen bei Seeigeleiern_, Inaug. Diss. 1908.

[368] Observations sur les Abeilles, _Mém. Acad. Sc. Paris_, 1712, p.
299.

[369] As explained by Leslie Ellis, in his essay “On the Form of Bees’
Cells,” in _Mathematical and other Writings_, 1853, p. 353; cf. O.
Terquem, _Nouv. Ann. Math._ 1856, p. 178.

[370] _Phil. Trans._ XLII, 1743, pp. 565–571.

[371] _Mém. de l’Acad. de Berlin_, 1781.

[372] Cf. Gregory, _Examples_, p. 106, Wood’s _Homes without Hands_,
1865, p. 428, Mach, _Science of Mechanics_, 1902, p. 453, etc., etc.

[373] _Origin of Species_, ch. VIII (6th ed., p. 221). The cells of
various bees, humble-bees and social wasps have been described and
mathematically investigated by K. Müllenhoff, _Pflüger’s Archiv_
XXXII, p. 589, 1883; but his many interesting results are too complex
to epitomise. For figures of various nests and combs see (e.g.) von
Büttel-Reepen, _Biol. Centralbl._ XXXIII, pp. 4, 89, 129, 183, 1903.

[374] Darwin had a somewhat similar idea, though he allowed more play
to the bee’s instinct or conscious intention. Thus, when he noticed
certain half-completed cell-walls to be concave on one side and convex
on the other, but to become perfectly flat when restored for a short
time to the hive, he says: “It was absolutely impossible, from the
extreme thinness of the little plate, that they could have effected
this by gnawing away the convex side; and I suspect that the bees
in such cases stand on opposite sides and push and bend the ductile
and warm wax (which as I have tried is easily done) into its proper
intermediate plane, and thus flatten it.”

[375] Since writing the above, I see that Müllenhoff gives the
same explanation, and declares that the waxen wall is actually a
_Flüssigkeitshäutchen_, or liquid film.

[376] Bonnet criticised Buffon’s explanation, on the ground that his
description was incomplete; for Buffon took no account of the Maraldi
pyramids.

[377] Buffon, _Histoire Naturelle_, IV, p. 99. Among many other papers
on the Bee’s cell, see Barclay, _Mem. Wernerian Soc._ II, p. 259
(1812), 1818; Sharpe, _Phil. Mag._ IV, 1828, pp. 19–21; L. Lalanne,
_Ann. Sci. Nat._ (2) Zool. XIII, pp. 358–374, 1840; Haughton, _Ann.
Mag. Nat. Hist._ (3), XI, pp. 415–429, 1863; A. R. Wallace, _ibid._
XII, p. 303, 1863; Jeffries Wyman. _Pr. Amer. Acad. of Arts and Sc._
VII, pp. 68–83, 1868; Chauncey Wright, _ibid._ IV, p. 432, 1860.

[378] Sir W. Thomson, On the Division of Space with Minimum Partitional
Area, _Phil. Mag._ (5), XXIV, pp. 503–514, Dec. 1887; cf. _Baltimore
Lectures_, 1904, p. 615.

[379] Also discovered independently by Sir David Brewster, _Trans.
R.S.E._ XXIV, p. 505, 1867, XXV, p. 115, 1869.

[380] Von Fedorow had already described (in Russian) the same figure,
under the name of cubo-octahedron, or hepta-parallelohedron, limited
however to the case where all the faces are plane. This figure,
together with the cube, the hexagonal prism, the rhombic dodecahedron
and the “elongated dodecahedron,” constituted the five plane-faced,
parallel-sided figures by which space is capable of being completely
filled and symmetrically partitioned; the series so forming the
foundation of Von Fedorow’s theory of crystalline structure. The
elongated dodecahedron is, essentially, the figure of the bee’s cell.

[381] F. R. Lillie, Embryology of the Unionidae, _Journ. of
Morphology_, X, p. 12, 1895.

[382] E. B. Wilson, The Cell-lineage of Nereis, _Journ. of Morphology_,
VI, p. 452, 1892.

[383] It is highly probable, and we may reasonably assume, that the two
little triangles do not actually meet at an apical _point_, but merge
into one another by a twist, or minute surface of complex curvature, so
as not to contravene the normal conditions of equilibrium.

[384] Professor Peddie has given me this interesting and important
result, but the mathematical reasoning is too lengthy to be set forth
here.

[385] Cf. Rhumbler, _Arch. f. Entw. Mech._ XIV, p. 401, 1902; Assheton,
_ibid._ XXXI, pp. 46–78, 1910.

[386] M. Robert (_l. c._ p. 305) has compiled a long list of cases
among the molluscs and the worms, where the initial segmentation of
the egg proceeds by equal or unequal division. The two cases are about
equally numerous. But like many other writers, he would ascribe this
equality or inequality rather to a provision for the future than to
a direct effect of immediate physical causation: “Il semble assez
probable, comme on l’a dit souvent, que la plus grande taille d’un
blastomère est liée à l’importance et au développement précoce des
parties du corps qui doivent en naître: il y aurait là une sorte de
reflet des stades postérieures du développement sur les premières
phénomènes, ce que M. Ray Lankester appelle _precocious segregation_.
Il faut avouer pourtant qu’on est parfois assez embarrassé pour
assigner une cause à pareilles différences.”

[387] The principle is well illustrated in an experiment of Sir David
Brewster’s (_Trans. R.S.E._ XXV, p. 111, 1869). A soap-film is drawn
over the rim of a wine-glass, and then covered by a watch-glass. The
film is inclined or shaken till it becomes attached to the glass
covering, and it then immediately changes place, leaving its transverse
position to take up that of a spherical segment extending from one side
of the wine-glass to its cover, and so enclosing the same volume of air
as formerly but with a great economy of surface, precisely as in the
case of our spherical partition cutting off one corner of a cube.

[388] Cf. Wildeman, _Attache des Cloisons_, etc., pls. 1, 2.

[389] _Nova Acta K. Leop. Akad._ XI, 1, pl. IV.

[390] Cf. _Protoplasmamechanik_, p. 229: “Insofern liegen also die
Verhältnisse hier wesentlich anders als bei der Zertheilung hohler
Körperformen durch flüssige Lamellen. Wenn die Membran bei der
Zelltheilung die von dem Prinzip der kleinsten Flächen geforderte Lage
und Krümmung annimmt, so werden wir den Grund dafür in andrer Weise
abzuleiten haben.”

[391] There is, I think, some ambiguity or disagreement among botanists
as to the use of this latter term: the sense in which I am using it,
viz. for any partition which meets the outer or peripheral wall at
right angles (the strictly _radial_ partition being for the present
excluded), is, however, clear.

[392] _Cit._ Plateau, _Statique des Liquides_, i, p. 358.

[393] Even in a Protozoon (_Euglena viridis_), when kept alive under
artificial compression, Ryder found a process of cell-division to occur
which he compares to the segmenting blastoderm of a fish’s egg, and
which corresponds in its essential features with that here described.
_Contrib. Zool. Lab. Univ. Pennsylvania_, I, pp. 37–50, 1893.

[394] This, like many similar figures, is manifestly drawn under the
influence of Sachs’s theoretical views, or assumptions, regarding
orthogonal trajectories, coaxial circles, confocal ellipses, etc.

[395] Such preconceptions as Rauber entertained were all in a direction
likely to lead him away from such phenomena as he has faithfully
depicted. Rauber had no idea whatsoever of the principles by which we
are guided in this discussion, nor does he introduce at all the analogy
of surface-tension, or any other purely physical concept. But he was
deeply under the influence of Sachs’s rule of rectangular intersection;
and he was accordingly disposed to look upon the configuration
represented above in Fig. 168, 6, as the most typical or most primitive.

[396] Cf. Rauber, Neue Grundlage z. K. der Zelle, _Morph. Jahrb._ VIII,
1883, pp. 273, 274:

“Ich betone noch, dass unter meinen Figuren diejenige gar nicht
enthalten ist, welche zum Typus der Batrachierfurchung gehörig am
meisten bekannt ist .... Es haben so ausgezeichnete Beobachter sie als
vorhanden beschrieben, dass es mir nicht einfallen kann, sie überhaupt
nicht anzuerkennen.”

[397] Roux’s experiments were performed with drops of paraffin
suspended in dilute alcohol, to which a little calcium acetate was
added to form a soapy pellicle over the drops and prevent them from
reuniting with one another.

[398] Cf. (e.g.) Clerk Maxwell, On Reciprocal Figures, etc., _Trans. R.
S. E._ XXVI, p. 9, 1870.

[399] See Greville, K. R., Monograph of the Genus Asterolampra,
_Q.J.M.S._ VIII, (Trans.), pp. 102–124, 1860; cf. IBID. (n.s.), II, pp.
41–55, 1862.

[400] The same is true of the insect’s wing; but in this case I do not
hazard a conjectural explanation.

[401] _Ann. Mag. N. H._ (2), III, p. 126, 1849.

[402] _Phil. Trans._ CLVII, pp. 643–656, 1867.

[403] Sachs, _Pflanzenphysiologie_ (_Vorlesung_ XXIV), 1882; cf.
Rauber, Neue Grundlage zur Kenntniss der Zelle, _Morphol. Jahrb._ VIII,
p. 303 _seq._, 1883; E. B. Wilson, Cell-lineage of Nereis, _Journ. of
Morphology_, VI, p. 448, 1892, etc.

[404] In the following account I follow closely on the lines laid down
by Berthold; _Protoplasmamechanik_, cap. vii. Many botanical phenomena
identical and similar to those here dealt with, are elaborately
discussed by Sachs in his _Physiology of Plants_ (chap. xxvii, pp.
431–459, Oxford, 1887); and in his earlier papers, Ueber die Anordnung
der Zellen in jüngsten Pflanzentheilen, and Ueber Zellenanordnung und
Wachsthum (_Arb. d. botan. Inst. Würzburg_, 1878, 1879). But Sachs’s
treatment differs entirely from that which I adopt and advocate here:
his explanations being based on his “law” of rectangular succession,
and involving complicated systems of confocal conics, with their
orthogonally intersecting ellipses and hyperbolas.

[405] Cf. p. 369.

[406] There is much information regarding the chemical composition and
mineralogical structure of shells and other organic products in H. C.
Sorby’s Presidential Address to the Geological Society (_Proc. Geol.
Soc._ 1879, pp. 56–93); but Sorby failed to recognise that association
with “organic” matter, or with colloid matter whether living or dead,
introduced a new series of purely physical phenomena.

[407] Vesque, _Ann. des Sc. Nat._ (_Bot._) (5), XIX, p. 310, 1874.

[408] Cf. Kölliker, _Icones Histiologicae_, 1864, pp. 119, etc.

[409] In an interesting paper by Irvine and Sims Woodhead on the
“Secretion of Carbonate of Lime by Animals” (_Proc. R. S. E._ XVI,
1889, p. 351) it is asserted that “lime salts, of whatever form, are
deposited _only_ in vitally inactive tissue.”

[410] The tube of Teredo shews no trace of organic matter, but consists
of irregular prismatic crystals: the whole structure “being identical
with that of small veins of calcite, such as are seen in thin sections
of rocks” (Sorby, _Proc. Geol. Soc._ 1879, p. 58). This, then, would
seem to be a somewhat exceptional case of a shell laid down completely
outside of the animal’s external layer of organic or colloid substance.

[411] _C. R. Soc. Biol. Paris_ (9), I, pp. 17–20, 1889; _C. R. Ac. Sc._
CVIII, pp. 196–8, 1889.

[412] Cf. Heron-Allen, _Phil. Trans._ (B), vol. CCVI, p. 262, 1915.

[413] See Leduc, _Mechanism of Life_ (1911), ch. X, for copious
references to other works on the artificial production of “organic”
forms.

[414] Lectures on the Molecular Asymmetry of Natural Organic Compounds,
_Chemical Soc. of Paris_, 1860, and also in Ostwald’s _Klassiker d. ex.
Wiss._ No. 28, and in _Alembic Club Reprints_, No. 14, Edinburgh, 1897;
cf. Richardson, G. M., _Foundations of Stereochemistry_, N. Y. 1901.

[415] Japp, Stereometry and Vitalism, _Brit. Ass. Rep._ (Bristol), p.
813, 1898; cf. also a voluminous discussion in _Nature_, 1898–9.

[416] They represent the general theorem of which particular cases are
found, for instance, in the asymmetry of the ferments (or _enzymes_)
which act upon asymmetrical bodies, the one fitting the other,
according to Emil Fischer’s well-known phrase, as lock and key. Cf.
his Bedeutung der Stereochemie für die Physiologie, _Z. f. physiol.
Chemie_, V, p. 60, 1899, and various papers in the _Ber. d. d. chem.
Ges._ from 1894.

[417] In accordance with Emil Fischer’s conception of “asymmetric
synthesis,” it is now held to be more likely that the process is
synthetic than analytic: more likely, that is to say, that the plant
builds up from the first one asymmetric body to the exclusion of the
other, than that it “selects” or “picks out” (as Japp supposed) the
right-handed or the left-handed molecules from an original, optically
inactive, mixture of the two; cf. A. McKenzie, Studies in Asymmetric
Synthesis, _Journ. Chem. Soc._ (Trans.), LXXXV, p. 1249, 1904.

[418] See for a fuller discussion, Hans Przibram, _Vitalität_, 1913,
Kap. iv, Stoffwechsel (Assimilation und Katalyse).

[419] Cf. Cotton, _Ann. de Chim. et de Phys._ (7), VIII, pp. 347–432
(cf. p. 373), 1896.

[420] Byk, A., Zur Frage der Spaltbarkeit von Razemverbindungen durch
Zirkularpolarisiertes Licht, ein Beitrag zur primären Entstehung
optisch-activer Substanzen, _Zeitsch. f. physikal. Chemie_, XLIX, p.
641, 1904. It must be admitted that further positive evidence on these
lines is still awanting.

[421] Cf. (_int. al._) Emil Fischer, _Untersuchungen über Aminosäuren,
Proteine_, etc. Berlin, 1906.

[422] Japp, _l. c._ p. 828.

[423] Rainey, G., On the Elementary Formation of the Skeletons of
Animals, and other Hard Structures formed in connection with Living
Tissue, _Brit. For. Med. Ch. Rev._ XX, pp. 451–476, 1857; published
separately with additions, 8vo. London, 1858. For other papers by
Rainey on kindred subjects see _Q. J. M. S._ VI (_Tr. Microsc. Soc._),
pp. 41–50, 1858, VII, pp. 212–225, 1859, VIII, pp. 1–10, 1860, I (n.
s.), pp. 23–32, 1861. Cf. also Ord, W. M., On Molecular Coalescence,
and on the influence exercised by Colloids upon the Forms of Inorganic
Matter, _Q. J. M. S._ XII, pp. 219–239, 1872; and also the early
but still interesting observations of Mr Charles Hatchett, Chemical
Experiments on Zoophytes; with some observations on the component parts
of Membrane, _Phil. Trans._ 1800. pp. 327–402.

[424] Cf. Quincke, Ueber unsichtbare Flüssigkeitsschichten, _Ann. der
Physik_, 1902.

[425] See for instance other excellent illustrations in Carpenter’s
article “Shell,” in Todd’s _Cyclopædia_, vol. IV. pp. 550–571,
1847–49. According to Carpenter, the shells of the mollusca (and also
of the crustacea) are “essentially composed of _cells_, consolidated
by a deposit of carbonate of lime in their interior.” That is to
say, Carpenter supposed that the spherulites, or calcospherites
of Harting, were, to begin with, just so many living protoplasmic
cells. Soon afterwards however, Huxley pointed out that the mode of
formation, while at first sight “irresistibly suggesting a cellular
structure, ... is in reality nothing of the kind,” but “is simply the
result of the concretionary manner in which the calcareous matter is
deposited”; _ibid._ art. “Tegumentary Organs,” vol. V, p. 487, 1859.
Quekett (_Lectures on Histology_, vol. II, p. 393, 1854, and _Q. J.
M. S._ XI, pp. 95–104, 1863) supported Carpenter; but Williamson
(Histological Features in the Shells of the Crustacea, _Q. J. M. S._
VIII, pp. 35–47, 1860) amply confirmed Huxley’s view, which in the
end Carpenter himself adopted (_The Microscope_, 1862, p. 604). A
like controversy arose later in regard to corals. Mrs Gordon (M. M.
Ogilvie) asserted that the coral was built up “of successive layers
of calcified cells, which hang together at first by their cell-walls,
and ultimately, as crystalline changes continue, form the individual
laminae of the skeletal structures” (_Phil. Trans._ CLXXXVII, p. 102,
1896): whereas v. Koch had figured the coral as formed out of a mass
of “Kalkconcremente” or “crystalline spheroids,” laid down outside
the ectoderm, and precisely similar both in their early rounded and
later polygonal stages (though von Koch was not aware of the fact) to
the calcospherites of Harting (Entw. d. Kalkskelettes von Asteroides,
_Mitth. Zool. St. Neapel_, III, pp. 284–290, pl. XX, 1882). Lastly
Duerden shewed that external to, and apparently secreted by the
ectoderm lies a homogeneous organic matrix or membrane, “in which the
minute calcareous crystals forming the skeleton are laid down” (The
Coral _Siderastraea radians_, etc., _Carnegie Inst. Washington_, 1904,
p. 34). Cf. also M. M. Ogilvie-Gordon, _Q. J. M. S._ XLIX, p. 203,
1905, etc.

[426] Cf. Claparède, _Z. f. w. Z._ XIX, p. 604, 1869.

[427] Spicules extremely like those of the Alcyonaria occur also in a
few sponges; cf. (e.g.), Vaughan Jennings, _Journ. Linn. Soc._ XXIII,
p. 531, pl. 13, fig. 8, 1891.

[428] _Mem. Manchester Lit. and Phil. Soc._ LX, p. 11, 1916.

[429] Mummery, J. H., On Calcification in Enamel and Dentine, _Phil.
Trans._ CCV (B), pp. 95–111, 1914.

[430] The artificial concretion represented in Fig. 202 is identical
in appearance with the concretions found in the kidney of Nautilus, as
figured by Willey (_Zoological Results_, p. lxxvi, Fig. 2, 1902).

