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  THE FOUNDATIONS OF GEOMETRY.




  London: C. J. CLAY AND SONS,

  CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
  AVE MARIA LANE.

  Glasgow: 263, ARGYLE STREET.

  [Illustration]

  Leipzig: F. A. BROCKHAUS.
  New York: THE MACMILLAN COMPANY.
  Bombay: GEORGE BELL AND SONS.




  AN ESSAY

  ON THE

  FOUNDATIONS OF GEOMETRY

  BY

  BERTRAND A. W. RUSSELL. M.A.

  FELLOW OF TRINITY COLLEGE, CAMBRIDGE.

  CAMBRIDGE:
  AT THE UNIVERSITY PRESS.

  1897

  [_All Rights reserved._]


  Cambridge:

  PRINTED BY J. AND C. F. CLAY,
  AT THE UNIVERSITY PRESS.




PREFACE.


The present work is based on a dissertation submitted at the
Fellowship Examination of Trinity College, Cambridge, in the year
1895. Section B of the third chapter is in the main a reprint, with
some serious alterations, of an article in _Mind_ (New Series, No.
17). The substance of the book has been given in the form of lectures
at the Johns Hopkins University, Baltimore, and at Bryn Mawr College,
Pennsylvania.

My chief obligation is to Professor Klein. Throughout the first
chapter, I have found his "Lectures on non-Euclidean Geometry"
an invaluable guide; I have accepted from him the division of
Metageometry into three periods, and have found my historical work
much lightened by his references to previous writers. In Logic, I
have learnt most from Mr Bradley, and next to him, from Sigwart and
Dr Bosanquet. On several important points, I have derived useful
suggestions from Professor James's "Principles of Psychology."

My thanks are due to Mr G. F. Stout and Mr A. N. Whitehead for
kindly reading my proofs, and helping me by many useful criticisms.
To Mr Whitehead I owe, also, the inestimable assistance of constant
criticism and suggestion throughout the course of construction,
especially as regards the philosophical importance of projective
Geometry.

  HASLEMERE.

  _May, 1897._




  TO

  JOHN McTAGGART ELLIS McTAGGART

  TO WHOSE DISCOURSE AND FRIENDSHIP IS OWING
  THE EXISTENCE OF THIS BOOK.




TABLE OF CONTENTS.


  INTRODUCTION.

  OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC,
  PSYCHOLOGY AND MATHEMATICS.
                                                                    PAGE

  1.  The problem first received a modern form through Kant, who
  connected the _à priori_ with the subjective                         1

  2.  A mental state is subjective, for Psychology, when its immediate
  cause does not lie in the outer world                                2

  3.  A piece of knowledge is _à priori_, for Epistemology, when
  without it knowledge would be impossible                             2

  4.  The subjective and the _à priori_ belong respectively to
  Psychology and to Epistemology. The latter alone will be
  investigated in this essay                                           3

  5.  My test of the _à priori_ will be purely logical: what knowledge
  is necessary for experience?                                         3

  6.  But since the necessary is hypothetical, we must include, in
  the _à priori_, the ground of necessity                              4

  7.  This may be the essential postulate of our science, or the
  element, in the subject-matter, which is necessary to
  experience;                                                          4

  8.  Which, however, are both at bottom the same ground               5

  9.  Forecast of the work                                             5


  CHAPTER I.

  A SHORT HISTORY OF METAGEOMETRY.

  10.  Metageometry began by rejecting the axiom of parallels          7

  11.  Its history may be divided into three periods: the synthetic,
  the metrical and the projective                                      7

  12.  The first period was inaugurated by Gauss,                     10

  13.  Whose suggestions were developed independently by
  Lobatchewsky                                                        10

  14.  And Bolyai                                                     11

  15.  The purpose of all three was to show that the axiom of parallels
  could not be deduced from the others, since its denial did
  not lead to contradictions                                          12

  16.  The second period had a more philosophical aim, and was
  inspired chiefly by Gauss and Herbart                               13

  17.  The first work of this period, that of Riemann, invented two
  new conceptions:                                                    14

  18.  The first, that of a manifold, is a class-conception, containing
  space as a species,                                                 14

  19.  And defined as such that its determinations form a collection
  of magnitudes                                                       15

  20.  The second, the measure of curvature of a manifold, grew out
  of curvature in curves and surfaces                                 16

  21.  By means of Gauss's analytical formula for the curvature of
  surfaces,                                                           19

  22.  Which enables us to define a _constant_ measure of curvature
  of a three-dimensional space without reference to a fourth
  dimension                                                           20

  23.  The main result of Riemann's mathematical work was to
  show that, if magnitudes are independent of place, the
  measure of curvature of space must be constant                      21

  24.  Helmholtz, who was more of a philosopher than a mathematician, 22

  25.  Gave a new but incorrect formulation of the essential axioms,  23

  26.  And deduced the quadratic formula for the infinitesimal arc,
  which Riemann had assumed                                           24

  27.  Beltrami gave Lobatchewsky's planimetry a Euclidean
  interpretation,                                                     25

  28.  Which is analogous to Cayley's theory of distance;             26

  29.  And dealt with _n_-dimensional spaces of constant negative
  curvature                                                           27

  30.  The third period abandons the metrical methods of the second,
  and extrudes the notion of spatial quantity                         27

  31.  Cayley reduced metrical properties to projective properties,
  relative to a certain conic or quadric, the Absolute;               28

  32.  And Klein showed that the Euclidean or non-Euclidean systems
  result, according to the nature of the Absolute;                    29

  33.  Hence Euclidean _space_ appeared to give rise to all the kinds
  of Geometry, and the question, which is true, appeared
  reduced to one of convention                                        30

  34.  But this view is due to a confusion as to the nature of the
  coordinates employed                                                30

  35.  Projective coordinates have been regarded as dependent on
  distance, and thus really metrical                                  31

  36.  But this is not the case, since anharmonic ratio can be
  projectively defined                                                32

  37.  Projective coordinates, being purely descriptive, can give no
  information as to metrical properties, and the reduction of
  metrical to projective properties is purely technical               33

  38.  The true connection of Cayley's measure of distance with
  non-Euclidean Geometry is that suggested by Beltrami's
  Saggio, and worked out by Sir R. Ball,                              36

  39.  Which provides a Euclidean equivalent for every non-Euclidean
  proposition, and so removes the possibility of contradictions
  in Metageometry                                                     38

  40.  Klein's elliptic Geometry has not been proved to have a
  corresponding variety of space                                      39

  41.  The geometrical use of imaginaries, of which Cayley demanded
  a philosophical discussion,                                         41

  42.  Has a merely technical validity,                               42

  43.  And is capable of giving geometrical results only when it
  begins and ends with real points and figures                        45

  44.  We have now seen that projective Geometry is logically prior
  to metrical Geometry, but cannot supersede it                       46

  45.  Sophus Lie has applied projective methods to Helmholtz's
  formulation of the axioms, and has shown the axiom of
  Monodromy to be superfluous                                         46

  46.  Metageometry has gradually grown independent of philosophy,
  but has grown continually more interesting to philosophy            50

  47.  Metrical Geometry has three indispensable axioms,              50

  48.  Which we shall find to be not results, but conditions, of
  measurement,                                                        51

  49.  And which are nearly equivalent to the three axioms of
  projective Geometry                                                 52

  50.  Both sets of axioms are necessitated, not by facts, but by
  logic                                                               52


  CHAPTER II.

  CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL
  THEORIES OF GEOMETRY.

  51.  A criticism of representative modern theories need not begin
  before Kant                                                         54

  52.  Kant's doctrine must be taken, in an argument about Geometry,
  on its purely logical side                                          55

  53.  Kant contends that since Geometry is apodeictic, space must
  be _à priori_ and subjective, while since space is _à priori_
  and subjective, Geometry must be apodeictic                         55

  54.  Metageometry has upset the first line of argument, not the
  second                                                              56

  55.  The second may be attacked by criticizing either the distinction
  of synthetic and analytic judgments, or the first two arguments
  of the metaphysical deduction of space                              57

  56.  Modern Logic regards every judgment as both synthetic and
  analytic,                                                           57

  57.  But leaves the _à priori_, as that which is presupposed in the
  possibility of experience                                           59

  58.  Kant's first two arguments as to space suffice to prove _some_
  form of externality, but not necessarily Euclidean space, a
  necessary condition of experience                                   60

  59.  Among the successors of Kant, Herbart alone advanced the
  theory of Geometry, by influencing Riemann                          62

  60.  Riemann regarded space as a particular kind of manifold, i.e.
  wholly quantitatively                                               63

  61.  He therefore unduly neglected the qualitative adjectives of
  space                                                               64

  62.  His philosophy rests on a vicious disjunction                  65

  63.  His definition of a manifold is obscure,                       66

  64.  And his definition of measurement applies only to space        67

  65.  Though mathematically invaluable, his view of space as a
  manifold is philosophically misleading                              69

  66.  Helmholtz attacked Kant both on the mathematical and on
  the psychological side;                                             70

  67.  But his criterion of apriority is changeable and often
  invalid;                                                            71

  68.  His proof that non-Euclidean spaces are imaginable is
  inconclusive;                                                       72

  69.  And his assertion of the dependence of measurement on rigid
  bodies, which may be taken in three senses,                         74

  70.  Is wholly false if it means that the axiom of Congruence
  actually asserts the existence of rigid bodies,                     75

  71.  Is untrue if it means that the necessary reference of geometrical
  propositions to matter renders pure Geometry empirical,             76

  72.  And is inadequate to his conclusion if it means, what is
  true, that _actual_ measurement involves approximately rigid
  bodies                                                              78

  73.  Geometry deals with an abstract matter, whose physical
  properties are disregarded; and Physics must presuppose
  Geometry                                                            80

  74.  Erdmann accepted the conclusions of Riemann and Helmholtz,     81

  75.  And regarded the axioms as necessarily successive steps in
  classifying space as a species of manifold                          82

  76.  His deduction involves four fallacious assumptions, namely:    82

  77.  That conceptions must be abstracted from a series of
  instances;                                                          83

  78.  That all definition is classification;                         83

  79.  That conceptions of magnitude can be applied to space as
  a whole;                                                            84

  80.  And that if conceptions of magnitude could be so applied, all
  the adjectives of space would result from their application         86

  81.  Erdmann regards Geometry alone as incapable of deciding on
  the truth of the axiom of Congruence,                               86

  82.  Which he affirms to be empirically proved by Mechanics.        88

  83.  The variety and inadequacy of Erdmann's tests of apriority     89

  84.  Invalidate his final conclusions on the theory of Geometry     90

  85.  Lotze has discussed two questions in the theory of Geometry:   93

  86.  (1) He regards the possibility of non-Euclidean spaces as
  suggested by the subjectivity of space,                             93

  87.  And rejects it owing to a mathematical misunderstanding,       96

  88.  Having missed the most important sense of their possibility,   96

  89.  Which is that they fulfil the logical conditions to which any
  form of externality must conform                                    97

  90.  (2) He attacks the mathematical procedure of Metageometry      98

  91.  The attack begins with a question-begging definition of
  parallels                                                           99

  92.  Lotze maintains that all apparent departures from Euclid
  could be physically explained, a view which really makes
  Euclid empirical                                                    99

  93.  His criticism of Helmholtz's analogies rests wholly on
    mathematical mistakes                                            101

  94.  His proof that space must have three dimensions rests on
  neglect of different orders of infinity                            104

  95.  He attacks non-Euclidean spaces on the mistaken ground
  that they are not homogeneous                                      107

  96.  Lotze's objections fall under four heads                      108

  97.  Two other semi-philosophical objections may be urged,         109

  98.  One of which, the absence of similarity, has been made the
  basis of attack by Delbœuf,                                        110

  99.  But does not form a valid ground of objection                 111

  100.  Recent French speculation on the foundations of Geometry
  has suggested few new views                                        112

  101.  All homogeneous spaces are _à priori_ possible, and the
  decision between them is empirical                                 114


  CHAPTER III.

  SECTION A. THE AXIOMS OF PROJECTIVE GEOMETRY.

  102.  Projective Geometry does not deal with magnitude, and
  applies to all spaces alike                                        117

  103.  It will be found wholly _à priori_                           117

  104.  Its axioms have not yet been formulated philosophically      118

  105.  Coordinates, in projective Geometry, are not spatial
  magnitudes, but convenient names for points                        118

  106.  The possibility of distinguishing various points is an axiom 119

  107.  The qualitative relations between points, dealt with by
  projective Geometry, are presupposed by the quantitative
  treatment                                                          119

  108.  The only qualitative relation between two points is the
  straight line, and all straight lines are qualitatively
  similar                                                            120

  109.  Hence follows, by extension, the principle of projective
  transformation                                                     121

  110.  By which figures qualitatively indistinguishable from a
  given figure are obtained                                          122

  111.  Anharmonic ratio may and must be descriptively defined       122

  112.  The quadrilateral construction is essential to the
  projective definition of points,                                   123

  113.  And can be projectively defined,                             124

  114.  By the general principle of projective transformation        126

  115.  The principle of duality is the mathematical form of a
  philosophical circle,                                              127

  116.  Which is an inevitable consequence of the relativity of
  space, and makes any definition of the point contradictory         128

  117.  We define the point as that which is spatial, but contains
  no space, whence other definitions follow                          128

  118.  What is meant by qualitative equivalence in Geometry?        129

  119.  Two pairs of points on one straight line, or two pairs of
  straight lines through one point, are qualitatively equivalent     129

  120.  This explains why _four_ collinear points are needed, to give
  an intrinsic relation by which the fourth can be descriptively
  defined when the first three are given                             130

  121.  Any two projectively related figures are qualitatively
  equivalent, i.e. differ in no non-quantitative conceptual
  property                                                           131

  122.  Three axioms are used by projective Geometry,                132

  123.  And are required for qualitative spatial comparison,         132

  124.  Which involves the homogeneity, relativity and passivity
  of space                                                           133

  125.  The conception of a form of externality,                     134

  126.  Being a creature of the intellect, can be dealt with by
  pure mathematics                                                   134

  127.  The resulting doctrine of extension will be, for the moment,
  hypothetical                                                       135

  128.  But is rendered assertorical by the necessity, for
  experience, of some form of externality                            136

  129.  Any such form must be relational                             136

  130.  And homogeneous                                              137

  131.  And the relations constituting it must appear infinitely
  divisible                                                          137

  132.  It must have a finite integral number of dimensions,         139

  133.  Owing to its passivity and homogeneity                       140

  134.  And to the systematic unity of the world                     140

  135.  A one-dimensional form alone would not suffice for
  experience                                                         141

  136.  Since its elements would be immovably fixed in a series      142

  137.  Two positions have a relation independent of other
  positions,                                                         143

  138.  Since positions are wholly defined by mutually independent
  relations                                                          143

  139.  Hence projective Geometry is wholly _à priori_,              146

  140.  Though metrical Geometry contains an empirical element       146


  SECTION B. THE AXIOMS OF METRICAL GEOMETRY.

  141.  Metrical Geometry is distinct from projective, but has the
  same fundamental postulate                                         147

  142.  It introduces the new idea of motion, and has three
  _à priori_ axioms                                                  148


  I. _The Axiom of Free Mobility._

  143.  Measurement requires a criterion of spatial equality         149

  144.  Which is given by superposition, and involves the axiom
  of Free Mobility                                                   150

  145.  The denial of this axiom involves an action of empty
  space on things                                                    151

  146.  There is a mathematically possible alternative to the axiom, 152

  147.  Which, however, is logically and philosophically untenable   153

  148.  Though Free Mobility is _à priori_, actual measurement is
  empirical                                                          154

  149.  Some objections remain to be answered, concerning--          154

  150.  (1) The comparison of volumes and of Kant's symmetrical
  objects                                                            154

  151.  (2) The measurement of time, where congruence is impossible  156

  152.  (3) The immediate perception of spatial magnitude; and       157

  153.  (4) The Geometry of non-congruent surfaces                   158

  154.  Free Mobility includes Helmholtz's Monodromy                 159

  155.  Free Mobility involves the relativity of space               159

  156.  From which, reciprocally, it can be deduced                  160

  157.  Our axiom is therefore _à priori_ in a double sense          160


  II. _The Axiom of Dimensions._

  158.  Space must have a finite integral number of dimensions       161

  159.  But the restriction to three is empirical                    162

  160.  The general axiom follows from the relativity of position    162

  161.  The limitation to three dimensions, unlike most empirical
  knowledge, is accurate and certain                                 163


  III. _The Axiom of Distance._

  162.  The axiom of distance corresponds, here, to that of the
  straight line in projective Geometry                               164

  163.  The possibility of spatial measurement involves a magnitude
  uniquely determined by two points,                                 164

  164.  Since two points must have some relation, and the passivity
  of space proves this to be independent of external reference       165

  165.  There can be only one such relation                          166

  166.  This must be measured by a curve joining the two points,     166

  167.  And the curve must be uniquely determined by the two
  points                                                             167

  168.  Spherical Geometry contains an exception to this axiom,      168

  169.  Which, however, is not quite equivalent to Euclid's          168

  170.  The exception is due to the fact that two points, in
  spherical space, may have an external relation unaltered by
  motion,                                                            169

  171.  Which, however, being a relation of linear magnitude,
  presupposes the possibility of linear magnitude                    170

  172.  A relation between two points must be a line joining them    170

  173.  Conversely, the existence of a unique line between two
  points can be deduced from the nature of a form of
  externality,                                                       171

  174.  And necessarily leads to distance, when quantity is applied
  to it                                                              172

  175.  Hence the axiom of distance, also, is _à priori_ in a double
  sense                                                              172

  176.  No metrical coordinate system can be set up without the
  straight line                                                      174

  177.  No axioms besides the above three are necessary to metrical
  Geometry                                                           175

  178.  But these three are necessary to the direct measurement
  of any continuum                                                   176

  179.  Two philosophical questions remain for a final chapter       177


  CHAPTER IV.

  PHILOSOPHICAL CONSEQUENCES.

  180.  What is the relation to experience of a form of externality
  in general?                                                        178

  181.  This form is the class-conception, containing every possible
  intuition of externality; and some such intuition is
  necessary to experience                                            178

  182.  What relation does this view bear to Kant's?                 179

  183.  It is less psychological, since it does not discuss whether
  space is given in sensation,                                       180

  184.  And maintains that not only space, but any form of
  externality which renders experience possible, must be given in
  sense-perception                                                   181

  185.  Externality should mean, not externality to the Self, but
  the mutual externality of presented things                         181

  186.  Would this be unknowable without a given form of
  externality?                                                       182

  187.  Bradley has proved that space and time preclude the
  existence of mere particulars,                                     182

  188.  And that knowledge requires the _This_ to be neither simple
  nor self-subsistent                                                183

  189.  To prove that experience requires a form of externality, I
  assume that all knowledge requires the recognition of identity in
  difference                                                         184

  190.  Such recognition involves time                               184

  191.  And some other form giving simultaneous diversity            185

  192.  The above argument has not deduced sense-perception from
  the categories, but has shown the former, unless it contains
  a certain element, to be unintelligible to the latter              186

  193.  How to account for the realization of this element, is a
  question for metaphysics                                           187

  194.  What are we to do with the contradictions in space?          188

  195.  Three contradictions will be discussed in what follows       188

  196.  (1) The antinomy of the Point proves the relativity of
  space,                                                             189

  197.  And shows that Geometry must have some reference to
  matter,                                                            190

  198.  By which means it is made to refer to spatial order, not
  to empty space                                                     191

  199.  The causal properties of matter are irrelevant to Geometry,
  which must regard it as composed of unextended atoms,
  by which points are replaced                                       191

  200.  (2) The circle in defining straight lines and planes is
  overcome by the same reference to matter                           192

  201.  (3) The antinomy that space is relational and yet more
  than relational,                                                   193

  202.  Seems to depend on the confusion of empty space with
  spatial order                                                      193

  203.  Kant regarded empty space as the subject-matter of Geometry, 194

  204.  But the arguments of the Aesthetic are inconclusive on this
  point,                                                             195

  205.  And are upset by the mathematical antinomies, which prove
  that spatial order should be the subject-matter of Geometry        196

  206.  The apparent thinghood of space is a psychological illusion,
  due to the fact that spatial relations are immediately given       196

  207.  The apparent divisibility of spatial relations is either an
  illusion, arising out of empty space, or the expression of the
 possibility of quantitatively different spatial relations           197

  208.  Externality is not a relation, but an aspect of relations.
  Spatial order, owing to its reference to matter, is a real
  relation                                                           198

  209.  Conclusion                                                   199




INTRODUCTION.

OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC, PSYCHOLOGY AND
MATHEMATICS.


=1.= Geometry, throughout the 17th and 18th centuries, remained, in the
war against empiricism, an impregnable fortress of the idealists.
Those who held--as was generally held on the Continent--that certain
knowledge, independent of experience, was possible about the real
world, had only to point to Geometry: none but a madman, they said,
would throw doubt on its validity, and none but a fool would deny its
objective reference. The English Empiricists, in this matter, had,
therefore, a somewhat difficult task; either they had to ignore the
problem, or if, like Hume and Mill, they ventured on the assault,
they were driven into the apparently paradoxical assertion that
Geometry, at bottom, had no certainty of a different _kind_ from that
of Mechanics--only the perpetual presence of spatial impressions,
they said, made our experience of the truth of the axioms so wide as
to seem absolute certainty.

Here, however, as in many other instances, merciless logic drove
these philosophers, whether they would or no, into glaring opposition
to the common sense of their day. It was only through Kant, the
creator of modern Epistemology, that the geometrical problem
received a modern form. He reduced the question to the following
hypotheticals: If Geometry has apodeictic certainty, its matter,
_i.e._ space, must be _à priori_, and as such must be purely
subjective; and conversely, if space is purely subjective, Geometry
must have apodeictic certainty. The latter hypothetical has more
weight with Kant, indeed it is ineradicably bound up with his whole
Epistemology; nevertheless it has, I think, much less force than
the former. Let us accept, however, for the moment, the Kantian
formulation, and endeavour to give precision to the terms _à priori_
and _subjective_.

=2.= One of the great difficulties, throughout this controversy,
is the extremely variable use to which these words, as well as
the word _empirical_, are put by different authors. To Kant, who
was nothing of a psychologist, _à priori_ and _subjective_ were
almost interchangeable terms[1]; in modern usage there is, on the
whole, a tendency to confine the word _subjective_ to Psychology,
leaving _à priori_ to do duty for Epistemology. If we accept this
differentiation, we may set up, corresponding to the problems of
these two sciences, the following provisional definitions: _à priori_
applies to any piece of knowledge which, though perhaps elicited by
experience, is _logically_ presupposed in experience: _subjective_
applies to any mental state whose immediate cause lies, not in the
external world, but within the limits of the subject. The latter
definition, of course, is framed exclusively for Psychology: from the
point of view of physical Science all mental states are subjective.
But for a Science whose matter, strictly speaking, is _only_ mental
states, we require, if we are to use the word to any purpose,
some differentia among mental states, as a mark of a more special
subjectivity on the part of those to which this term is applied.

Now the only mental states whose immediate causes lie in the external
world are _sensations_. A pure sensation is, of course, an impossible
abstraction--we are never wholly passive under the action of an
external stimulus--but for the purposes of Psychology the abstraction
is a useful one. Whatever, then, is not sensation, we shall, in
Psychology, call subjective. It is in sensation alone that we are
directly affected by the external world, and only here does it give
us direct information about itself.

=3.= Let us now consider the epistemological question, as to the sort
of knowledge which can be called _à priori_. Here we have nothing to
do--in the first instance, at any rate--with the cause or genesis of
a piece of knowledge; we accept knowledge as a datum to be analysed
and classified. Such analysis will reveal a formal and a material
element in knowledge. The formal element will consist of postulates
which are required to make knowledge possible at all, and of all
that can be deduced from these postulates; the material element, on
the other hand, will consist of all that comes to fill in the form
given by the formal postulates--all that is contingent or dependent
on experience, all that might have been otherwise without rendering
knowledge impossible. We shall then call the formal element _à
priori_, the material element empirical.

=4.= Now what is the connection between the subjective and the _à
priori_? It is a connection, obviously--if it exists at all--from the
outside, _i.e._ not deducible directly from the nature of either,
but provable--if it can be proved--only by a general view of the
conditions of both. The question, what knowledge is _à priori_, must,
on the above definition, depend on a logical analysis of knowledge,
by which the conditions of possible experience may be revealed; but
the question, what elements of a cognitive state are subjective, is
to be investigated by pure Psychology, which has to determine what,
in our perceptions, belongs to sensation, and what is the work of
thought or of association. Since, then, these two questions belong
to different sciences, and can be settled independently, will it not
be wise to conduct the two investigations separately? To decree that
the _à priori_ shall always be subjective, seems dangerous, when we
reflect that such a view places our results, as to the _à priori_, at
the mercy of empirical psychology. How serious this danger is, the
controversy as to Kant's pure intuition sufficiently shows.

=5.= I shall, therefore, throughout the present Essay, use the
word _à priori_ without any psychological implication. My test of
apriority will be purely logical: Would experience be impossible, if
a certain axiom or postulate were denied? Or, in a more restricted
sense, which gives apriority only within a particular science: Would
experience as to the subject-matter of that science be impossible,
without a certain axiom or postulate? My results also, therefore,
will be purely logical. If Psychology declares that some things,
which I have declared _à priori_, are not subjective, then, failing
an error of detail in my proofs, the connection of the _à priori_ and
the subjective, so far as those things are concerned, must be given
up. There will be no discussion, accordingly, throughout this Essay,
of the relation of the _à priori_ to the subjective--a relation which
cannot determine what pieces of knowledge are _à priori_, but rather
depends on that determination, and belongs, in any case, rather to
Metaphysics than to Epistemology.

=6.= As I have ventured to use the word _à priori_ in a slightly
unconventional sense, I will give a few elucidatory remarks of a
general nature.

The _à priori_, since Kant at any rate, has generally stood for the
necessary or apodeictic element in knowledge. But modern logic has
shown that necessary propositions are always, in one aspect at least,
hypothetical. There may be, and usually is, an implication that the
connection, of which necessity is predicated, has some existence,
but still, necessity always points beyond itself to a _ground_ of
necessity, and asserts this ground rather than the actual connection.
As Bradley points out, "arsenic poisons" remains true, even if it
is poisoning no one. If, therefore, the _à priori_ in knowledge be
primarily the necessary, it must be the necessary on some hypothesis,
and the _ground_ of necessity must be included as _à priori_. But
the ground of necessity is, so far as the necessary connection in
question can show, a mere fact, a merely categorical judgment. Hence
necessity alone is an insufficient criterion of apriority.

To supplement this criterion, we must supply the hypothesis or
ground, on which alone the necessity holds, and this ground will
vary from one science to another, and even, with the progress of
knowledge, in the same science at different times. For as knowledge
becomes more developed and articulate, more and more necessary
connections are perceived, and the merely categorical truths, though
they remain the foundation of apodeictic judgments, diminish in
relative number. Nevertheless, in a fairly advanced science such as
Geometry, we can, I think, pretty completely supply the appropriate
ground, and establish, within the limits of the isolated science, the
distinction between the necessary and the merely assertorical.

=7.= There are two grounds, I think, on which necessity may be sought
within any science. These may be (very roughly) distinguished as
the ground which Kant seeks in the _Prolegomena_, and that which
he seeks in the _Pure Reason_. We may start from the existence of
our science as a fact, and analyse the reasoning employed with a
view to discovering the fundamental postulate on which its logical
possibility depends; in this case, the postulate, and all which
follows from it alone, will be _à priori_. Or we may accept the
existence of the subject-matter of our science as our basis of fact,
and deduce dogmatically whatever principles we can from the essential
nature of this subject-matter. In this latter case, however, it is
not the whole empirical nature of the subject-matter, as revealed by
the subsequent researches of our science, which forms our ground; for
if it were, the whole science would, of course, be _à priori_. Rather
it is that element, in the subject-matter, which makes _possible_
the branch of experience dealt with by the science in question[2].
The importance of this distinction will appear more clearly as we
proceed[3].

=8.= These two grounds of necessity, in ultimate analysis, fall
together. The _methods_ of investigation in the two cases differ
widely, but the _results_ cannot differ. For in the first case, by
analysis of the science, we discover the postulate on which alone
its reasonings are possible. Now if reasoning in the science is
impossible without some postulate, this postulate must be essential
to experience of the subject-matter of the science, and thus we
get the second ground. Nevertheless, the two methods are useful as
supplementing one another, and the first, as starting from the actual
science, is the safest and easiest method of investigation, though
the second seems the more convincing for exposition.

=9.= The course of my argument, therefore, will be as follows: In
the first chapter, as a preliminary to the logical analysis of
Geometry, I shall give a brief history of the rise and development
of non-Euclidean systems. The second chapter will prepare the ground
for a constructive theory of Geometry, by a criticism of some
previous philosophical views; in this chapter, I shall endeavour
to exhibit such views as partly true, partly false, and so to
establish, by preliminary polemics, the truth of such parts of my own
theory as are to be found in former writers. A large part of this
theory, however, cannot be so introduced, since the whole field of
projective Geometry, so far as I am aware, has been hitherto unknown
to philosophers. Passing, in the third chapter, from criticism to
construction, I shall deal first with projective Geometry. This, I
shall maintain, is necessarily true of any form of externality, and
is, since some such form is necessary to experience, completely _à
priori_. In metrical Geometry, however, which I shall next consider,
the axioms will fall into two classes: (1) Those common to Euclidean
and non-Euclidean spaces. These will be found, on the one hand,
essential to the possibility of measurement in any continuum, and
on the other hand, necessary properties of any form of externality
with more than one dimension. They will, therefore, be declared
_à priori_. (2) Those axioms which distinguish Euclidean from
non-Euclidean spaces. These will be regarded as wholly empirical.
The axiom that the number of dimensions is three, however, though
empirical, will be declared, since small errors are here impossible,
exactly and certainly true of our actual world; while the two
remaining axioms, which determine the value of the space-constant,
will be regarded as only approximately known, and certain only
within the errors of observation[4]. The fourth chapter, finally,
will endeavour to prove, what was assumed in Chapter III., that some
form of externality is necessary to experience, and will conclude by
exhibiting the logical impossibility, if knowledge of such a form is
to be freed from contradictions, of wholly abstracting this knowledge
from all reference to the matter contained in the form.

I shall hope to have touched, with this discussion, on all the main
points relating to the Foundations of Geometry.

FOOTNOTES:

[1] Cf. Erdmann, Axiome der Geometrie, p. 111: "Für Kant sind
Apriorität und ausschliessliche Subjectivität allerdings
Wechselbegriffe."

[2] I use "experience" here in the widest possible sense, the sense
in which the word is used by Bradley.

[3] Where the branch of experience in question is essential to all
experience, the resulting apriority may be regarded as absolute;
where it is necessary only to some special science, as relative to
that science.

[4] I have given no account of these empirical proofs, as they seem
to be constituted by the whole body of physical science. Everything
in physical science, from the law of gravitation to the building of
bridges, from the spectroscope to the art of navigation, would be
profoundly modified by any considerable inaccuracy in the hypothesis
that our actual space is Euclidean. The observed truth of physical
science, therefore, constitutes overwhelming empirical evidence that
this hypothesis is very approximately correct, even if not rigidly
true.




CHAPTER I.

A SHORT HISTORY OF METAGEOMETRY.


=10.= When a long established system is attacked, it usually happens
that the attack begins only at a single point, where the weakness of
the established doctrine is peculiarly evident. But criticism, when
once invited, is apt to extend much further than the most daring, at
first, would have wished.

      "First cut the liquefaction, what comes last,
      But Fichte's clever cut at God himself?"

So it has been with Geometry. The liquefaction of Euclidean orthodoxy
is the axiom of parallels, and it was by the refusal to admit this
axiom without proof that Metageometry began. The first effort in
this direction, that of Legendre[5], was inspired by the hope of
deducing this axiom from the others--a hope which, as we now know,
was doomed to inevitable failure. Parallels are defined by Legendre
as lines in the same plane, such that, if a third line cut them, it
makes the sum of the interior and opposite angles equal to two right
angles. He proves without difficulty that such lines would not meet,
but is unable to prove that non-parallel lines in a plane must meet.
Similarly he can prove that the sum of the angles of a triangle
cannot exceed two right angles, and that if any one triangle has a
sum equal to two right angles, all triangles have the same sum; but
he is unable to prove the existence of this one triangle.

=11.= Thus Legendre's attempt broke down; but mere failure could
prove nothing. A bolder method, suggested by Gauss, was carried out
by Lobatchewsky and Bolyai[6]. If the axiom of parallels is logically
deducible from the others, we shall, by denying it and maintaining
the rest, be led to contradictions. These three mathematicians,
accordingly, attacked the problem indirectly: they denied the axiom
of parallels, and yet obtained a logically consistent Geometry. They
inferred that the axiom was logically independent of the others, and
essential to the Euclidean system. Their works, being all inspired by
this motive, may be distinguished as forming the first period in the
development of Metageometry.

The second period, inaugurated by Riemann, had a much deeper import:
it was largely philosophical in its aims and constructive in its
methods. It aimed at no less than a logical analysis of all the
essential axioms of Geometry, and regarded space as a particular
case of the more general conception of a _manifold_. Taking its
stand on the methods of analytical metrical Geometry, it established
two non-Euclidean systems, the first that of Lobatchewsky, the
second--in which the axiom of the straight line, in Euclid's form,
was also denied--a new variety, by analogy called spherical. The
leading conception in this period is the _measure of curvature_, a
term invented by Gauss, but applied by him only to surfaces. Gauss
had shown that free mobility on surfaces was only possible when the
measure of curvature was constant; Riemann and Helmholtz extended
this proposition to _n_ dimensions, and made it the fundamental
property of space.

In the third period, which begins with Cayley, the philosophical
motive, which had moved the first pioneers, is less apparent, and
is replaced by a more technical and mathematical spirit. This
period is chiefly distinguished from the second, in a mathematical
point of view, by its method, which is projective instead of
metrical. The leading mathematical conception here is the Absolute
(_Grundgebild_), a figure by relation to which all metrical
properties become projective. Cayley's work, which was very brief,
and attracted little attention, has been perfected and elaborated by
F. Klein, and through him has found general acceptance. Klein has
added to the two kinds of non-Euclidean Geometry already known, a
third, which he calls elliptic; this third kind closely resembles
Helmholtz's spherical Geometry, but is distinguished by the important
difference that, in it, two straight lines meet in only one point[7].
The distinctive mark of the spaces represented by both is that,
like the surface of a sphere, they are finite but unbounded. The
reduction of metrical to projective properties, as will be proved
hereafter, has only a technical importance; at the same time,
projective Geometry is able to deal directly with those purely
descriptive or qualitative properties of space which are common to
Euclid and Metageometry alike. The third period has, therefore, great
philosophical importance, while its method has, mathematically, much
greater beauty and unity than that of the second; it is able to
treat all kinds of space at once, so that every symbolic proposition
is, according to the meaning given to the symbols, a proposition
in whichever Geometry we choose. This has the advantage of proving
that further research cannot lead to contradictions in non-Euclidean
systems, unless it at the same moment reveals contradictions in
Euclid. These systems, therefore, are logically as sound as that of
Euclid himself.

After this brief sketch of the characteristics of the three periods,
I will proceed to a more detailed account. It will be my aim to
avoid, as far as possible, all technical mathematics, and bring
into relief only those fundamental points in the mathematical
development, which seem of logical or philosophical importance.


First Period.

=12.= The originator of the whole system, _Gauss_, does not appear,
as regards strictly non-Euclidean Geometry, in any of his hitherto
published papers, to have given more than results; his proofs remain
unknown to us. Nevertheless he was the first to investigate the
consequences of denying the axiom of parallels[8], and in his letters
he communicated these consequences to some of his friends, among whom
was Wolfgang Bolyai. The first mention of the subject in his letters
occurs when he was only 18; four years later, in 1799, writing to
W. Bolyai, he enunciates the important theorem that, in hyperbolic
Geometry, there is a maximum to the area of a triangle. From later
writings it appears that he had worked out a system nearly, if not
quite, as complete as those of Lobatchewsky and Bolyai[9].

It is important to remember, however, that Gauss's work on curvature,
which _was_ published, laid the foundation for the whole method of
the second period, and was undertaken, according to Riemann and
Helmholtz[10], with a view to an (unpublished) investigation of the
foundations of Geometry. His work in this direction will, owing to
its method, be better treated of under the second period, but it is
interesting to observe that he stood, like many pioneers, at the head
of two tendencies which afterwards diverged.

=13.= _Lobatchewsky_, a professor in the University of Kasan, first
published his results, in their native Russian, in the proceedings
of that learned body for the years 1829-1830. Owing to this double
obscurity of language and place, they attracted little attention,
until he translated them into French[11] and German[12]: even then,
they do not appear to have obtained the notice they deserved, until,
in 1868, Beltrami unearthed the article in Crelle, and made it the
theme of a brilliant interpretation.

In the introduction to his little German book, Lobatchewsky laments
the slight interest shown in his writings by his compatriots, and the
inattention of mathematicians, since Legendre's abortive attempt, to
the difficulties in the theory of parallels. The body of the work
begins with the enunciation of several important propositions which
hold good in the system proposed as well as in Euclid: of these, some
are in any case independent of the axiom of parallels, while others
are rendered so by substituting, for the word "parallel," the phrase
"not intersecting, however far produced." Then follows a definition,
intentionally framed so as to contradict Euclid's: With respect to
a given straight line, all others in the same plane may be divided
into two classes, those which cut the given straight line, and those
which do not cut it; a line which is the limit between the two
classes is called _parallel_ to the given straight line. It follows
that, from any external point, two parallels can be drawn, one in
each direction. From this starting-point, by the Euclidean synthetic
method, a series of propositions are deduced; the most important of
these is, that in a triangle the sum of the angles is always less
than, or always equal to two right angles, while in the latter case
the whole system becomes orthodox. A certain analogy with spherical
Geometry--whose meaning and extent will appear later--is also proved,
consisting roughly in the substitution of hyperbolic for circular
functions.

=14.= Very similar is the system of _Johann Bolyai_, so similar,
indeed, as to make the independence of the two works, though a
well-authenticated fact, seem all but incredible. Johann Bolyai
first published his results in 1832, in an appendix to a work by his
father Wolfgang, entitled; "Appendix, scientiam spatii absolute veram
exhibens: a veritate aut falsitate Axiomatis XI. Euclidei (a priori
haud unquam decidenda) independentem; adjecta ad casum falsitatis,
quadratura circuli geometrica." Gauss, whose bosom friend he became
at college and remained through life, was, as we have seen, the
inspirer of Wolfgang Bolyai, and used to say that the latter was the
only man who appreciated his philosophical speculations on the axioms
of Geometry; nevertheless, Wolfgang appears to have left to his son
Johann the detailed working out of the hyperbolic system. The works
of both the Bolyai are very rare, and their method and results are
known to me only through the renderings of Frischauf and Halsted[13].
Both as to method and as to results, the system is very similar to
Lobatchewsky's, so that neither need detain us here. Only the initial
postulates, which are more explicit than Lobatchewsky's, demand a
brief attention. Frischauf's introduction, which has a philosophical
and Newtonian air, begins by setting forth that Geometry deals with
absolute (empty) space, obtained by abstracting from the bodies in
it, that two figures are called congruent when they differ only in
position, and that the axiom of Congruence is indispensable in all
determination of spatial magnitudes. Congruence was to refer to
geometrical bodies, with none of the properties of ordinary bodies
except impenetrability (Erdmann, Axiome der Geometrie, p. 26). A
straight line is defined as determined by two of its points[14],
and a plane as determined by three. These premisses, with a slight
exception as to the straight line, we shall hereafter find essential
to every Geometry. I have drawn attention to them, as it is often
supposed that non-Euclideans deny the axiom of Congruence, which,
here and elsewhere, is never the case. The stress laid on this axiom
by Bolyai is probably due to the influence of Gauss, whose work on
the curvature of surfaces laid the foundation for the use made of
congruence by Helmholtz.

=15.= It is important to remember that, throughout the period we have
just reviewed, the purpose of hyperbolic Geometry is indirect: not
the truth of the latter, but the logical independence of the axiom
of parallels from the rest, is the guiding motive of the work. If,
by denying the axiom of parallels while retaining the rest, we can
obtain a system free from logical contradictions, it follows that
the axiom of parallels cannot be implicitly contained in the others.
If this be so, attempts to dispense with the axiom, like Legendre's,
cannot be successful; Euclid must stand or fall with the suspected
axiom. Of course, it remained possible that, by further development,
latent contradictions might have been revealed in these systems. This
possibility, however, was removed by the more direct and constructive
work of the second period, to which we must now turn our attention.


Second Period.

=16.= The work of Lobatchewsky and Bolyai remained, for nearly a
quarter of a century, without issue--indeed, the investigations
of Riemann and Helmholtz, when they came, appear to have been
inspired, not by these men, but rather by Gauss[15] and Herbart. We
find, accordingly, very great difference, both of aim and method,
between the first period and the second. The former, beginning
with a criticism of one point in Euclid's system, preserved his
synthetic method, while it threw over one of his axioms. The latter,
on the contrary, being guided by a philosophical rather than a
mathematical spirit, endeavoured to classify the conception of
space as a species of a more general conception: it treated space
algebraically, and the properties it gave to space were expressed
in terms, not of intuition, but of algebra. The aim of Riemann and
Helmholtz was to show, by the exhibition of logically possible
alternatives, the empirical nature of the received axioms. For this
purpose, they conceived space as a particular case of a manifold,
and showed that various relations of magnitude (_Massverhältnisse_)
were mathematically possible in an extended manifold. Their
philosophy, which seems to me not always irreproachable, will be
discussed in Chapter II.; here, while it is important to remember
the philosophical motive of Riemann and Helmholtz, we shall confine
our attention to the mathematical side of their work. In so doing,
while we shall, I fear, somewhat maim the system of their thoughts,
we shall secure a closer unity of subject, and a more compact
account of the purely mathematical development. But there is, in my
opinion, a further reason for separating their philosophy from their
mathematics. While their philosophical purpose was, to prove that all
the axioms of Geometry are empirical, and that a different content of
our experience might have changed them all, the unintended result of
their mathematical work was, if I am not mistaken, to afford material
for an _à priori_ proof of certain axioms. These axioms, though they
believed them to be unnecessary, were always introduced in their
mathematical works, before laying the foundations of non-Euclidean
systems. I shall contend, in Chapter III., that this retention was
logically inevitable, and was not merely due, as they supposed, to a
desire for conformity with experience. If I am right in this, there
is a divergence between Riemann and Helmholtz the philosophers, and
Riemann and Helmholtz the mathematicians. This divergence makes it
the more desirable to trace the mathematical development apart from
the accompanying philosophy.

=17.= _Riemann's_ epoch-making work, "_Ueber die Hypothesen, welche
der Geometrie zu Grande liegen_[16]", was written, and read to a
small circle, in 1854; owing, however, to some changes which he
desired to make in it, it remained unpublished till 1867, when it
was published by his executors. The two fundamental conceptions, on
whose invention rests the historic importance of this dissertation,
are that of a _manifold_, and that of the _measure of curvature_
of a manifold. The former conception serves a mainly philosophical
purpose, and is designed, principally, to exhibit space as an
instance of a more general conception. On this aspect of the
manifold, I shall have much to say in Chapter II.; its mathematical
aspect, which alone concerns us here, is less complicated and less
fruitful of controversy. The latter conception also serves a double
purpose, but its mathematical use is the more prominent. We will
consider these two conceptions successively.

=18.= (1) _Conception of a manifold[17]._ The general purpose of
Riemann's dissertation is, to exhibit the axioms as successive steps
in the classification of the species space. The axioms of Geometry,
like the marks of a scholastic definition, appear as successive
determinations of class-conceptions, ending with Euclidean space.
We have thus, from the analytical point of view, about as logical
and precise a formulation as can be desired--a formulation in which,
from its classificatory character, we seem certain of having nothing
superfluous or redundant, and obtain the axioms explicitly in the
most desirable form, namely as adjectives of the conception of
space. At the same time, it is a pity that Riemann, in accordance
with the metrical bias of his time, regarded space as primarily a
magnitude[18], or assemblage of magnitudes, in which the main problem
consists in assigning quantities to the different elements or points,
without regard to the qualitative nature of the quantities assigned.
Considerable obscurity thus arises as to the whole nature of
magnitude[19]. This view of Geometry underlies the definition of the
manifold, as the general conception of which space forms a special
case. This definition, which is not very clear, may be rendered as
follows.

=19.= Conceptions of magnitude, according to Riemann, are possible
there only, where we have a general conception, capable of various
determinations (_Bestimmungsweisen_). The various determinations of
such a conception together form a _manifold_, which is continuous
or discrete, according as the passage from one determination to
another is continuous or discrete. Particular bits of a manifold, or
quanta, can be compared by counting when discrete, and by measurement
when continuous. "Measurement consists in a superposition of the
magnitudes to be compared. If this be absent, magnitudes can only be
compared when one is part of another, and then only the more or less,
not the how much, can be decided" (p. 256). We thus reach the general
conception of a manifold of several dimensions, of which space and
colours are mentioned as special cases.

To the absence of this conception Riemann attributes the "obscurity"
which, on the subject of the axioms, "lasted from Euclid to
Legendre" (p. 254). And Riemann certainly has succeeded, from an
algebraic point of view, in exhibiting, far more clearly than any
of his predecessors, the axioms which distinguish spatial quantity
from other quantities with which mathematics is conversant. But
by the assumption, from the start, that space can be regarded as
a quantity, he has been led to state the problem as: What sort of
magnitude is space? rather than: What must space be in order that
we may be able to regard it as a magnitude at all? He does not
realise, either--indeed in his day there were few who realized--that
an elaborate Geometry is possible which does not deal with space
as a quantity at all. His definition of space as a species of
manifold, therefore, though for analytical purposes it defines, most
satisfactorily, the nature of spatial magnitudes, leaves obscure the
true ground for this nature, which lies in the nature of space as a
system of relations, and is anterior to the possibility of regarding
it as a system of magnitudes at all.

But to proceed with the mathematical development of Riemann's
ideas. We have seen that he declared measurement to consist in
a superposition of the magnitudes to be compared. But in order
that this may be a possible means of determining magnitudes, he
continues, these magnitudes must be independent of their position
in the manifold (p. 259). This can occur, he says, in several ways,
as the simplest of which, he assumes that the lengths of lines are
independent of their position. One would be glad to know what other
ways are possible: for my part, I am unable to imagine any other
hypothesis on which magnitude would be independent of place. Setting
this aside, however, the problem, owing to the fact that measurement
consists in superposition, becomes identical with the determination
of the most general manifold in which magnitudes are independent of
place. This brings us to Riemann's other fundamental conception,
which seems to me even more fruitful than that of a manifold.

=20.= (2) _Measure of curvature._ This conception is due to Gauss,
but was applied by him only to surfaces; the novelty in Riemann's
dissertation was its extension to a manifold of _n_ dimensions.
This extension, however, is rather briefly and obscurely expressed,
and has been further obscured by Helmholtz's attempts at popular
exposition. The term _curvature_, also, is misleading, so that
the phrase has been the source of more misunderstanding, even
among mathematicians, than any other in Pangeometry. It is often
forgotten, in spite of Helmholtz's explicit statement[20], that the
"measure of curvature" of an _n_-dimensional manifold is a purely
analytical expression, which has only a symbolic affinity to ordinary
curvature. As applied to three-dimensional space, the implication
of a four-dimensional "plane" space is wholly misleading; I shall,
therefore, generally use the term space-constant instead[21].
Nevertheless, as the conception grew, historically, out of that of
curvature, I will give a very brief exposition of the historical
development of theories of curvature.

Just as the notion of _length_ was originally derived from the
straight line, and extended to other curves by dividing them into
infinitesimal straight lines, so the notion of _curvature_ was
derived from the circle, and extended to other curves by dividing
them into infinitesimal circular arcs. Curvature may be regarded,
originally, as a measure of the amount by which a curve departs from
a straight line; in a circle, which is similar throughout, this
amount is evidently constant, and is measured by the reciprocal
of the radius. But in all other curves, the amount of curvature
varies from point to point, so that it cannot be measured without
infinitesimals. The measure which at once suggests itself is, the
curvature of the circle most nearly coinciding with the curve at the
point considered. Since a circle is determined by three points, this
circle will pass through three consecutive points of the curve. We
have thus defined the curvature of any curve, plane or tortuous; for,
since any three points lie in a plane, such a circle can always be
described.