[431] Cf. Taylor’s _Chemistry of Colloids_, p. 18, etc., 1915.

[432] This rule, undreamed of by Errera, supports and justifies the
cardinal assumption (of which we have had so much to say in discussing
the forms of cells and tissues) that the _incipient_ cell-wall behaves
as, and indeed actually is, a liquid film (cf. p. 306).

[433] Cf. p. 254.

[434] Cf. Harting, _op. cit._, pp. 22, 50: “J’avais cru d’abord que ces
couches concentriques étaient produites par l’alternance de la chaleur
ou de la lumière, pendant le jour et la nuit. Mais l’expérience,
expressément instituée pour examiner cette question, y a répondu
négativement.”

[435] Liesegang, R. E., _Ueber die Schichtungen bei Diffusionen_,
Leipzig, 1907, and other earlier papers.

[436] Cf. Taylor’s _Chemistry of Colloids_, pp. 146–148, 1915.

[437] Cf. S. C. Bradford, The Liesegang Phenomenon and Concretionary
Structure in Rocks, _Nature_, XCVII, p. 80, 1916; cf. _Sci. Progress_,
X, p. 369, 1916.

[438] Cf. Faraday, On Ice of Irregular Fusibility, _Phil. Trans._,
1858, p. 228; _Researches in Chemistry, etc._, 1859, p. 374; Tyndall,
_Forms of Water_, p. 178, 1872; Tomlinson, C., On some effects of small
Quantities of Foreign Matter on Crystallisation, _Phil. Mag._ (5) XXXI,
p. 393, 1891, and other papers.

[439] A Study in Crystallisation, _J. of Soc. of Chem. Industry_, XXV,
p. 143, 1906.

[440] _Ueber Zonenbildung in kolloidalen Medien_, Jena, 1913.

[441] _Verh. d. d. Zool. Gesellsch._ p. 179, 1912.

[442] _Descent of Man_, II, pp. 132–153, 1871.

[443] As a matter of fact, the phenomena associated with the
development of an “ocellus” are or may be of great complexity, inasmuch
as they involve not only a graded distribution of pigment, but also,
in “optical” coloration, a symmetrical distribution of structure or
form. The subject therefore deserves very careful discussion, such
as Bateson gives to it (_Variation_, chap. xii). This, by the way,
is one of the very rare cases in which Bateson appears inclined to
suggest a purely physical explanation of an organic phenomenon: “The
suggestion is strong that the whole series of rings (in _Morpho_) may
have been formed by some one central disturbance, somewhat as a series
of concentric waves may be formed by the splash of a stone thrown into
a pool, etc.”

[444] Cf. also Sir D. Brewster, On optical properties of Mother of
Pearl, _Phil. Trans._ 1814, p. 397.

[445] Biedermann, W., Ueber die Bedeutung von Kristallisationsprozessen
der Skelette wirbelloser Thiere, namentlich der Molluskenschalen,
_Z. f. allg. Physiol._ I, p. 154, 1902; Ueber Bau und Entstehung der
Molluskenschale, _Jen. Zeitschr._ XXXVI, pp. 1–164, 1902. Cf. also
Steinmann, Ueber Schale und Kalksteinbildungen, _Ber. Naturf. Ges.
Freiburg i. Br_ IV, 1889; Liesegang, _Naturw. Wochenschr._ p. 641, 1910.

[446] Cf. Bütschli, Ueber die Herstellung künstlicher Stärkekörner oder
von Sphärokrystallen der Stärke, _Verh. nat. med. Ver. Heidelberg_, V,
pp. 457–472, 1896.

[447] _Untersuchungen über die Stärkekörner_, Jena, 1905.

[448] Cf. Winge, _Meddel. fra Komm. for Havundersögelse_ (_Fiskeri_),
IV, p. 20, Copenhagen, 1915.

[449] The anhydrite is sulphate of lime (CaSO_{4}); the polyhalite is
a triple sulphate of lime, magnesia and potash (2 CaSO_{4}. MgSO_{4}.
K_{2}SO_{4} + 2 H_{2}O).

[450] Cf. van’t Hoff, _Physical Chemistry in the Service of the
Sciences_, p. 99 seq. Chicago, 1903.

[451] Sphärocrystalle von Kalkoxalat bei Kakteen, _Ber. d. d. Bot.
Gesellsch._ p. 178, 1885.

[452] Pauli, W. u. Samec, M., Ueber Löslichkeitsbeeinflüssung von
Elektrolyten durch Eiweisskörper, _Biochem. Zeitschr._ XVII, p. 235,
1910. Some of these results were known much earlier; cf. Fokker in
_Pflüger’s Archiv_, VII, p. 274, 1873; also Irvine and Sims Woodhead,
_op. cit._ p. 347.

[453] Which, in 1000 parts of ash, contains about 840 parts of
phosphate and 76 parts of calcium carbonate.

[454] Cf. Dreyer, Fr., Die Principien der Gerüstbildung bei Rhizopoden,
Spongien und Echinodermen, _Jen. Zeitschr._ XXVI, pp. 204–468, 1892.

[455] In an anomalous and very remarkable Australian sponge, just
described by Professor Dendy (_Nature_, May 18, 1916, p. 253) under the
name of _Collosclerophora_, the spicules are “gelatinous,” consisting
of a gel of colloid silica with a high percentage of water. It is not
stated whether an organic colloid is present together with the silica.
These gelatinous spicules arise as exudations on the outer surface of
cells, and come to lie in intercellular spaces or vesicles.

[456] Lister, in Willey’s _Zoological Results_, pt IV, p. 459, 1900.

[457] The peculiar spicules of Astrosclera are now said to consist of
spherules, or calcospherites, of aragonite, spores of a certain red
seaweed forming the nuclei, or starting-points, of the concretions (R.
Kirkpatrick, _Proc. R. S._ LXXXIV (B), p. 579, 1911).

[458] See for instance the plates in Théel’s Monograph of the
Challenger Holothuroidea; also Sollas’s Tetractinellida, p. lxi.

[459] For very numerous illustrations of the triradiate and
quadriradiate spicules of the calcareous sponges, see (_int. al._),
papers by Dendy (_Q. J. M. S._ XXXV, 1893), Minchin (_P. Z. S._ 1904),
Jenkin (_P. Z. S._ 1908), etc.

[460] Cf. again Bénard’s _Tourbillons cellulaires_, _Ann. de Chimie_,
1901, p. 84.

[461] Léger, Stolc and others, in Doflein’s _Lehrbuch d.
Protozoenkunde_, 1911, p. 912.

[462] See, for instance, the figures of the segmenting egg of
Synapta (after Selenka), in Korschelt and Heider’s _Vergleichende
Entwicklungsgeschichte_ (Allgem. Th., 3^{te} Lief.), p. 19, 1909. On
the spiral type of segmentation as a secondary derivative, due to
mechanical causes, of the “radial” type of segmentation, see E. B.
Wilson, Cell-lineage of Nereis, _Journ. of Morphology_, VI, p. 450,
1892.

[463] Korschelt and Heider, p. 16.

[464] _Chall. Rep. Hexactinellida_, pls. xvi, liii, lxxvi, lxxxviii.

[465] “Hierbei nahm der kohlensaure Kalk eine halb-krystallinische
Beschaffenheit an, und gestaltete sich unter Aufnahme von
Krystallwasser und in Verbindung mit einer geringen Quantität von
organischer Substanz zu jenen individuellen, festen Körpern, welche
durch die natürliche Züchtung als _Spicula_ zur Skeletbildung benützt,
und späterhin durch die Wechselwirkung von Anpassung und Vererbung im
Kampfe ums Dasein auf das Vielfältigste umgebildet und differenziert
wurden.” _Die Kalkschwämme_, I, p. 377, 1872; cf. also pp. 482, 483.

[466] _Op. cit._ p. 483. “Die geordnete, oft so sehr regelmässige und
zierliche Zusammensetzung des Skeletsystems ist zum grössten Theile
unmittelbares Product der Wasserströmung; die characteristische
Lagerung der Spicula ist von der constanten Richtung des Wasserstroms
hervorgebracht; zum kleinsten Theile ist sie die Folge von Anpassungen
an untergeordnete äussere Existenzbedingungen.”

[467] Materials for a Monograph of the Ascones, _Q. J. M. S._ XL. pp.
469–587, 1898.

[468] Haeckel, in his _Challenger Monograph_, p. clxxxviii (1887)
estimated the number of known forms at 4314 species, included in 739
genera. Of these, 3508 species were described for the first time in
that work.

[469] Cf. Gamble, _Radiolaria_ (Lankester’s _Treatise on Zoology_),
vol. I, p. 131, 1909. Cf. also papers by V. Häcker, in _Jen. Zeitschr._
XXXIX, p. 581, 1905, _Z. f. wiss. Zool._ LXXXIII, p. 336, 1905, _Arch.
f. Protistenkunde_, IX, p. 139, 1907, etc.

[470] Bütschli, Ueber die chemische Natur der Skeletsubstanz der
Acantharia, _Zool. Anz._ XXX, p. 784, 1906.

[471] For figures of these crystals see Brandt, _F. u. Fl. d. Golfes
von Neapel_, XIII, _Radiolaria_, 1885, pl. v. Cf. J. Müller, Ueber die
Thalassicollen, etc. _Abh. K. Akad. Wiss. Berlin_, 1858.

[472] Celestine, or celestite, is SrSO_{4} with some BaO replacing SrO.

[473] With the colloid chemists, we may adopt (as Rhumbler has done)
the terms _spumoid_ or _emulsoid_ to denote an agglomeration of
fluid-filled vesicles, restricting the name _froth_ to such vesicles
when filled with air or some other gas.

[474] Cf. Koltzoff, Zur Frage der Zellgestalt, _Anat. Anzeiger_, XLI,
p. 190, 1912.

[475] _Mém. de l’Acad. des Sci., St. Pétersbourg_, XII, Nr. 10, 1902.

[476] The manner in which the minute spicules of Raphidiophrys arrange
themselves round the bases of the pseudopodial rays is a similar
phenomenon.

[477] Rhumbler, Physikalische Analyse von Lebenserscheinungen der
Zelle, _Arch. f. Entw. Mech._ VII, p. 103, 1898.

[478] The whole phenomenon is described by biologists as a “surprising
exhibition of constructive and selective activity,” and is ascribed,
in varying phraseology, to intelligence, skill, purpose, psychical
activity, or “microscopic mentality”: that is to say, to Galen’s
τεχνικὴ φύσις, or “artistic creativeness” (cf. Brock’s _Galen_, 1916,
p. xxix). Cf. Carpenter, _Mental Physiology_, 1874, p. 41; Norman,
Architectural achievements of Little Masons, etc., _Ann. Mag. Nat.
Hist._ (5), I, p. 284, 1878; Heron-Allen, Contributions ... to the
Study of the Foraminifera, _Phil. Trans._ (B), CCVI, pp. 227–279,
1915; Theory and Phenomena of Purpose and Intelligence exhibited
by the Protozoa, as illustrated by selection and behaviour in the
Foraminifera, _Journ. R. Microscop. Soc._ pp. 547–557, 1915; _ibid._,
pp. 137–140, 1916. Prof. J. A. Thomson (_New Statesman_, Oct. 23,
1915) describes a certain little foraminifer, whose protoplasmic body
is overlaid by a crust of sponge-spicules, as “a psycho-physical
individuality whose experiments in self-expression include a masterly
treatment of sponge-spicules, and illustrate that organic skill which
came before the dawn of Art.” Sir Ray Lankester finds it “not difficult
to conceive of the existence of a mechanism in the protoplasm of the
Protozoa which selects and rejects building-material, and determines
the shapes of the structures built, comparable to that mechanism
which is assumed to exist in the nervous system of insects and other
animals which ‘automatically’ go through wonderfully elaborate series
of complicated actions.” And he agrees with “Darwin and others [who]
have attributed the building up of these inherited mechanisms to
the age-long action of Natural Selection, and the survival of those
individuals possessing qualities or ‘tricks’ of life-saving value,” _J.
R. Microsc. Soc._ April, 1916, p. 136.

[479] Rhumbler, _Das Protoplasma als physikalisches System_, Jena, p.
591, 1914; also in _Arch. f. Entwickelungsmech._ VII, pp. 279–335, 1898.

[480] Verworn, _Psycho-physiologische Protisten-Studien_, Jena, 1889
(219 pp.).

[481] Leidy, J., _Fresh-water Rhizopods of N. America_, 1879, p. 262,
pl. xli, figs. 11, 12.

[482] Carnoy, _Biologie Cellulaire_, p. 244, fig. 108; cf. Dreyer, _op.
cit._ 1892, fig. 185.

[483] In all these latter cases we recognise a relation to, or
extension of, the principle of Plateau’s _bourrelet_, or van der
Mensbrugghe’s _masse annulaire_, of which we have already spoken (p.
297).

[484] Apart from the fact that the apex of each pyramid is interrupted,
or truncated, by the presence of the little central cell, it is also
possible that the solid angles are not precisely equivalent to those of
Maraldi’s pyramids, owing to the fact that there is a certain amount of
distortion, or axial asymmetry, in the Nassellarian system. In other
words (to judge from Haeckel’s figures), the tetrahedral symmetry in
Nassellaria is not absolutely regular, but has a main axis about which
three of the trihedral pyramids are symmetrical, the fourth having its
solid angle somewhat diminished.

[485] Cf. Faraday’s beautiful experiments, On the Moving Groups of
Particles found on Vibrating Elastic Surfaces, etc., _Phil. Trans._
1831, p. 299; _Researches in Chem. and Phys._ 1859, pp. 314–358.

[486] We need not go so far as to suppose that the external layer
of cells wholly lacked the power of secreting a skeleton. In many
of the Nassellariae figured by Haeckel (for there are many variant
forms or species besides that represented here), the skeleton of the
partition-walls is very slightly and scantily developed. In such a
case, if we imagine its few and scanty strands to be broken away, the
central tetrahedral figure would be set free, and would have all the
appearance of a complete and independent structure.

[487] The “bourrelet” is not only, as Plateau expresses it, a “surface
of continuity,” but we also recognise that it tends (so far as
material is available for its production) to further lessen the free
surface-area. On its relation to vapour-pressure and to the stability
of foam, see FitzGerald’s interesting note in _Nature_, Feb. 1, 1894
(_Works_, p. 309).

[488] Of the many thousand figures in the hundred and forty plates of
this beautifully illustrated book, there is scarcely one which does
not depict, now patently, now in pregnant suggestion, some subtle and
elegant geometrical configuration.

[489] They were known (of course) long before Plato: Πλάτων δὲ καὶ ἐν
τούτοις πυθαγορίζει.

[490] If the equation of any plane face of a crystal be written in
the form _h_ _x_ + _k_ _y_ + _l_ _z_ = 1, then _h_, _k_, _l_ are
the indices of which we are speaking. They are the reciprocals of
the parameters, or reciprocals of the distances from the origin
at which the plane meets the several axes. In the case of the
regular or pentagonal dodecahedron these indices are 2, 1 + √5, 0.
Kepler described as follows, briefly but adequately, the common
characteristics of the dodecahedron and icosahedron: “Duo sunt corpora
regularia, dodecaedron et icosaedron, quorum illud quinquangulis
figuratur expresse, hoc triangulis quidem sed in quinquanguli formam
coaptatis. Utriusque horum corporum ipsiusque adeo quinquanguli
_structura perfici non potest sine proportione illa, quam hodierni
geometrae divinam appellant_” (_De nive sexangula_ (1611), Opera,
ed. Frisch, VII, p. 723). Here Kepler was dealing, somewhat after
the manner of Sir Thomas Browne, with the mysteries of the quincunx,
and also of the hexagon; and was seeking for an explanation of the
mysterious or even mystical beauty of the 5-petalled or 3-petalled
flower,—_pulchritudinis aut proprietatis figurae, quae animam harum
plantarum characterisavit_.

[491] Cf. Tutton, _Crystallography_, p. 932, 1911.

[492] However, we can often recognise, in a small artery for instance,
that the so-called “circular” fibres tend to take a slightly oblique,
or spiral, course.

[493] The spiral fibres, or a large portion of them, constitute what
Searle called “the rope of the heart” (Todd’s _Cyclopaedia_, II, p.
621, 1836). The “twisted sinews of the heart” were known to early
anatomists, and have been frequently and elaborately studied: for
instance, by Gerdy (_Bull. Fac. Med. Paris_, 1820, pp. 40–148), and by
Pettigrew (_Phil. Trans._ 1864), and of late by J. B. Macallum (_Johns
Hopkins Hospital Report_, IX, 1900) and by Franklin P. Mall (_Amer. J.
of Anat._ XI, 1911).

[494] Cf. Bütschli, “Protozoa,” in Bronn’s _Thierreich_, II, p. 848,
III, p. 1785, etc., 1883–87; Jennings, _Amer. Nat._ XXXV, p. 369, 1901;
Pütter, Thigmotaxie bei Protisten, _Arch. f. Anat. u. Phys._ (_Phys.
Abth. Suppl._), pp. 243–302, 1900.

[495] A great number of spiral forms, both organic and artificial, are
described and beautifully illustrated in Sir T. A. Cook’s _Curves of
Life_, 1914, and _Spirals in Nature and Art_, 1903.

[496] Cf. Vines, The History of the Scorpioid Cyme, _Journ. of Botany_
(n.s.), X, pp. 3–9, 1881.

[497] Leslie’s _Geometry of Curved Lines_, p. 417, 1821. This is
practically identical with Archimedes’ own definition (ed. Torelli, p.
219); cf. Cantor, _Geschichte der Mathematik_, I, p. 262, 1880.

[498] See an interesting paper by Whitworth, W. A., “The Equiangular
Spiral, its chief properties proved geometrically,” in the _Messenger
of Mathematics_ (1), I, p. 5, 1862.

[499] I am well aware that the debt of Greek science to Egypt and the
East is vigorously denied by many scholars, some of whom go so far as
to believe that the Egyptians never had any science, save only some
“rough rules of thumb for measuring fields and pyramids” (Burnet’s
_Greek Philosophy_, 1914, p. 5).