If we now pass to a surface, what we want is, by analogy, a measure
of its departure from a plane. The curvature, as above defined, has
become indeterminate, for through any point of the surface we can
draw an infinite number of arcs, which will not, in general, all
have the same curvature. Let us, then, draw all the geodesics joining
the point in question to neighbouring points of the surface in all
directions. Since these arcs form a singly infinite manifold, there
will be among them, if they have not all the same curvature, one arc
of maximum, and one of minimum curvature[22]. The product of these
maximum and minimum curvatures is called the _measure of curvature_
of the surface at the point under consideration. To illustrate by a
few simple examples: on a sphere, the curvatures of all such lines
are equal to the reciprocal of the radius of the sphere, hence the
measure of curvature everywhere is the square of the reciprocal
of the radius of the sphere. On any surface, such as a cone or
a cylinder, on which straight lines can be drawn, these have no
curvature, so that the measure of curvature is everywhere zero--this
is the case, in particular, with the plane. In general, however, the
measure of curvature of a surface varies from point to point.

Gauss, the inventor of this conception[23], proved that, in order
that two surfaces may be developable upon each other--_i.e._ may
be such that one can be bent into the shape of the other without
stretching or tearing--it is necessary that the two surfaces should
have equal measures of curvature at corresponding points. When
this is the case, every figure which is possible on the one is, in
general, possible on the other, and the two have practically the same
Geometry[24]. As a corollary, it follows that a necessary condition,
for the free mobility of figures on any surface, is the constancy
of the measure of curvature[25]. This condition was proved to be
sufficient, as well as necessary, by Minding[26].

=21.= So far, all has been plain sailing--we have been dealing with
purely geometrical ideas in a purely geometrical manner--but we have
not, as yet, found any sense of the measure of curvature, in which it
can be extended to space, still less to an _n_-dimensional manifold.
For this purpose, we must examine Gauss's method, which enables us to
determine the measure of curvature of a surface at any point as an
inherent property, quite independent of any reference to the third
dimension.

The method of determining the measure of curvature from within is,
briefly, as follows: If any point on the surface be determined by two
coordinates, _u_, _v_, then small arcs of the surface are given by
the formula

    ds^{2} = Edu^{2} + 2Fdu dv + Gdv^{2},

where _E_, _F_, _G_ are, in general, functions of _u_, _v_.[27]
From this formula alone, without reference to any space outside the
surface, we can determine the measure of curvature at the point _u_,
_v_, as a function of _E_, _F_, _G_ and their differentials with
respect to _u_ and _v_. Thus we may regard the measure of curvature
of a surface as an inherent property, and the above geometrical
definition, which involved a reference to the third dimension, may
be dropped. But at this point a caution is necessary. It will appear
in Chap. III. (§ 176), that it is logically impossible to set up a
precise coordinate system, in which the coordinates represent spatial
magnitudes, without the axiom of Free Mobility, and this axiom, as we
have just seen, holds on surfaces only when the measure of curvature
is constant. Hence our definition of the measure of curvature will
only be _really_ free from reference to the third dimension, when we
are dealing with a surface of constant measure of curvature--a point
which Riemann entirely overlooks. This caution, however, applies
only in space, and if we take the coordinate system as presupposed
in the conception of a manifold, we may neglect the caution
altogether--while remembering that the possibility of a coordinate
system in space involves axioms to be investigated later. We can thus
see how a meaning might be found, without reference to any higher
dimension, for a constant measure of curvature of three-dimensional
space, or for any measure of curvature of an _n_-dimensional manifold
in general.

=22.= Such a meaning is supplied by Riemann's dissertation, to which,
after this long digression, we can now return. We may define a
continuous manifold as any continuum of elements, such that a single
element is defined by _n_ continuously variable magnitudes. This
definition does not really include space, for coordinates in space do
not define a point, but its relations to the origin, which is itself
arbitrary. It includes, however, the analytical conception of space
with which Riemann deals, and may, therefore, be allowed to stand for
the moment. Riemann then assumes that the difference--or distance,
as it may be loosely called--between any two elements is comparable,
as regards magnitude, to the difference between any other two. He
assumes further, what it is Helmholtz's merit to have proved, that
the difference _ds_ between two consecutive elements can be expressed
as the square root of a quadratic function of the differences of the
coordinates: _i.e._

    ds^{2} = Σ{1}^{n} Σ{1}^{n} a{ik} dx{i}.dx{k},

where the coefficients _a{ik}_ are, in general, functions of the
coordinates _x{1} x{2} ... x{n}_.[28] The question is: How are
we to obtain a definition of the measure of curvature out of this
formula? It is noticeable, in the first place, that, just as in a
surface we found an infinite number of _radii_ of curvature at a
point, so in a manifold of three or more dimensions we must find an
infinite number of _measures_ of curvature at a point, one for every
two-dimensional manifold passing through the point, and contained
in the higher manifold. What we have first to do, therefore, is to
define such two-dimensional manifolds. They must consist, as we saw
on the surface, of a singly infinite series of geodesics through the
point. Now a geodesic is completely determined by one point and its
direction at that point, or by one point and the next consecutive
point. Hence a geodesic through the point considered is determined
by the ratios of the increments of coordinates, _dx{1} dx{2} ...
dx{n}_. Suppose we have two such geodesics, in which the _i_′th
increments are respectively _d′x{i}_ and _d″x{i}_. Then all the
geodesics given by

    dx{i} = λ′d′x{i} + λ″d″x{i}

[Illustration]

form a singly infinite series, since they contain one parameter,
namely λ′: λ″. Such a series of geodesics, therefore, must form a
two-dimensional manifold, with a measure of curvature in the ordinary
Gaussian sense. This measure of curvature can be determined from
the above formula for the elementary arc, by the help of Gauss's
general formula alluded to above. We thus obtain an infinite number
of measures of curvature at a point, but from n.(n - 1)/2 of these,
the rest can be deduced (Riemann, Gesammelte Werke, p. 262). When all
the measures of curvature at a point are constant, and equal to all
the measures of curvature at any other point, we get what Riemann
calls a manifold of constant curvature. In such a manifold free
mobility is possible, and positions do not differ intrinsically from
one another. If _a_ be the measure of curvature, the formula for the
arc becomes, in this case,

    ds^{2} = Σdx^{2}/(1 + a/4 Σx^{2})^{2}.

In this case only, as I pointed out above, can the term "measure of
curvature" be properly applied to space without reference to a higher
dimension, since free mobility is logically indispensable to the
existence of quantitative or metrical Geometry.

=23.= The mathematical result of Riemann's dissertation may be summed
up as follows. Assuming it possible to apply magnitude to space,
_i.e._ to determine its elements and figures by means of algebraical
quantities, it follows that space can be brought under the conception
of a manifold, as a system of quantitatively determinable elements.
Owing, however, to the peculiar nature of spatial measurement, the
quantitative determination of space demands that magnitudes shall
be independent of place--in so far as this is not the case, our
measurement will be necessarily inaccurate. If we now assume, as
the quantitative relation of distance between two elements, the
square root of a quadratic function of the coordinates--a formula
subsequently proved by Helmholtz and Lie--then it follows, since
magnitudes are to be independent of place, that space must, within
the limits of observation, have a constant measure of curvature,
or must, in other words, be homogeneous in all its parts. In the
infinitesimal, Riemann says (p. 267), observation could not detect
a departure from constancy on the part of the measure of curvature;
but he makes no attempt to show how Geometry could remain possible
under such circumstances, and the only Geometry he has constructed
is based entirely on Free Mobility. I shall endeavour to prove, in
Chapter III., that any metrical Geometry, which should endeavour to
dispense with this axiom, would be logically impossible. At present
I will only point out that Riemann, in spite of his desire to prove
that all the axioms can be dispensed with, has nevertheless, in his
mathematical work, retained three fundamental axioms, namely, Free
Mobility, the finite integral number of dimensions, and the axiom
that two points have a unique relation, namely distance. These, as we
shall see hereafter, are retained, in actual mathematical work, by
all metrical Metageometers, even when they believe, like Riemann and
Helmholtz, that no axioms are philosophically indispensable.

=24.= _Helmholtz_, the historically nearest follower of Riemann, was
guided by a similar empirical philosophy, and arrived independently
at a very similar method of formulating the axioms. Although
Helmholtz published nothing on the subject until after Riemann's
death, he had then only just seen Riemann's dissertation (which was
published posthumously), and had worked out his results, so far as
they were then completed, in entire independence both of Riemann
and of Lobatchewsky. Helmholtz is by far the most widely read of all
writers on Metageometry, and his writings, almost alone, represent
to philosophers the modern mathematical standpoint on this subject.
But his importance is much greater, in this domain, as a philosopher
than as a mathematician; almost his only original mathematical
result, as regards Geometry, is his proof of Riemann's formula for
the infinitesimal arc, and even this proof was far from rigid, until
Lie reformed it by his method of continuous groups. In this chapter,
therefore, only two of his writings need occupy us, namely the two
articles in the _Wissenschaftliche Abhandlungen_, Vol. II., entitled
respectively "Ueber die thatsächlichen Grundlagen der Geometrie,"
1866 (p. 610 ff.), and "Ueber die Thatsachen, die der Geometrie zum
Grunde liegen," 1868 (p. 618 ff.).

=25.= In the first of these, which is chiefly philosophical,
Helmholtz gives hints of his then uncompleted mathematical work,
but in the main contents himself with a statement of results. He
announces that he will prove Riemann's quadratic formula for the
infinitesimal arc; but for this purpose, he says, we have to _start_
with Congruence, since without it spatial measurement is impossible.
Nevertheless, he maintains that Congruence is proved by experience.
How we could, without the help of measurement, discover lapses
from Congruence, is a point which he leaves undiscussed. He then
enunciates the four axioms which he considers essential to Geometry,
as follows:

(1) _As regards continuity and dimensions._ In a space of _n_
dimensions, a point is uniquely determined by the measurement of _n_
continuous variables (coordinates).

(2) _As regards the existence of moveable rigid bodies._ Between the
2_n_ coordinates of any point-pair of a rigid body, there exists
an equation which is the same for all congruent point-pairs. By
considering a sufficient number of point-pairs, we get more equations
than unknown quantities: this gives us a method of determining the
form of these equations, so as to make it possible for them all to be
satisfied.

(3) _As regards free mobility._ Every point can pass freely and
continuously from one position to another. From (2) and (3) it
follows, that if two systems _A_ and _B_ can be brought into
congruence in any one position, this is also possible in every other
position.

(4) _As regards independence of rotation in rigid bodies_
(Monodromy). If (_n_ - 1) points of a body remain fixed, so that every
other point can only describe a certain curve, then that curve is
closed.

These axioms, says Helmholtz, suffice to give, with the axiom of
three dimensions, the Euclidean and non-Euclidean systems as the only
alternatives. That they _suffice_, mathematically, cannot be denied,
but they seem, in some respects, to go too far. In the first place,
there is no necessity to make the axiom of Congruence apply to actual
rigid bodies--on this subject I have enlarged in Chapter II.[29]
Again, Free Mobility, as distinct from Congruence, hardly needs to
be specially formulated: what barrier could empty space offer to a
point's progress? The axiom is involved in the homogeneity of space,
which is the same thing as the axiom of Congruence. Monodromy, also,
has been severely criticized; not only is it evident that it might
have been included in Congruence, but even from the purely analytical
point of view, Sophus Lie has proved it to be superfluous[30]. Thus
the axiom of Congruence, rightly formulated, includes Helmholtz's
third and fourth axioms and part of his second axiom. All the
four, or rather, as much of them as is relevant to Geometry, are
consequences, as we shall see hereafter, of the one fundamental
principle of the relativity of position.

=26.= The second article, which is mainly mathematical, supplies
the promised proof of the arc-formula, which is Helmholtz's most
important contribution to Geometry. Riemann had _assumed_ this
formula, as the simplest of a number of alternatives: Helmholtz
proved it to be a necessary consequence of his axioms. The present
paper begins with a short repetition of the first, including the
statement of the axioms, to which, at the end of the paper, two more
are added, (5) that space has three dimensions, and (6) that space is
infinite. It is supposed in the text, as also in the first paper,
that the measure of curvature cannot be negative, and, consequently,
that an infinite space must be Euclidean. This error in both papers
is corrected in notes, added after the appearance of Beltrami's paper
on negative curvature. It is a sample of the slightly unprofessional
nature of Helmholtz's mathematical work on this subject, which
elicits from Klein the following remarks[31]: "Helmholtz is not a
mathematician by profession, but a physicist and physiologist....
From this non-mathematical quality of Helmholtz, it follows naturally
that he does not treat the mathematical portion of his work with the
thoroughness which one would demand of a mathematician by trade (_von
Fach_)." He tells us himself that it was the physiological study of
vision which led him to the question of the axioms, and it is as a
physicist that he makes his axioms refer to actual rigid bodies.
Accordingly, we find errors in his mathematics, such as the axiom
of Monodromy, and the assumption that the measure of curvature must
be positive. Nevertheless, the proof of Riemann's arc-formula is
extremely able, and has, on the whole, been substantiated by Lie's
more thorough investigations.

=27.= Helmholtz's other writings on Geometry are almost wholly
philosophical, and will be discussed at length in Chapter II. For the
present, we may pass to the only other important writer of the second
period, _Beltrami_. As his work is purely mathematical, and contains
few controverted points, it need not, despite its great importance,
detain us long.

The "Saggio di Interpretazione della Geometria non-Euclidea[32],"
which is principally confined to two dimensions, interprets
Lobatchewsky's results by the characteristic method of the second
period. It shows, by a development of the work of Gauss and
Minding[33], that all the propositions in plane Geometry, which
Lobatchewsky had set forth, hold, within ordinary Euclidean space,
on surfaces of constant negative curvature. It is strange, as Klein
points out[34], that this interpretation, which was known to Riemann
and perhaps even to Gauss, should have remained so long without
explicit statement. This is the more strange, as Lobatchewsky's
"Géométrie Imaginaire" had appeared in Crelle, Vol. XVII.[35],
and Minding's article, from which the interpretation follows at
once, had appeared in Crelle, Vol. XIX. Minding had shewn that the
Geometry of surfaces of constant negative curvature, in particular
as regards geodesic triangles, could be deduced from that of the
sphere by giving the radius a purely imaginary value _ia_[36]. This
result, as we have seen, had also been obtained by Lobatchewsky for
his Geometry, and yet it took thirty years for the connection to be
brought to general notice.

=28.= In Beltrami's Saggio, straight lines are, of course, replaced
by geodesics; his coordinates are obtained through a point-by-point
correspondence with an auxiliary plane, in which straight lines
correspond to geodesics on the surface. Thus geodesics have linear
equations, and are always uniquely determined by two points.
Distances on the surface, however, are not equal to distances on
the plane; thus while the surface is infinite, the corresponding
portion of the plane is contained within a certain finite circle.
The distance of two points on the surface is a certain function of
the coordinates, not the ordinary function of elementary Geometry.
These relations of plane and surface are important in connection with
Cayley's theory of distance, which we shall have to consider next.
If we were to define distance on the plane as that function of the
coordinates which gives the corresponding distance on the surface, we
should obtain what Klein calls "a plane with a hyperbolic system of
measurement (_Massbestimmung_)" in which Cayley's theory of distance
would hold. It is evident, however, that the ordinary notion of
distance has been presupposed in setting up the coordinate system, so
that we do not really get alternative Geometries on one and the same
plane. The bearing of these remarks will appear more fully when we
come to consider Cayley and Klein.

=29.= The value of Beltrami's Saggio, in his own eyes, lies in
the intelligible Euclidean sense which it gives to Lobatchewsky's
planimetry: the corresponding system of Solid Geometry, since it has
no meaning for Euclidean space, is barely mentioned in this work.
In a second paper[37], however, almost contemporaneous with the
first, he proceeds to consider the general theory of _n_-dimensional
manifolds of constant negative curvature. This paper is greatly
influenced by Riemann's dissertation; it begins with the formula
for the linear element, and proves from this first, that Congruence
holds for such spaces, and next, that they have, according to
Riemann's definition, a constant negative measure of curvature. (It
is instructive to observe, that both in this and in the former Essay,
great stress is laid on the necessity of the Axiom of Congruence.)

This work has less philosophical interest than the former, since it
does little more than repeat, in a general form, the results which
the Saggio had obtained for two dimensions--results which sink,
when extended to _n_ dimensions, to the level of mere mathematical
constructions. Nevertheless, the paper is important, both as a
restoration of negative curvature, which had been overlooked
by Helmholtz, and as an analytical treatment of Lobatchewsky's
results--a treatment which, together with the Saggio, at last
restored to them the prominence they deserved.


Third Period.

=30.= The third period differs radically, alike in its methods and
aims, and in the underlying philosophical ideas, from the period
which it replaced. Whereas everything, in the second period, turned
on measurement, with its apparatus of Congruence, Free Mobility,
Rigid Bodies, and the rest, these vanish completely in the third
period, which, swinging to the opposite extreme, regards quantity
as a perfectly irrelevant category in Geometry, and dispenses with
congruence and the method of superposition. The ideas of this period,
unfortunately, have found no exponent so philosophical as Riemann or
Helmholtz, but have been set forth only by technical mathematicians.
Moreover the change of fundamental ideas, which is immense, has
not brought about an equally great change in actual procedure; for
though spatial quantity is no longer a part of projective Geometry,
quantity is still employed, and we still have equations, algebraic
transformations, and so on. This is apt to give rise to confusion,
especially in the mind of the student, who fails to realise that the
quantities used, so far as the propositions are really projective,
are mere names for points, and not, as in metrical Geometry, actual
spatial magnitudes.

Nevertheless, the fundamental difference between this period
and the former must strike any one at once. Whereas Riemann
and Helmholtz dealt with metrical ideas, and took, as their
foundations, the measure of curvature and the formula for the linear
element--both purely metrical--the new method is erected on the
formulae for transformation of coordinates required to express a
given collineation. It begins by reducing all so-called metrical
notions--distance, angle, etc.--to projective forms, and obtains,
from this reduction, a methodological unity and simplicity before
impossible. This reduction depends, however, except where the
space-constant is negative, upon imaginary figures--in Euclid, the
circular points at infinity; it is moreover purely symbolic and
analytical, and must be regarded as philosophically irrelevant.
As the question concerning the import of this reduction is of
fundamental importance to our theory of Geometry, and as Cayley, in
his Presidential Address to the British Association in 1883, formally
challenged philosophers to discuss the use of imaginaries, on which
it depends, I will treat this question at some length. But first let
us see how, as a matter of mathematics, the reduction is effected.

=31.= We shall find, throughout this period, that almost
every important proposition, though misleading in its obvious
interpretation, has nevertheless, when rightly interpreted, a wide
philosophical bearing. So it is with the work of _Cayley_, the
pioneer of the projective method.

The projective formula for angles, in Euclidean Geometry, was first
obtained by Laguerre, in 1853. This formula had, however, a perfectly
Euclidean character, and it was left for Cayley to generalize
it so as to include both angles and distances in Euclidean and
non-Euclidean systems alike[38].

_Cayley_ was, to the last, a staunch supporter of Euclidean
_space_, though he believed that non-Euclidean _Geometries_ could
be applied, within Euclidean space, by a change in the definition
of distance[39]. He has thus, in spite of his Euclidean orthodoxy,
provided the believers in the possibility of non-Euclidean spaces
with one of their most powerful weapons. In his "Sixth Memoir upon
Quantics" (1859), he set himself the task of "establishing the notion
of distance upon purely descriptive principles." He showed that, with
the ordinary notion of distance, it can be rendered projective by
reference to the circular points and the line at infinity, and that
the same is true of angles[40]. Not content with this, he suggested
a new definition of distance, as the inverse sine or cosine of a
certain function of the coordinates; with this definition, the
properties usually known as metrical become projective properties,
having reference to a certain conic, called by Cayley the Absolute.
(The circular points are, analytically, a degenerate conic, so that
ordinary Geometry forms a particular case of the above.) He proves
that, when the Absolute is an _imaginary_ conic, the Geometry so
obtained for two dimensions is spherical Geometry. The correspondence
with Lobatchewsky, in the case where the Absolute is _real_, is not
worked out: indeed there is, throughout, no evidence of acquaintance
with non-Euclidean systems. The importance of the memoir, to Cayley,
lies entirely in its proof that metrical is only a branch of
descriptive Geometry.

=32.= The connection of Cayley's Theory of Distance with Metageometry
was first pointed out by Klein[41]. Klein showed in detail that, if
the Absolute be real, we get Lobatchewsky's (hyperbolic) system; if
it be imaginary, we get either spherical Geometry or a new system,
analogous to that of Helmholtz, called by Klein elliptic; if the
Absolute be an imaginary point-pair, we get parabolic Geometry, and
if, in particular, the point-pair be the circular points, we get
ordinary Euclid. In elliptic Geometry, two straight lines in the same
plane meet in only one point, not two as in Helmholtz's system. The
distinction between the two kinds of Geometry is difficult, and will
be discussed later.

=33.= Since these systems are all obtained from a Euclidean plane,
by a mere alteration in the definition of distance, Cayley and
Klein tend to regard the whole question as one, not of the nature
of space, but of the definition of distance. Since this definition,
on their view, is perfectly arbitrary, the philosophical problem
vanishes--Euclidean _space_ is left in undisputed possession, and
the only problem remaining is one of convention and mathematical
convenience[42]. This view has been forcibly expressed by Poincaré:
"What ought one to think," he says, "of this question: Is the
Euclidean Geometry true? The question is nonsense." Geometrical
axioms, according to him, are mere conventions: they are "definitions
in disguise[43]." Thus Klein blames Beltrami for regarding his
auxiliary plane as merely auxiliary, and remarks that, if he had
known Cayley's Memoir, he would have seen the relation between
the plane and the pseudosphere to be far more intimate than he
supposed[44]. A view which removes the problem entirely from the
arena of philosophy demands, plainly, a full discussion. To this
discussion we will now proceed.

=34.= The view in question has arisen, it would seem, from a natural
confusion as to the nature of the coordinates employed. Those who
hold the view have not adequately realised, I believe, that their
coordinates are not _spatial_ quantities, as in metrical Geometry,
but mere conventional signs, by which different points can be
distinctly designated. There is no reason, therefore, until we
already have metrical Geometry, for regarding one function of the
coordinates as a better expression of distance than another, so
long as the fundamental addition-equation[45] is preserved. Hence,
if our coordinates are regarded as adequate for all Geometry, an
indeterminateness arises in the expression of distance, which can
only be avoided by a convention. But projective coordinates--so our
argument will contend--though perfectly adequate for all projective
properties, and entirely free from any metrical presupposition, are
inadequate to express metrical properties, just because they have
no metrical presupposition. Thus where metrical properties are in
question, Beltrami remains justified as against Klein; the reduction
of metrical to projective properties is only apparent, though
the independence of these last, as against metrical Geometry, is
perfectly real.

=35.= But what are projective coordinates, and how are they
introduced? This question was not touched upon in Cayley's Memoir,
and it seemed, therefore, as if a logical error were involved in
using coordinates to define distance. For coordinates, in all
previous systems, had been deduced from distance; to use any existing
coordinate system in defining distance was, accordingly, to incur
a vicious circle. Cayley mentions this difficulty in a note, where
he only remarks, however, that he had regarded his coordinates
as numbers arbitrarily assigned, on some system not further
investigated, to different points. The difficulty has been treated at
length by Sir R. Ball (Theory of the Content, Trans. R. I. A. 1889),
who urges that if the values of our coordinates already involve the
usual measure of distance, then to give a new definition, while
retaining the usual coordinates, is to incur a contradiction. He
says (op. cit. p. 1): "In the study of non-Euclidean Geometry I have
often felt a difficulty which has, I know, been shared by others. In
that theory it seems as if we try to replace our ordinary notion of
distance between two points by the logarithm of a certain anharmonic
ratio[46]. But this ratio itself involves the notion of distance
measured in the ordinary way. How, then, can we supersede our old
notion of distance by the non-Euclidean notion, inasmuch as the very
definition of the latter involves the former?"

=36.= This objection is valid, we must admit, so long as anharmonic
ratio is defined in the ordinary metrical manner. It would be
valid, for example, against any attempt to found a new definition
of distance on Cremona's account of anharmonic ratio[47], in
which it appears as a metrical property unaltered by projective
transformation. If a logical error is to be avoided, in fact, all
reference to spatial magnitude of any kind must be avoided; for all
spatial magnitude, as will be shown hereafter[48], is logically
dependent on the fundamental magnitude of distance. Anharmonic
ratio and coordinates must alike be defined by purely descriptive
properties, if the use afterwards made of them is to be free from
metrical presuppositions, and therefore from the objections of Sir R.
Ball.

Such a definition has been satisfactorily given by Klein[49],
who appeals, for the purpose, to v. Staudt's quadrilateral
construction[50]. By this construction, which I have reproduced in
outline in Chapter III. Section A, § 112 ff., we obtain a purely
descriptive definition of harmonic and anharmonic ratio, and, given
a pair of points, we can obtain the harmonic conjugate to any
third point on the same straight line. On this construction, the
introduction of projective coordinates is based. Starting with any
three points on a straight line, we assign to them arbitrarily the
numbers 0, 1, ∞. We then find the harmonic conjugate to the first
with respect to 1, ∞, and assign to it the number 2. The object of
assigning this number rather than any other, is to obtain the value
-1 for the anharmonic ratio of the four numbers corresponding to the
four points[51]. We then find the harmonic conjugate to the point
1, with respect to 2, ∞, and assign to it the number 3; and so on.
Klein has shown that by this construction, we can obtain any number
of points, and can construct a point corresponding to any given
number, fractional or negative. Moreover, when two sets of four
points have the same anharmonic ratio, descriptively defined[52],
the corresponding numbers also have the same anharmonic ratio. By
introducing such a numerical system on two straight lines, or on
three, we obtain the coordinates of any point in a plane, or in
space. By this construction, which is of fundamental importance
to projective Geometry, the logical error, upon which Sir R. Ball
bases his criticism, is satisfactorily avoided. Our coordinates
are introduced by a purely descriptive method, and involve no
presupposition whatever as to the measurement of distance.

=37.= With this coordinate system, then, to define distance as a
certain function of the coordinates is not to be guilty of a vicious
circle. But it by no means follows that the definition of distance is
arbitrary. All reference to distance has been hitherto excluded, to
avoid metrical ideas; but when distance is introduced, metrical ideas
inevitably reappear, and we have to remember that our coordinates
give no information, _primâ facie_, as to any of these metrical
ideas. It is open to us, of course, if we choose, to continue to
exclude distance in the ordinary sense, as the quantity of a finite
straight line, and to define the _word_ distance in any way we
please. But the conception, for which the word has hitherto stood,
will then require a new name, and the only result will be a confusion
between the _apparent_ meaning of our propositions, to those who
retain the associations belonging to the old sense of the word, and
the _real_ meaning, resulting from the new sense in which the word is
used.

This confusion, I believe, has actually occurred, in the case of
those who regard the question between Euclid and Metageometry as
one of the definition of distance. Distance is a quantitative
relation, and as such presupposes identity of quality. But projective
Geometry deals only with quality--for which reason it is called
descriptive--and cannot distinguish between two figures which are
qualitatively alike. Now the meaning of qualitative likeness,
in Geometry, is the possibility of mutual transformation by a
collineation[53]. Any two pairs of points on the same straight line,
therefore, are qualitatively alike; their only qualitative relation
is the straight line, which both pairs have in common; and it is
exactly the qualitative identity of the relations of the two pairs,
which enables the difference of their relations to be exhaustively
dealt with by quantity, as a difference of distance. But where
quantity is excluded, any two pairs of points on the same straight
line appear as alike, and even any two sets of three: for any three
points on a straight line can be projectively transformed into any
other three. It is only with _four_ points in a line that we acquire
a projective property distinguishing them from other sets of four,
and this property is anharmonic ratio, descriptively defined. The
projective Geometer, therefore, sees no reason to give a name to the
relation between two points, in so far as this relation is anything
over and above the unlimited straight line on which they lie; and
when he introduces the notion of distance, he defines it, in the
only way in which projective principles allow him to define it, as a
relation between _four_ points. As he nevertheless wishes the word
to give him the power of distinguishing between different _pairs_
of points, he agrees to take two out of the four points as fixed.
In this way, the only variables in distance are the two remaining
points, and distance appears, therefore, as a function of _two_
variables, namely the coordinates of the two variable points. When we
have further defined our function so that distance may be additive,
we have a function with many of the properties of distance in the
ordinary sense. This function, therefore, the projective Geometer
regards as the only proper definition of distance.

We can see, in fact, from the manner in which our projective
coordinates were introduced, that _some_ function of these
coordinates must express distance in the ordinary sense. For
they were introduced serially, so that, as we proceeded from the
zero-point towards the infinity-point, our coordinates continually
grew. To every point, a definite coordinate corresponded: to the
distance between two variable points, therefore, as a function
dependent on no other variables, must correspond some definite
function of the coordinates, since these are themselves functions of
their points. The function discussed above, therefore, must certainly
include distance in the ordinary sense.

But the arbitrary and conventional nature of distance, as maintained
by Poincaré and Klein, arises from the fact that the two fixed
points, required to determine our distance in the projective sense,
may be arbitrarily chosen, and although, when our choice is once
made, any two points have a definite distance, yet, according as we
make that choice, distance will become a different function of the
two variable points. The ambiguity thus introduced is unavoidable on
projective principles; but are we to conclude, from this, that it
is really unavoidable? Must we not rather conclude that projective
Geometry cannot adequately deal with distance? If _A_, _B_, _C_, be
three different points on a line, there must be _some_ difference
between the relation of _A_ to _B_ and of _A_ to _C_, for otherwise,
owing to the qualitative identity of all points, _B_ and _C_ could
not be distinguished. But such a difference involves a relation,
between _A_ and _B_, which is independent of other points on the
line; for unless we have such a relation, the other points cannot
be distinguished as different. Before we can distinguish the two
fixed points, therefore, from which the projective definition
starts, we must already suppose some relation, between any two
points on our line, in which they are independent of other points;
and this relation is distance in the ordinary sense[54]. When we
have measured this quantitative relation by the ordinary methods
of metrical Geometry, we can proceed to decide what base-points
must be chosen, on our line, in order that the projective function
discussed above may have the same value as ordinary distance. But
the choice of these base-points, when we are discussing distance
in the ordinary sense, is not arbitrary, and their introduction is
only a technical device. Distance, in the ordinary sense, remains
a relation between _two_ points, not between _four_; and it is the
failure to perceive that the projective sense differs from, and
cannot supersede, the ordinary sense, which has given rise to the
views of Klein and Poincaré. The question is not one of convention,
but of the irreducible metrical properties of space. To sum up:
Quantities, as used in projective Geometry, do not stand for spatial
magnitudes, but are conventional symbols for purely qualitative
spatial relations. But distance, _quâ_ quantity, presupposes identity
of quality, as the condition of quantitative comparison. Distance in
the ordinary sense is, in short, that quantitative relation, between
two points on a line, by which their difference from other points
can be defined. The projective definition, however, being unable to
distinguish a collection of less than four points from any other on
the same straight line, makes distance depend on two other points
besides those whose relation it defines. No name remains, therefore,
for distance in the ordinary sense, and many projective Geometers,
having abolished the name, believe the thing to be abolished also,
and are inclined to deny that _two_ points have a unique relation at
all. This confusion, in projective Geometry, shows the importance of
a name, and should make us chary of allowing new meanings to obscure
one of the fundamental properties of space.

=38.= It remains to discuss the manner in which non-Euclidean
Geometries result from the projective definition of distance, as also
the true interpretation to be given to this view of Metageometry. It
is to be observed that the projective methods which follow Cayley
deal throughout with a Euclidean plane, on which they introduce
different measures of distance. Hence arises, in any interpretation
of these methods, an apparent subordination of the non-Euclidean
spaces, as though these were less self-subsistent than Euclid's.
This subordination is not intended in what follows; on the contrary,
the correlation with Euclidean space is regarded as valuable,
first, because Euclidean space has been longer studied and is more
familiar, but secondly, because this correlation proves, when truly
interpreted, that the other spaces are self-subsistent. We may
confine ourselves chiefly, in discussing this interpretation, to
distances measured along a single straight line. But we must be
careful to remember that the metrical definition of distance--which,
according to the view here advocated, is the only adequate
definition--is the same in Euclidean and in non-Euclidean spaces; to
argue in its favour is not, therefore, to argue in favour of Euclid.

The projective scheme of coordinates consists of a series of numbers,
of which each represents a certain anharmonic ratio and denotes one
and only one point, and which increase uniformly with the distance
from a fixed origin, until they become infinite on reaching a certain
point. Now Cayley showed that, in Euclidean Geometry, distance may
be expressed as the limit of the logarithm of the anharmonic ratio
of the two points and the (coincident) points at infinity on their
straight line; while, if we assumed that the points at infinity were
distinct, we obtained the formula for distance in hyperbolic or
spherical Geometry, according as these points were real or imaginary.
Hence it follows that, with the projective definition of distance,
we shall obtain precisely the formulae of hyperbolic, parabolic or
spherical Geometry, according as we choose the point, to which the
value +∞ is assigned, at a finite, infinite or imaginary distance
(in the ordinary sense) from the point to which we assign the value
0. Our straight line remains, all the while, an ordinary Euclidean
straight line. But we have seen that the projective definition of
distance fits with the true definition only when the two fixed
points to which it refers are suitably chosen. Now the ordinary
meaning of distance is required in non-Euclidean as in Euclidean
Geometries--indeed, it is only in metrical properties that these
Geometries differ. Hence our _Euclidean_ straight line, though it
may serve to illustrate other Geometries than Euclid's, can only be
dealt with correctly by Euclid. Where we give a different definition
of distance from Euclid's, we are still in the domain of purely
projective properties, and derive no information as to the metrical
properties of our straight line. But the importance, to Metageometry,
of this new interpretation, lies in the fact that, having
independently established the metrical formulae of non-Euclidean
spaces, we find, as in Beltrami's Saggio, that these spaces can
be related, by a homographic correspondence, with the points of
Euclidean space; and that this can be effected in such a manner as
to give, for the distance between two points of our non-Euclidean
space, the hyperbolic or spherical measure of distance for the
corresponding points of Euclidean space.

=39.= On the whole, then, a modification of Sir R. Ball's view,
which is practically a generalized statement of Beltrami's method,
seems the most tenable. He imagines what, with Grassmann, he calls
a Content, _i.e._ a perfectly general three-dimensional manifold,
and then correlates its elements, one by one, with points in
Euclidean space. Thus every element of the Content acquires, as its
coordinates, the ordinary Euclidean coordinates of the corresponding
point in Euclidean space. By means of this correlation, our
calculations, though they refer to the Content, are carried on, as in
Beltrami's Saggio, in ordinary Euclidean space. Thus the confusion
disappears, but with it, the supposed Euclidean interpretation also
disappears. Sir R. Ball's Content, if it is to be a space at all,
must be a space radically different from Euclid's[55]; to speak, as
Klein does, of ordinary planes with hyperbolic or elliptic measures
of distance, is either to incur a contradiction, or to forego any
metrical meaning of distance. Instead of ordinary planes, we have
surfaces like Beltrami's, of constant measure of curvature; instead
of Euclid's space, we have hyperbolic or spherical space. At the
same time, it remains true that we can, by Klein's method, give a
Euclidean meaning to every symbolic proposition in non-Euclidean
Geometry. For by substituting, for distance, the logarithm above
alluded to, we obtain, from the non-Euclidean result, a result which
follows from the ordinary Euclidean axioms. This correspondence
removes, once for all, the possibility of a lurking contradiction in
Metageometry, since, to a proposition in the one, corresponds one
and only one proposition in the other, and contradictory results in
one system, therefore, would correspond to contradictory results in
the other. Hence Metageometry cannot lead to contradictions, unless
Euclidean Geometry, at the same moment, leads to corresponding
contradictions. Thus the Euclidean plane with hyperbolic or elliptic
measure of distance, though either contradictory or not metrical
as an independent notion, has, as a help in the interpretation of
non-Euclidean results, a very high degree of utility.

=40.= We have still to discuss Klein's third kind of non-Euclidean
Geometry, which he calls elliptic. The difference between this and
spherical Geometry is difficult to grasp, but it may be illustrated
by a simpler example. A plane, as every one knows, can be wrapped,
without stretching, on a cylinder, and straight lines in the plane
become, by this operation, geodesics on the cylinder. The Geometries
of the plane and the cylinder, therefore, have much in common.
But since the generating circle of the cylinder, which is one of
its geodesics, is finite, only a portion of the plane is used up
in wrapping it once round the cylinder. Hence, if we endeavour to
establish a point-to-point correspondence between the plane and the
cylinder, we shall find an infinite series of points on the plane
for a single point on the cylinder. Thus it happens that geodesics,
though on the plane they have only one point in common, may on
the cylinder have an infinite number of intersections. Somewhat
similar to this is the relation between the spherical and elliptic
Geometries. To any one point in elliptic space, two points correspond
in spherical space. Thus geodesics, which in spherical space may have
two points in common, can never, in elliptic space, have more than
one intersection.

But Klein's method can only prove that elliptic Geometry holds of
the ordinary Euclidean plane with elliptic measure of distance.
Klein has made great endeavours to enforce the distinction between
the spherical and elliptic Geometries[56], but it is not immediately
evident that the latter, as distinct from the former, is valid.

In the first place, Klein's elliptic Geometry, which arises as
one of the alternative metrical systems on a Euclidean plane or
in a Euclidean space, does not by itself suffice, if the above
discussion has been correct, to prove the possibility of an elliptic
space, _i.e._ of a space having a point-to-point correspondence
with the Euclidean space, and having as the ordinary distance
between two of its points the elliptic definition of the distance
between corresponding points of the Euclidean space. To prove this
possibility, we must adopt the direct method of Newcomb (Crelle's
Journal, Vol. 83). Now in the first place Newcomb has not proved that
his postulates are self-consistent; he has only failed to prove that
they are contradictory[57]. This would leave elliptic space in the
same position in which Lobatchewsky and Bolyai left hyperbolic space.
But further there seems to be, at first sight, in _two_-dimensional
elliptic space, a positive contradiction. To explain this, however,
some account of the peculiarities of the elliptic plane will be
necessary.

[Illustration]

The elliptic plane, regarded as a figure in three-dimensional
elliptic space, is what is called a double surface[58], _i.e._ as
Newcomb says (_loc. cit._ p. 298): "The two sides of a complete plane
are not distinct, as in a Euclidean surface.... If ... a being should
travel to distance 2_D_, he would, on his return, find himself on
the opposite surface to that on which he started, and would have
to repeat his journey in order to return to his original position
without leaving the surface." Now if we imagine a _two_-dimensional
elliptic space, the distinction between the sides of a plane becomes
unmeaning, since it only acquires significance by reference to the
third dimension. Nevertheless, some such distinction would be forced
upon us. Suppose, for example, that we took a small circle provided
with an arrow, as in the figure, and moved this circle once round the
universe. Then the sense of the arrow would be reversed. We should
thus be forced, either to regard the new position as distinct from
the former, which transforms our plane into a spherical plane, or to
attribute the reversal of the arrow to the action of a motion which
restores our circle to its original place. It is to be observed that
nothing short of moving round the universe would suffice to reverse
the sense of the arrow. This reversal _seems_ like an action of
empty space, which would force us to regard the points which, from a
three-dimensional point of view, are coincident though opposite, as
really distinct, and so reduce the elliptic to the spherical plane.
But motion, not space, really causes the change, and the elliptic
plane is therefore not proved to be impossible. The question is not,
however, of any great philosophic importance.

=41.= In connection with the reduction of metrical to projective
Geometry, we have one more topic for discussion. This is the
geometrical use of imaginaries, by means of which, except in the
case of hyperbolic space, the reduction is effected. I have already
contended, on other grounds, that this reduction, in spite of its
immense technical importance, and in spite of the complete logical
freedom of projective Geometry from metrical ideas, is _purely_
technical, and is not philosophically valid. The same conclusion will
appear, if we take up Cayley's challenge at the British Association,
in his Presidential Address of 1883.

In this address, Professor Cayley devoted most of his time to
non-Euclidean systems. Non-Euclidean _spaces_, he declared, seemed to
him mistaken _à priori_[59]; but non-Euclidean _Geometries_, here as
in his mathematical works, were accepted as flowing from a change in
the definition of distance. This view has been already discussed, and
need not, therefore, be further criticised here. What I wish to speak
about, is the question with which Cayley himself opened his address,
namely, the geometrical use and meaning of imaginary quantities. From
the manner in which he spoke of this question, it becomes imperative
to treat it somewhat at length. For he said (pp. 8-9):

"... The notion which is the really fundamental one (and I cannot
too strongly emphasize the assertion) underlying and pervading the
whole notion of modern analysis and Geometry, [is] that of imaginary
magnitude in analysis, and of imaginary space (or space as the _locus
in quo_ of imaginary points and figures) in Geometry: I use in each
case the word imaginary as including real.... Say even the conclusion
were that the notion belongs to mere technical mathematics, or has
reference to nonentities in regard to which no science is possible,
still it seems to me that (as a subject of philosophical discussion)
the notion ought not to be thus ignored; it should at least be shown
that there is a right to ignore it."

=42.= This right it is now my purpose to demonstrate. But for fear
non-mathematicians should miss the point of Cayley's remark (which
has sometimes been erroneously supposed to refer to non-Euclidean
spaces), I may as well explain, at the outset, that this question
is radically distinct from, and only indirectly connected with, the
validity or import of Metageometry. An imaginary quantity is one
which involves √-1: its most general form is _a_ + √-1_b_ where _a_
and _b_ are real; Cayley uses the word imaginary so as to include
real, in order to cover the special case where _b_ = 0. It will
be convenient, in what follows, to exclude this wider meaning,
and assume that _b_ is not zero. An imaginary point is one whose
coordinates involve √-1, _i.e._ whose coordinates are imaginary
quantities. An imaginary curve is one whose points are imaginary--or,
in some special uses, one whose equation contains imaginary
coefficients. The mathematical subtleties to which this notion leads
need not be here discussed; the reader who is interested in them will
find an excellent elementary account of their geometrical uses in
Klein's Nicht-Euklid, II. pp. 38-46. But for our present purpose, we
may confine ourselves to imaginary points. If these are found to have
a merely technical import, and to be destitute of any philosophical
meaning, then the same will hold of any collection of imaginary
points, _i.e._ of any imaginary curve or surface.

That the notion of imaginary points is of supreme importance in
Geometry, will be seen by any one who reflects that the circular
points are imaginary, and that the reduction of metrical to
projective Geometry, which is one of Cayley's greatest achievements,
depends on these points. But to discuss adequately their
philosophical import is difficult to me, since I am unacquainted with
any satisfactory philosophy of imaginaries in pure Algebra. I will
therefore adopt the most favourable hypothesis, and assume that no
objection can be successfully urged against this use. Even on this
hypothesis, I think, no case can be made out for imaginary points in
Geometry.

In the first place, we must exclude, from the imaginary points
considered, those whose coordinates are only imaginary with certain
special systems of coordinates. For example, if one of a point's
coordinates be the tangent from it to a sphere, this coordinate will
be imaginary for any point inside the sphere, and yet the point is
perfectly real. A point, then, is only to be called imaginary, when,
whatever real system of coordinates we adopt, one or more of the
quantities expressing these coordinates remains imaginary. For this
purpose, it is mathematically sufficient to suppose our coordinates
Cartesian--a point whose Cartesian coordinates are imaginary, is a
true imaginary point in the above sense.

To discuss the meaning of such a point, it is necessary to consider
briefly the fundamental nature of the correspondence between a
point and its coordinates. Assuming that elementary Geometry has
proved--what I think it does satisfactorily prove--that spatial
relations are susceptible of quantitative measurement, then a given
point will have, with a suitable system of coordinates, in a space
of _n_ dimensions, _n_ quantitative relations to the fixed spatial
figure forming the axes of coordinates, and these _n_ quantitative
relations will, under certain reservations, be unique--_i.e._, no
other point will have the same quantities assigned to it. (With many
possible coordinate systems, this latter condition is not realized:
but for that very reason they are inconvenient, and employed only in
special problems.) Thus given a coordinate system, and given any set
of quantities, these quantities, _if they determine a point at all_,
determine it uniquely. But, by a natural extension of the method, the
above reservation is dropped, and it is assumed that to _every_ set
of quantities some point must correspond. For this assumption there
seems to me no vestige of evidence. As well might a postman assume
that, because every house in a street is uniquely determined by its
number, therefore there must be a house for every imaginable number.
We must know, in fact, that a given set of quantities can be the
coordinates of some point in space, before it is legitimate to give
any spatial significance to these quantities: and this knowledge,
obviously, cannot be derived from operations with coordinates alone,
on pain of a vicious circle. We must, to return to the above analogy,
know the number of houses in Piccadilly, before we know whether a
given number has a corresponding house or not; and arithmetic alone,
however subtly employed, will never give us this information.

Thus the distinction which is important is, not the distinction
between real and imaginary quantities, but between quantities to
which points correspond and quantities to which no points correspond.
We can conventionally agree to denote real points by imaginary
coordinates, as in the Gaussian method of denoting by the single
quantity (_a_ + √-1_b_) the point whose ordinary coordinates are
_a_, _b_. But this does not touch Cayley's meaning. Cayley means
that it is of great utility in mathematics to regard, as points
with a real existence in space, the assumed spatial correlates of
quantities which, with the coordinate system employed, have no
correlates in every-day space; and that this utility is supposed,
by many mathematicians, to indicate the validity of so fruitful an
assumption. To fix our ideas, let us consider Cartesian axes in
three-dimensional Euclidean space. Then it appears, by inspection,
that a point may be situated at any distance to right or left of any
of the three coordinate planes; taking this distance as a coordinate,
therefore, it appears that real points correspond to all quantities
from -∞ to +∞. The same appears for the other two coordinates;
and since elementary Geometry proves their variations mutually
independent, we know that one and only one real point corresponds
to any three real quantities. But we also know, from the exhaustive
method pursued, that all space is covered by the range of these three
variable quantities: a fresh set of quantities, therefore, such as is
introduced by the use of imaginaries, possesses no spatial correlate,
and can be supposed to possess one only by a convenient fiction.

=43.= The fact that the fiction _is_ convenient, however, may
be thought to indicate that it is more than a fiction. But this
presumption, I think, can be easily explained away. For all the
fruitful uses of imaginaries, in Geometry, are those which begin
and end with real quantities, and use imaginaries only for the
intermediate steps. Now in all such cases, we have a real spatial
interpretation at the beginning and end of our argument, where alone
the spatial interpretation is important: in the intermediate links,
we are dealing in a purely algebraical manner with purely algebraical
quantities, and may perform any operations which are algebraically
permissible. If the quantities with which we end are capable of
spatial interpretation, then, and only then, our result may be
regarded as geometrical. To use geometrical language, in any other
case, is only a convenient help to the imagination. To speak, for
example, of projective properties which refer to the circular points,
is a mere _memoria technica_ for purely algebraical properties;
the circular points are not to be found in space, but only in the
auxiliary quantities by which geometrical equations are transformed.
That no contradictions arise from the geometrical interpretation of
imaginaries, is not wonderful: for they are interpreted solely by the
rules of Algebra, which we may admit as valid in their application
to imaginaries. The perception of space being wholly absent,
Algebra rules supreme, and no inconsistency can arise. Wherever,
for a moment, we allow our ordinary spatial notions to intrude, the
grossest absurdities do arise--every one can see that a circle, being
a closed curve, cannot get to infinity. The metaphysician, who should
invent anything so preposterous as the circular points, would be
hooted from the field. But the mathematician may steal the horse with
impunity.

Finally, then, only a knowledge of space, not a knowledge of Algebra,
can assure us that any given set of quantities will have a spatial
correlate, and in the absence of such a correlate, operations with
these quantities have no geometrical import. This is the case with
imaginaries in Cayley's sense, and their use in Geometry, great as
are its technical advantages, and rigid as is its technical validity,
is wholly destitute of philosophical importance.

=44.= We have now, I think, discussed most of the questions
concerning the scope and validity of the projective method. We
have seen that it is independent of all metrical presuppositions,
and that its use of coordinates does not involve the assumption
that spatial magnitudes are measured or expressed by them. We have
seen that it is able to deal, by its own methods alone, with the
question of the qualitative likeness of geometrical figures, which
is logically prior to any comparison as to quantity, since quantity
presupposes qualitative likeness. We have seen also that, so far as
its legitimate use extends, it applies equally to all homogeneous
spaces, and that its criterion of an independently possible
space--the determination of a straight line by two points[60]--is
not subject to the qualifications and limitations which belong, as
we have seen in the case of the cylinder, to the metrical criterion
of constant curvature. But we have also seen that, when projective
Geometry endeavours to grapple with spatial magnitude, and bring
distance and the measurement of angles beneath its sway, its success,
though technically valid and important, is philosophically an
apparent success only. Metrical Geometry, therefore, if quantity is
to be applied to space at all, remains a separate, though logically
subsequent branch of Mathematics.