[500] Euclid (II, def. 2).

[501] Cf. Treutlein, _Z. f. Math. u. Phys._ (_Hist. litt. Abth._),
XXVIII, p. 209, 1883.

[502] This is the so-called _Dreifachgleichschenkelige Dreieck_; cf.
Naber, _op. infra cit._ The ratio 1 : 0·618 is again not hard to find
in this construction.

[503] See, on the mathematical history of the Gnomon, Heath’s _Euclid_,
I, _passim_, 1908; Zeuthen, _Theorème de Pythagore_, Genève, 1904; also
a curious and interesting book, _Das Theorem des Pythagoras_, by Dr. H.
A. Naber, Haarlem, 1908.

[504] For many beautiful geometrical constructions based on the
molluscan shell, see Colman, S. and Coan, C. A., _Nature’s Harmonic
Unity_ (ch. ix, Conchology), New York, 1912.

[505] The Rev. H. Moseley, On the Geometrical Forms of Turbinated and
Discoid Shells, _Phil. Trans._ pp. 351–370. 1838.

[506] It will be observed that here Moseley, speaking as a
mathematician and considering the _linear_ spiral, speaks of _whorls_
when he means the linear boundaries, or lines traced by the revolving
radius vector; while the conchologist usually applies the term _whorl_
to the whole space between the two boundaries. As conchologists,
therefore, we call the _breadth of a whorl_ what Moseley looked upon
as the _distance between two consecutive whorls_. But this latter
nomenclature Moseley himself often uses.

[507] In the case of Turbo, and all other “turbinate” shells, we are
dealing not with a plane logarithmic spiral, as in Nautilus, but with
a “gauche” spiral, such that the radius vector no longer revolves
in a plane perpendicular to the axis of the system, but is inclined
to that axis at some constant angle (θ). The figure still preserves
its continued similarity, and may with strict accuracy be called a
logarithmic spiral in space. It is evident that its envelope will
be a right circular cone; and indeed it is commonly spoken of as a
logarithmic spiral _wrapped upon a cone_, its pole coinciding with the
apex of the cone. It follows that the distances of successive whorls
of the spiral measured on the same straight line passing through the
apex of the cone, are in geometrical progression, and conversely just
as in the former case. But the ratio between any two consecutive
interspaces (i.e. _R__{3} − _R__{2}/_R__{2} − _R__{1}) is now equal to
ε^{2π sin θ cot α}, θ being the semi-angle of the enveloping cone. (Cf.
Moseley, _Phil. Mag._ XXI, p. 300, 1842.)

[508] As the successive increments evidently constitute similar
figures, similarly related to the pole (_P_), it follows that their
linear dimensions are to one another as the radii vectores drawn to
similar points in them: for instance as _P_ _P__{1}, _P_ _P__{2}, which
(in Fig. 264, 1) are radii vectores drawn to the points where they meet
the common boundary.

[509] The equation to the surface of a turbinate shell is discussed
by Moseley (_Phil. Trans._ tom. cit. p. 370), both in terms of polar
coordinates and of the rectangular coordinates _x_, _y_, _z_. A more
elegant representation can be given in vector notation, by the method
of quaternions.

[510] J. C. M. Reinecke, _Maris protogaei Nautilos, etc._, Coburg,
1818. Leopold von Buch, Ueber die Ammoniten in den älteren
Gebirgsschichten, _Abh. Berlin. Akad., Phys. Kl._ pp. 135–158, 1830;
_Ann. Sc. Nat._ XXVIII, pp. 5–43, 1833; cf. Elie de Beaumont, Sur
l’enroulement des Ammonites, _Soc. Philom., Pr. verb._ pp. 45–48, 1841.

[511] _Biblia Naturae sive Historia Insectorum_, Leydae, 1737, p. 152.

[512] Alcide D’Orbigny, _Bull. de la soc. géol. Fr._ XIII, p. 200,
1842; _Cours élém. de Paléontologie_, II, p. 5, 1851. A somewhat
similar instrument was described by Boubée. in _Bull. soc. géol._ I,
p. 232, 1831. Naumann’s Conchyliometer (_Poggend. Ann._ LIV, p. 544,
1845) was an application of the screw-micrometer; it was provided also
with a rotating stage, for angular measurement. It was adapted for the
Study of a discoid or ammonitoid shell, while D’Orbigny’s instrument
was meant for the study of a turbinate shell.

[513] It is obvious that the ratios of opposite whorls, or of radii
180° apart, are represented by the square roots of these values; and
the ratios of whorls or radii 90° apart, by the square roots of these
again.

[514] For the correction to be applied in the case of the helicoid, or
“turbinate” shells, see p. 557.

[515] On the Measurement of the Curves formed by Cephalopods and other
Mollusks. _Phil. Mag._ (5), VI, pp. 241–263, 1878.

[516] For an example of this method, see Blake, _l.c._ p. 251.

[517] Naumann, C. F., Ueber die Spiralen von Conchylien, _Abh. k.
sächs_. Ges. pp. 153–196, 1846; Ueber die cyclocentrische Conchospirale
u. über das Windungsgesetz von _Planorbis corneus_, _ibid._ I, pp.
171–195, 1849; Spirale von Nautilus u. _Ammonites galeatus_, _Ber. k.
sächs. Ges._ II, p. 26, 1848; Spirale von _Amm. Ramsaueri_, _ibid._
XVI, p. 21, 1864; see also _Poggendorff’s Annalen_, L, p. 223, 1840;
LI, p. 245, 1841; LIV, p. 541, 1845, etc.

[518] Sandberger, G., Spiralen des _Ammonites Amaltheus_, _A. Gaytani_,
und _Goniatites intumescens_, _Zeitschr. d. d. Geol. Gesellsch._ X, pp.
446–449, 1858.

[519] Grabau, A. H., _Ueber die Naumannsche Conchospirale_, etc.
Inauguraldiss. Leipzig, 1872; _Die Spiralen von Conchylien_, etc.
Programm, Nr. 502, Leipzig, 1882.

[520] It has been pointed out to me that it does not follow at once
and obviously that, because the interspace _AB_ is a mean proportional
between the breadths of the adjacent whorls, therefore the whole
distance _OB_ is a mean proportional between _OA_ and _OC_. This is a
corollary which requires to be proved; but the proof is easy.

[521] A beautiful construction: _stupendum Naturae artificium_,
Linnaeus.

[522] English edition, p. 537, 1900. The chapter is revised by
Prof. Alpheus Hyatt, to whom the nomenclature is largely due. For
a more copious terminology, see Hyatt, _Phylogeny of an Acquired
Characteristic_, p. 422 _seq._, 1894.

[523] This latter conclusion is adopted by Willey, _Zoological
Results_, p. 747, 1902.

[524] See Moseley, _op. cit._ pp. 361 _seq._

[525] In Nautilus, the “hood” has somewhat different dimensions in the
two sexes, and these differences are impressed upon the shell, that is
to say upon its “generating curve.” The latter constitutes a somewhat
broader ellipse in the male than in the female. But this difference
is not to be detected in the young; in other words, the form of the
generating curve perceptibly alters with advancing age. Somewhat
similar differences in the shells of Ammonites were long ago suspected,
by D’Orbigny, to be due to sexual differences. (Cf. Willey, _Natural
Science_, VI, p. 411, 1895; _Zoological Results_, p. 742, 1902.)

[526] Macalister, Alex., Observations on the Mode of Growth of Discoid
and Turbinated Shells, _P. R. S._ XVIII, pp. 529–532, 1870.

[527] See figures in Arnold Lang’s _Comparative Anatomy_ (English
translation), II, p. 161, 1902.

[528] Kappers, C. U. A., Die Bildung künstlicher Molluskenschalen,
_Zeitschr. f. allg. Physiol._ VII, p. 166, 1908.

[529] We need not assume a _close_ relationship, nor indeed any more
than such a one as permits us to compare the shell of a Nautilus with
that of a Gastropod.

[530] Cf. Owen, “These shells [Nautilus and Ammonites] are revolutely
spiral or coiled over the back of the animal, not involute like
Spirula”: _Palaeontology_, 1861, p. 97; cf. _Mem. on the Pearly
Nautilus_, 1832; also _P.Z.S._ 1878, p. 955.

[531] The case of Terebratula or of Gryphaea would be closely
analogous, if the smaller valve were less closely connected and
co-articulated with the larger.

[532] “It has been suggested, and I think in some quarters adopted
as a dogma, that the formation of successive septa [in Nautilus] is
correlated with the recurrence of reproductive periods. This is not the
case, since, according to my observations, propagation only takes place
after the last septum is formed;” Willey, _Zoological Results_, p. 746,
1902.

[533] Cf. Woodward, Henry, On the Structure of Camerated Shells, _Pop.
Sci. Rev._ XI, pp. 113–120, 1872.

[534] See Willey, Contributions to the Natural History of the Pearly
Nautilus, _Zoological Results_, etc. p. 749, 1902. Cf. also Bather,
Shell-growth in Cephalopoda, _Ann. Mag. N. H._ (6), I, pp 298–310,
1888; _ibid._ pp. 421–427, and other papers by Blake, Riefstahl, etc.
quoted therein.

[535] It was this that led James Bernoulli, in imitation of Archimedes,
to have the logarithmic spiral graven on his tomb, with the pious
motto, _Eadem mutata resurgam_. On Goodsir’s grave the same symbol is
reinscribed.

[536] The “lobes” and “saddles” which arise in this manner, and on
whose arrangement the modern classification of the nautiloid and
ammonitoid shells largely depends, were first recognised and named by
Leopold von Buch, _Ann. Sci. Nat._ XXVII, XXVIII, 1829.

[537] Blake has remarked upon the fact (_op. cit._ p. 248) that in
some Cyrtocerata we may have a curved shell in which the ornaments
approximately run at a constant angular distance from the pole, while
the septa approximate to a radial direction; and that “thus one law
of growth is illustrated by the inside, and another by the outside.”
In this there is nothing at which we need wonder. It is merely a
case where the generating curve is set very obliquely to the axis of
the shell; but where the septa, which have no necessary relation to
the _mouth_ of the shell, take their places, as usual, at a certain
definite angle to the _walls_ of the tube. This relation of the septa
to the walls of the tube arises after the tube itself is fully formed,
and the obliquity of growth of the open end of the tube has no relation
to the matter.

[538] Cf. pp. 255, 463, etc.

[539] In a few cases, according to Awerinzew and Rhumbler, where the
chambers are added on in concentric series, as in Orbitolites, we have
the crystalline structure arranged radially in the radial walls but
tangentially in the concentric ones: whereby we tend to obtain, on a
minute scale, a system of orthogonal trajectories, comparable to that
which we shall presently study in connection with the structure of
bone. Cf. S. Awerinzew, Kalkschale der Rhizopoden, _Z. f. w. Z._ LXXIV,
pp. 478–490, 1903.

[540] Rhumbler, L., Die Doppelschalen von Orbitolites und anderer
Foraminiferen, etc., _Arch. f. Protistenkunde_, I, pp. 193–296, 1902;
and other papers. Also _Die Foraminiferen der Planktonexpedition_, I,
1911, pp. 50–56.

[541] Bénard, H, Les tourbillons cellulaires, _Ann. de Chimie_ (8),
XXIV, 1901. Cf. also the pattern of cilia on an Infusorian, as figured
by Bütschli in Bronn’s _Protozoa_, III, p. 1281, 1887.

[542] A similar hexagonal pattern is obtained by the mutual repulsion
of floating magnets in Mr R. W. Wood’s experiments, _Phil. Mag._ XLVI,
pp. 162–164, 1898.

[543] Cf. D’Orbigny, Alc., Tableau méthodique de la classe des
Céphalopodes, _Ann. des Sci. Nat._ (1), VII, pp. 245–315, 1826;
Dujardin. Félix, Observations nouvelles sur les prétendus Céphalopodes
microscopiques, _ibid._ (2), III, pp. 108, 109, 312–315, 1835;
Recherches sur les organismes inférieurs, _ibid._ IV, pp. 343–377,
1835, etc.

[544] It is obvious that the actual _outline_ of a foraminiferal, just
as of a molluscan shell, may depart widely from a logarithmic spiral.
When we say here, for short, that the shell _is_ a logarithmic spiral,
we merely mean that it is essentially related to one: that it can be
inscribed in such a spiral, or that corresponding points (such, for
instance, as the centres of gravity of successive chambers, or the
extremities of successive septa) wall always be found to lie upon such
a spiral.

[545] von Möller, V., Die spiral-gewundenen Foraminifera des russischen
Kohlenkalks, _Mém. de l’Acad. Imp. Sci., St Pétersbourg_ (7), XXV, 1878.

[546] As von Möller is careful to explain, Naumann’s formula for the
“cyclocentric conchospiral” is appropriate to this and other spiral
Foraminifera, since we have in all these cases a central or initial
chamber, approximately spherical, about which the logarithmic spiral
is coiled (cf. Fig. 309). In species where the central chamber is
especially large, Naumann’s formula is all the more advantageous.
But it is plain that it is only required when we are dealing with
diameters, or with radii; so long as we are merely comparing the
breadths of _successive whorls_, the two formulae come to the same
thing.

[547] Van Iterson, G., _Mathem. u. mikrosk.-anat. Studien über
Blattstellungen, nebst Betrachtungen über den Schalenbau der
Miliolinen_, 331 pp., Jena, 1907.

[548] Hans Przibram asserts that the linear ratio of successive
chambers tends in many Foraminifera to approximate to 1·26, which = ∛2;
in other words, that the volumes of successive chambers tend to double.
This Przibram would bring into relation with another law, viz. that
insects and other arthropods tend to moult, or to metamorphose, just
when they double their weights, or increase their linear dimensions
in the ratio of 1 : ∛2. (Die Kammerprogression der Foraminiferen als
Parallele zur Häutungsprogression der Mantiden, _Arch. f. Entw. Mech._
XXXIV p. 680, 1813.) Neither rule seems to me to be well grounded.

[549] Cf. Schacko, G., Ueber Globigerina-Einschluss bei Orbulina,
_Wiegmann’s Archiv_, XLIX, p. 428, 1883; Brady, _Chall. Rep._, p. 607,
1884.

[550] Cf. Brady, H. B., _Challenger Rep._, _Foraminifera_, 1884, p.
203, pl. XIII.

[551] Brady, _op. cit._, p. 206; Batsch, one of the earliest writers
on Foraminifera, had already noticed that this whole series of
ear-shaped and crozier-shaped shells was filled in by gradational
forms; _Conchylien des Seesandes_, 1791, p. 4, pl. VI, fig. 15_a_–_f_.
See also, in particular, Dreyer, _Peneroplis_; _eine Studie zur
biologischen Morphologie und zur Speciesfrage_, Leipzig, 1898; also
Eimer und Fickert, Artbildung und Verwandschaft bei den Foraminiferen,
_Tübinger zool. Arbeiten_, III, p. 35, 1899.

[552] Doflein, _Protozoenkunde_, 1911, p. 263; “Was diese Art
veranlässt in dieser Weise gelegentlich zu varüren, ist vorläufig noch
ganz räthselhaft.”

[553] In the case of Globigerina, some fourteen species (out of a very
much larger number of described forms) were allowed by Brady (in 1884)
to be distinct; and this list has been, I believe, rather added to than
diminished. But these so-called species depend for the most part on
slight differences of degree, differences in the angle of the spiral,
in the ratio of magnitude of the segments, or in their area of contact
one with another. Moreover with the exception of one or two “dwarf”
forms, said to be limited to Arctic and Antarctic waters, there is no
principle of geographical distribution to be discerned amongst them.
A species found fossil in New Britain turns up in the North Atlantic:
a species described from the West Indies is rediscovered at the
ice-barrier of the Antarctic.

[554] Dreyer, F., Principien der Gerüstbildung bei Rhizopoden, etc.,
_Jen. Zeitschr._ XXVI, pp. 204–468, 1892.

[555] A difficulty arises in the case of forms (like Peneroplis) where
the young shell appears to be more complex than the old, the first
formed portion being closely coiled while the later additions become
straight and simple: “die biformen Arten verhalten sich, kurz gesagt.
gerade umgekehrt als man nach dem biogenetischen Grundgesetz erwarten
sollte,” Rhumbler, _op. cit._, p. 33 etc.

[556] “Das Festigkeitsprinzip als Movens der Weiterentwicklung ist zu
interessant und für die Aufstellung meines Systems zu wichtig um die
Frage unerörtert zu lassen, warum diese Bevorzügung der Festigkeit
stattgefunden hat. Meiner Ansicht nach lautet die Antwort auf diese
Frage einfach, weil die Foraminiferen meistens unter Verhältnissen
leben, die ihre Schalen in hohem Grade der Gefahr des Zerbrechens
aussetzen; es muss also eine fortwahrende Auslese des Festeren
stattfinden,” Rhumbler, _op. cit._, p. 22.

[557] “Die Foraminiferen kiesige oder grobsandige Gebiete des
Meeresbodens _nicht lieben_, u.s.w.”: where the last two words have no
particular meaning, save only that (as M. Aurelius says) “of things
that use to be, we say commonly that they love to be.”

[558] In regard to the Foraminifera, “die Palaeontologie lässt uns
leider an Anfang der Stammesgeschichte fast gänzlich im Stiche,”
Rhumbler, _op. cit._, p. 14.

[559] The evolutionist theory, as Bergson puts it, “consists above all
in establishing relations of ideal kinship, and in maintaining that
wherever there is this relation of, so to speak, _logical_ affiliation
between forms, _there is also a relation of chronological succession
between the species in which these forms are materialised_”: _Creative
Evolution_, 1911, p. 26. Cf. _supra_, p. 251.

[560] In the case of the ram’s horn, the assumption that the rings are
annual is probably justified. In cattle they are much less conspicuous,
but are sometimes well-marked in the cow; and in Sweden they are then
called “calf-rings,” from a belief that they record the number of
offspring. That is to say, the growth of the horn is supposed to be
retarded during gestation, and to be accelerated after parturition,
when superfluous nourishment seeks a new outlet. (Cf. Lönnberg,
_P.Z.S._, p. 689, 1900.)

[561] Cf. Sir V. Brooke, On the Large Sheep of the Thian Shan,
_P.Z.S._, p. 511, 1875.

[562] Cf. Lönnberg, E., On the Structure of the Musk Ox, _P.Z.S._, pp.
686–718, 1900.

[563] St Venant, De la torsion des prismes, avec des considérations
sur leur flexion, etc., _Mém. des Savants Étrangers_, Paris, XIV, pp.
233–560, 1856.

[564] This is not difficult to do, with considerable accuracy, if the
clay be kept well wetted, or semi-fluid, and the smoothing be done with
a large wet brush.

[565] The curves are well shewn in most of Sir V. Brooke’s figures of
the various species of Argali, in the paper quoted on p. 614.

[566] _Climbing Plants_, 1865 (2nd edit. 1875); _Power of Movement in
Plants_, 1880.

[567] Palm, _Ueber das Winden der Pflanzen_, 1827; von Mohl, _Bau
und Winden der Ranken_, etc., 1827; Dutrochet, Mouvements révolutifs
spontanés, _C.R._ 1843, etc.

[568] Cf. (e.g.) Lepeschkin, Zur Kenntnis des Mechanismus der
Variationsbewegungen, _Ber. d. d. Bot. Gesellsch._ XXVI A, pp. 724–735,
1908; also A. Tröndle, Der Einfluss des Lichtes auf die Permeabilität
des Plasmahaut, _Jahrb. wiss. Bot._ XLVIII, pp. 171–282, 1910.

[569] For an elaborate study of antlers, see Rörig, A., _Arch. f.
Entw. Mech._ X, pp. 525–644, 1900, XI, pp. 65–148, 225–309, 1901;
Hoffmann, C., _Zur Morphologie der rezenten Hirschen_, 75 pp., 23 pls.,
1901: also Sir Victor Brooke, On the Classification of the Cervidae,
_P.Z.S._, pp. 883–928, 1878. For a discussion of the development of
horns and antlers, see Gadow, H., _P.Z.S._, pp. 206–222, 1902, and
works quoted therein.

[570] Cf. Rhumbler, L., Ueber die Abhängigkeit des Geweihwachstums der
Hirsche, speziell des Edelhirsches, vom Verlauf der Blutgefässe im
Kolbengeweih, _Zeitschr. f. Forst. und Jagdwesen_, 1911, pp. 295–314.

[571] The fact that in one very small deer, the little South American
Coassus, the antler is reduced to a simple short spike, does not
preclude the general distinction which I have drawn. In Coassus we have
the beginnings of an antler, which has not yet manifested its tendency
to expand; and in the many allied species of the American genus
Cariacus, we find the expansion manifested in various simple modes of
ramification or bifurcation. (Cf. Sir V. Brooke, Classification of the
Cervidae, p. 897.)

[572] Cf. also the immense range of variation in elks’ horns, as
described by Lönnberg, _P.Z.S._ II, pp. 352–360, 1902.

[573] Besides papers referred to below, and many others quoted in
Sach’s _Botany_ and elsewhere, the following are important: Braun,
Alex., Vergl. Untersuchung über die Ordnung der Schuppen an den
Tannenzapfen, etc., _Verh. Car. Leop. Akad._ XV, pp. 199–401, 1831; Dr
C. Schimper’s Vorträge über die Möglichkeit eines wissenschaftlichen
Verständnisses der Blattstellung, etc., _Flora_, XVIII, pp. 145–191,
737–756, 1835; Schimper, C. F., Geometrische Anordnung der um eine Axe
peripherische Blattgebilde, _Verhandl. Schweiz. Ges._, pp. 113–117,
1836; Bravais, L. and A., Essai sur la disposition des feuilles
curvisériées, _Ann. Sci. Nat._ (2), VII, pp. 42–110, 1837; Sur la
disposition symmétrique des inflorescences, _ibid._, pp. 193–221,
291–348, VIII, pp. 11–42, 1838; Sur la disposition générale des
feuilles rectisériées, _ibid._ XII, pp. 5–41, 65–77, 1839; Zeising,
_Normalverhältniss der chemischen und morphologischen Proportionen_,
Leipzig, 1856; Naumann, C. F., Ueber den Quincunx als Gesetz der
Blattstellung bei Sigillaria, etc., _Neues Jahrb. f. Miner._ 1842,
pp. 410–417; Lestiboudois, T., _Phyllotaxie anatomique_, Paris, 1848;
Henslow, G., _Phyllotaxis_, London, 1871; Wiesner, Bemerkungen über
rationale und irrationale Divergenzen, _Flora_, LVIII, pp. 113–115,
139–143, 1875; Airy, H., On Leaf Arrangement, _Proc. R. S._ XXI, p.
176, 1873; Schwendener, S., _Mechanische Theorie der Blattstellungen_,
Leipzig, 1878; Delpino, F., _Causa meccanica della filotassi
quincunciale_, Genova, 1880; de Candolle, C., _Étude de Phyllotaxie_,
Genève, 1881.

[574] _Allgemeine Morphologie der Gewächse_, p. 442, etc. 1868.

[575] _Relation of Phyllotaxis to Mechanical Laws_, Oxford, 1901–1903;
cf. _Ann. of Botany_, XV, p. 481, 1901.

[576] “The proposition is that the genetic spiral is a logarithmic
spiral, homologous with the line of current-flow in a spiral vortex;
and that in such a system the action of orthogonal forces will be
mapped out by other orthogonally intersecting logarithmic spirals—the
‘parastichies’ ”; Church, _op. cit._ I, p. 42.

[577] Mr Church’s whole theory, if it be not based upon, is interwoven
with, Sachs’s theory of the orthogonal intersection of cell-walls, and
the elaborate theories of the symmetry of a growing point or apical
cell which are connected therewith. According to Mr Church, “the law
of the orthogonal intersection of cell-walls at a growing apex may be
taken as generally accepted” (p. 32); but I have taken a very different
view of Sachs’s law, in the eighth chapter of the present book. With
regard to his own and Sachs’s hypotheses, Mr Church makes the following
curious remark (p. 42): “Nor are the hypotheses here put forward more
imaginative than that of the paraboloid apex of Sachs which remains
incapable of proof, or his construction for the apical cell of Pteris
which does not satisfy the evidence of his own drawings.”

[578] _Amer. Naturalist_, VII, p. 449, 1873.

[579] This celebrated series, which appears in the continued fraction
(1/1) + (1/(1 + )) etc. and is closely connected with the _Sectio
aurea_ or Golden Mean, is commonly called the Fibonacci series, after a
very learned twelfth century arithmetician (known also as Leonardo of
Pisa), who has some claims to be considered the introducer of Arabic
numerals into christian Europe. It is called Lami’s series by some,
after Father Bernard Lami, a contemporary of Newton’s, and one of
the co-discoverers of the parallelogram of forces. It was well-known
to Kepler, who, in his paper _De nive sexangula_ (cf. _supra_, p.
480), discussed it in connection with the form of the dodecahedron
and icosahedron, and with the ternary or quinary symmetry of the
flower. (Cf. Ludwig, F., Kepler über das Vorkommen der Fibonaccireihe
im Pflanzenreich, _Bot. Centralbl._ LXVIII, p. 7, 1896). Professor
William Allman, Professor of Botany in Dublin (father of the historian
of Greek geometry), speculating on the same facts, put forward the
curious suggestion that the cellular tissue of the dicotyledons,
or exogens, would be found to consist of dodecahedra. and that of
the monocotyledons or endogens of icosahedra (_On the mathematical
connexion between the parts of Vegetables_: abstract of a Memoir read
before the Royal Society in the year 1811 (privately printed, _n.d._).
Cf. De Candolle, _Organogénie végétale_, I, p. 534).

[580] _Proc. Roy. Soc. Edin._ VII, p. 391, 1872.

[581] The necessary existence of these recurring spirals is also
proved, in a somewhat different way, by Leslie Ellis, On the Theory
of Vegetable Spirals, in _Mathematical and other Writings_, 1853, pp.
358–372.

[582] _Proc. Roy. Soc. Edin._ VII, p. 397, 1872; _Trans. Roy. Soc.
Edin._ XXVI, p. 505, 1870–71.

[583] A common form of pail-shaped waste-paper basket, with wide
rhomboidal meshes of cane, is well-nigh as good a model as is required.

[584] _Deutsche Vierteljahrsschrift_, p. 261, 1868.

[585] _Memoirs of Amer. Acad._ IX, p. 389.

[586] _De avibus circa aquas Danubii vagantibus et de ipsarum Nidis_
(Vol. V of the _Danubius Pannonico-mysicus_), Hagae Com., 1726.

[587] Sir Thomas Browne had a collection of eggs at Norwich, according
to Evelyn, in 1671.

[588] Cf. Lapierre, in Buffon’s _Histoire Naturelle_, ed. Sonnini, 1800.

[589] _Eier der Vögel Deutschlands_, 1818–28 (_cit._ des Murs, p. 36).

[590] _Traité d’Oologie_, 1860.

[591] Lafresnaye, F. de, Comparaison des œufs des Oiseaux avec
leurs squelettes, comme seul moven de reconnaître la cause de leurs
différentes formes, _Rev. Zool._, 1845, pp. 180–187, 239–244.

[592] Cf. Des Murs, p. 67: “Elle devait encore penser au moment où ce
germe aurait besoin de l’espace nécessaire à son accroissement, à ce
moment où ... il devra remplir exactement l’intervalle circonscrit par
sa fragile prison, etc.”

[593] Thienemann, F. A. L., _Syst. Darstellung der Fortpflanzung der
Vögel Europas_. Leipzig, 1825–38.

[594] Cf. Newton’s _Dictionary of Birds_, 1893, p. 191; Szielasko,
Gestalt der Vogeleier, _J. f. Ornith._ LIII, pp. 273–297, 1905.

[595] Jacob Steiner suggested a Cartesian oval, _r_ + _mr′_ = _c_, as a
general formula for all eggs (cf. Fechner, _Ber. sächs. Ges._, 1849, p.
57); but this formula (which fails in such a case as the guillemot), is
purely empirical, and has no mechanical foundation.

[596] Günther, F. C., _Sammlung von Nestern und Eyern verschiedener
Vögel_, Nürnb. 1772. Cf. also Raymond Pearl, Morphogenetic Activity of
the Oviduct, _J. Exp. Zool._ VI, pp. 339–359, 1909.

[597] The following account is in part reprinted from _Nature_, June 4,
1908.

[598] In so far as our explanation involves a shaping or moulding of
the egg by the uterus or “oviduct” (an agency supplemented by the
proper tensions of the egg), it is curious to note that this is very
much the same as that old view of Telesius regarding the formation
of the embryo (_De rerum natura_, VI, cc. 4 and 10), which he had
inherited from Galen, and of which Bacon speaks (_Nov. Org._ cap. 50;
cf. Ellis’s note). Bacon expressly remarks that “Telesius should have
been able to shew the like formation in the shells of eggs.” This old
theory of embryonic modelling survives only in our usage of the term
“matrix” for a “mould.”

[599] _Journal of Tropical Medicine_, 15th June, 1911. I leave this
paragraph as it was written, though it is now once more asserted that
the terminal and lateral-spined eggs belong to separate and distinct
species of Bilharzia (Leiper, _Brit. Med. Journ._, 18th March, 1916, p.
411).

[600] Cf. Bashforth and Adams, _Theoretical Forms of Drops, etc._,
Cambridge, 1883.

[601] Woods, R. H., On a Physical Theorem applied to tense Membranes,
_Journ. of Anat. and Phys._ XXVI, pp. 362–371, 1892. A similar
investigation of the tensions in the uterine wall, and of the varying
thickness of its muscles, was attempted by Haughton in his _Animal
Mechanics_, pp. 151–158, 1873.

[602] This corresponds with a determination of the normal pressures (in
systole) by Krohl, as being in the ratio of 1 : 6·8.

[603] Cf. Schwalbe, G., Ueber Wechselbeziehungen und ihr Einfluss
auf die Gestaltung des Arteriensystem, _Jen. Zeitschr._ XII, p. 267,
1878, Roux, Ueber die Verzweigungen der Blutgefässen des Menschen,
_ibid._ XII, p. 205, 1878; Ueber die Bedeutung der Ablenkung des
Arterienstämmen bei der Astaufgabe, _ibid._ XIII, p. 301, 1879;
Hess, Walter, Eine mechanisch bedingte Gesetzmässigkeit im Bau des
Blutgefässsystems, _A. f. Entw. Mech._ XVI, p. 632, 1903; Thoma, R.,
_Ueber die Histogenese und Histomechanik des Blutgefässsystems_, 1893.

[604] _Essays_, etc., edited by Owen, I, p. 134, 1861.

[605] On the Functions of the Heart and Arteries, _Phil. Trans._ 1809,
pp. 1–31, cf. 1808, pp. 164–186; _Collected Works_, I, pp. 511–534,
1855. The same lesson is conveyed by all such work as that of Volkmann,
E. H. Weber and Poiseuille. Cf. Stephen Hales’ _Statical Essays_, II,
_Introduction_: “Especially considering that they [i.e. animal Bodies]
are in a manner framed of one continued Maze of innumerable Canals,
in which Fluids are incessantly circulating, some with great Force
and Rapidity, others with very different Degrees of rebated Velocity:
Hence, _etc._”

[606] “Sizes” is Owen’s editorial emendation, which seems amply
justified.

[607] For a more elaborate classification, into colours cryptic,
procryptic, anticryptic, apatetic, epigamic, sematic, episematic,
aposematic, etc., see Poulton’s _Colours of Animals_ (Int. Scientific
Series, LXVIII), 1890; cf. also Meldola, R., Variable Protective
Colouring in Insects, _P.Z.S._ 1873, pp. 153–162, etc.

[608] Dendy, _Evolutionary Biology_, p. 336, 1912.

[609] Delight in beauty is one of the pleasures of the imagination;
there is no limit to its indulgence, and no end to the results which
we may ascribe to its exercise. But as for the particular “standard of
beauty” which the bird (for instance) admires and selects (as Darwin
says in the _Origin_, p. 70, edit. 1884), we are very much in the
dark, and we run the risk of arguing in a circle: for wellnigh all we
can safely say is what Addison says (in the 412th _Spectator_)—that
each different species “is most affected with the beauties of its own
kind .... Hinc merula in nigro se oblectat nigra marito; ... hinc
noctua tetram Canitiem alarum et glaucos miratur ocellos.”

[610] Cf. Bridge, T. W., _Cambridge Natural History_ (Fishes), VII,
p. 173, 1904; also Frisch, K. v., Ueber farbige Anpassung bei Fische,
_Zool. Jahrb._ (_Abt. Allg. Zool._), XXXII, pp. 171–230, 1914.

[611] _Nature_, L, p. 572; LI, pp. 33, 57, 533, 1894–95.

[612] They are “wonderfully fitted for ‘vanishment’ against the
flushed, rich-coloured skies of early morning and evening .... their
chief feeding-times”; and “look like a real sunset or dawn, repeated on
the opposite side of the heavens,—either east or west as the case may
be”: Thayer, _Concealing-coloration in the Animal Kingdom_, New York,
1909, pp. 154–155. This hypothesis, like the rest, is not free from
difficulty. Twilight is apt to be short in the homes of the flamingo:
and moreover, Mr Abel Chapman, who watched them on the Guadalquivir,
tells us that they _feed by day_.

[613] Principal Galloway, _Philosophy of Religion_, p. 344, 1914.

[614] Cf. Professor Flint, in his Preface to Affleck’s translation
of Janet’s _Causes finales_: “We are, no doubt, still a long way
from a mechanical theory of organic growth, but it may be said to be
the _quaesitum_ of modern science, and no one can say that it is a
chimaera.”

[615] Cf. Sir Donald MacAlister, How a Bone is Built, _Engl. Ill. Mag._
1884.

[616] Professor Claxton Fidler, _On Bridge Construction_, p. 22 (4th
ed.), 1909; cf. (_int. al._) Love’s _Elasticity_, p. 20 (_Historical
Introduction_), 2nd ed., 1906.

[617] In preparing or “macerating” a skeleton, the naturalist nowadays
carries on the process till nothing is left but the whitened bones. But
the old anatomists, whose object was not the study of “comparative”
morphology but the wider theme of comparative physiology, were wont
to macerate by easy stages; and in many of their most instructive
preparations, the ligaments were intentionally left in connection with
the bones, and as part of the “skeleton.”

[618] In a few anatomical diagrams, for instance in some of the
drawings in Schmaltz’s _Atlas der Anatomie des Pferdes_, we may see
the system of “ties” diagrammatically inserted in the figure of the
skeleton. Cf. Gregory, On the principles of Quadrupedal Locomotion,
_Ann. N. Y. Acad. of Sciences_, XXII, p. 289, 1912.

[619] Galileo, _Dialogues concerning Two New Sciences_ (1638), Crew
and Salvio’s translation, New York, 1914, p. 150; _Opere_, ed. Favaro,
VIII, p. 186. Cf. Borelli, _De Motu Animalium_, I, prop. CLXXX, 1685.
Cf. also Camper, P., La structure des os dans les oiseaux, _Opp._ III,
p. 459, ed. 1803; Rauber, A., Galileo über Knochenformen, _Morphol.
Jahrb._ VII, pp. 327, 328, 1881; Paolo Enriques, Della economia di
sostanza nelle osse cave, _Arch. f. Ent. Mech._ XX, pp. 427–465, 1906.

[620] _Das mechanische Prinzip. im anatomischen Bau der Monocotylen_,
Leipzig, 1874.

[621] For further botanical illustrations, see (_int. al._) Hegler,
Einfluss der Zugkraften auf die Festigkeit und die Ausbildung
mechanischer Gewebe in Pflanzen, _SB. sächs. Ges. d. Wiss._ p.
638, 1891; Kny, L., Einfluss von Zug und Druck auf die Richtung
der Scheidewande in sich teilenden Pflanzenzellen, _Ber. d. bot.
Gesellsch._ XIV, 1896; Sachs, Mechanomorphose und Phylogenie, _Flora_,
LXXVIII, 1894; cf. also Pflüger, Einwirkung der Schwerkraft, etc., über
die Richtung der Zelltheilung, _Archiv_, XXXIV, 1884.