=45.= It only remains to say a few words about _Sophus Lie_. As a
mathematician, as the inventor of a new and immensely powerful method
of analysis, he cannot be too highly praised. Geometry is only one
of the numerous subjects to which his theory of continuous groups
applies, but its application to Geometry has made a revolution in
method, and has rendered possible, in such problems as Helmholtz's, a
treatment infinitely more precise and exhaustive than any which was
possible before.

The general definition of a group is as follows: If we have any
number of independent variables _x{1}x{2}...x{n}_, and
any series of transformations of these into new variables--the
transformations being defined by equations of specified forms,
with parameters varying from one transformation to another--then
the series of transformations form a _group_, if the successive
application of any two is equivalent to a single member of the
original series of transformations. The group is _continuous_, when
we can pass, by infinitesimal gradations within the group, from any
one of the transformations to any other.

Now, in Geometry, the result of two successive motions or
collineations of a figure can always be obtained by a single motion
or collineation, and any motion or collineation can be built up
of a series of infinitesimal motions or collineations. Moreover
the analytical expression of either is a certain transformation
of the coordinates of all the points of the figure[61]. Hence the
transformations determining a motion or a collineation are such
as to form a continuous group. But the question of the projective
equivalence of two figures, to which all projective Geometry is
reducible, must always be dealt with by a collineation; and the
question of the equality of two figures, to which all metrical
Geometry is reducible, must always be decided by a motion such as
to cause superposition; hence the whole subject of Geometry may
be regarded as a theory of the continuous groups which define all
possible collineations and motions.

Now Sophus Lie has developed, at great length, the purely analytical
theory of groups; he has therefore, by this method of formulating
the problem, a very powerful weapon ready for the attack. In two
papers "On the foundations of Geometry[62]," undertaken at Klein's
urgent request, he takes premisses which roughly correspond to those
of Helmholtz, omitting Monodromy, and applies the theory of groups to
the deduction of their consequences[63]. Helmholtz's work, he says,
can hardly be looked upon as _proving_ its conclusions, and indeed
the more searching analysis of the group-theory reveals several
possibilities unknown to Helmholtz. Nevertheless, as a pioneer,
devoid of Lie's machinery, Helmholtz deserves, I think, more praise
than Lie is willing to give him[64].

Lie's method is perfectly exhaustive; omitting the premiss of
Monodromy, the others show that a body has six degrees of freedom,
_i.e._ that the group giving all possible motions of a body will
have six independent members; if we keep one point fixed, the
number of independent members is reduced to three. He then, from
his general theory, enumerates all the groups which satisfy this
condition. In order that such a group should give possible motions,
it is necessary, by Helmholtz's second axiom, that it should leave
invariant some function of the coordinates of any two points. This
eliminates several of the groups previously enumerated, each of which
he discusses in turn. He is thus led to the following results:

I. _In two dimensions_, if free mobility is to hold _universally_,
there are no groups satisfying Helmholtz's first three axioms, except
those which give the ordinary Euclidean and non-Euclidean motions;
but if it is to hold only _within a certain region_, there is also
a possible group in which the curve described by any point in a
rotation is not closed, but an equiangular spiral. To exclude this
possibility, Helmholtz's axiom of Monodromy is required.

II. _In three dimensions_, the results go still more against
Helmholtz. Assuming free mobility only _within a certain region_, we
have to distinguish two cases: _Either_ free mobility holds, within
that region, absolutely without exception, _i.e._ when one point is
held fast, _every_ other point within the region can move freely
over a surface: in this case the axiom of Monodromy is unnecessary,
and the first three axioms suffice to define our group as that of
Euclidean and non-Euclidean motions. _Or_ free mobility, within the
specified region, holds only of every point _of general position_,
while the points of a certain line, when one point is fixed, are only
able to move on that line, not on a surface: when this is the case,
other groups are possible, and can only be excluded by Helmholtz's
fourth axiom.

Having now stated the purely mathematical results of Lie's
investigations, we may return to philosophical considerations, by
which Helmholtz's work was mainly motived. It becomes obvious,
not only that exceptions within a certain region, but also that
limitation to a certain region, of the axiom of Free Mobility,
are philosophically quite impossible and inconceivable. How can a
certain line, or a certain surface, form an impassable barrier in
space, or have any mobility different in kind from that of all other
lines or surfaces? The notion cannot, in philosophy, be permitted
for a moment, since it destroys that most fundamental of all the
axioms, the homogeneity of space. We not only may, therefore, but
must take Helmholtz's axiom of Free Mobility in its very strictest
sense; the axiom of Monodromy thus becomes mathematically, as well
as philosophically, superfluous. This is, from a philosophical
standpoint, the most important of Lie's results.

=46.= I have now come to the end of my history of Metageometry.
It has not been my aim to give an exhaustive account of even the
important works on the subject--in the third period, especially, the
names of Poincaré, Pasch, Cremona, Veronese, and others who might be
mentioned, would have cried shame upon me, had I had any such object.
But I have tried to set forth, as clearly as I could, the principles
at work in the various periods, the motives and results of successive
theories. We have seen how the philosophical motive, at first
predominant, has been gradually extruded by the purely mathematical
and technical spirit of most recent Geometers. At first, to discredit
the Transcendental Aesthetic seemed, to Metageometers, as important
as to advance their science; but from the works of Cayley, Klein
or Lie, no reader could gather that Kant had ever lived. We have
also seen, however, that as the interest _in_ philosophy waned, the
interest _for_ philosophy increased: as the mathematical results
shook themselves free from philosophical controversies, they
assumed gradually a stable form, from which further development,
we may reasonably hope, will take the form of growth, rather than
transformation. The same gradual development out of philosophy might,
I believe, be traced in the infancy of most branches of mathematics;
when philosophical motives cease to operate, this is, in general, a
sign that the stage of uncertainty as to premisses is past, so that
the future belongs entirely to mathematical technique. When this
stable stage has been attained, it is time for Philosophy to borrow
of Science, accepting its final premisses as those imposed by a real
necessity of fact or logic.

=47.= Now in discussing the systems of Metageometry, we have found
two kinds, radically distinct and subject to different axioms. The
historically prior kind, which deals with metrical ideas, discusses,
to begin with, the conditions of Free Mobility, which is essential to
all measurement of space. It finds the analytical expression of these
conditions in the existence of a space-constant, or constant measure
of curvature, which is equivalent to the homogeneity of space. This
is its first axiom.

Its second axiom states that space has a finite integral number of
dimensions, _i.e._ in metrical terms, that the position of a point,
relative to any other figure in space, is uniquely determined by a
finite number of spatial magnitudes, called coordinates.

The third axiom of metrical Geometry may be called, to distinguish it
from the corresponding projective axiom, the axiom of distance. There
exists one relation, it says, between any two points, which can be
preserved unaltered in a combined motion of both points, and which,
in any motion of a system as one rigid body, is always unaltered.
This relation we call distance.

The above statement of the three essential axioms of metrical
Geometry is taken from Helmholtz as amended by Lie. Lie's own
statement of the axioms, as quoted above, has been too much
influenced by projective methods to give a historically correct
rendering of the spirit of the second period; Helmholtz's statement,
on the other hand, requires, as Lie has shewn, very considerable
modifications. The above compromise may, therefore, I hope be taken
as accepting Lie's corrections while retaining Helmholtz's spirit.

=48.= But metrical Geometry, though it is historically prior, is
logically subsequent to projective Geometry. For projective Geometry
deals directly with that qualitative likeness, which the judgment of
quantitative comparison requires as its basis. Now the above three
axioms of metrical Geometry, as we shall see in Chapter III. Section
B, do not presuppose measurement, but are, on the contrary, the
conditions presupposed by measurement. Without these axioms, which
are common to all three spaces, measurement would be impossible;
with them, so I shall contend, measurement is able, though only
empirically, to decide approximately which of the three spaces is
valid of our actual world. But if these three axioms themselves
express, not results, but conditions, of measurement, must they not
be equivalent to the statement of that qualitative likeness on which
quantitative comparison depends? And if so, must we not expect to
find the same axioms, though perhaps under a different form, in
projective Geometry?

=49.= This expectation will not be disappointed. The above three
axioms, as we shall see hereafter, are one and all philosophically
equivalent to the homogeneity of space, and this in turn is
equivalent to the axioms of projective Geometry. The axioms of
projective Geometry, in fact, may be roughly stated thus:

I. Space is continuous and infinitely divisible; the zero of
extension, resulting from infinite division, is called a Point. All
points are qualitatively similar, and distinguished by the mere fact
that they lie outside one another.

II. Any two points determine a unique figure, the straight line;
two straight lines, like two points, are qualitatively similar, and
distinguished by the mere fact that they are mutually external.

III. Three points not in one straight line determine a unique figure,
the plane, and four points not in one plane determine a figure of
three dimensions. This process may, so far as can be seen _à priori_,
be continued, without in any way interfering with the possibility
of projective Geometry, to five or to _n_ points. But projective
Geometry requires, as an axiom, that the process should stop with
some positive integral number of points, after which, any fresh point
is contained in the figure determined by those already given. If the
process stops with (_n_ + 1) points, our space is said to have _n_
dimensions.

These three axioms, it will be seen, are the equivalents of the
three axioms of metrical Geometry[65], expressed without reference
to quantity. We shall find them to be deducible, as before, from
the homogeneity of space, or, more generally still, from the
possibility of experiencing externality. They will therefore appear
as _à priori_, as essential to the existence of any Geometry and to
experience of an external world as such.

=50.= That some logical necessity is involved in these axioms might,
I think, be inferred as probable, from their historical development
alone. For the systems of Metageometry have not, in general, been set
up as more likely to fit facts than the system of Euclid; with the
exception of Zöllner, for example, I know of no one who has regarded
the fourth dimension as required to explain phenomena. As regards the
space-constant again, though a _small_ space-constant is regarded
as empirically possible, it is not usually regarded as probable;
and the finite space-constants, with which Metageometry is equally
conversant, are not usually thought even possible, as explanations
of empirical fact[66]. Thus the motive has been throughout not one
of fact, but one of logic. Does not this give a strong presumption,
that those axioms which are retained, are retained because they are
logically indispensable? If this be so, the axioms common to Euclid
and Metageometry will be _à priori_, while those peculiar to Euclid
will be empirical. After a criticism of some differing theories of
Geometry, I shall proceed, in Chapters III. and IV., to the proof and
consequences of this thesis, which will form the remainder of the
present work.

FOOTNOTES:

[5] V. Mémoires de l'Académie royale des Sciences de l'lnstitut de
France, T. XII. 1833, for a full statement of his results, with
references to former writings.

[6] This bolder method, it appears, had been suggested, nearly a
century earlier, by an Italian, Saccheri. His work, which seems to
have remained completely unknown until Beltrami rediscovered it in
1889, is called "Euclides ab omni naevo vindicatus, etc." Mediolani,
1733. (See Veronese, Grundzüge der Geometrie, German translation,
Leipzig, 1894, p. 636.) His results included spherical as well as
hyperbolic space; but they alarmed him to such an extent that he
devoted the last half of his book to disproving them.

[7] Klein's first account of elliptic Geometry, as a result of
Cayley's projective theory of distance, appeared in two articles
entitled "Ueber die sogenannte Nicht-Euklidische Geometrie, I,
II," Math. Annalen 4, 6 (1871-2). It was afterwards independently
discovered by Newcomb, in an article entitled "Elementary Theorems
relating to the geometry of a space of three dimensions, and of
uniform positive curvature in the fourth dimension," Crelle's
Journal für die reine und angewandte Mathematik, Vol. 83 (1877). For
an account of the mathematical controversies concerning elliptic
Geometry, see Klein's "Vorlesungen über Nicht-Euklidische Geometrie,"
Göttingen 1893, I. p. 284 ff. A bibliography of the relevant
literature up to the year 1878 was given by Halsted in the American
Journal of Mathematics, Vols. 1, 2.

[8] Veronese (op. cit. p. 638) denies the priority of Gauss in the
invention of a non-Euclidean system, though he admits him to have
been the first to regard the axiom of parallels as indemonstrable.
His grounds for the former assertion seem scarcely adequate: on the
evidence against it, see Klein, Nicht-Euklid, I. pp. 171-174.

[9] V. Briefwechsel mit Schumacher, Bd. II. p. 268.

[10] f. Helmholtz, Wiss. Abh. II. p. 611.

[11] Crelle's Journal, 1837.

[12] Theorie der Parallellinien, Berlin, 1840. Republished, Berlin,
1887. Translated by Halsted, Austin, Texas, U.S.A. 4th edition, 1892.

[13] Frischauf, Absolute Geometrie, nach Johann Bolyai, Leipzig,
1872. Halsted, The Science Absolute of Space, translated from the
Latin, 4th edition, Austin, Texas, U.S.A. 1896.

[14] Both Lobatchewsky and Bolyai, as Veronese remarks, start rather
from the point-pair than from distance. See Frischauf, Absolute
Geometrie, Anhang.

[15] Compare Stallo, Concepts of Modern Physics, p. 248.

[16] Gesammelte Werke, pp. 255-268.

[17] On the history of this word, see Stallo, Concepts of Modern
Physics, p. 258. It was used by Kant, and adapted by Herbart to
almost the same meaning as it bears in Riemann. Herbart, however,
also uses the word _Reihenform_ to express a similar idea. See
Psychologie als Wissenschaft, I. § 100 and II. § 139, where Riemann's
analogy with colours is also suggested.

[18] Compare Erdmann's "Grössenbegriff vom Raum."

[19] Compare Veronese, op. cit. p. 642: "Riemann ist in seiner
Definition des Begriffs Grösse dunkel." See also Veronese's whole
following criticism.

[20] Vorträge und Reden, Vol. II. p. 18.

[21] Cf. Klein, Nicht-Euklid, I. p. 160.

[22] Since we are considering the curvature at a point, we are only
concerned with the first infinitesimal elements of the geodesics that
start from such a point.

[23] Disquisitiones generales circa superficies curvas, Werke, Bd.
IV. SS. 219-258, 1827.

[24] Nevertheless, the Geometries of different surfaces of equal
curvature are liable to important differences. For example, the
cylinder is a surface of zero curvature, but since its lines of
curvature in one direction are finite, its Geometry coincides with
that of the plane only for lengths smaller than the circumference of
its generating circle (see Veronese, op. cit. p. 644). Two geodesics
on a cylinder may meet in many points. For surfaces of zero curvature
on which this is not possible, the identity with the plane may
be allowed to stand. Otherwise, the identity extends only to the
properties of figures not exceeding a certain size.

[25] For we may consider two different parts of the same surface as
corresponding parts of different surfaces; the above proposition then
shows that a figure can be reproduced in one part when it has been
drawn in another, if the measures of curvature correspond in the two
parts.

[26] Crelle, Vols, XIX., XX., 1839-40.

[27] In this formula, _u_, _v_ may be the lengths of lines, or the
angles between lines, drawn on the surface, and having thus no
necessary reference to a third dimension.

[28] In what follows, I have given rather Klein's exposition of
Riemann, than Riemann's own account. The former is much clearer
and fuller, and not substantially different in any way. V. Klein,
Nicht-Euklid, I. pp. 206 ff.

[29] See §§ 69-73.

[30] Grundlagen der Geometrie, I. and II., Leipziger Berichte, 1890;
v. end of present chapter, § 45.

[31] Nicht-Euklid, I. pp. 258-9.

[32] Giornale di Matematiche, Vol. VI., 1868. Translated into
French by J. Hoüel in the "Annales Scientifiques de l'École Normale
Supérieure," Vol. VI. 1869.

[33] Crelle's Journal, Vols. XIX. XX., 1839-40.

[34] Nicht-Euklid, I. p. 190.

[35] This article is more trigonometrical and analytical than the
German book, and therefore makes the above interpretation peculiarly
evident.

[36] Such surfaces are by no means particularly remote. One of them,
for example, is formed by the revolution of the common Tractrix

    x = asin φ,    y = a(log tan φ/2 + cos φ).

[37] "Teoria fondamentale degli spazii di curvatura costanta," Annali
di Matematica, II. Vol. 2, 1868-9. Also translated by J. Hoüel, _loc.
cit._

[38] See Klein, Nicht-Euklid, I. p. 47 ff., and the references there
given.

[39] See quotation below, from his British Association Address.

[40] Compare the opening sentence, due to Cayley, of Salmon's Higher
Plane Curves.

[41] V. Nicht-Euklid, I. Chaps. I. and II.

[42] See p. 9 of Cayley's address to the Brit. Ass. 1883. Also a
quotation from Klein in Erdmann's Axiome der Geometrie, p. 124 note.

[43] Nature, Vol. XLV. p. 407.

[44] Nicht-Euklid, I. p. 200.

[45] I.e. the equation _AB_ + _BC_ = _AC_, for three points in one
straight line.

[46] The formula substituted by Klein for Cayley's inverse sine or
cosine. The two are equivalent, but Klein's is mathematically much
the more convenient.

[47] Elements of Projective Geometry, Second Edition, Oxford, 1893,
Chap. IX.

[48] Chap. III. Section B.

[49] See Nicht-Euklid, I. p. 338 ff.

[50] See his Geometrie der Lage, § 8, Harmonische Gebilde.

[51] The anharmonic ratio of four numbers, _p_, _q_, _r_, _s_, is
defined as

    (p - q).(r - s) / (p - r).(q - s).

[52] _I.e._ as transformable into each other by a collineation. See
Chap. III. Sec. A, § 110.

[53] See Chap. III. Sec. A.

[54] It follows from this, that the reduction of metrical to
projective properties, even when, as in hyperbolic Geometry, the
Absolute is real, is only apparent, and has a merely technical
validity.

[55] Sir R. Ball does not regard his non-Euclidean content as a
possible space (_v. op. cit._ p. 151). In this important point I
disagree with his interpretation, holding such a content to be a
space as possible, _à priori_, as Euclid's, and perhaps actually true
within the margin due to errors of observation.

[56] See Nicht-Euklid, I. p. 97 ff. and p. 292 ff.

[57] Newcomb says (_loc. cit._ p. 293): "The system here set forth is
founded on the following three postulates.

"1. I assume that space is triply extended, unbounded, without
properties dependent either on position or direction, and possessing
such planeness in its smallest parts that both the postulates of the
Euclidean Geometry, and our common conceptions of the relations of
the parts of space are true for every indefinitely small region in
space.

"2. I assume that this space is affected with such curvature that a
right line shall always return into itself at the end of a finite and
real distance 2_D_ without losing, in any part of its course, that
symmetry with respect to space on all sides of it which constitutes
the fundamental property of our conception of it.

"3. I assume that if two right lines emanate from the same point,
making the indefinitely small angle _a_ with each other, their
distance apart at the distance _r_ from the point of intersection
will be given by the equation

    s = 2aD/π sin rπ/2D.

The right line thus has this property in common with the Euclidean
right line that two such lines intersect only in a single point.
It may be that the number of points in which two such lines can
intersect admit of being determined from the laws of curvature,
but not being able so to determine it, I assume as a postulate the
fundamental property of the Euclidean right line."

It is plain that in the absence of the determination spoken of, the
possibility of elliptic space is not established. It may be possible,
for example, to prove that, in a space where there is a maximum to
distance, there must be an infinite number of straight lines joining
two points of maximum distance. In this event, elliptic space would
become impossible.

[58] For an elucidation of this term, see Klein, Nicht-Euklid, I. p.
99 ff.

[59] Cf. p. 9 of Report: "My own view is that Euclid's twelfth axiom,
in Playfair's form of it, does not need demonstration, but is part
of our notion of space, of the physical space of our experience,
but which is the representation lying at the bottom of all external
experience."

[60] The exception to this axiom, in spherical space, presupposes
metrical Geometry, and does not destroy the validity of the axiom for
projective Geometry. See Chap. III. Sec. B, § 171.

[61] Mathematicians of Lie's school have a habit, at first somewhat
confusing, of speaking of motions of space instead of motions of
bodies, as though space as a whole could move. All that is meant is,
of course, the equivalent motion of the coordinate axes, _i.e._ a
change of axes in the usual elementary sense.

[62] "Ueber die Grundlagen der Geometrie," Leipziger Berichte, 1890.
The problem of these two papers is really metrical, since it is
concerned, not with collineations in general, but with motions. The
problem, however, is dealt with by the projective method, motions
being regarded as collineations which leave the Absolute unchanged.
It seemed impossible, therefore, to discuss Lie's work, until some
account had been given of the projective method.

[63] Lie's premisses, to be accurate, are the following:

Let

    x{1} = f(x, y, z, a{1}, a{2}...)
    x{2} = φ(x, y, z, a{1}, a{2}...)
    x{3} = ψ(x, y, z, a{1}, a{2}...)

give an infinite family of real transformations of space, as to which
we make the following hypotheses:

A. The functions f, φ, ψ, are _analytical_ functions of

    x, y, z, a{1}, a{2}....

B. Two points x{1}y{1}z{1}, x{2}y{2}z{2} possess an
invariant, _i.e._

    Ω(x{1}, y{1}, z{1}, x{2}, y{2}, z{2}) =
        Ω(x{1′}, y{1′}, z{1′}, x{2′}, y{2′}, z{2′})

where x{1′}..., x{2′}..., are the transformed coordinates of
the two points.

C. Free Mobility: _i.e._, any point can be moved into any other
position; when one point is fixed, any other point of general
position can take up ∞^{2} positions; when two points are fixed, any
other of general position can take up ∞^{1} positions; when three, no
motion is possible--these limitations being results of the equations
given by the invariant Ω.

[64] On this point, cf. Klein, Höhere Geometrie, Göttingen, 1893, II.
pp. 225-244, especially pp. 230-1.

[65] Axiom II. of the metrical triad corresponds to Axiom III. of the
projective, and _vice versâ_.

[66] Cf. Helmholtz, Wiss. Abh. Vol. II. p. 640, note: "Die Bearbeiter
der Nicht-Euklidischen Geometrie (haben) deren objective Wahrheit nie
behauptet."




CHAPTER II.

CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.


=51.= We have now traced the mathematical development of the theory
of geometrical axioms, from the first revolt against Euclid to the
present day. We may hope, therefore, to have at our command the
technical knowledge required for the philosophy of the subject. The
importance of Geometry, in the theories of knowledge which have
arisen in the past, can scarcely be exaggerated. In Descartes, we
find the whole theory of method dominated by analytical Geometry, of
whose fruitfulness he was justly proud. In Spinoza, the paramount
influence of Geometry is too obvious to require comment. Among
mathematicians, Newton's belief in absolute space was long supreme,
and is still responsible for the current formulation of the laws of
motion. Against this belief on the one hand, and against Leibnitz's
theory of space on the other, and not, as Caird has pointed out[67],
against Hume's empiricism, was directed that keystone of the Critical
Philosophy, the Kantian doctrine of space. Thus Geometry has been,
throughout, of supreme importance in the theory of knowledge.

But in a criticism of representative modern theories of Geometry,
which is designed to be, not a history of the subject, but an
introduction to, and defence of, the views of the author, it will
not be necessary to discuss any more ancient theory than that of
Kant. Kant's views on this subject, true or false, have so dominated
subsequent thought, that whether they were accepted or rejected,
they seemed equally potent in forming the opinions, and the manner of
exposition, of almost all later writers.


Kant.

=52.= It is not my purpose, in this chapter, to add to the voluminous
literature of Kantian criticism, but only to discuss the bearing of
Metageometry on the argument of the Transcendental Aesthetic, and
the aspect under which this argument must be viewed in a discussion
of Geometry[68]. On this point several misunderstandings seem to me
to have had wide prevalence, both among friends and foes, and these
misunderstandings I shall endeavour, if I can, to remove.