[622] Among other works on the mechanical construction of bone
see: Bourgery, _Traité de l’anatomie_ (_I. Ostéologie_), 1832
(with admirable illustrations of trabecular structure); Fick, L.,
_Die Ursachen der Knochenformen_, Göttingen, 1857; Meyer, H., Die
Architektur der Spongiosa, _Archiv f. Anat. und Physiol._ XLVII, pp.
615–628, 1867; _Statik u. Mechanik des menschlichen Knochengerüstes_,
Leipzig, 1873; Wolff, J., Die innere Architektur der Knochen, _Arch.
f. Anat, und Phys._ L, 1870; _Das Gesetz der Transformation bei
Knochen_, 1892; von Ebner, V., Der feinere Bau der Knochensubstanz,
_Wiener Bericht_, LXXII, 1875; Rauber, Anton, _Elastizität und
Festigkeit der Knochen_, Leipzig, 1876; O. Meserer, _Elast, u.
Festigk. d. menschlichen Knochen_, Stuttgart, 1880; MacAlister, Sir
Donald, How a Bone is Built, _English Illustr. Mag._ pp. 640–649,
1884; Rasumowsky, Architektonik des Fussskelets, _Int. Monatsschr. f.
Anat._ p. 197, 1889; Zschokke, _Weitere Unters. über das Verhältniss
der Knochenbildung zur Statik und Mechanik des Vertebratenskelets_,
Zürich, 1892; Roux, W., _Ges. Abhandlungen über Entwicklungsmechanik
der Organismen, Bd. I, Funktionelle Anpassung_, Leipzig, 1895; Triepel,
H., Die Stossfestigkeit der Knochen, _Arch. f. Anat. u. Phys._ 1900;
Gebhardt, Funktionell wichtige Anordnungsweisen der feineren und
gröberen Bauelemente des Wirbelthierknochens, etc., _Arch. f. Entw.
Mech._ 1900–1910; Kirchner. A., Architektur der Metatarsalien, _A.
f. E. M._ XXIV, 1907; Triepel, Herm., Die trajectorielle Structuren
(in _Einf. in die Physikalische Anatomie_, 1908); Dixon, A. F.,
Architecture of the Cancellous Tissue forming the Upper End of the
Femur, _Journ. of Anat. and Phys._ (3) XLIV, pp. 223–230, 1910.

[623] Sédillot, De l’influence des fonctions sur la structure et la
forme des organes; _C. R._ LIX, p. 539, 1864; cf. LX, p. 97, 1865,
LXVIII. p. 1444. 1869.

[624] E.g. (1) the head, nodding backwards and forwards on a fulcrum,
represented by the atlas vertebra, lying between the weight and the
power; (2) the foot, raising on tip-toe the weight of the body against
the fulcrum of the ground, where the weight is between the fulcrum and
the power, the latter being represented by the _tendo Achillis_; (3)
the arm, lifting a weight in the hand, with the power (i.e. the biceps
muscle) between the fulcrum and the weight. (The second case, by the
way, has been much disputed; cf. Haycraft in Schäfer’s _Textbook of
Physiology_, p. 251, 1900.)

[625] Our problem is analogous to Dr Thomas Young’s problem of the best
disposition of the timbers in a wooden ship (_Phil. Trans._ 1814, p.
303). He was not long of finding that the forces which may act upon the
fabric are very numerous and very variable, and that the best mode of
resisting them, or best structural arrangement for ultimate strength,
becomes an immensely complicated problem.

[626] In like manner, Clerk Maxwell could not help employing the term
“skeleton” in defining the mathematical conception of a “frame,”
constituted by points and their interconnecting lines: in studying the
equilibrium of which, we consider its different points as mutually
acting on each other with forces whose directions are those of the
lines joining each pair of points. Hence (says Maxwell), “in order
to exhibit the mechanical action of the frame in the most elementary
manner, we may draw it as a _skeleton_, in which the different points
are joined by straight lines, and we may indicate by numbers attached
to these lines the tensions or compressions in the corresponding
pieces of the frame” (_Trans. R. S. E._ XXVI, p. 1, 1870). It follows
that the diagram so constructed represents a “diagram of forces,” in
this limited sense that it is geometrical as regards the position and
direction of the forces, but arithmetical as regards their magnitude.
It is to just such a diagram that the animal’s skeleton tends to
approximate.

[627] When the jockey crouches over the neck of his race-horse, and
when Tod Sloan introduced the “American seat,” the object in both cases
is to relieve the hind-legs of weight, and so leave them free for the
work of propulsion. Nevertheless, we must not exaggerate the share
taken by the hind-limbs in this latter duty; cf. Stillman, _The Horse
in Motion_, p. 69, 1882.

[628] This and the following diagrams are borrowed and adapted from
Professor Fidler’s _Bridge Construction_.

[629] The method of constructing _reciprocal diagrams_, in which one
should represent the outlines of a frame, and the other the system of
forces necessary to keep it in equilibrium, was first indicated in
Culmann’s _Graphische Statik_; it was greatly developed soon afterwards
by Macquorn Rankine (_Phil. Mag._ Feb. 1864, and _Applied Mechanics_,
passim), to whom is mainly due the general application of the principle
to engineering practice.

[630] _Dialogues concerning Two New Sciences_ (1638): Crew and Salvio’s
translation, p. 140 _seq._

[631] The form and direction of the vertebral spines have been
frequently and elaborately described; cf. (e.g.) Gottlieb, H., Die
Anticlinie der Wirbelsäule der Säugethiere, _Morphol. Jahrb._ LXIX,
pp. 179–220, 1915, and many works quoted therein. According to
Morita, Ueber die Ursachen der Richtung und Gestalt der thoracalen
Dornfortsätze der Säugethierwirbelsäule (_ibi cit._ p. 201), various
changes take place in the direction or inclination of these processes
in rabbits, after section of the interspinous ligaments and muscles.
These changes seem to be very much what we should expect, on simple
mechanical grounds. See also Fischer, O., _Theoretische Grundlagen für
eine Mechanik der lebenden Körper_, Leipzig, pp. 3, 372, 1906.

[632] I owe the first four of these determinations to the kindness
of Dr Chalmers Mitchell, who had them made for me at the Zoological
Society’s Gardens; while the great Clydesdale carthorse was weighed for
me by a friend in Dundee.

[633] This pose of Diplodocus, and of other Sauropodous reptiles, has
been much discussed. Cf. (_int. al._) Abel, O., _Abh. k. k. zool.
bot. Ges. Wien_, V. 1909–10 (60 pp.); Tornier, _SB. Ges. Naturf.
Fr. Berlin_, pp. 193–209, 1909; Hay, O. P., _Amer. Nat._ Oct. 1908;
_Tr. Wash. Acad. Sci._ XLII, pp. 1–25, 1910; Holland, _Amer. Nat._
May, 1910, pp. 259–283; Matthew, _ibid._ pp. 547–560; Gilmore, C. W.
(_Restoration of Stegosaurus_). _Pr. U.S. Nat. Museum_, 1915.

[634] The form of the cantilever is much less typical in the small
flying birds, where the strength of the pelvic region is insured in
another way, with which we need not here stop to deal.

[635] The motto was Macquorn Rankine’s.

[636] John Hunter was seldom wrong; but I cannot believe that he was
right when he said (_Scientific Works_, ed. Owen, I, p. 371), “The
bones, in a mechanical view, appear to be the first that are to be
considered. We can study their shape, connexions, number, uses, etc.,
_without considering any other part of the body_.”

[637] _Origin of Species_, 6th ed. p. 118.

[638] _Amer. Naturalist_, April, 1915, p. 198, etc. Cf. _infra_, p. 727.

[639] Driesch sees in “Entelechy” that something which differentiates
the whole from the sum of its parts in the case of the organism: “The
organism, we know, is a system the single constituents of which are
inorganic in themselves; only the whole constituted by them in their
typical order or arrangement owes its specificity to ‘Entelechy’ ”
(_Gifford Lectures_, p. 229, 1908): and I think it could be shewn that
many other philosophers have said precisely the same thing. So far
as the argument goes, I fail to see how _this_ Entelechy is shewn to
be peculiarly or specifically related to the _living_ organism. The
conception that the whole is _always_ something very different from its
parts is a very ancient doctrine. The reader will perhaps remember how,
in another vein, the theme is treated by Martinus Scriblerus: “In every
Jack there is a _meat-roasting_ Quality, which neither resides in the
fly, nor in the weight, nor in any particular wheel of the Jack, but is
the result of the whole composition; etc., etc.”

[640] “There can be no doubt that Fraas is correct in regarding this
type (_Procetus_) as an annectant form between the Zeuglodonts and
the Creodonta, but, although the origin of the Zeuglodonts is thus
made clear, it still seems to be by no means so certain as that author
believes, that they may not themselves be the ancestral forms of the
Odontoceti”; Andrews, _Tertiary Vertebrata of the Fayum_, 1906, p. 235.

[641] Reprinted, with some changes and additions, from a paper in the
_Trans. Roy. Soc. Edin._ L, pp. 857–95, 1915.

[642] M. Bergson repudiates, with peculiar confidence, the
application of mathematics to biology. Cf. _Creative Evolution_,
p. 21, “Calculation touches, at most, certain phenomena of organic
destruction. Organic creation, on the contrary, the evolutionary
phenomena which properly constitute life, we cannot in any way subject
to a mathematical treatment.”

[643] In this there lies a certain justification for a saying of
Minot’s, of the greater part of which, nevertheless, I am heartily
inclined to disapprove. “We biologists,” he says, “cannot deplore
too frequently or too emphatically the great mathematical delusion
by which men often of great if limited ability have been misled into
becoming advocates of an erroneous conception of accuracy. The delusion
is that no science is accurate until its results can be expressed
mathematically. The error comes from the assumption that mathematics
can express complex relations. Unfortunately mathematics have a
very limited scope, and are based upon a few extremely rudimentary
experiences, which we make as very little children and of which no
adult has any recollection. The fact that from this basis men of
genius have evolved wonderful methods of dealing with numerical
relations should not blind us to another fact, namely, that the
observational basis of mathematics is, psychologically speaking, very
minute compared with the observational basis of even a single minor
branch of biology .... While therefore here and there the mathematical
methods may aid us, _we need a kind and degree of accuracy of which
mathematics is absolutely incapable_ .... With human minds constituted
as they actually are, we cannot anticipate that there will ever be a
mathematical expression for any organ or even a single cell, although
formulae will continue to be useful for dealing now and then with
isolated details...” (_op. cit._, p. 19, 1911). It were easy to discuss
and criticise these sweeping assertions, which perhaps had their
origin and parentage in an _obiter dictum_ of Huxley’s, to the effect
that “Mathematics is that study which knows nothing of observation,
nothing of experiment, nothing of induction, nothing of causation”
(_cit._ Cajori, _Hist of Elem. Mathematics_, p. 283). But Gauss called
mathematics “a science of the eye”; and Sylvester assures us that
“most, if not all, of the great ideas of modern mathematics have had
their origin in observation” (_Brit. Ass. Address_, 1869, and _Laws of
Verse_, p. 120, 1870).

[644] _Historia Animalium_ I, 1.

[645] Cf. _supra_, p. 714.

[646] Cf. Osborn, H. F., On the Origin of Single Characters, as
observed in fossil and living Animals and Plants, _Amer. Nat._ XLIX,
pp. 193–239, 1915 (and other papers); _ibid._ p. 194, “Each individual
is composed of a vast number of somewhat similar new or old characters,
each character has its independent and separate history, each character
is in a certain stage of evolution, each character is correlated with
the other characters of the individual .... The real problem has
always been that of the origin and development of characters. Since
the _Origin of Species_ appeared, the terms variation and variability
have always referred to single characters; if a species is said to be
variable, we mean that a considerable number of the single characters
or groups of characters of which it is composed are variable,” etc.

[647] Cf. Sorby, _Quart. Journ. Geol. Soc._ (_Proc._), 1879, p. 88.

[648] Cf. D’Orbigny, Alc., _Cours élém. de Paléontologie_, etc., I, pp.
144–148, 1849; see also Sharpe, Daniel, On Slaty Cleavage, _Q.J.G.S._
III, p. 74, 1847.

[649] Thus _Ammonites erugatus_, when compressed, has been described as
_A. planorbis_: cf. Blake, J. F., _Phil. Mag._ (5), VI, p. 260, 1878.
Wettstein has shewn that several species of the fish-genus _Lepidopus_
have been based on specimens artificially deformed in various ways:
Ueber die Fischfauna des Tertiären Glarnerschiefers, _Abh. Schw.
Palaeont. Gesellsch._ XIII, 1886 (see especially pp. 23–38, pl. I).
The whole subject, interesting as it is, has been little studied: both
Blake and Wettstein deal with it mathematically.

[650] Cf. Sir Thomas Browne, in _The Garden of Cyrus_: “But why
ofttimes one side of the leaf is unequall unto the other, as in
Hazell and Oaks, why on either side the master vein the lesser and
derivative channels stand not directly opposite, nor at equall angles,
respectively unto the adverse side, but those of one side do often
exceed the other, as the Wallnut and many more, deserves another
enquiry.”

[651] Where gourds are common, the glass-blower is still apt to take
them for a prototype, as the prehistoric potter also did. For instance,
a tall, annulated Florence oil-flask is an exact but no longer a
conscious imitation of a gourd which has been converted into a bottle
in the manner described.

[652] Cf. _Elsie Venner_, chap. ii.

[653] This significance is particularly remarkable in connection with
the development of speed, for the metacarpal region is the seat of very
important leverage in the propulsion of the body. In the Museum of
the Royal College of Surgeons in Edinburgh, there stand side by side
the skeleton of an immense carthorse (celebrated for having drawn all
the stones of the Bell Rock Lighthouse to the shore), and a beautiful
skeleton of a racehorse, which (though the fact is disputed) there is
good reason to believe is the actual skeleton of Eclipse. When I was a
boy my grandfather used to point out to me that the cannon-bone of the
little racer is not only relatively, but actually, longer than that of
the great Clydesdale.

[654] Cf. Vitruvius, III, 1.

[655] _Les quatres livres d’Albert Dürer de la proportion des parties
et pourtraicts des corps humains_, Arnheim, 1613, folio (and earlier
editions). Cf. also Lavater, _Essays on Physiognomy_, III, p. 271, 1799.

[656] It was these very drawings of Dürer’s that gave to Peter Camper
his notion of the “facial angle.” Camper’s method of comparison was the
very same as ours, save that he only drew the axes, without filling in
the network, of his coordinate system; he saw clearly the essential
fact, that the skull _varies as a whole_, and that the “facial angle”
is the index to a general deformation. “The great object was to shew
that natural differences might be reduced to rules, of which the
direction of the facial line forms the _norma_ or canon; and that these
directions and inclinations are always accompanied by correspondent
form, size and position of the other parts of the cranium,” etc.; from
Dr T. Cogan’s preface to Camper’s work _On the Connexion between the
Science of Anatomy and the Arts of Drawing, Painting and Sculpture_
(1768?), quoted in Dr R. Hamilton’s Memoir of Camper, in _Lives of
Eminent Naturalists_ (_Nat. Libr._), Edin. 1840.

[657] The co-ordinate system of Fig. 382 is somewhat different from
that which I drew and published in my former paper. It is not unlikely
that further investigation will further simplify the comparison, and
shew it to involve a still more symmetrical system.

[658] _Dinosaurs of North America_, pl. LXXXI, etc. 1896.

[659] _Mem. Amer. Mus. of Nat. Hist._ I, III, 1898.

[660] These and also other coordinate diagrams will be found in Mr G.
Heilmann’s book _Fuglenes Afstamning_, 398 pp., Copenhagen, 1916; see
especially pp. 368–380.

[661] Cf. W. B. Scott (_Amer. Journ. of Science_, XLVIII, pp. 335–374,
1894), “We find that any mammalian series at all complete, such as
that of the horses, is remarkably continuous, and that the progress
of discovery is steadily filling up what few gaps remain. So closely
do successive stages follow upon one another that it is sometimes
extremely difficult to arrange them all in order, and to distinguish
clearly those members which belong in the main line of descent, and
those which represent incipient branches. Some phylogenies actually
suffer from an embarrassment of riches.”

[662] Cf. Dwight, T., The Range of Variation of the Human Scapula,
_Amer. Nat._ XXI, pp. 627–638, 1887. Cf. also Turner, _Challenger Rep._
XLVII, on Human Skeletons, p. 86, 1886: “I gather both from my own
measurements, and those of other observers, that the range of variation
in the relative length and breadth of the scapula is very considerable
in the same race, so that it needs a large number of bones to enable
one to obtain an accurate idea of the mean of the race.”

[663] There is a paper on the mathematical study of organic forms
and organic processes by the learned and celebrated Gustav Theodor
Fechner, which I have only lately read, but which would have been of
no little use and help to our argument had I known it before. (Ueber
die mathematische Behandlung organischer Gestalten und Processe,
_Berichte d. k. sächs. Gesellsch._, _Math.-phys. Cl._, Leipzig,
1849, pp. 50–64.) Fechner’s treatment is more purely mathematical
and less physical in its scope and bearing than ours, and his paper
is but a short one; but the conclusions to which he is led differ
little from our own. Let me quote a single sentence which, together
with its context, runs precisely on the lines of the discussion with
which this chapter of ours began. “So ist also die mathematische
Bestimmbarkeit im Gebiete des Organischen ganz eben so gut vorhanden
als in dem des Unorganischen, und in letzterem eben solchen oder
äquivalenten Beschränkungen unterworfen als in ersterem; und nur sofern
die unorganischen Formen und das unorganische Geschehen sich einer
einfacheren Gesetzlichkeit mehr nähern als die organischen, kann die
Approximation im unorganischen Gebiet leichter und weiter getrieben
werden als im organischen. Dies wäre der ganze, sonach rein relative,
Unterschied.” Here in a nutshell, in words written some seventy years
ago, is the gist of the whole matter.

An interesting little book of Schiaparelli’s (which I ought to have
known long ago)—_Forme organiche naturali e forme geometriche pure_,
Milano, Hoepli, 1898—has likewise come into my hands too late for
discussion.

{780}




INDEX.


 Abbe’s diffraction plates, 323

 Abel, O., 706

 Abonyi, A., 127

 Acantharia, spicules of, 458

 Acanthometridae, 462

 Acceleration, 64

 Aceratherium, 761

 Achlya, 244

 Acromegaly, 135

 Actinomma, 469

 Actinomyxidia, 452

 Actinophrys, 165, 197, 264, 298

 Actinosphaerium, 197, 266, 298, 468

 Adams, J. C., 663

 Adaptation, 670

 Addison, Joseph, 671

 Adiantum, 408

 Adsorption, 192, 208, 241, 277, 357;
   orientirte, 440, 590;
   pseudo, 282

 Agglutination, 201

 Aglaophenia, 748

 Airy, H., 636

 Albumin molecule, 41

 Alcyonaria, 387, 413, 424, 459

 Alexeieff, A., 157, 165

 Allmann, W., 643

 Alpheus, claws of, 150

 Alpine plants, 124

 Altmann’s granules, 285

 Alveolar meshwork, 170

 Ammonites, 526, 530, 537, 539, 550, 552, 576, 583, 584, 728

 Amoeba, 12, 165, 209, 212, 245, 255, 288, 463, 605

 Amphidiscs, 440

 Amphioxus, 311

 Ampullaria, 560

 Anabaena, 300

 Anaxagoras, 8

 Ancyloceras, 550

 Andrews, G. F., 164;
   C. W., 716

 Anhydrite, 433

 Anikin, W. P., 130

 Anisonema, 126

 Anisotropy, 241, 357

 Anomia, 565, 567

 Antelopes, horns of, 614, 671

 Antheridia, 303, 403, 405, 409

 Anthoceros, spore of, 397

 Anthogorgia, spicules of, 413

 Anthropometry, 51

 Anticline, 360

 Antigonia, 750, 775

 Antlers, 628

 Apatornis, 757

 Apocynum, pollen of, 396

 Aptychus, 576

 Arachnoidiscus, 387

 Arachnophyllum, 325

 Arcella, 323

 Arcestes, 539, 540

 Archaeopteryx, 757

 Archimedes, 580;
   spiral of, 503, 524, 552

 Argali, horns of, 617

 Argiope, 561

 Argonauta, 546, 561

 Argus pheasant, 431, 631

 Argyropelecus, 748

 Aristotle, 3, 4, 5, 8, 15, 138, 149, 158, 509, 653, 714, 725, 726

 Arizona trees, 121

 Arrhenius, Sv., 28, 48, 171

 Artemia, 127

 Artemis, 561

 Ascaris megalocephala, 180, 195

 Aschemonella, 255

 Assheton, R., 344

 Asterina, 342

 Asteroides, 423

 Asterolampra, 386

 Asters, 167, 174

 Asthenosoma, 664

 Astrorhiza, 255, 463, 587, 607

 Astrosclera, 436

 Asymmetric substances, 416

 Asymmetry, 241

 Atrypa, 569

 Auerbach, F., 9

 Aulacantha, 460

 Aulastrum, 471

 Aulonia, 468

 Auricular height, 93

 Autocatalysis, 131

 Auximones, 135

 Awerinzew, S., 589

 Babak, E., 32

 Babirussa, teeth of, 634

 Baboon, skull of, 771

 Bacillus, 39;
   B. ramosus, 133

 Bacon, Lord, 4, 5, 51, 53, 131, 656, 716

 Bacteria, 245, 250

 Baer, K. E., von, 3, 55, 57, 155

 Balancement, 714, 776

 Balfour, F. M., 57, 348

 Baltzer, Fr., 327

 Bamboo, growth of, 77

 Barclay, J., 334

 Barfurth, D., 85

 Barlow, W., 202

 Barratt, J. O. W., 285

 Bartholinus, E., 329

 Bashforth, Fr., 663

 Bast-fibres, strength of, 679

 Baster, Job, 138

 Bateson, W., 104, 431

 Bather, F. A., 578

 Batsch, A. J. G. K., 606

 Baudrimont, A., and St Ange, 124

 Baumann and Roos, 136

 Bayliss, W. M., 135, 277

 Beads or globules, 234

 Beak, shape of, 632

 Beal, W. J., 643

 Beam, loaded, 674

 Bee’s cell, 327, 779

 Begonia, 412, 733

 Beisa antelope, horns of, 616, 621

 Bellerophon, 550

 Bénard, H., 259, 319, 448, 590

 Bending moments, 19, 677, 696

 Beneden, Ed. van, 153, 170, 198

 Bergson, H., 7, 103, 251, 611, 721

 Bernard, Claude, 2, 13, 127

 Bernoulli, James, 580;
   John, 30, 54

 Berthold, G., 8, 234, 298, 306, 322, 346, 351, 357, 358, 372, 399

 Bethe, A., 276

 Bialaszewicz, K., 114, 125

 Biedermann, W., 431

 Bilharzia, egg of, 656

 Binuclearity, 286

 Biocrystallisation, 454

 Biogenetisches Grundgesetz, 608

 Biometrics, 78

 Bird, flight of, 24;
   form of, 673

 Bisection of solids, 352, etc.

 Bishop, John 31

 Bivalve shells, 561

 Bjerknes, V. 186

 Blackman, F. F. 108, 110, 114, 124, 131, 132

 Blackwall, J. 234

 Blake, J. F. 536, 547, 553, 578, 583, 728

 Blastosphere, 56, 344

 Blood-corpuscles, form of, 270;
   size of, 36

 Blood-vessels, 665

 Boas, Fr., 79

 Bodo, 230, 269

 Boerhaave, Hermann, 380

 Bonanni, F., 318

 Bone, 425, 435;
   repair of, 687;
   structure of, 673, 680

 Bonnet, Ch., 108, 138, 334, 635

 Borelli, J. A., 8, 27, 29, 318, 677, 690

 Bosanquet, B., 5

 Boscovich, Father R. J., S.J., 8

 Bose, J. C., 87

 Bostryx, 502

 Bottazzi, F., 127

 Bottomley, J. T., 135

 Boubée, N., 529

 Bourgery, J. M., 683

 Bourne, G. C., 199

 Bourrelet, Plateau’s, 297, 339, 446, 470, 477

 Boveri, Th., 38, 147, 170, 198

 Bowditch, H. P., 61, 79

 Bower, F. O., 406

 Bowman, J. H., 428

 Boyd, R., 61

 Boys, C. V., 233

 Brachiopods, 561, 568, 577

 Bradford, S. C., 428

 Brady, H. B., 255, 606

 Brain, growth of, 89;
   weight of, 90

 Branchipus, 128, 342

 Brandt, K., 459, 482

 Brauer, A., 180

 Braun, A., 636

 Bravais, L. and A., 202, 502, 636

 Bredig, G., 178

 Brewster, Sir D., 209, 337, 350, 431

 Bridge, T. W., 671

 Bridge construction, 18, 691

 Brine shrimps, 127

 Brooke, Sir V., 614, 624, 628, 631

 Browne, Sir T., 324, 329, 480, 650, 652, 733

 Brownian movement, 45, 279, 421

 Brücke, C., 160, 199

 Buccinum, 520, 527

 Buch, Leopold von, 528, 583

 Buchner, Hans, 133

 Budding, 213, 399

 Buffon, on the bee’s cell, 333

 Bühle, C. A., 653

 Bulimus, 549, 556

 Burnet, J., 509

 Bütschli, O., 165, 170, 171, 204, 432, 434, 458, 492

 Büttel-Reepen, H. von, 332

 Byk, A., 419

 Cactus, sphaerocrystals, in 434

 Cadets, growth of German, 119

 Calandrini, G. L., 636

 Calcospherites, 421, 434

 Callimitra, 472

 Callithamnion, spore of, 396

 Calman, T. W., 149

 Calyptraea, 556

 Camel, 703, 704

 Campanularia, 237, 262, 747

 Campbell, D. H., 302, 397, 402

 Camper, P., 742

 Camptosaurus, 754

 Cannon bone, 730

 Cantilever, 678, 694

 Cantor, Moritz, 503

 Caprella, 743

 Caprinella, 567, 577

 Carapace of crabs, 744

 Cardium, 561

 Cariacus, 629

 Carlier, E. W., 211

 Carnoy, J. B., 468

 Carpenter, W. B., 45, 422, 465

 Caryokinesis, 14, 157, etc.

 Cassini, D., 329

 Cassis, 559

 Catabolic products, 435

 Catalytic action, 130

 Catenoid, 218, 223, 227, 252

 Causation, 6

 Cavolinia, 573

 Cayley, A., 385

 Celestite, 459

 Cell-theory, 197, 199

 Cells, forms of, 201;
   sizes of, 35

 Cellular pathology, 200;
   tissue, artificial, 320

 Cenosphaera, 470

 Centres of force, 156, 196

 Centrosome, 167, 168, 173

 Cephalopods, 548, etc.;
   eggs of, 378

 Ceratophyllum, growth of, 97

 Ceratorhinus, 612

 Cerebratulus, egg of, 189

 Cerianthus, 125

 Cerithium, 530, 557, 559

 Chabrier, J., 25

 Chabry, L., 30, 306, 415

 Chaetodont fishes, 671, 749

 Chaetopterus, egg of, 195

 Chamois, horns of, 615

 Chapman, Abel, 672

 Chara, 303

 Characters, biological, 196, 727

 Chevron bones, 709

 Chick, hatching of, 108

 Chilomonas, 114

 Chladni figures, 386, 475

 Chlorophyll, 291

 Choanoflagellates, 253

 Chodat, R., 78, 132

 Cholesterin, 272

 Chondriosomes, 285

 Chorinus, 744

 Chree, C., 19

 Chromatin, 153

 Chromidia, 286

 Chromosomes, 157, 173, 179, 181, 190, 195

 Church, A. H., 639

 Cicero, 62

 Cicinnus, 502

 Cidaris, 664

 Circogonia, 479

 Cladocarpus, 748

 Claparède, E. R, 423

 Clathrulina, 470

 Clausilia, 520, 549

 Claws, 149, 632

 Cleland, John, 4

 Cleodora, 570–575

 Climate and growth, 121

 Clio, 570

 Close packing, 453

 Clytia, 747

 Coan, C. A., 514

 Coassus, 629

 Cod, otoliths of, 432;
   skeleton of, 710

 Codonella, 248

 Codosiga, 253

 Coe, W. R., 189

 Coefficient of growth, 153;
   of temperature, 109

 Coelopleurus, 664

 Cogan, Dr T., 742

 Cohen, A., 110

 Cohesion figures, 259

 Collar-cells, 253

 Colloids, 162, 178, 201, 279, 412, 421, etc.

 Collosclerophora, 436

 Collosphaera, 459

 Colman, S., 514

 Comoseris, 327

 Compensation, law of, 714, 776

 Conchospiral, 531, 539, 594

 Conchyliometer, 529

 Concretions, 410, etc.

 Conjugate curves, 561, 613

 Conklin, E. G., 36, 191, 310, 340, 377

 Conostats, 427

 Continuous girder, 700

 Contractile vacuole, 165, 264

 Conus, 557, 559, 560

 Cook, Sir T. A., 493, 635, 639, 650

 Co-ordinates, 723

 Corals, 325, 388, 423

 Cornevin, Ch., 102

 Cornuspira, 594

 Correlation, 78, 727

 Corystes, 744

 Cotton, A., 418

 Cox, J., 46

 Crane-head, 682

 Crayfish, sperm-cells of, 273

 Creodonta, 716

 Crepidula, 36, 310, 340

 Creseis, 570

 Cristellaria, 515, 600

 Crocodile, 704, 752

 Crocus, growth of, 88

 Crookes, Sir W., 32

 Cryptocleidus, 755

 Crystals, 202, 250, 429, 444, 480, 601

 Ctenophora, 391

 Cube, partition of, 346

 Cucumis, growth of, 109

 Culmann, Professor C., 682, 697

 Cultellus, 564

 Curlew, eggs of, 652

 Cushman, J. A., 323

 Cuvier, 727

 Cuvierina, 258, 570

 Cyamus, 743

 Cyathophyllum, 325, 391

 Cyclammina, 595, 596, 602

 Cyclas, 561

 Cyclostoma, 554

 Cylinder, 218, 227, 377

 Cymba, 559

 Cyme, 502

 Cypraea, 547, 554, 560, 561

 Cyrtina, 569

 Cyrtocerata, 583

 Cystoliths, 412

 Daday de Dees, E. v., 130

 Daffner, Fr., 61, 118

 Dalyell, Sir John G., 146

 Danilewsky, B., 135

 Darling, C. R., 219, 257, 664

 D’Arsonval, A., 192, 281

 Darwin, C., 4, 44, 57, 332, 431, 465, 549, 624, 671, 714

 Dastre, A., 136

 Davenport, C. B., 107, 123, 125, 126, 211

 De Candolle, A., 108, 643;
   A. P., 20;
   C., 636

 Decapod Crustacea, sperm-cells of, 273

 Deer, antlers of, 628

 Deformation, 638, 728, etc.

 Degree, differences of, 586, 725

 Delage, Yves, 153

 Delaunay, C. E., 218

 Delisle, 31

 Dellinger, O. P., 212

 Delphinula, 557

 Delpino, F., 636

 Democritus, 44

 Dendy, A., 137, 436, 440, 671

 Dentalium, 535, 537, 546, 555, 556, 561

 Dentine, 425

 Descartes, R., 185, 723

 Des Murs, O., 653

 Devaux, H., 43

 De Vries, H., 108

 Diatoms, 214, 386, 426

 Diceras, 567

 Dickson, Alex., 647

 Dictyota, 303, 356, 474

 Diet and growth, 134

 Difflugia, 463, 466

 Diffusion figures, 259, 430

 Dimorphism of earwigs, 105

 Dimorphodon, 756

 Dinenympha, 252

 Dinobryon, 248

 Dinosaurs, 702, 704, 754

 Diodon, 751, 777

 Dionaea, 734

 Diplodocus, 702, 706, 710

 Disc, segmentation of a, 367

 Discorbina, 602

 Distigma, 246

 Distribution, geographical, 457, 606

 Ditrupa, 586

 Dixon, A. F., 684

 Dobell, C. C., 286

 Dodecahedron, 336, 478, etc.

 Doflein, F. J., 46, 267, 606

 Dog’s skull, 773

 Dolium, 526, 528, 530, 557, 559, 560

 Dolphin, skeleton of, 709

 Donaldson, H. H., 82, 93

 Dorataspis, 481

 D’Orbigny, Alc., 529, 555, 591, 728

 Douglass, A. E., 121

 Draper, J. W., 165, 264

 Dreyer, F. R., 435, 447, 455, 468, 606, 608

 Driesch, H., 4, 35, 157, 306, 310, 312, 377, 378, 714

 Dromia, 275

 Drops, 44, 257, 587

 Du Bois-Reymond, Emil, 1, 92

 Duerden, J. E., 423

 Dufour, Louis, 219

 Dujardin, F., 257, 591

 Dunan, 7

 Duncan, P. Martin, 388

 Dupré, Athanase, 279

 Durbin, Marion L., 138

 Dürer, A., 55, 740, 742

 Dutrochet, R. J. H., 212, 624

 Dwight, T., 769

 Dynamical similarity, 17

 Earthworm, calcospheres in, 423

 Earwigs, dimorphism in, 104

 Ebner, V. von, 444, 683

 Echinoderms, larval, 392;
   spicules of, 449

 Echinus, 377, 378, 664

 Eclipse, skeleton of, 739

 Ectosarc, 281

 Eel, growth of, 85

 Efficiency, mechanical, 670

 Efficient cause, 6, 158, 248

 Eggs of birds, 652

 Eiffel tower, 20

 Eight cells, grouping of, 381, etc.

 Eimer, Th., 606

 Einstein formula, 47

 Elastic curve, 219, 265, 271

 Elaters, 489

 Electrical convection, 187;
   stimulation of growth, 153

 Elephant, 21, 633, 703, 704

 Elk, antlers of, 629, 632

 Ellipsolithes, 728

 Ellis, R. Leslie, 4, 329, 647;
   M. M., 147, 656

 Elodea, 322

 Emarginula, 556

 Emmel, V. E., 149

 Empedocles, 8

 Emperor Moth, 431

 Encystment, 213, 283

 Engelmann, T. W., 210, 285

 Enriques, P., 4, 36, 64, 133, 134, 677

 Entelechy, 4, 714

 Entosolenia, 449

 Enzymes, 135

 Epeira, 233

 Epicurus, 47

 Epidermis, 314, 370

 Epilobium, pollen of, 396

 Epipolic force, 212

 Equatorial plate, 174

 Equiangular spiral, 50, 505

 Equilibrium, figures of, 227

 Equipotential lines, 640

 Equisetum, spores of, 290, 489

 Errera, Leo, 8, 40, 110, 111, 213, 306, 346, 348, 426

 Erythrotrichia, 358, 372, 390

 Ethmosphaera, 470

 Euastrum, 214

 Eucharis, 391

 Euclid, 509

 Euglena, 376

 Euglypha, 189

 Euler, L., 3, 208, 385, 484, 690

 Eulima, 559

 Eunicea, spicules of, 424

 Euomphalus, 557, 559

 Evelyn, John, 652

 Evolution, 549, 610, etc.