In the first place, what does Kant's doctrine mean for Geometry?
Obviously not the aspect of the doctrine which has been attacked by
psychologists, the "Kantian machine-shop" as James calls it--at any
rate, if this can be clearly separated from the logical aspect. The
question whether space is given in sensation, or whether, as Kant
maintained, it is given by an intuition to which no external matter
corresponds, may for the present be disregarded. If, indeed, we
held the view which seems crudely to sum up the standpoint of the
Critique, the view that all certain knowledge is self-knowledge,
then we should be committed, if we had decided that Geometry was
apodeictic, to the view that space is subjective. But even then, the
psychological question could only arise when the epistemological
question had been solved, and could not, therefore, be taken into
account in our first investigation. The question before us is
precisely the question whether, or how far, Geometry is apodeictic,
and for the moment we have only to investigate this question, without
fear of psychological consequences.

=53.= Now on this question, as on almost all questions in the
Aesthetic or the Analytic, Kant's argument is twofold. On the one
hand, he says, Geometry is known to have apodeictic certainty:
therefore space must be _à priori_ and subjective. On the other hand,
it follows, from grounds independent of Geometry, that space is
subjective and _à priori_; therefore Geometry must have apodeictic
certainty. These two arguments are not clearly distinguished in the
Aesthetic, but a little analysis, I think, will disentangle them.
Thus in the first edition, the first two arguments deduce, from
non-geometrical grounds, the apriority of space; the third deduces
the apodeictic certainty of Geometry, and maintains, conversely,
that no other view can account for this certainty[69]; the last two
arguments only maintain that space is an intuition, not a concept.
In the second edition, the double argument is clearer, the apriority
of space being proved independently of Geometry in the metaphysical
deduction, and deduced from the certainty of Geometry, as the only
possible explanation of this, in the transcendental deduction. In the
Prolegomena, the latter argument alone is used, but in the Critique
both are employed.

=54.= Now it must be admitted, I think, that Metageometry has
destroyed the legitimacy of the argument from Geometry to space; we
can no longer affirm, on purely geometrical grounds, the apodeictic
certainty of Euclid. But unless Metageometry has done more than
this--unless it has proved, what I believe it alone cannot prove,
that Euclid has _not_ apodeictic certainty--then Kant's other line
of argument retains what force it may ever have had. The actual
space we know, it may say, is admittedly Euclidean, and is proved,
without any reference to Geometry, to be _à priori_; _hence_ Euclid
has apodeictic certainty, and non-Euclid stands condemned. To this
it is no answer to urge, with the Metageometers, that non-Euclidean
systems are _logically_ self-consistent; for Kant is careful to argue
that geometrical reasoning, by virtue of our intuition of space, is
synthetic, and cannot, though _à priori_, be upheld by the principle
of contradiction alone[70]. Unless non-Euclideans can prove, what
they have certainly failed to prove up to the present, that we can
frame an _intuition_ of non-Euclidean spaces, Kant's position cannot
be upset by Metageometry alone, but must also be attacked, if it is
to be successfully attacked, on its purely philosophical side.

=55.= For such an attack, two roads lie open: either we may disprove
the first two arguments of the Aesthetic, or we may criticize, from
the standpoint of general logic, the Kantian doctrine of synthetic
_à priori_ judgments and their connection with subjectivity. Both
these attacks, I believe, could be conducted with some success;
but if we are to disprove the apodeictic certainty of Geometry,
one or other is essential, and both, I believe, will be found only
partially successful. It will be my aim to prove, in discussing
these two lines of attack, (1) that the distinction of synthetic and
analytic judgments is untenable, and further, that the principle of
contradiction can only give fruitful results on the assumption that
experience in general, or, in a particular science, some special
branch of experience, is to be formally possible; (2) that the first
two arguments of the Transcendental Aesthetic suffice to prove,
not Euclidean space, but _some_ form of externality--which may be
sensational or intuitional, but not merely conceptual--a necessary
prerequisite of experience of an external world. In the third and
fourth chapters, I shall contend, as a result of these conclusions,
that those axioms, which Euclid and Metageometry have in common,
coincide with those properties of any form of externality which are
deducible, by the principle of contradiction, from the possibility of
experience of an external world. These properties, then, may be said,
though not quite in the Kantian sense, to be _à priori_ properties of
space, and as to these, I think, a modified Kantian position may be
maintained. But the question of the subjective or objective nature
of space may be left wholly out of account during the course of this
discussion, which will gain by dealing exclusively with logical, as
opposed to psychological points of view.

=56.= (1) _Kant's logical position._ The doctrine of synthetic and
analytic judgments--at any rate if this is taken as the corner-stone
of Epistemology--has been so completely rejected by most modern
logicians[71], that it would demand little attention here, but
for the fact that an enthusiastic French Kantian, M. Renouvier,
has recently appealed to it, with perfect confidence, on the very
question of Geometry[72]. And it must be owned, with M. Renouvier,
that if such judgments existed, in the Kantian sense, non-Euclidean
Geometry, which makes no appeal to intuition, could have nothing to
say against them. M. Renouvier's contention, therefore, forces us
briefly to review the arguments against Kant's doctrine, and briefly
to discuss what logical canon is to replace it.

Every judgment--so modern logic contends--is both synthetic and
analytic; it combines parts into a whole, and analyses a whole into
parts[73]. If this be so, the distinction of analysis and synthesis,
whatever may be its importance in pure Logic, can have no value in
Epistemology. But such a doctrine, it must be observed, allows full
scope to the principle of contradiction: this criterion, since all
judgments, in one aspect at least, are analytic, is applicable to
all judgments alike. On the other hand, the whole which is analysed
must be supposed already given, before the parts can be mutually
contradictory: for only by connection in a given whole can two parts
or adjectives be incompatible. Thus the principle of contradiction
remains barren until we already have some judgments, and even some
inference: for the parts may be regarded, to some extent, as an
inference from the whole, or _vice versâ_. When once the arch of
knowledge is constructed, the parts support one another, and the
principle of contradiction is the keystone: but until the arch is
built, the keystone remains suspended, unsupported and unsupporting,
in the empty air. In other words, knowledge once existent can be
analysed, but knowledge which should have to win every inch of the
way against a critical scepticism, could never begin, and could never
attain that circular condition in which alone it can stand.

But Kant's doctrine, if true, is designed to restrain a critical
scepticism even where it might be effective. Certain fundamental
propositions, he says, are not deducible from logic, _i.e._
their contradictories are not self-contradictory; they combine a
subject and predicate which cannot, in any purely logical way,
be shewn to have any connection, and yet these judgments have
apodeictic certainty. But concerning such judgments, Kant is
generally careful not to rely upon the mere subjective conviction
that they are undeniable: he proves, with every precaution, that
without them experience would be impossible. Experience consists
in the combination of terms which formal logic leaves apart, and
presupposes, therefore, certain judgments by which a framework is
made for bringing such terms together. Without these judgments--so
Kant contends--all synthesis and all experience would be impossible.
If, therefore, the detail of the Kantian reasoning be sound, his
results may be obtained by the principle of contradiction _plus_ the
possibility of experience, as well as by his distinction of synthetic
and analytic judgments.

Logic, at the present day, arrogates to itself at once a wider and a
narrower sphere than Kant allowed to it. Wider, because it believes
itself capable of condemning any false principle or postulate;
narrower, because it believes that its law of contradiction,
without a given whole or a given hypothesis, is powerless, and
that two terms, _per se_, though they may be different, cannot be
contradictories, but acquire this relation only by combination
in a whole about which something is known, or by connection with
a postulate which, for some reason, must be preserved. Thus no
judgment, _per se_, is either analytic or synthetic, for the
severance of a judgment from its context robs it of its vitality, and
makes it not truly a judgment at all. But in its proper context it
is neither purely synthetic nor purely analytic; for while it is the
further determination of a given whole, and thus in so far analytic,
it also involves the emergence of _new_ relations within this whole,
and is so far synthetic.

=57.= We may retain, however, a distinction roughly corresponding to
the Kantian _à priori_ and _à posteriori_, though less rigid, and
more liable to change with the degree of organisation of knowledge.
Kant usually endeavoured to prove, as observed above, that his
synthetic _à priori_ propositions were necessary prerequisites of
experience; now although we cannot retain the term synthetic, we
can retain the term _à priori_, for those assumptions, or those
postulates, from which alone the possibility of experience follows.
Whatever can be deduced from these postulates, without the aid of the
matter of experience, will also, of course, be _à priori_. From the
standpoint of general logic, the laws of thought and the categories,
with the indispensable conditions of their applicability, will be
alone _à priori_; but from the standpoint of any special science,
we may call _à priori_ whatever renders possible the experience
which forms the subject-matter of our science. In Geometry, to
particularize, we may call _à priori_ whatever renders possible
experience of externality as such.

It is to be observed that this use of the term is at once more
rationalistic and less precise than that of Kant. Kant would seem
to have supposed himself immediately aware, by inspection, that
some knowledge was apodeictic, and its subject-matter, therefore,
_à priori_: but he did not always deduce its apriority from any
further principle. Here, however, it is to be shown, before admitting
apriority, that the falsehood of the judgment in question would not
be effected by a mere change in the _matter_ of experience, but only
by a change which should render some branch of experience formally
impossible, _i.e._ inaccessible to our methods of cognition. The
above use is also less precise, for it varies according to the
specialization of the experience we are assuming possible, and with
every progress of knowledge some new connection is perceived, two
previously isolated judgments are brought into logical relation, and
the _à priori_ may thus, at any moment, enlarge its sphere, as more
is found deducible from fundamental postulates.

=58.= (2) _Kant's arguments for the apriority of space._ Having now
discussed the logical canon to be used as regards the _à priori_, we
may proceed to test Kant's arguments as regards space. The argument
from Geometry, as remarked above, is upset by Metageometry, at least
so far as those properties are concerned, which belong to Euclid
but not to non-Euclidean spaces; as regards the common properties
of both kinds of space, we cannot decide on their apriority till we
have discussed the consequences of denying them, which will be done
in Chapter III. As regards the two arguments which prove that space
is an intuition, not a concept, they would call for much discussion
in a special criticism of Kant, but here they may be passed by
with the obvious comment that infinite homogeneous Euclidean space
is a concept, not an intuition--a concept invented to explain an
intuition, it is true, but still a pure concept[74]. And it is this
pure concept which, in all discussions of Geometry, is primarily to
be dealt with; the intuition need only be referred to where it throws
light on the functions or the nature of the concept. The second
of Kant's arguments, that we can imagine empty space, though not
the absence of space, is false if it means a space without matter
anywhere, and irrelevant if it merely means a space between matters
and regarded as empty[75]. The only argument of importance, then,
is the first argument. But I must insist, at the outset, that our
problem is purely logical, and that all psychological implications
must be excluded to the utmost possible extent. Moreover, as will be
proved in Chapter IV., the proper function of space is to distinguish
between different presented things, not between the Self and the
object of sensation or perception. The argument then becomes the
following: consciousness of a world of mutually external things
demands, in presentations, a cognitive but non-inferential element
leading to the discrimination of the objects presented. This element
must be non-inferential, for from whatever number or combination of
presentations, which did not of themselves demand diversity in their
objects, I could never be led to infer the mutual externality of
their objects. Kant says: "In order that sensations may be ascribed
to something external to me ... and similarly in order that I may
be able to present them as outside and beside one another, ... the
presentation of space must be already present." But this goes rather
too far: in the first place, the question should be only as to the
mutual externality of presented things, not as to their externality
to the Self[76]; and in the second place, things will appear mutually
external if I have the presentation of _any_ form of externality,
whether Euclidean or non-Euclidean. Whatever may be true of the
_psychological_ scope of this argument--whose validity is here
irrelevant--the _logical_ scope extends, not to Euclidean space, but
only to any form of externality which could exist intuitively, and
permit knowledge, in beings with our laws of thought, of a world of
diverse but interrelated things.

Moreover externality, to render the scope of the argument wholly
logical, must not be left with a sensational or intuitional meaning,
though it must be supposed given in sensation or intuition. It must
mean, in this argument, the fact of Otherness[77], the fact of being
different from some other thing: it must involve the distinction
between different things, and must be that element, in a cognitive
state, which leads us to discriminate constituent parts in its
object. So much, then, would appear to result from Kant's argument,
that experience of diverse but interrelated things demands, as a
necessary prerequisite, some sensational or intuitional element, in
perception, by which we are led to attribute complexity to objects
of perception[78]; that this element, in its isolation may be called
a form of externality; and that those properties of this form, if
any such be found, which can be deduced from its mere function of
rendering experience of interrelated diversity possible, are to
be regarded as _à priori_. What these properties are, and how the
various lines of argument here suggested converge to a single result,
we shall see in Chapters III. and IV.

=59.= In the philosophers who followed Kant, Metaphysics, for the
most part, so predominated over Epistemology, that little was added
to the theory of Geometry. What was added, came indirectly from
the one philosopher who stood out against the purely ontological
speculations of his time, namely _Herbart_. Herbart's actual views on
Geometry, which are to be found chiefly in the first section of his
_Synechologie_, are not of any great value, and have borne no great
fruit in the development of the subject. But his psychological theory
of space, his construction of extension out of series of points, his
comparison of space with the tone and colour-series, his general
preference for the discrete above the continuous, and finally his
belief in the great importance of classifying space with other forms
of series (_Reihenformen_[79]), gave rise to many of Riemann's
epoch-making speculations, and encouraged the attempt to explain the
nature of space by its analytical and quantitative aspect alone[80].
Through his influence on Riemann, he acquired, indirectly, a great
importance in geometrical philosophy. To Riemann's dissertation,
which we have already discussed in its mathematical aspect, we must
now return, considering, this time, only its philosophical views.


Riemann.

=60.= The aim of Riemann's dissertation, as we saw in Chapter I.,
was to define space as a species of manifold, _i.e._ as a particular
kind of collection of magnitudes. It was thus assumed, to begin with,
that spatial figures could be regarded as magnitudes, and the axioms
which emerged, accordingly, determined only the particular place of
these among the many algebraically possible varieties of magnitudes.
The resulting formulation of the axioms--while, from the mathematical
standpoint of metrical Geometry, it was almost wholly laudable--must,
from the standpoint of philosophy, be regarded, in my opinion, as a
_petitio principii_. For when we have arrived at regarding spatial
figures as magnitudes, we have already traversed the most difficult
part of the ground. The axioms of metrical Geometry--and it is
metrical Geometry, exclusively, which is considered in Riemann's
Essay--will appear, in Chapter III., to be divisible into two
classes. Of these, the first class--which contains the axioms common
to Euclid and Metageometry, the only axioms seriously discussed by
Riemann--are not the results of measurement, nor of any conception
of magnitude, but are conditions to be fulfilled before measurement
becomes possible. The second class only--those which express the
difference between Euclidean and non-Euclidean spaces--can be
deduced as results of measurement or of conceptions of magnitude.
As regards the first class, on the contrary, we shall see that
the relativity of position--by which space is distinguished from
all other known manifolds, except time--leads logically to the
necessity of three of the most distinctive axioms of Geometry, and
yet this relativity cannot be called a deduction from conceptions of
magnitude. In analytical Geometry, owing to the fact that coordinate
systems start from points, and hence build up lines and surfaces, it
is easy to suppose that points can be given independently of lines
and of each other, and thus the relativity of position is lost sight
of. The error thus suggested by mathematics was probably reinforced
by Herbart's theory of space, which, by its serial character, as
we have seen, appeared to him to facilitate a construction out of
successive points, and to which Riemann acknowledges his indebtedness
both in his Dissertation and elsewhere. The same error reappears
in Helmholtz, in whom it is probably due wholly to the methods of
analytical Geometry. It is a striking fact that, throughout the
writings of these two men, there is not, so far as I know, one
allusion to the relativity of position, that property of space
from which, as our next chapter will shew, the richest quarry of
consequences can be extracted. This is not a result of any conception
of magnitude, but follows from the nature of our space-intuition; yet
no one, surely, could call it empirical, since it is bound up in the
very possibility of locating things _there_ as opposed to _here_.

=61.= Indeed we can see, from a purely logical consideration of
the judgment of quantity, that Riemann's manner of approaching the
problem can never, by legitimate methods, attain to a philosophically
sound formulation of the axioms. For quantity is a result of
comparison of two qualitatively similar objects, and the judgment of
quantity neglects altogether the qualitative aspect of the objects
compared. Hence a knowledge of the essential properties of space can
never be obtained from judgments of quantity, which neglect these
properties, while they yet presuppose them. As well might one hope
to learn the nature of man from a census. Moreover, the judgment of
quantity is the result of comparison, and therefore presupposes
the possibility of comparison. To know whether, or by what means,
comparison is possible, we must know the qualities of the things
compared and of the medium in which comparison is effected; while to
know that _quantitative_ comparison is possible, we must know that
there is a qualitative identity between the things compared, which
again involves a previous qualitative knowledge. When spatial figures
have once been reduced to quantity, their quality has already been
neglected, as known and similar to the quality of other figures. To
hope, therefore, for the qualities of space, from a comparison of its
expression as pure quantity with other pure quantities, is an error
natural to an analytical geometer, but an error, none the less, from
which there is no return to the qualitative basis of spatial quantity.

=62.= We must entirely dissent, therefore, from the disjunction which
underlies Riemann's philosophy of space. Either the axioms must be
consequences of general conceptions of magnitude, he thinks, or
else they can only be proved by experience (p. 255). Whatever _can_
be derived from general conceptions of magnitude, we may retort,
cannot be an _à priori_ adjective of space: for all the necessary
adjectives of space are presupposed in any judgment of spatial
quantity, and cannot, therefore, be consequences of such a judgment.
Riemann's disjunction, accordingly, since one of its alternatives is
obviously impossible, really begs the question. In formulating the
axioms of metrical Geometry, our question should be: What axioms,
_i.e._ what adjectives of space, must be presupposed, in order that
quantitative comparison of the parts of space may be possible at
all? And only when we have determined these conditions, which are
_à priori_ necessary to any quantitative science of space, does the
second question arise: what inferences can we draw, as to space,
from the observed results of this quantitative science, _i.e._ of
this measurement of spatial figures? The conditions of measurement
themselves, though not results of any conception of magnitude, will
be _à priori_, if it can be shown that, without them, experience of
externality would be impossible.

After this initial protest against Riemann's general philosophical
position, let us proceed to examine, in detail, his use of the notion
of a manifold.

=63.= In the first place there is, if I am not mistaken, considerable
obscurity in the definition of a manifold, of which an almost verbal
rendering was given in Chapter I. What is meant, to begin with, by
a general conception capable of various determinations? Does not
this property belong to all conceptions? It affords, certainly,
a basis for counting, but if continuous quantity is to arise, we
must, surely, have some less discrete formulation. It might afford
a basis, for example, for the distinction of points in projective
Geometry, but projective Geometry has nothing to do with quantity.
Something more fluid and flexible than a conception, one would think,
is necessary as the basis of continua. Then, again, what is meant by
a quantum of a manifold? In space, the answer is obvious: what is
meant is a piece of volume. But how about Riemann's other continuous
manifold, colour? Does a quantum of colour mean a single line in the
spectrum, or a band of finite thickness? In either case, what are
the magnitudes to be compared? And how is superposition necessary,
or even possible? A colour is fixed by its position in the spectrum:
two lines in the same spectrum cannot be superposed, and two lines in
different spectra need not be--their positions in their respective
spectra suffice, or even, roughly, their immediate sense-quality. The
fact is, Riemann had space in his mind from the start, and many of
the properties, which he enunciates as belonging to all manifolds,
belong, as a matter of fact, only to space. It is far from clear what
the magnitudes are which the various determinations make possible.
Do these magnitudes measure the elements of the manifold, or the
relations between elements? This is surely a very fundamental point,
but it is one which Riemann never touches on. In the former case,
the superposition which he speaks of becomes unnecessary, since the
magnitude is inherent in the element considered. We do not require
superposition to measure quantities corresponding to different tones
or colours; these can be discovered by analysis of single tones or
colours. With space, on the other hand, if we seek for elements,
we can find none except points, and no analysis of a point will
find magnitudes inherent in it--such magnitudes are a fiction of
coordinate Geometry. The magnitudes which space deals with, as we
shall see in Chapter III., are relations between points, and it is
for this reason that superposition is essential to space-measurement.
There is no inherent quality in a single point, as there is in a
single colour, by which it can be quantitatively distinguished from
another. Thus the conception of a manifold, as defined by Riemann,
either does not include colours, or does not involve superposition as
the only means of measurement. From this dilemma there is no escape.

=64.= But if "measurement _consists_ in a superposition of the
magnitudes compared" (p. 256), does it not follow immediately that
measurement is logically possible _only_ where such superposition
leaves the magnitudes unchanged? And therefore that measurement, as
above defined, involves, as an _à priori_ condition, that magnitudes
are unchanged by motion? This consequence is not drawn by Riemann;
indeed he proceeds immediately (pp. 256-7) to consider what he calls
a general portion of the doctrine of magnitude (_Grössenlehre_),
independent of measurement. But how is any doctrine of magnitude
possible, in which the magnitudes cannot be measured? The reason
of the confusion is, that Riemann's definition of measurement is
applicable to no single manifold except space, since it depends
on the noteworthy property that what we measure in Geometry is
not points, but relations between points, and the latter, though
not the former, may of course be unaltered by motion. Let us try,
in illustration, to apply Riemann's definition of measurement to
colours. We must remember that motion, in dealing with the colour
manifold, means--not motion in space but--motion in the colour
manifold itself. Now since every point of the colour manifold is
completely determined by three magnitudes, which are given in fact,
and cannot be arbitrarily chosen, it is plain that measurement
by superposition--involving, as it does, motion, and therefore
change in these determining magnitudes--is totally out of the
question. The superposition of one colour on another, as a means
of measurement, is sheer nonsense. And yet measurement is possible
in the colour-manifold, by means of Helmholtz's law of mixture
(_Mischungsgesetz_); but the measurement is of every separate
element, not of the relations between elements, and is thus radically
different from space-measurement[81]. The elements are not, like
points in space, qualitatively alike, and distinguished by the
mere fact of their mutual externality. What we have, in colours, is
three fundamental qualitatively distinct elements, out of certain
proportions of which we can build up all the other elements of the
manifold--each of the resulting elements having the same combination
of qualitative diversity and similarity as the three original
elements. But in space, what could we make of such a procedure? Given
three points, how are we to combine them in certain proportions? The
phrase is meaningless. If some one makes the obvious retort, that we
have to combine lines, not points, my rejoinder is equally obvious.
To begin with, lines are not elements. Metaphysically, space has _no_
elements, being, as the sequel will show, mere relations between
non-spatial elements. Mathematically, this fact exhibits itself in
the self-contradictory notion of the point, or zero magnitude in
space, as the limit in our vain search for spatial elements. But
even if we allow the line to pass as the spatial element, what does
the combination of three lines in definite proportions give us? It
gives us, simply, the coordinates of a _point_. Here again we see a
great difference between the colour and space-manifolds. In colours,
the combination of magnitudes gives a new magnitude of the same
kind; in space, it defines, not a magnitude at all, but a would-be
element of a different kind from the defining magnitudes. In the
tone-manifold, we should find still different conditions. Here, no
one of the measuring magnitudes can vanish without the tone vanishing
too, and all three are so bound up together, in the single resulting
sensation, that none can exist without a finite quantity of the
others. They are all qualitatively different, both from each other,
and from any possible tone, being constituents of it, as mass and
velocity are constituents of momentum. All these different conditions
require to be examined, before a manifold can be completely defined;
and until we have conducted such an examination in detail, we cannot
pronounce as to the _à priori_ or empirical nature of the laws of the
manifold. As regards space, I have attempted such an examination in
the third and fourth chapters of this Essay.

=65.= I do not wish to deny, however, the great value of the
conception of space as a manifold. On the contrary, this conception
seems to have become essential to any treatment of the question.
I only wish to urge that the purely algebraical treatment of any
manifold, important as it may be in deducing fresh consequences from
known premisses, tends rather to conceal than to make clear the
basis of the premisses themselves, and is therefore misleading in a
philosophical investigation. For mathematics, where quantity reigns
supreme, Riemann's conception has proved itself abundantly fruitful;
for philosophy, on the contrary, where quantity appears rather as
a cloak to conceal the qualities it abstracts from, the conception
seems to me more productive of error and confusion than of sound
doctrine.