 Ewart, A. J., 20

 Fabre, J. H., 64, 779

 Facial angle, 742, 770, 772

 Faraday, M., 163, 167, 428, 475

 Farmer, J. B. and Digby, 190

 Fatigue, molecular, 689

 Faucon, A., 88

 Favosites, 325

 Fechner, G. T., 654, 777

 Fedorow, E. S. von, 338

 Fehling, H., 76, 126

 Ferns, spores of, 396

 Fertilisation, 193

 Fezzan-worms, 127

 Fibonacci, 643

 Fibrillenkonus, 285

 Fick, R., 57, 683

 Fickert, C., 606

 Fidler, Prof. T. Claxton, 691, 674, 696

 Films, liquid, 215, 217, 426

 Filter-passers, 39

 Final cause, 3, 248, 714

 Fir-cone, 635, 647

 Fischel, Alfred, 88

 Fischer, Alfred, 40, 172;
   Emil, 417, 418;
   Otto, 30, 699

 Fishes, forms of, 748

 Fission, multiplication by, 151

 Fissurella, 556

 FitzGerald, G. F., 158, 281, 323, 440, 477

 Flagellum, 246, 267, 291

 Flemming, W., 170, 172, 180

 Flight, 24

 Flint, Professor, 673

 Fluid crystals, 204, 272, 485

 Fluted pattern, 260

 Fly’s cornea, 324

 Fol, Hermann, 168, 194

 Folliculina, 249

 Foraminifera, 214, 255, 415, 495, 515

 Forth Bridge, 694, 699, 700

 Fossula, 390

 Foster, M., 185

 Fraas, E., 716

 Frankenheim, M. L., 202

 Frazee, O. E., 153

 Frédéricq, L., 127, 130

 Free cell formation, 396

 Friedenthal, H., 64

 Frisch, K. von, 671

 Frog, egg of, 310, 363, 378, 382;
   growth of, 93, 126

 Froth or foam, 171, 205, 305, 314, 322, 343

 Froude, W., 22

 Fucus, 355

 Fundulus, 125

 Fusulina, 593, 594

 Fusus, 527, 557

 Gadow, H. F., 628

 Galathea, 273

 Galen, 3, 465, 656

 Galileo, 8, 19, 28, 562, 677, 720

 Gallardo, A., 163

 Galloway, Principal, 672

 Gamble, F. A., 458

 Ganglion-cells, size of, 37

 Gans, R., 46

 Garden of Cyrus, 324, 329

 Gastrula, 344

 Gauss, K. F., 207, 278, 723

 Gebhardt, W., 430, 683

 Gelatination, water of, 203

 Generating curves and spirals, 526, 561, 615, 637, 641

 Geodetics, 440, 488

 Geoffroy St Hilaire, Et. de, 714

 Geotropism, 211

 Gerassimow, J. J., 35

 Gerdy, P. N., 491

 Geryon, 744

 Gestaltungskraft, 485

 Giard, A., 156

 Gilmore, C. W., 707

 Giraffe, 705, 730, 738

 Girardia, 321, 408

 Glaisher, J., 250

 Glassblowing, 238, 737

 Gley, E., 135, 136

 Globigerina, 214, 234, 440, 495, 589, 602, 604, 606

 Gnomon, 509, 515, 591

 Goat, horns of, 613

 Goat moth, wings of, 430

 Goebel, K., 321, 397, 408

 Goethe, 20, 38, 199, 714, 719

 Golden Mean, 511, 643, 649

 Goldschmidt, R., 286

 Goniatites, 550, 728

 Gonothyraea, 747

 Goodsir, John, 156, 196, 580

 Gottlieb, H., 699

 Gourd, form of, 737

 Grabau, A. H., 531, 539, 550

 Graham, Thomas, 162, 201, 203

 Grant, Kerr, 259

 Grantia, 445

 Graphic statics, 682

 Gravitation, 12, 32

 Gray, J., 188

 Greenhill, Sir A. G., 19

 Gregory, D. F., 330, 675

 Greville, R. K., 386

 Gromia, 234, 257

 Gruber, A., 165

 Gryphaea, 546, 576, 577

 Guard-cells, 394

 Gudernatsch, J. F., 136

 Guillemot, egg of, 652

 Gulliver, G., 36

 Günther, F. C., 633, 654

 Gurwitsch, A., 285

 Häcker, V., 458

 Haddock, 774

 Haeckel, E., 199, 445, 454, 455, 457, 467, 480, 481

 Hair, pigmentation of, 430

 Hales, Stephen, 36, 59, 95, 669

 Haliotis, 514, 527, 546, 547, 554, 555, 557, 561

 Hall, C. E., 119

 Haller, A. von, 2, 54, 56, 59, 64, 68

 Hardesty, Irving, 37

 Hardy, W. B., 160, 162, 172, 187, 287

 Harlé, N., 28

 Harmozones, 135

 Harpa, 526, 528, 559

 Harper, R. A., 283

 Harpinia, 746

 Harting, P., 282, 420, 426, 434

 Hartog, M., 163, 327

 Harvey, E. N. and H. W., 187

 Hatai, S., 132, 135

 Hatchett, C., 420

 Hatschek, B., 180

 Haughton, Rev. S., 334, 666

 Haüy, R. J., 720

 Hay, O. P., 707

 Haycraft, J. B., 211, 690

 Head, length of, 93

 Heart, growth of, 89;
   muscles of, 490

 Heath, Sir T., 511

 Hegel, G. W. F., 4

 Hegler, 680, 688

 Heidenhain, M., 170, 212

 Heilmann, Gerhard, 757, 768, 772

 Helicoid, 230;
   cyme, 502, 605

 Helicometer, 529

 Helicostyla, 557

 Heliolites, 326

 Heliozoa, 264, 460

 Helix, 528, 557

 Helmholtz, H. von, 2, 9, 25

 Henderson, W. P., 323

 Henslow, G., 636

 Heredity, 158, 286, 715

 Hermann, F., 170

 Hero of Alexandria, 509

 Heron-Allen, E., 257, 415, 465

 Herpetomonas, 268

 Hertwig, O., 56, 114, 153, 199, 310;
   R., 170, 285

 Hertzog, R. O., 109

 Hess, W., 666, 668

 Heteronymous horns, 619

 Heterophyllia, 388

 Hexactinellids, 429, 452, 453

 Hexagonal symmetry, 319, 323, 471, 513

 Hickson, S. J., 424

 Hippopus, 561

 His, W., 55, 56, 74, 75

 Hobbes, Thomas, 159

 Höber, R., 1, 126, 130, 172

 Hodograph, 516

 Hoffmann, C., 628

 Hofmeister, F., 41; W., 87, 210, 234, 304, 306, 636, 639

 Holland, W. J., 707

 Holmes, O. W., 62, 737

 Holothuroid spicules, 440, 451

 Homonymous horns, 619

 Homoplasy, 251

 Hooke, Robert, 205

 Hop, growth of, 118;
   stem of, 627

 Horace, 44

 Hormones, 135

 Horns, 612

 Horse, 694, 701, 703, 764

 Houssay, F., 21

 Huber, P., 332

 Huia bird, 633

 Humboldt, A. von, 127

 Hume, David, 6

 Hunter, John, 667, 669, 713, 715

 Huxley, T. H., 423, 722, 752

 Hyacinth, 322, 394

 Hyalaea, 571–577

 Hyalonema, 442

 Hyatt, A., 548

 Hyde, Ida H., 125, 163, 184, 188

 Hydra, 252;
   egg of, 164

 Hydractinia, 342

 Hydraulics, 669

 Hydrocharis, 234

 Hyperia, 746

 Hyrachyus, 760, 765

 Hyracotherium, 766, 768

 Ibex, 617

 Ice, structure of, 428

 Ichthyosaurus, 755

 Icosahedron, 478

 Iguanodon, 706, 708

 Inachus, sperm-cells of, 273

 Infusoria, 246, 489

 Intussusception, 202

 Inulin, 432

 Invagination, 56, 344

 Iodine, 136

 Irvine, Robert, 414, 434

 Isocardia, 561, 577

 Isoperimetrical problems, 208, 346

 Isotonic solutions, 130, 274

 Iterson, G. van, 595

 Jackson, C. M., 75, 88, 106

 Jamin, J. C., 418

 Janet, Paul, 5, 18, 673

 Japp, F. R., 417

 Jellett, J. H., 1

 Jenkin, C. F., 444

 Jenkinson, J. W., 94, 114, 170

 Jennings, H. S., 212, 492;
   Vaughan, 424

 Jensen, P., 211

 Johnson, Dr S., 62

 Joly, John, 9, 63

 Jost, L., 110, 111

 Juncus, pith of, 335

 Jungermannia, 404

 Kangaroo, 705, 706, 709

 Kanitz, Al., 109

 Kant, Immanuel, 1, 3, 714

 Kappers, C. U. A., 566

 Kellicott, W. E., 91

 Kelvin, Lord, 9, 49, 188, 202, 336, 453

 Kepler, 328, 480, 486, 643, 650

 Kienitz-Gerloff, F., 404, 408

 Kirby and Spence, 28, 30, 127

 Kirchner, A., 683

 Kirkpatrick, R., 437

 Klebs, G., 306

 Kny, L., 680

 Koch, G. von, 423

 Koenig, Samuel, 330

 Kofoid, C. A., 268

 Kölliker, A. von, 413

 Kollmann, M., 170

 Koltzoff, N. K., 273, 462

 Koninckina, 570

 Koodoo, horns of, 624

 Köppen, Wladimir, 111

 Korotneff, A., 377

 Kraus, G., 77

 Krogh, A., 109

 Krohl, 666

 Kühne, W., 235

 Küster, E., 430

 Lafresnaye, F. de, 653

 Lagena, 251, 256, 260, 587

 Lagrange, J. L., 649

 Lalanne, L., 334

 Lamarck, J. B. de, 549, 716

 Lamb, A. B., 186

 Lamellaria, 554

 Lamellibranchs, 561

 Lami, B., 296, 643

 Laminaria, 315

 Lammel, R., 100

 Lanchester, F. W., 26

 Lang, Arnold, 561

 Lankester, Sir E. Ray, 4, 251, 348, 465

 Laplace, P. S. de, 1, 207, 217

 Larmor, Sir J., 9, 259

 Lavater, J. C., 740

 Law, Borelli’s, 29;
   Brandt’s, 482;
   of Constant Angle, 599;
   Errera’s, 213, 306;
   Froude’s, 22;
   Lamarle’s, 309;
   of Mass, 130;
   Maupertuis’s, 208;
   Müller’s, 481;
   of Optimum, 110;
   van’t Hoff’s, 109;
   Willard-Gibbs’, 280;
   Wolff’s, 3, 51, 155

 Leaping, 29

 Leaves, arrangement of, 635;
   form of, 731

 Ledingham, J. C. G., 211

 Leduc, Stéphane, 162, 167, 185, 219, 259, 415, 428, 431, 590

 Leeuwenhoek, A. van, 36, 209

 Leger, L., 452

 Le Hello, P., 30

 Lehmann, O., 203, 272, 440, 485, 590

 Leibniz, G. W. von, 3, 5, 159, 385

 Leidenfrost, J. G., 279

 Leidy, J., 252, 468

 Leiper, R. T., 660

 Leitch, I., 112

 Leitgeb, H., 305

 Length-weight coefficient, 98–103, 775

 Leonardo da Vinci, 27, 635;
   of Pisa, 643

 Lepeschkin, 625

 Leptocephalus, 87

 Leray, Ad., 18

 Lesage, G. L., 18

 Leslie, Sir John, 163, 503

 Lestiboudois, T., 636

 Leucocytes, 211

 Levers, Orders of, 690

 Levi, G., 35, 37

 Lewis, C. M., 280

 Lhuilier, S. A. J., 330

 Liesegang’s rings, 427, 475

 Light, pressure of, 48

 Lillie, F. R., 4, 147, 341;
   R. S., 180, 187, 192

 Lima, 565

 Limacina, 571

 Lines of force, 163;
   of growth, 562

 Lingula, 251, 567

 Linnaeus, 28, 250, 547, 720

 Lion, brain of, 91

 Liquid veins, 265

 Lister, Martin, 318;
   J. J., 436

 Listing, J. B., 385

 Lithostrotion, 325

 Littorina, 524

 Lituites, 546, 550

 Llama, 703

 Lobsters’ claws, 149

 Locke, John, 6

 Loeb, J., 125, 132, 135, 136, 147, 157, 191, 193

 Loewy, A., 281

 Logarithmic spiral, 493, etc.

 Loisel, G., 88

 Loligo, shell of, 575

 Lo Monaco, 83

 Lönnberg, E., 614, 632

 Looss, A., 660

 Lotze, R. H., 55

 Love, A. E. H., 674

 Lucas, F. A., 138

 Luciani, L., 83

 Lucretius, 47, 71, 137, 160

 Ludwig, Carl, 2;
   F., 643;
   H. J., 342

 Lupa, 744

 Lupinus, growth of, 109, 112

 Macalister, A., 557

 MacAlister, Sir D., 673, 683

 Macallum, A. B., 277, 287, 357, 395;
   J. B., 492

 McCoy, F., 388

 Mach, Ernst, 209, 330

 Machaerodus, teeth of, 633

 McKendrick, J. G., 42

 McKenzie, A., 418

 Mackinnon, D. L., 268

 Maclaurin, Colin, 330, 779

 Macroscaphites, 550

 Mactra, 562

 Magnitude, 16

 Maillard, L., 163

 Maize, growth of, 109, 111, 298

 Mall, F. P., 492

 Maltaux, Mlle, 114

 Mammoth, 634, 705

 Man, growth of, 61;
   skull of, 770

 Maraldi, J. P., 329, 473

 Marbled papers, 736

 Marcus Aurelius, 609

 Markhor, horns of, 619

 Marsh, O. C., 706, 754

 Marsigli, Comte L. F. de, 652

 Massart, J., 114

 Mastodon, 634

 Mathematics, 719, 778, etc.

 Mathews, A., 285

 Matrix, 656

 Matter and energy, 11

 Matthew, W. D., 707

 Matuta, 744

 Maupas, M., 133

 Maupertuis, 3, 5, 208

 Maxwell, J. Clerk, 9, 18, 40, 44, 160, 207, 385, 691

 Mechanical efficiency, 670

 Mechanism, 5, 161, 185, etc.

 Meek, C. F. U., 190

 Melanchthon, 4

 Melanopsis, 557

 Meldola, R., 670

 Melipona, 332

 Mellor, J. W., 134

 Melo, 525

 Melobesia, 412

 Melsens, L. H. F., 282

 Membrane-formation, 281

 Mensbrugghe, G. van der, 212, 298, 470

 Meserer, O., 683

 Mesocarpus, 289

 Mesohippus, 766

 Metamorphosis, 82

 Meves, F., 163, 285

 Meyer, Arthur, 432;
   G. H., 8, 682, 683

 Micellae, 157

 Michaelis, L., 277

 Microchemistry, 288

 Micrococci, 39, 245, 250

 Micromonas, 38

 Miliolidae, 595, 604

 Milner, R. S., 280

 Milton, John, 779

 Mimicry, 671

 Minchin, E. A., 267, 444, 449, 455

 Minimal areas, 208, 215, 225, 293, 306, 336, 349

 Minot, C. S., 37, 72, 722

 Miohippus, 767

 Mitchell, P. Chalmers, 703

 Mitosis, 170

 Mitra, 557, 559

 Möbius, K., 449

 Modiola, 562

 Mohl, H. von, 624

 Molar and molecular forces, 53

 Mole-cricket, chromosomes of, 181

 Molecular asymmetry, 416

 Molecules, 41

 Möller, V. von, 593

 Monnier, A., 78, 132

 Monticulipora, 326

 Moore, B., 272

 Morey, S., 264

 Morgan, T. H., 126, 134, 138, 147

 Morita, 699

 Morphodynamique, 156

 Morphologie synthétique, 420

 Morphology, 719, etc.

 Morse, Max, 136

 Moseley, H., 8, 518, 521, 538, 553, 555, 592

 Moss, embryo of, 374;
   gemma of, 403;
   rhizoids of, 356

 Mouillard, L. P., 27

 Mouse, growth of, 82

 Mucor, sporangium of, 303

 Müllenhof, K. von, 25, 332

 Müller, Fritz, 3;
   Johannes, 459, 481

 Mummery, J. H., 425

 Munro, H., 323

 Musk-ox, horns of, 615

 Mya, 422, 561

 Myonemes, 562

 Naber, H. A., 511, 650

 Nägeli, C., 124, 159, 210

 Nassellaria, 472

 Natica, 554, 557, 559

 Natural selection, 4, 58, 137, 456, 586, 609, 651, 653

 Naumann, C. F., 529, 531, 539, 550, 577, 594, 636;
   J. F., 653

 Nautilus, 355, 494, 501, 515, 518, 532, 535, 546, 552, 557, 575, 577,
     580, 592, 633;
   hood of, 554;
   kidney of, 425;
   N. umbilicatus, 542, 547, 554

 Nebenkern, 285

 Neottia, pollen of, 396

 Nereis, egg of, 342, 378, 453

 Nerita, 522, 555

 Neumayr, M., 608

 Neutral zone, 674, 676, 686

 Newton, 1, 6, 158, 643, 721

 Nicholson, H. A., 325, 327

 Noctiluca, 246

 Nodoid, 218, 223

 Nodosaria, 262, 535, 604

 Norman, A. M., 465

 Norris, Richard, 272

 Nostoc, 300, 313

 Notosuchus, 753

 Nuclear spindle, 170;
   structure, 166

 Nummulites, 504, 552, 591

 Nussbaum, M., 198

 Oekotraustes, 550

 Ogilvie-Gordon, M. M., 423

 Oil-globules, Plateau’s, 219

 Oithona, 742

 Oken, L., 4, 635

 Oliva, 554

 Ootype, 660

 Operculina, 594

 Operculum of gastropods, 521

 Oppel, A., 88

 Optimum temperature, 110

 Orbitolites, 605

 Orbulina, 59, 225, 257, 587, 598, 604, 607

 Organs, growth of, 88

 Orthagoriscus, 751, 775, 777

 Orthis, 561, 567

 Orthoceras, 515, 548, 551, 556, 579, 735

 Orthogenesis, 549

 Orthogonal trajectories, 305, 377, 400, 640, 678

 Orthostichies, 649

 Orthotoluidene, 219

 Oryx, horns of, 616

 Osborn, H. F., 714, 727, 760

 Oscillatoria, 300

 Osmosis, 124, 287, etc.

 Osmunda, 396, 406

 Ostrea, 562

 Ostrich, 25, 707, 708

 Ostwald, Wilhelm, 44, 131, 426;
   Wolfgang, 32, 77, 82, 132, 277, 281

 Otoliths, 425, 432

 Ovis Ammon, 614

 Owen, Sir R., 20, 575, 654, 669, 715

 Ox, cannon-bone of, 730, 738;
   growth of, 102

 Oxalate, calcium, 412, 434

 Palaeechinus, 663

 Palm, 624

 Pander, C. H., 55

 Pangenesis, 44, 157

 Papillon, Fernand, 10

 Pappus of Alexandria, 328

 Parabolic girder, 693, 696

 Parahippus, 767

 Paralomis, 744

 Paraphyses of mosses, 351

 Parastichies, 640, 641

 Passiflora, pollen of, 396

 Pasteur, L., 416

 Patella, 561

 Pauli, W., 211, 434

 Pearl, Raymond, 90, 97, 654

 Pearls, 425, 431

 Pearson, Karl, 36, 78

 Peas, growth of, 112

 Pecten, 562

 Peddie, W., 182, 272, 344, 448

 Pellia, spore of, 302

 Pelseneer, P., 570

 Pendulum, 30

 Peneroplis, 606

 Percentage-curves, Minot’s, 72

 Pericline, 360

 Periploca, pollen of, 396

 Peristome, 239

 Permeability, magnetic, 177, 182

 Perrin, J., 43, 46

 Peter, Karl, 117

 Pettigrew, J. B., 490

 Pfeffer, W., 111, 273, 688

 Pflüger, E., 680

 Phagocytosis, 211

 Phascum, 408

 Phase of curve, 68, 81, etc.

 Phasianella, 557, 559

 Phatnaspis, 482

 Phillipsastraea, 327

 Philolaus, 779

 Pholas, 561

 Phormosoma, 664

 Phractaspis, 484

 Phyllotaxis, 635

 Phylogeny, 196, 251, 548, 716

 Pike, F. H., 110

 Pileopsis, 555

 Pinacoceras, 584

 Pithecanthropus, 772

 Pith of rush, 335

 Plaice, 98, 105, 117, 432, 710, 774

 Planorbis, 539, 547, 554, 557, 559

 Plateau, F., 30, 232;
   J. A. F., 192, 212, 218, 239, 275, 297, 374, 477

 Plato, 2, 478, 720;
   Platonic bodies, 478

 Plesiosaurs, 755

 Pleurocarpus, 289

 Pleuropus, 573

 Pleurotomaria, 557

 Plumulariidae, 747

 Pluteus larva, 392, 415

 Podocoryne, 342

 Poincaré, H., 134

 Poiseuille, J. L. M., 669

 Polar bodies, 179;
   furrow, 310, 340

 Polarised light, 418

 Polarity, morphological, 166, 168, 246, 295, 284

 Pollen, 396, 399

 Polyhalite, 433

 Polyprion, 749, 776

 Polyspermy, 193

 Polytrichum, 355

 Pomacanthus, 749

 Popoff, M., 286

 Potamides, 554

 Potassium, in living cells, 288

 Potential energy, 208, 294, 601, etc.

 Potter’s wheel, 238

 Potts, R., 126

 Pouchet, G., 415

 Poulton, E. B., 670

 Poynting, J. H., 235

 Precocious segregation, 348

 Preformation, 54, 159

 Prenant, A., 163, 104, 189, 286, 289

 Prévost, Pierre, 18

 Pringsheim, N., 377

 Probabilities, theory of, 61

 Productus, 567

 Protective colouration, 671

 Protococcus, 59, 300, 410

 Protoconch, 531

 Protohippus, 767

 Protoplasm, structure of, 172

 Przibram, Hans, 16, 82, 107, 149, 204, 211, 418, 595;
   Karl, 46

 Psammobia, 564

 Pseuopriacauthus, 749

 Pteranodon, 756

 Pteris, antheridia of, 409

 Pteropods of, 258, 570

 Pulvinulina, 514, 595, 600, 602

 Pupa, 530, 549, 556

 Pütter, A., 110, 211, 492

 Pyrosoma, egg of, 377

 Pythagoras, 2, 509, 651, 720, 779

 Quadrant, bisection of, 359

 Quekett, J. T., 423

 Quetelet, A., 61, 78, 93

 Quincke, G. H., 187, 191, 279, 421

 Rabbit, skull of, 764

 Rabl, K., 36, 310

 Radial co-ordinates, 730

 Radiolaria, 252, 264, 457, 467, 588, 607

 Rainey, George, 7, 420, 431, 434

 Rainfall and growth, 121

 Ram, horns of, 613–624

 Ramsden, W., 282

 Ramulina, 255

 Rankine, W. J. Macquorn, 697, 712

 Ransom’s waves, 164

 Raphides, 412, 429, 434

 Raphidiophrys, 460, 463

 Rasumowsky, 683

 Rat, growth of, 106

 Rath, O. vom, 181

 Rauber, A., 200, 305, 310, 380, 382, 398, 677, 683

 Ray, John, 3

 Rayleigh, Lord, 43, 44

 Réaumur, R. A. de, 8, 108, 329

 Reciprocal diagrams, 697

 Rees, R. van, 374

 Regeneration, 138

 Reid, E. Waymouth, 272

 Reinecke, J. C. M., 528

 Reinke, J., 303, 305, 355, 356

 Reniform shape, 735

 Reticularia, 569

 Reticulated patterns, 258

 Réticulum plasmatique, 468

 Rhabdammina, 589

 Rheophax, 263

 Rhinoceros, 612, 760

 Rhumbler, L., 162, 165, 260, 322, 344, 465, 466, 589, 590, 595, 599,
     608, 628

 Rhynchonella, 561

 Riccia, 372, 403, 405

 Rice, J., 242, 273

 Richardson, G. M., 416

 Riefstahl, E., 578

 Riemann, B., 385

 Ripples, 33, 261, 323

 Rivularia, 300

 Roaf, H. C., 272

 Robert, A., 306, 339, 348, 377

 Roberts, C., 61

 Robertson, T. B., 82, 132, 191, 192

 Robinson, A., 681

 Rörig, A., 628

 Rose, Gustav, 421

 Rossbach, M. J., 165

 Rotalia, 214, 535, 602

 Rotifera, cells of, 38

 Roulettes, 218

 Roux, W., 8, 55, 57, 157, 194, 378, 383, 666, 683

 Ruled surfaces, 230, 270, 582

 Ruskin, John, 20

 Russow, ——, 73, 75

 Ryder, J. A., 376

 Sachs, J., 35, 38, 95, 108, 110, 111, 200, 360, 398, 399, 624, 635,
     640, 651, 680

 Sachs’s rule, 297, 300, 305, 347, 376

 Saddles, of ammonites, 583

 Sagrina, 263

 St Venant, Barré de, 621, 627

 Salamander, sperm-cells of, 179

 Salpingoeca, 248

 Salt, crystals of, 429

 Salvinia, 377

 Samec, M., 434

 Samter, M. and Heymons, 130

 Sandberger, G., 539

 Sapphirina, 742

 Saville Kent, W., 246, 247, 248

 Scalaria, 526, 547, 554, 557, 559

 Scale, effect of, 17, 438

 Scaphites, 550

 Scapula, human, 769

 Scarus, 749

 Schacko, G., 604

 Schaper, A. A., 83

 Schaudinn, F., 46, 286

 Scheerenumkehr, 149

 Schewiakoff, W., 189, 462

 Schimper, C. F., 502, 636

 Schmaltz, A., 675

 Schmankewitsch, W., 130

 Schmidt, Johann, 85, 87, 118

 Schönflies, A., 202

 Schultze, F. E., 452, 454

 Schwalbe, G., 666

 Schwann, Theodor, 199, 380, 591

 Schwartz, Fr., 172

 Schwendener, S., 210, 305, 636, 678

 Scorpaena, 749

 Scorpioid cyme, 502

 Scott, E. L., 110;
   W. B., 768

 Scyromathia, 744

 Searle, H., 491

 Sea urchins, 661;
   egg of, 173;
   growth of, 117, 147

 Sebastes, 749

 Sectio aurea, 511, 643, 649

 Sedgwick, A., 197, 199

 Sédillot, Charles E., 688

 Segmentation of egg, 57, 310, 344, 382, etc.;
   spiral do., 371, 453

 Segner, J. A. von, 205

 Selaginella, 404

 Semi-permeable membranes, 272

 Sepia, 575, 577

 Septa, 577, 592

 Serpula, 603

 Sexual characters, 135

 Sharpe, D., 728

 Shearing stress, 684, 730, etc.

 Sheep, 613, 730, 738

 Shell, formation of, 422

 Sigaretus, 554

 Silkworm, growth of, 83

 Similitude, principle of, 17

 Sims Woodhead, G., 414, 434

 Siphonogorgia, 413

 Skeleton, 19, 438, 675, 691, etc.

 Snow crystals, 250, 480, 611

 Soap-bubbles, 43, 219, 299, 307, etc.

 Socrates, 8

 Sohncke, L. A., 202

 Solanum, 625

 Solarium, 547, 554, 557, 559

 Solecurtus, 564

 Solen, 565

 Sollas, W. J., 440, 450, 455

 Solubility of salts, 434

 Sorby, H. C., 412, 414, 728

 Spallanzani, L., 138

 Span of arms, 63, 93

 Spangenberg, Fr., 342

 Specific characters, 246, 380;
   inductive capacity, 177;
   surface, 32, 215

 Spencer, Herbert, 18, 22

 Spermatozoon, path of, 193

 Sperm-cells of Crustacea, 273

 Sphacelaria, 351

 Sphaerechinus, 117, 147

 Sphagnum, 402, 407

 Sphere, 218, 225

 Spherocrystals, 434

 Spherulites, 422

 Spicules, 282, 411, etc.

 Spider’s web, 231

 Spindle, nuclear, 169, 174

 Spinning of protoplasm, 164

 Spiral, geodetic, 488;
   logarithmic, 493, etc.;
   segmentation, 371, 453

 Spireme, 173, 180

 Spirifer, 561, 568

 Spirillum, 46, 253

 Spirochaetes, 46, 230, 266

 Spirographis, 586

 Spirogyra, 12, 221, 227, 242, 244, 275, 287, 289

 Spirorbis, 586, 603

 Spirula, 528, 547, 554, 575, 577

 Spitzka, E. A., 92

 Splashes, 235, 236, 254, 260

 Sponge-spicules, 436, 440

 Spontaneous generation, 420

 Sporangium, 406

 Spottiswoode, W., 779

 Spray, 236

 Stallo, J. B., 1

 Standard deviation, 78

 Starch, 432

 Starling, E. H., 135

 Stassfurt salt, 433

 Stegocephalus, 746

 Stegosaurus, 706, 707, 710, 754

 Steiner, Jacob, 654

 Steinmann, G., 431

 Stellate cells, 335

 Stentor, 147

 Stereometry, 417

 Sternoptyx, 748

 Stillmann, J. D. B., 695

 St Loup, R., 82

 Stokes, Sir G. G., 44

 Stolc, Ant., 452

 Stomach, muscles of, 490

 Stomata, 393

 Stomatella, 554

 Strasbürger, E., 35, 283, 409

 Straus-Dürckheim, H. E., 30

 Stream-lines, 250, 673, 736

 Strength of materials, 676, 679

 Streptoplasma, 391

 Strophomena, 567

 Studer, T., 413

 Stylonichia, 133

 Succinea, 556

 Sunflower, 494, 635, 639, 688

 Surface energy, 32, 34, 191, 207, 278, 293, 460, 599

 Survival of species, 251

 Sutures of cephalopods, 583

 Swammerdam, J., 8, 87, 380, 528, 585

 Swezy, Olive, 268

 Sylvester, J. J., 723

 Symmetry, meaning of, 209

 Synapta, egg of, 453

 Syncytium, 200

 Synhelia, 327

 Szielasko, A., 654

 Tadpole, growth of, 83, 114, 138, 153

 Tait, P. G., 35, 43, 207, 644

 Taonia, 355, 356

 Tapetum, 407

 Tapir, 741, 763

 Taylor, W. W., 277, 282, 426, 428

 Teeth, 424, 612, 632

 Telescopium, 557

 Telesius, Bernardinus, 656

 Tellina, 562

 Temperature coefficient, 109

 Terebra, 529, 557, 559

 Terebratula, 568, 574, 576

 Teredo, 414

 Terni, T., 35

 Terquem, O., 329

 Tesch, J. J., 573

 Tetractinellida, 443, 450

 Tetrahedral symmetry, 315, 396, 476

 Tetrakaidecahedron, 337

 Tetraspores, 396

 Textularia, 604

 Thamnastraea, 327

 Thayer, J. E., 672

 Thecidium, 570

 Thecosmilia, 325

 Théel, H., 451

 Thienemann, F. A. L., 653

 Thistle, capitulum of, 639

 Thoma, R., 666

 Thomson, James, 18, 259;
   J. A., 465;
   J. J., 235, 280;
   Wyville, 466

 Thurammina, 256

 Thyroid gland, 136

 Time-element, 51, 496, etc.;
   time-energy diagram, 63

 Tintinnus, 248

 Tissues, forms of, 293

 Titanotherium, 704, 762

 Tomistoma, 753

 Tomlinson, C., 259, 428

 Tornier, G., 707

 Torsion, 621, 624

 Trachelophyllum, 249

 Transformations, theory of, 562, 719

 Traube, M., 287

 Trees, growth of, 119;
   height of, 19

 Trembley, Abraham, 138, 146

 Treutlein, P., 510

 Trianea, hairs of, 234

 Triangle, properties of, 508;
   of forces, 295

 Triasters, 327

 Trichodina, 252

 Trichomastix, 267

 Triepel, H., 683, 684

 Triloculina, 595

 Triton, 554

 Trochus, 377, 557, 560;
   embryology of, 340

 Tröndle, A., 625

 Trophon, 526

 Trout, growth of, 94

 Trypanosomes, 245, 266, 269

 Tubularia, 125, 126, 146

 Turbinate shells, 534

 Turbo, 518, 555

 Turgor, 125

 Turner, Sir W., 769

 Turritella, 489, 524, 527, 555, 557, 559

 Tusks, 515, 612

 Tutton, A. E. H., 202

 Twining plants, 624

 Tyndall, John, 428

 Umbilicus of shell, 547

 Underfeeding, effect of, 106

 Undulatory membrane, 266

 Unduloid, 218, 222, 229, 246, 256

 Unio, 341

 Univalve shells, 553

 Urechinus, 664

 Vaginicola, 248

 Vallisneri, Ant., 138

 Van Iterson, G., 595

 Van Rees, R., 374

 Van’t Hoff, J. H., 1, 110, 433

 Variability, 78, 103

 Venation of wings, 385

 Verhaeren, Emile, 778

 Verworn, M., 198, 211, 467, 605

 Vesque, J., 412

 Vierordt, K., 73

 Villi, 32

 Vincent, J. H., 323

 Vines, S. H., 502

 Virchow, R., 200, 286

 Vital phenomena, 14, 417, etc.

 Vitruvius, 740

 Volkmann, A. W., 669

 Voltaire, 4, 146

 Vorticella, 237, 246, 291

 Wager, H. W. T., 259

 Walking, 30

 Wallace, A. R., 5, 432, 549

 Wallich-Martius, 77

 Warburg, O., 161

 Warburton, C., 233

 Ward, H. Marshall, 133

 Warnecke, P., 93

 Watase, S., 378

 Water, in growth, 125

 Watson, F. R., 323

 Weber, E. H., 210, 259, 669;
   E. H. and W. E., 30;
   Max, 91

 Weight, curve of, 64, etc.

 Weismann, A., 158

 Werner, A. G., 19

 Wettstein, R. von, 728

 Whale, affinities, 716;
   size, 21;
   structure, 708

 Whipple, I. L., 123

 Whitman, C. O., 157, 164, 193, 194, 199, 200

 Whitworth, W. A., 506, 512

 Wiener, A. F., 45

 Wildeman, E. de, 307, 355

 Willey, A., 425, 548, 555, 578

 Williamson, W. C., 423, 609

 Willughby, Fr., 318

 Wilson, E. B., 150, 163, 173, 195, 199, 311, 341, 342, 398, 453

 Winge, O., 433

 Winter eggs, 283

 Wissler, Clark, 79

 Wissner, J., 636

 Wöhler, Fr., 416, 420

 Wolff, J., 683;
   J. C. F., 3, 51, 155

 Wood, R. W., 590

 Woods, R. H., 666

 Woodward, H., 578;
   S. P., 554, 567

 Worthington, A. M., 235, 254

 Wreszneowski, A., 249

 Wright, Chauncey, 335

 Wright, T. Strethill, 210

 Wyman, Jeffrey, 335

 Yeast cell, 213, 242

 Yield-point, 679

 Yolk of egg, 165, 660

 Young, Thomas, 9, 36, 669, 691

 Zangger, H., 282

 Zeising, A., 636, 650

 Zeleny, C., 149

 Zeuglodon, 716

 Zeuthen, H. G., 511

 Ziehen, Ch., 92

 Zittel, K. A. von, 325, 327, 548, 584

 Zoogloea, 282

 Zschokke, F., 683

 Zsigmondy, 39

 Zuelzer, M., 165

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TRANSCRIBER’S NOTE

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Page 106. Changed “it we could believe” to “if we could believe”.

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Page 117. Both “_Q__{10}” and “_Q_^{10}” appear on the page as
originally printed.

Page 272n. Changed “_Proc. R y. Soc._ XII” to “_Proc. Roy. Soc._ XII”.

Page 368. Perhaps, the original “The area of the enlarged sector,
_p′OA′_” should read “The area of the enlarged sector, _P′OA′_”.

Page 428n. Changed “Phenonemon” to “Phenomenon”.

Page 463n. Changed “Raphidophrys” to “Raphidiophrys”.

Page 543. The Unicode character [⪌ u+2a8c greater-than above
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Page 748. Changed “Fig. 474” to “Fig. 374”.

Page 768. Changed “in the case of _Pro ohippus_” to “in the case of
_Protohippus_”. ¶ There were three footnotes on this page, but only two
footnote anchors. The second footnote, missing an anchor, said “† Cf.
_Zittel, Grundzüge d. Palaeontologie_, p. 463, 1911.”















End of Project Gutenberg's On Growth and Form, by D'Arcy Wentworth Thompson