We are thus brought back to the point from which we started, namely,
the falsity of Riemann's initial disjunction, and the consequent
fallacy in his proof of the empirical nature of the axioms. His
philosophy is chiefly vitiated, to my mind, by this fallacy, and
by the uncritical assumption that a metrical coordinate system can
be set up independently of any axioms as to space-measurement[82].
Riemann has failed to observe, what I have endeavoured to prove in
the next chapter, that, unless space had a strictly constant measure
of curvature, Geometry would become impossible; also that the absence
of constant measure of curvature involves absolute position, which is
an absurdity. Hence he is led to the conclusion that all geometrical
axioms are empirical, and may not hold in the infinitesimal, where
observation is impossible. Thus he says (p. 267): "Now the empirical
conceptions, on which spatial measurements are based, the conceptions
of the rigid body and the light-ray, appear to lose their validity
in the infinitesimal: it is therefore quite conceivable that
the relations of spatial magnitudes in the infinitesimal do not
correspond to the presuppositions of Geometry, and this would, in
fact, have to be assumed, as soon as it would enable us to explain
the phenomena more simply." From this conclusion I must entirely
dissent. In very large spaces, there might be a departure from
Euclid; for they depend upon the axiom of parallels, which is not
contained in the axiom of Free Mobility; but in the infinitesimal,
departures from Euclid could only be due to the absence of Free
Mobility, which, as I hope my third chapter will show, is once for
all impossible.


Helmholtz.

=66.= Helmholtz, like Riemann, was important both in the mathematics
and in the philosophy of Geometry. From the mathematical point
of view, his work has been already considered in Chapter I.; the
consideration of his philosophy, which must occupy us here, will
be a more serious task. Like Riemann, he endeavoured to prove that
all the axioms are empirical, and like Riemann, he based his proof
chiefly on Metageometry. He had an additional resource, however,
in the physiology of the senses, which first led him to reject the
Transcendental Aesthetic, and enabled him to attack Kant from the
psychological as well as the mathematical side[83].

The principal topics, for a criticism of Helmholtz, are three: First,
his criterion of the _à priori_; second, his discussion with Land as
to the "imaginability" of non-Euclidean spaces; third--and this is by
far the most important of the three--his theory of the dependence of
Geometry on Mechanics. Let us discuss these three points successively.

=67.= Helmholtz's criterion of apriority is difficult to discover,
as he never, to my knowledge, gives a precise statement of it.
From his discussion of physical and transcendental Geometry[84],
however, it would appear that he regards as empirical whatever
applies to empirical matter. For he there maintains, that even if
space were an _à priori_ form, yet any Geometry, which aimed at an
application to Physics, would, since the actual places of bodies are
not known _à priori_, be necessarily empirical[85]. It seems the
more probable that he regards this as a possible criterion, as it
is adopted, in several passages, by his disciple Erdmann[86], and
so strange a test could hardly be accepted by a philosopher, unless
he had found it in his master. I have called this a strange test,
because it seems to me completely to ignore the work of the Critical
Philosophy. For if there is one thing which, one might have hoped,
had been made sufficiently clear by Kant's Critique, it is this,
that knowledge which is _à priori_, being itself the condition of
possible experience, applies--and in Kant's view, applies only--to
empirical matter. Helmholtz and Erdmann, therefore, in setting up
this test without discussion, simply ignore the existence of Kant
and the possibility of a transcendental argument. Helmholtz assumes
always that empirical knowledge must be wholly empirical, that there
can be no _à priori_ conditions of the experience in question,
that experience will always be possible, and may give any kind of
result. Thus in discussing "physical" Geometry, he assumes that the
possibility of empirical measurement involves no _à priori_ axioms,
and that no _à priori_ element can be contained in the process. This
assumption, as we shall see in Chapter III., is quite unwarrantable:
certain properties of _space_, in fact, are involved in the
possibility of measuring _matter_. In spite of the fact, therefore,
that we apply measurement to empirical matter, and that our results
are therefore empirical, there may well be an _à priori_ element
in measurement, which is presupposed in its possibility. Such a
criterion, therefore, must pronounce everything empirical, but must
itself be pronounced worthless.

Another and a better criterion, it is true, is also to be found in
Helmholtz, and has also been adopted by Erdmann. Whatever might,
by a different experience, have been rendered different--so this
criterion contends--must itself be dependent on experience, and so
empirical. This criterion seems perfectly sound, but Helmholtz's use
of it is usually vitiated by his neglecting to prove the possibility
of the different experience in question. He says, for example, that
if our experience showed us only bodies which changed their shapes
in motion, we should not arrive at the axiom of Congruence, which
he pronounces accordingly to be empirical. But I shall endeavour
to prove, in Chapter III., that without the axiom of Congruence,
experience of spatial magnitude would be impossible. If my proof
be correct, it follows that no experience can ever reveal spatial
magnitudes which contradict this axiom--a possibility which Helmholtz
nowhere discusses, in setting up his hypothetical experience. Thus
this second criterion, though perfectly sound, requires always
an accompanying transcendental argument, as to the conditions of
possible experience. But this accompaniment is seldom to be found in
Helmholtz.

=68.= One of the few cases, in which Helmholtz has attempted such
an accompaniment, occurs in connection with our second point, the
imaginability of non-Euclidean spaces. The argument on this point
was elicited by Helmholtz's Kantian opponents, who maintained that
the merely logical possibility of these spaces was irrelevant, since
the basis of Geometry was not logic, but intuition. The axioms,
they said, are synthetic propositions, and their contraries are,
therefore, not self-contradictory; they are nevertheless apodeictic
propositions, since no other _intuition_ than the Euclidean is
possible to us[87]. I have already criticized this line of argument
in the beginning of the present chapter. Helmholtz's criticism,
however, was different: admitting the internal consistency of
the argument, he denied one of its premisses. We _can_ imagine
non-Euclidean spaces, he said, though their unfamiliarity makes
this difficult. From this view it followed, of course, that Kant's
argument, even if it were formally valid, could not prove the
apriority of Euclidean space in particular, but only of that general
space which included Euclid and non-Euclid alike[88].

Although I agree with Helmholtz in thinking the distinction between
Euclidean and non-Euclidean spaces empirical, I cannot think his
argument on the "imaginability" of the latter a very happy one. The
validity of any proof must turn, obviously, on the definition of
imaginability. The definition which Helmholtz gives in his answer
to Land is as follows: Imaginability requires "die vollständige
Vorstellbarkeit derjenigen Sinneseindrücke, welche das betreffende
Object in uns nach den bekannten Gesetzen unserer Sinnesorgane
unter allen denkbaren Bedingungen der Beobachtung erregen, und
wodurch es sich von anderen ähnlichen Objecten unterscheiden würde"
(Wiss. Abh. II. p. 644). This definition is not very clear, owing
to the ambiguity of the word "_Vorstellbarkeit_." The following
definition seems less ambiguous: "Wenn die Reihe der Sinneseindrücke
vollständig und eindeutig angegeben werden kann, muss man m. E.
die Sache für _anschaulich vorstellbar_ erklären" (Vorträge und
Reden, II. p. 234). This makes clear, what also appears from his
manner of proof, that he regards things as imaginable which can be
_described_ in conceptual terms. Such, as Land remarks (Mind, Vol.
II. p. 45), "is not the sense required for argumentation in this
case." That Land's criticism is just, is shown by Helmholtz's proof
for non-Euclidean spaces, for it consists only in an analogy to
the volume inside a sphere, which is mathematically obtained thus:
We take the symbols representing magnitudes in "pseudo-spherical"
(hyperbolic) space, and give them a new Euclidean meaning; thus all
our symbolic propositions become capable of two interpretations,
one for pseudo-spherical space, and one for the volume inside a
sphere. It is, however, sufficiently obvious that this procedure,
though it enables us to _describe_ our new space, does not enable
us to _imagine_ it, in the sense of calling up images of the way
things would look in it. We really derive, from this analogy, no
more knowledge than a man born blind may derive, as to light,
from an analogy with heat. The dictum "Nihil est in intellectu
quod non fuerit ante in sensu," would unquestionably be true, if
for _intellect_ we were to substitute _imagination_; it is vain,
therefore, _if_ our actual space be Euclidean, to hope for a power of
_imagining_ a non-Euclidean space. What Helmholtz might, I believe
with perfect truth, have urged against Land, is that the image we
actually have of space is not sufficiently accurate to exclude, in
the actual space we know, all possibility of a slight departure from
the Euclidean type. But in maintaining that we cannot imagine, though
we can conceive and describe, a space different from that we actually
have, Land is, in my opinion, unquestionably in the right. For a
pure Kantian, who maintains, with Land, that none of the axioms can
be proved, this question is of great importance. But if, as I have
maintained, some of the axioms are susceptible of a transcendental
proof, while the others can be verified empirically, the question
is freed from psychological implications, and the imaginability or
non-imaginability of metageometrical spaces becomes unimportant.

=69.= We come now to the third and most important question, the
relation of Geometry to Mechanics. There are three senses in which
Helmholtz's appeal to rigid bodies may be taken: the first, I think,
is the sense in which he originally intended it; the second seems
to be the sense which he adopted in his defence against Land; while
the third is admitted by Land, and will be admitted in the following
argument. These three senses are as follows:

(1) It may be asserted that the actual meaning of the axiom of Free
Mobility lies in the assertion of empirical rigid bodies, and that
the two propositions are equivalent to one another. This is certainly
false.

(2) The axiom of Free Mobility, it may be said, is logically
distinguishable from the assertion of rigid bodies, and may even be
not empirical; but it is barren, even for pure Geometry, without the
aid of measures, which must themselves be empirical rigid bodies.
This sense is more plausible than the first, but I believe we can
show that, in this sense also, the proposition is false.

(3) For pure Geometry and the abstract study of space, it may be
said, Free Mobility, as applied to an abstract geometrical matter,
gives a sufficient possibility of quantitative comparison; but
the moment we extend our results to mixed mathematics, and apply
them to empirically given matter, we require also, as measures,
empirically given rigid bodies, or bodies, at least, whose departures
from rigidity are empirically known. In this sense, I admit, the
proposition is correct[89].

In discussing these three meanings, I shall not confine myself
strictly to the text of Helmholtz or Land: if I endeavoured to do so,
I should be met by the difficulty that neither of them defines the
_à priori_, and that each is too much inclined, in my opinion, to
test it by psychological criteria. I shall, therefore, take the three
meanings in turn, without laying stress on their historical adequacy
to the views of Land or Helmholtz.

=70.= (1) Congruence may be taken to mean--as Helmholtz would
certainly seem to desire--that we find actual bodies, in our
mechanical experience, to preserve their shapes with approximate
constancy, and that we infer, from this experience, the homogeneity
of space. This view, in my opinion, radically misconceives the
nature of measurement, and of the axioms involved in it. For what
is meant by the non-rigidity of a body? We mean, simply, that
it has changed its shape. But this involves the possibility of
comparison with its former shape, in other words, of measurement.
In order, therefore, that there may be any question of rigidity
or non-rigidity, the measurement of spatial magnitudes must be
already possible. It follows that measurement cannot, without a
vicious circle, be itself derived from experience of rigid bodies.
Geometrical measurement, in fact, is the comparison of spatial
magnitudes, and such comparison involves, as will be proved at length
in Chapter III., the homogeneity of space. This is, therefore, the
logical prerequisite of all experience of rigid bodies, and cannot
be the result of such experience. Without the homogeneity of space,
the very notion of rigidity or non-rigidity could not exist, since
these mean, respectively, the constancy or inconstancy of spatial
magnitude in pieces of matter, and both alike, therefore, presuppose
the possibility of spatial measurement. From the homogeneity of
space, we learn that a body, when it moves, will not change its shape
without some physical cause; that it actually does not change its
shape, is never asserted, and is indeed known to be false. As soon as
measurement is possible, actual changes of shape can be estimated,
and their empirical causes can be sought. But if space were not
homogeneous, measurement would be impossible, constant shape would
be a meaningless phrase, and rigidity could never be experienced.
Congruence asserts, in short, that a body can, so far as mere space
is concerned, move without change of shape; rigidity asserts that
it actually does so move--a very different proposition, involving
obviously, as its logical prius, the former geometrical proposition.

This argument may be summed up by the following disjunction: If
bodies change their shapes in motion--and to some extent, since no
body is perfectly rigid, they must all do so--then one of two cases
must occur. _Either_ the changes of shape, as bodies move from
place to place, follow no geometrical law, are not, for instance,
functions of the amount or direction of motion; in which case the
law of causation requires that they should not be effects of the
change of place, but of some simultaneous non-geometrical change,
such as temperature. _Or_ the changes are regular, and the shape _S_
becomes, in a new position _p_, _Sf_(_p_). In this case, the law of
concomitant variations leads us to attribute the change of shape
to the mere motion, and shape thus becomes a function of absolute
position. But this is absurd, for position _means_ merely a relation
or set of relations; it is impossible, therefore, that mere position
should be able to effect changes in a body. Position is one term in a
relation, not a thing _per se_; it cannot, therefore, act on a thing,
nor exist by itself, apart from the other terms of the relation. Thus
Helmholtz's view, that Congruence depends on the existence of rigid
bodies, must, since it involves absolute position, be condemned as
a logical fallacy. Congruence, in fact, as I shall prove more fully
in Chapter III., is an _à priori_ deduction from the relativity of
position.

=71.= (2) The above argument seems to me to answer satisfactorily
Helmholtz's contention in the precise form which he first gave it.
The axiom of Congruence, we must agree, is logically distinguishable
from the existence of rigid bodies. Nevertheless some reference
to matter is logically involved in Geometry[90], but whether this
reference makes Geometry empirical, or does not, rather, show an _à
priori_ element in dynamics, is a further question.

The reference to matter is necessitated by the homogeneity of empty
space. For so long as we leave matter out of account, one position
is perfectly indistinguishable from another, and a science of
the relations of positions is impossible. Indeed, before spatial
relations can arise at all, the homogeneity of empty space must
be destroyed, and this destruction must be effected by matter.
The blank page is useless to the geometer until he defaces its
homogeneity by lines in ink or pencil. No spatial figures, in short,
are conceivable, without a reference to a not purely spatial matter.
Again, if Congruence is ever to be used, there must be motion: but
a purely geometrical point, being defined solely by its spatial
attributes, cannot be supposed to move without a contradiction in
terms. What moves, therefore, must be matter. Hence, in order that
motion may afford a test of equality, we must have some _matter_
which is known to be unaffected throughout the motion, that is, we
must have some rigid bodies. And the difficulty is, that these bodies
must not only undergo no change due solely to the nature of space,
but must, further, be unchanged by their changing relation to other
bodies. And here we have a requisite which can no longer be fulfilled
_à priori_: which, indeed, we know to be, in strictness, untrue.
For the forces acting on a body depend upon its spatial relations
to other bodies, and changing forces are liable to produce changing
configuration. Hence, it would seem, actual measurement must be
purely empirical, and must depend on the degree of rigidity to be
obtained, during the process of measurement, in the bodies with which
we are conversant.

This conclusion, I believe, is valid of all actual measurement.
But the possibility of such empirical and approximate rigidity,
I must insist, depends on the _à priori_ law that _mere_ motion,
apart from the action of other matter, cannot effect a change of
shape. For without this law, the effect of other matter would not be
discoverable; the laws of motion would be absurd, and Physics would
be impossible. Consider the second law, for example: How could we
measure the change of motion, if motion itself produced a change
in our measures? Or consider the law of gravitation: How could we
establish the inverse square, unless we were able, independently
of Dynamics, to measure distances? The whole science of Dynamics,
in short, is fundamentally dependent on Geometry, and but for the
independent possibility of measuring spatial magnitudes, none of the
magnitudes of Dynamics could be measured. Time, force, and mass are
alike measured by spatial correlates: these correlates are given, for
time, by the first law, for force and mass, by the second and third.
It is true, then, that an empirical element appears unavoidably in
all actual measurement, inasmuch as we can only know empirically
that a given piece of matter preserves its shape throughout the
necessary change of dynamical relations to other matter involved in
motion; but it is further true that, for Geometry--which regards
matter simply as supplying the necessary breach in the homogeneity
of space, and the necessary term for spatial relations, not as the
bearer of forces which change the configuration of other material
systems--for Geometry, which deals with this abstract and merely
kinematical matter, rigidity is _à priori_, in so far as the only
changes with which it is cognizant--changes of mere position,
namely--are incapable of affecting the shapes of the imaginary and
abstract bodies with which it deals. To use a scholastic distinction,
we may say that matter is the _causa essendi_ of space, but Geometry
is the _causa cognoscendi_ of Physics. Without a Geometry independent
of Physics, Physics itself, which necessarily assumes the results of
Geometry, could never arise; but when Geometry is used in Physics, it
loses some of its _à priori_ certainty, and acquires the empirical
and approximate character which belongs to all accounts of actual
phenomena.

=72.= (3) This argument leads us to Land's distinction of physical
and geometrical rigidity. The distinction may be expressed--and
I think it is better expressed--by distinguishing between
the conceptions of matter proper to Dynamics and to Geometry
respectively. In Dynamics, we are concerned with matter as subject
to and as causing motion, as affected by and as exerting _force_. We
are therefore concerned with the changes of spatial configuration to
which material systems are liable: the description and explanation
of these changes is the proper subject-matter of all Dynamics. But
in order that such a science may exist, it is obviously necessary
that spatial configuration should be already measurable. If this
were not the case, motion, acceleration and force would remain
perfectly indeterminate. Geometry, therefore, must already exist
before Dynamics becomes possible: to make Geometry dependent for its
possibility on the laws of motion or any of their consequences, is
a gross ὕστερον πρότερον. Nevertheless, as we have seen, some sort
of matter is essential to Geometry. But this geometrical matter is a
more abstract and wholly different matter from that of Dynamics. In
order to study space by itself, we reduce the properties of matter
to a bare minimum: we avoid entirely the category of causation,
so essential to Dynamics, and retain nothing, in our matter, but
its spatial adjectives[91]. The kind of rigidity affirmed of this
abstract matter--a kind which suffices for the theory of our
science, though not for its application to the objects of daily
life--is purely geometrical, and asserts no more than this: That
since our matter is devoid, _ex hypothesi_, of causal properties,
there remains nothing, in mere empty space, which is capable of
changing the configuration of any geometrical system. A change of
absolute position, it asserts, is nothing; therefore the only real
change involved in motion is a change of relation to other matter;
but such other matter, for the purposes of our science, is regarded
as destitute of causal powers; hence no change can occur, in the
configuration of our system, by the mere effect of motion through
empty space. The necessity of such a principle may be shown by a
simple _reductio ad absurdum_, as follows. A motion of translation
of the universe as a whole, with constant direction and velocity, is
dynamically negligeable; indeed it is, philosophically, no motion at
all, for it involves no change in the condition or mutual relations
of the things in the universe. But if our geometrical rigidity
were denied, the change in the parameter of space might cause all
bodies to change their shapes owing to the mere change of absolute
position, which is obviously absurd.

To make quite plain the function of rigid bodies in Geometry, let us
suppose a liquid geometer in a liquid world. We cannot suppose the
liquid perfectly homogeneous and undifferentiated, in the first place
because such a liquid would be indistinguishable from empty space,
in the second place because our geometer's body--unless he be a
disembodied spirit--will itself constitute a differentiation for him.
We may therefore assume

                    "dim beams,
          Which amid the streams
      Weave a network of coloured light,"

and we may suppose this network to form the occasion for our
geometer's reflections. Then he will be able to imagine a network
in which the lines are straight, or circular, or parabolic, or any
other shape, and he will be able to infer that such a network, if it
can be woven in one part of the fluid, can be woven in another. This
will form sufficient basis for his deductions. The superposition he
is concerned with--since not actual equality, but only the formal
conditions of equality, are the subject-matter of Geometry--is purely
ideal, and is unaffected by the impossibility of congealing any
actual network. But in order to apply his Geometry to the exigencies
of life, he would need some standard of comparison between actual
networks, and here, it is true, he would need either a rigid body,
or a knowledge of the conditions under which similar networks arose.
Moreover these conditions, being necessarily empirical, could hardly
be known apart from previous measurement. Hence for applied, though
not for pure Geometry, one rigid body at least seems essential.

=73.= The utility, for Dynamics, of our abstract geometrical matter,
is sufficiently evident. For having, by its means, a power of
determining the configurations of material systems in whatever part
of space, and knowing that changes of configuration are not due to
mere change of place, we are able to attribute these changes to the
action of other matter, and thus to establish the notion of force,
which would be impossible if change of shape might be due to empty
space.

Thus, to conclude: Geometry requires, if it is to be _practically_
possible, some body or bodies which are either rigid (in the
dynamical sense), or known to undergo some definite changes of shape
according to some definite law. (These changes, we may suppose,
are known by the laws of Physics, which have been experimentally
established, and which throughout assume the truth of Geometry.)
One or more such bodies are necessary to applied Geometry--but only
in the sense in which rulers and compasses are necessary. They are
necessary as, in making the Ordnance Survey, an elaborate apparatus
was necessary for measuring the base line on Salisbury Plain. But
for the _theory_ of Geometry, geometrical rigidity suffices, and
geometrical rigidity means only that a shape, which is possible in
one part of space, is possible in any other. The empirical element
in practice, arising from the purely empirical nature of physical
rigidity, is comparable to the empirical inaccuracies arising from
the failure to find straight lines or circles in the world--which no
one but Mill has regarded as rendering Geometry itself empirical or
inaccurate. But to make Geometry await the perfection of Physics,
is to make Physics, which depends throughout on Geometry, forever
impossible. As well might we leave the formation of numbers until we
had counted the houses in Piccadilly.


Erdmann.

=74.= In connection with Riemann and Helmholtz, it is natural to
consider Erdmann's philosophical work on their theories[92]. This is
certainly the most important book on the subject which has appeared
from the philosophical side, and in spite of the fact that, like
the whole theory of Riemann and Helmholtz, it is inapplicable to
projective Geometry, it still deserves a very full discussion.

Erdmann agrees throughout with the conclusions of Riemann and
Helmholtz, except on a few points of minor importance; and his views,
as this agreement would lead one to expect, are ultra-empirical.
Indeed his logic seems--though I say this with hesitation--to be
incompatible with any system but that of Mill: there is apparently
no distinction, to him, between the general and the universal, and
consequently no concept not embodied in a series of instances. Such
a theory of logic, to my mind, vitiates most of his work, as it
vitiated Riemann's philosophy[93]. This general criticism will find
abundant illustration in the course of our account of Erdmann's views.

=75.= After a general introduction, and a short history of the
development of Metageometry, Erdmann proceeds, in his second
chapter, to discuss what are the axioms of Euclidean Geometry. The
arithmetical axioms, as they are called, he leaves aside, as applying
to magnitude in general; what we want here, he says, is a definition
of space, for which the geometrical axioms are alone relevant.
But a definition of space, he says--following Riemann--demands a
genus of which space shall be a species, and this, since our space
is psychologically unique, can only be furnished by analytical
mathematics (p. 36). Now the space-forms dealt with by Geometry are
magnitudes, and conceptions of magnitude are everywhere applied in
Geometry. But before Riemann, only particular determinations of space
could be exhibited as magnitudes, and thus the desired definition
was impossible to obtain. Now, however, we can subsume space as a
whole under a general conception of magnitude, and thus obtain,
besides the space-intuition and the space-conception, a third form,
namely, the conception of space as a magnitude (_Grössenbegriff
vom Raum_, pp. 38-39). The definition of this will give us the
complete, but not redundant, system of axioms, which could not be
obtained by transforming the general intuition of space into the
space-conception, for want of a plurality of instances (p. 40).

=76.= Before considering the subsequent method of definition, let
us reflect on the theories involved in the above account of the
conception of space as a magnitude. In the first place, it is assumed
that conceptions cannot be formed unless we have a series of separate
objects from which to abstract a common property--in other words,
that the universal is always the general. In the second place, it
is assumed that all definition is classification under a genus. In
the third place, the conception of magnitude, if I am not mistaken,
is fundamentally misunderstood when it is supposed applicable to
space as a whole. But in the fourth place, even if such a conception
existed, it could give none of the essential properties of space. Let
us consider these four points successively.

=77.= As regards the first point, it is to be observed that people
certainly had some conception of space before Riemann invented
the notion of a manifold, and that this conception was certainly
something other than the common qualities of all the points, lines
or figures in space. In the second place, Erdmann's view would
make it impossible to conceive God, unless one were a polytheist,
or the universe--unless, like Leibnitz, one imagined a series of
possible worlds, set over against God, and none of them, therefore,
a true Universe--or, to take an instance more likely to appeal to an
empiricist, the necessarily unique centre of mass of the material
universe. Any universal, in short, which is a bond or unity between
things, and not merely a common property among independent objects,
becomes impossible on Erdmann's view. We cannot, therefore, unless
we adopt Mill's philosophy intact, regard the conception of space
as demanding a series of instances from which to abstract. But even
if we did so regard it, Riemann's manifolds would leave us without
resources. For Euclidean space still appears as unique, at the end of
his series of determinations. We have instances of manifolds, but not
instances of Euclidean space. Thus if Erdmann's theory of conceptions
were correct, he would still be left searching in vain for the
conception of Euclidean space.

=78.= The second point, the view that all definition is
classification, is closely allied to the first, and the two together
plunge us into the depths of scholastic formal logic. The same
instances of things which could not, on Erdmann's view, be conceived,
may now be adduced as things which cannot be defined. Whatever was
said above applies here also, and the point need not, therefore, be
further discussed[94].

=79.= As regards the third point, the impossibility of applying
conceptions of magnitude to space as a whole, a longer argument will
be necessary, for we are concerned, here, with the whole question
of the logical nature of judgments of magnitude. As we had before
too much comparison for our needs, so we have now too little. I will
endeavour to explain this point, which is of great importance, and
underlies, I think, most of the philosophical fallacies of Riemann's
school.

A judgment of magnitude is always a judgment of comparison, and what
is more, the comparison is never concerned with quality, but only
with quantity. Quality, in the judgment of magnitude, is supposed
identical, in the object whose magnitude is stated, and in the unit
with which it is compared. But quality, except in pure number, and in
pure quantity as dealt with by the Calculus, is always present, and
is partly absorbed into quantity, partly untouched by the judgment of
magnitude. As Bosanquet says (Logic, Vol. I. p. 124); "Quantitative
comparison is not _prima facie_ coordinate with qualitative,
but rather stands in its place as the _effect of comparison on
quality_, which so far as comparable _becomes quantity_, and so
far as incomparable furnishes the distinction of parts essential
to the quantitative whole" (italics in the original). Thus, if we
are to regard space as a magnitude, we must be able to adduce all
those series of instances of which Erdmann speaks, and which, for
the conception of space, seemed irrelevant. But it remains to be
proved that the comparison, which we _can_ institute between various
spaces, is capable of expression in a quantitative form. Rather it
would seem that the difference of quality is such as to preclude
quantitative comparison between different spaces, and therefore also
to preclude all judgments of magnitude about space as a whole. Here
an exception might seem to be demanded by non-Euclidean spaces,
whose space-constants give a definite magnitude, inherent in space
as a whole, and therefore, one might think, characterizing space as
a magnitude. But this is a mistake. For the space-constant, in such
spaces, is the ultimate unit, the fixed term in all quantitative
comparison; it is itself, therefore, destitute of quantity, since
there is no independently given magnitude with which to compare it.
A non-Euclidean world, in which the space-constant and all lines
and figures were suddenly multiplied in a constant ratio, would be
wholly unchanged; the lines, as measured against the space-constant,
would have the same magnitude as before, and the space-constant
itself, having no outside standard of comparison, would be destitute
of quantity, and therefore not subject to change of quantity. Such an
enlargement of a non-Euclidean world, in other words, is unmeaning;
and this proves how inapplicable is the notion of quantity to space
as a whole.

It might be objected that this only proves the absence of
quantitative difference between different spaces of positive
space-constant, or between those of negative space-constant: the
quantitative difference persists, it might be said, between those
of positive curvature in general and those of negative curvature
in general, or between both together and Euclidean space. This I
entirely deny. There is no qualitatively similar unit, in the three
kinds of space, by which quantitative comparison could be effected.
The straight lines of one space cannot be put into the other: the
two straight lines, in one space, whose product is the reciprocal
of the measure of curvature, have no corresponding curves in the
other space, and the measures of curvature cannot, therefore,
be quantitatively compared with each other. That the one may be
regarded as positive, the other negative, I admit, but their values
are indeterminate, and the units in the two cases are qualitatively
different. A debt of £300 may be represented as the asset of -£300,
and the height of the Eiffel Tower is +300 metres; but it does
not follow that the two are quantitatively comparable. So with
space-constants: the space-constant is itself the unit for magnitudes
in its own space, and differs qualitatively from the space-constant
of another kind of space.

Again, to proceed to a more philosophical argument, two different
spaces cannot co-exist in the same world: we may be unable to decide
between the alternatives of the disjunction, but they remain, none
the less, absolutely incompatible alternatives. Hence we cannot get
that coexistence of two spaces which is essential to comparison. The
fact seems to be that Erdmann, in his admiration for Riemann and
Helmholtz, has fallen in with their mathematical bias, and assumed,
as mathematicians naturally tend to assume, that quantity is
everywhere and always applicable and adequate, and can deal with more
than the mere comparison of things whose qualities are already known
as similar[95].

=80.= This suggests the fourth and last of the above points, that the
_qualities_ of space, even if space could be successfully regarded
as a magnitude, would have to be entirely omitted in such a manner
of regarding it, and that, therefore, none of its important or
essential properties would emerge from such treatment. For to regard
space as a magnitude involves, as we saw, a comparison with something
qualitatively similar, and an abstraction from the similar qualities.
To some extent and by the help of certain doubtful arguments, such a
comparison is instituted by Riemann and Erdmann; but when they have
instituted it, they forget all about the common qualities on which
its possibility depends. But these are precisely the fundamental
properties of space, and those from which, as I shall endeavour to
prove in Chapter III., the axioms common to Euclid and Metageometry
follow _à priori_. Such are the dangers of the quantitative bias.

=81.= After this protest against the initial assumptions in Erdmann's
deduction of space, let us return to consider the manner, in which
this deduction is carried out. Here there will be less ground
for criticism, as the deduction, given its presuppositions, is,
I think, as good as such a deduction can be. To define space as
a magnitude, he says, let us start with two of its most obvious
properties, continuity and the three dimensions. Tones and colours
afford other instances of a manifold with these two properties, but
differ from space in that their dimensions are not homogeneous and
interchangeable. To designate this difference, Erdmann introduces
a useful pair of terms: in the general case, he calls a manifold
_n_-determined (n-_bestimmt_); in the case where, as in space, the
dimensions are homogeneous, he calls the manifold _n_-extended
(n-_ausgedehnt_). Manifolds of the latter sort he calls extents
(_Ausgedehntheiten_). That the difference between the two kinds
is one of quality, not of quantity, he seems not to perceive; he
also overlooks the fact that, in the second kind, from its very
definition, the axiom of Congruence must hold, on account of the
qualitative similarity of different parts. In spite of this fact, he
defines space as an extent, and then regards Congruence as empirical,
and as possibly false in the infinitesimal. This is the more strange,
as he actually proves (p. 50) that measurement is impossible, in an
extent, unless the parts are independent of their place, and can be
carried about unaltered as measures. In spite of this, he proceeds
immediately to discuss whether the measure of curvature is constant
or variable, without investigating how, in the latter case, Geometry
could exist. We cannot know, he says, from geometrical superposition,
that geometrical bodies are independent of place, for if their
dimensions altered in motion according to any fixed law, two bodies
which could be superposed in one place could be superposed in any
other. That such a hypothesis involves absolute position, and denies
the qualitative similarity of the parts of space, which he declares
(p. 171) to be the principle of his theory of Geometry, is nowhere
perceived. But what is more, his notion that magnitude is something
absolute, independent of comparison, has prevented him from seeing
that such a hypothesis is unmeaning. He says himself that, even on
this hypothesis, a geometrical body can be defined as one whose
points retain constant distances from each other, for, since we have
no absolute measure, measurement could not reveal to us the change of
absolute magnitude (p. 60). But is not this a _reductio ad absurdum_?
For magnitude is nothing apart from comparison, and the comparison
here can only be effected by superposition; if, then, as on the above
hypothesis, superposition always gives the same result, by whatever
motion it is effected, there is no sense in speaking of magnitudes as
no longer equal when separated: absolute magnitude is an absurdity,
and the magnitude resulting from comparison does not differ from that
which would result if the dimensions of bodies were unchanged in
motion. Therefore, since magnitude is only intelligible as the result
of comparison, the dimensions of bodies _are_ unchanged in motion,
and the suggested hypothesis is unmeaning. On this subject I shall
have more to say in Chapter III.[96]

=82.= This hypothesis, however, is not introduced for its own sake,
but only to usher in the Helmholtzian _deus ex machina_, Mechanics.
For Mechanics proves--so Erdmann confidently continues--that
rigidity must hold, not merely as to ratios, in the above restricted
geometrical sense, but as to absolute magnitudes (p. 62). Hence we
get at last true Congruence, empirical as Mechanics is empirical, and
impossible to prove apart from Mechanics. I have already criticized
Helmholtz's view of the dependence of Geometry on Mechanics, and
need not here speak of it at length. It is a pity that Erdmann has
in no way specified the procedure by which Mechanics decides the
geometrical alternatives--indeed he seems to rely on the _ipse dixit_
of Helmholtz. How, if Geometry would be totally unable to discover a
change in dimensions of the kind suggested, the Laws of Motion, which
throughout depend on Geometry, should be able to discover it if it
existed, I am wholly at a loss to understand. Uniform motion in a
straight line, for example, presupposes geometrical measurement; if
this measurement is mistaken, what Mechanics imagines to be uniform
motion is not really such, but Mechanics can never discover the
discrepancy. If the Laws of Motion had been regarded as _à priori_,
Geometry might possibly have been reinforced by them; but so long
as they are empirical, they presuppose geometrical measurement, and
cannot therefore condition or affect it.

Erdmann's conclusion, in the second chapter, is that Congruence is
probable, but cannot be verified in the infinitesimal; that its truth
involves the actual existence of rigid bodies (though, by the way,
we know these to be, strictly speaking, non-existent), that rigid
bodies are freely moveable, and do not alter their size in rotation
(Helmholtz's Monodromy); that the axiom of three dimensions is
certain, since small errors are impossible; and that the remaining
axioms of Euclid--those of the straight line and of parallels--are
approximately, if not accurately, true of our actual space (pp.
78, 83). He does not discuss how Congruence, on the above view,
is compatible with the atomic theory, or even with the observed
deformations of approximately rigid bodies; nor how, if space, as he
assumes, is homogeneous, rigid bodies can fail to be freely moveable
through space. The axioms are all lumped together as empirical, and
it appears, in the following chapters, that Erdmann regards their
empirical nature as sufficiently proved by their applicability to
empirical material (cf. pp. 159, 165)--a strange criterion, which
would prove the same conclusion, with equal facility, of Arithmetic
and of the laws of thought.

=83.= The third chapter, on the philosophical consequences of
Metageometry, need not be discussed at length, since it deals rather
with space than with Geometry. At the same time, it will be worth
while to treat briefly of Erdmann's criterion of apriority. On this
subject it is very difficult to discover his meaning, since it
seems to vary with the topic he is discussing. Thus at one time (p.
147) he rejects most emphatically the Kantian connection of the _à
priori_ and the subjective[97], and yet at another time (p. 96) he
regards every presentation of external things as partly _à priori_,
partly empirical, merely because such a presentation is due to an
interaction between ourselves and things, and is therefore partly
due to subjective activity, partly due to outside objects. Hence,
he says, the distinction is not between different presentations,
but between different aspects of one and the same presentation.
This seems to return wholly to the Kantian psychological criterion
of subjectivity, with the added disadvantage that it makes the
distinction, like that of analytic and synthetic, epistemologically
worthless. And yet he never hesitates to pronounce every piece
of knowledge in turn empirical. The fact seems to be, that where
he wants a more logical criterion, he adopts a modification of
Helmholtz's criterion for sensations. If space be an _à priori_
form, he says, no experience could possibly change it (p. 108);
but this Metageometry has proved not to be the case, since we can
intuit the perceptions which non-Euclidean space would give us (p.
115). I have criticised this argument in discussing Helmholtz; at
present we are concerned with Erdmann's criterion of apriority. The
subjectivity-criterion--though he certainly uses it in discussing the
apriority of space, and solemnly decides, by its means, that space
is both _à priori_ and empirical since a change either in us or in
the outer world could change it (p. 97)--would seem, like several of
his other tests, to be a lapse on his part: the criterion which he
means to use is Helmholtz's. This criterion, I think, with a slight
change of wording, might be accepted; it seems to me a necessary,
but not a sufficient condition. The _à priori_, we may say, is not
only that which no experience can change, but that without which
experience would become impossible. It is the omission to discuss
the conditions which render geometrical (and mechanical) experience
possible, to my mind, which vitiates the empirical conclusions of
Helmholtz and Erdmann. Why certain conditions should be necessary
for experience--whether on account of the constitution of the mind,
or for some other reason--is a further question, which introduces
the relation of the _à priori_ to the subjective. But in discussing
the question as to what knowledge is _à priori_, as opposed to the
question concerning the further consequences of apriority, it is
well to keep to the purely logical criterion, and so preserve our
independence of psychological controversies. The fact, if it be a
fact, that the world might be such as to defy our attempts to know
it, will not, with the above criterion, invalidate the conclusion
that certain elements in knowledge are _à priori_; for whether
fulfilled or not, they remain necessary conditions for the existence
of any knowledge at all.

=84.= With this caution as to the meaning of apriority, we shall
find, I think, that the conclusions of Erdmann's final chapter, on
the principles of a theory of Geometry, are largely invalidated
by the diversity and inadequacy of his tests of the _à priori_.
He begins by asserting, in conformity with the quantitative bias
noticed above, that the question as to the nature of geometrical
axioms is completely analogous to the corresponding question of the
foundations of pure mathematics (p. 138). This is, I think, a radical
error: for the function of the axioms seems to be, to establish that
qualitative basis on which, as we saw, all qualitative comparison
must rest. But in pure mathematics, this qualitative basis is
irrelevant, for we deal there with pure quantity, _i.e._ with the
merely quantitative result of quantitative comparison, wherever it is
possible, independently of the qualities underlying the comparison.
Geometry, as Grassmann insists[98], ought not to be classed with
pure mathematics, for it deals with a matter which is given to the
intellect, not created by it. The axioms give the means by which
this matter is made amenable to quantity, and cannot, therefore, be
themselves deduced from purely quantitative considerations.

Leaving this point aside, however, let us return to Erdmann.
He distinguishes, within space, a form and a matter: the form
is to contain the properties common to all extents, the matter
the properties which distinguish space from other extents. This
distinction, he says, is purely logical, and does not correspond
with Kant's: matter and form, for Erdmann, are alike empirical. The
axioms and definitions of Geometry, he says, deal exclusively with
the matter of space. It seems a pity, having made this distinction,
to put it to so little use: after a few pages, it is dropped, and
no epistemological consequences are drawn from it. The reason is, I
think, that Erdmann has not perceived how much can be deduced from
his definition of an extent, as a manifold in which the dimensions
are homogeneous and interchangeable. For this property suffices to
prove the complete homogeneity of an extent, and hence--from the
absence of qualitative differences among elements--the relativity of
position and the axiom of Congruence. This deduction will be made at
length in the sequel[99]; at present, I have only to observe that
every extent, on this view, possesses all the properties (except
the three dimensions) common to Euclidean and non-Euclidean spaces.
The axioms which express these properties, therefore, apply to the
form of space, and follow from homogeneity alone, which Erdmann
allows (p. 171) as the principle of any theory of space. The above
distinction of form and matter, therefore, corresponds, when its
full consequences are deduced, to the distinction between the axioms
which follow from the homogeneity of space and those which do not.
Since, then, homogeneity is equivalent to the relativity of position,
and the relativity of position is of the very essence of a form of
externality, it would seem that his distinction of form and matter
can also be made coextensive with the distinction of the _à priori_
and empirical in Geometry. On this subject, I shall have more to say
in Chapter III.

In the remainder of the chapter, Erdmann insists that the straight
line, etc., though not abstracted from experience, which nowhere
presents straight lines, must yet, as applicable to admittedly
empirical sciences, be empirical (p. 159)--a criterion which he
appears to employ only when all other grounds for an empirical
opinion fail, and one which, obviously, can never refuse to do its
work, since all elements of knowledge are susceptible of employment
on some empirical material. He also defines the straight line (p.
155) as a line of constant curvature zero, as though curvature could
be measured independently of the straight line. Even the arithmetical
axioms are declared empirical (p. 165), since in a world where things
were all hopelessly different from one another, these axioms could
not be applied. After this reminder of Mill, we are not surprised, a
few pages later (p. 172), at a vague appeal to "English logicians"
as having proved Geometry to be an inductive science. Nevertheless,
Erdmann declares, almost on the last page of his book (p. 173), that
Geometry is distinguished from all other sciences by the homogeneity
of its material: a principle of which no single application occurs
throughout his book, and which, as we shall see in Chapter III.,
flatly contradicts the philosophical theories advocated throughout
his preceding pages.

On the whole, then, it cannot be said that Erdmann has done much to
strengthen the philosophical position of Riemann and Helmholtz. I
have criticized him at length, because his book has the appearance
of great thoroughness, and because it is undoubtedly the best
defence extant of the position which it takes up. We shall now have
the opposite task to perform, in defending Metageometry, on its
mathematical side, from the attacks of Lotze and others, and in
vindicating for it that measure of philosophical importance--far
inferior, indeed, to the hopes of Erdmann--which it seems really to
possess.


Lotze.

=85.= Lotze's argument as regards Geometry[100]--which follows a
metaphysical argument as to the ontological nature of space, and
assumes the results of this argument--consists of two parts: the
first discusses the various meanings logically assignable (pp.
233-247) to the proposition that other spaces than Euclid's are
possible, and the second criticizes, in detail, the procedure of
Metageometry. The first of these questions is very important, and
demands considerable care as to the logical import of a judgment
of possibility. Although Lotze's discussion is excellent in many
respects, I cannot persuade myself that he has hit on the only true
sense in which non-Euclidean spaces are possible. I shall endeavour
to make good this statement in the following pages.

=86.= Lotze opens with a somewhat startling statement, which,
though philosophically worthy to be true, does not appear to be
historically borne out. Euclidean Geometry has been chiefly shaken,
he says, by the Kantian notion of the exclusive subjectivity of
space--if space is only our private form of intuition, to which
there exists no analogue in the objective world, then other beings
may have other spaces, without supposing any difference in the world
which they arrange in these spaces (p. 233). This certainly seems
a legitimate deduction from the subjectivity of space, which, so
far from establishing the universal validity of Euclid, establishes
his validity only after an empirical investigation of the nature of
space as intuited by Tom, Dick or Harry. But as a matter of fact,
those who have done most to further non-Euclidean Geometry--with the
exception of Riemann, who was a disciple of Herbart--have usually
inherited from Newton a naïve realism as regards absolute space.
I might instance the passage quoted from Bolyai in Chapter I., or
Clifford, who seems to have thought that we actually see the images
of things on the retina[101], or again Helmholtz's belief in the
dependence of Geometry on the behaviour of rigid bodies. This belief
led to the view that Geometry, like Physics, is an experimental
science, in which objective truth can be attained, it is true, but
only by empirical methods. However, Lotze's ground for uncertainty
about Euclid is a philosophically tenable ground, and it will be
instructive to observe the various possibilities which arise from it.

If space is only a subjective form--so Lotze opens his
argument--other beings may have a different form. If this corresponds
to a different world, the difference, he says, is uninteresting:
for our world alone is relevant to any metaphysical discussion. But
if this different space corresponds to the same world which we know
under the Euclidean form, then, in his opinion, we get a question of
genuine philosophic interest. And here he distinguishes two cases:
_either_ the relations between things, which are presented to these
hypothetical beings under the form of some different space, are
relations which do not appear to us, or at any rate do not appear
spatial; _or_ they are the same relations which appear to us as
figures in Euclidean space (p. 235). The first possibility would be
illustrated, he says, by beings to whom the tone or colour-manifolds
appeared extended; but we cannot, in his opinion, imagine a manifold,
such as is required for this case, to have its dimensions homogeneous
and comparable _inter se_, and therefore the contents of the various
presentations constituting such a manifold could not be combined
into a single content containing them all. But the possibility of
such a combination is of the essence of anything worth calling a
space: therefore the first of the above possibilities is unmotived
and uninteresting. Lotze's conclusion on this point, I think, is
undeniable, but I doubt whether his argument is very cogent. However,
as this possibility has no connection with that contemplated by
non-Euclideans, it is not worth while to discuss it further.

The second possibility also, Lotze thinks, is not that of
Metageometry, but in truth it comes nearer to it than any of the
other possibilities discussed. If a non-Euclidean were at the same
time a believer in the subjectivity of space, he would have to be
an adherent of this view. Let us see more precisely what the view
is. In Book II., Chapter I., Lotze has accepted the argument of the
Transcendental Aesthetic, but rejected that of the mathematical
antinomies: he has decided that space is, as Kant believed,
subjective, but possesses nevertheless, what Kant denied it, an
objective counterpart. The relation of presented space to its
objective counterpart, as conceived by Lotze, is rather hard to
understand. It seems scarcely to resemble the relation of sensation
to its object--_e.g._ of light to ether-vibrations--for if it did,
space would not be in any peculiar sense subjective. It seems rather
to resemble the relation of a perceived bodily motion to the state
of mind of the person willing the motion. However this may be, the
objective counterpart of space is supposed to consist of certain
immediate interactions of monads, who experience the interactions
as modifications of their internal states. Such interactions, it
is plain, do not form the subject-matter of Geometry, which deals
only with our resulting perceptions of spatial figures. Now if
Lotze's construction of space be correct, there seems certainly
no reason why these resulting perceptions should not, for one and
the same interaction between monads, be very different in beings
differently constituted from ourselves. But if they were different,
says Lotze, they would have to be utterly different--as different,
for example, as the interval between two notes is from a straight
line. The possibility is, therefore, in his opinion, one about which
we can know nothing, and one which must remain always a mere empty
idea. This seems to me to go too far: for whatever the objective
counterpart may be, any argument which gives us information about
it must, when reversed, give us information about any possible form
of intuition in which this counterpart is presented. The argument
which Lotze has used in his former chapter, for example, deducing,
from the relativity of position, the merely relational nature of
the objective counterpart, allows us, conversely, to infer, from
this relational nature, the complete relativity of position in any
possible space-intuition--unless, indeed, it bore a wholly deceitful
relation to those interactions of monads which form its objective
counterpart. But the complete relativity of position, as I shall
endeavour to establish in Chapter III., suffices to prove that our
Geometry must be Euclidean, elliptic, spherical or pseudo-spherical.
We have, therefore, it would seem, very considerable knowledge,
on Lotze's theory of space, of the manner in which what appears
to us as space _must_ appear to any beings with our laws of
thought. We cannot know, it is true, what _psychological_ theory
of space-perception would apply to such beings: they might have a
sense different from any of ours, and they might have no sense in
any way resembling ours, but yet their Geometry would have points of
resemblance to ours, as that of the blind coincides with that of the
seeing. If space has any objective counterpart whatever, in short,
and if any inference is possible, as Lotze holds it to be, from space
to its counterpart, then a converse argument is also possible, though
it may give some only of the qualities of Euclidean space, since some
only of these qualities may be found to have a necessary analogue in
the counterpart.

=87.= Admitting, then, in Lotze's sense, the subjectivity of space,
the above possibility does not seem so empty as he imagines. He
discusses it briefly, however, in order to pass on to what he regards
as the real meaning of Metageometry. In this he is guilty of a
mathematical mistake, which causes much irrelevant reasoning. For
he believes that Metageometry constructs its spaces out of straight
lines and angles in all respects similar to Euclid's, whence he
derives an easy victory in proving that these elements can lead only
to the one space. In this he has been misled by the phraseology of
non-Euclideans, as well as by Euclid's separation of definitions
and axioms. For the fact is, of course, that straight lines are
only fully defined when we add to the formal definition the axioms
of the straight line and of parallels. Within Euclidean space,
Euclid's definition suffices to distinguish the straight line from
all other curves; the two axioms referred to are then absorbed into
the definition of space. But apart from the restriction to Euclidean
space, the definition has to be supplemented by the two axioms, in
order to define completely the Euclidean straight line. Thus Lotze
has misconceived the bearing of non-Euclidean constructions, and
has simply missed the point in arguing as he does. The possibility
contemplated by a non-Euclidean, if it fell under any of Lotze's
cases, would fall under the second case discussed above.

=88.= But the bearing of Metageometry is really, I think, different
from anything imagined by Lotze; and as few writers seem clear on
this point, I will enter somewhat fully into what I conceive to be
its purpose.

In the first place, there are some writers--notably Clifford--who,
being naïve realists as regards space, hold that our evidence is
wholly insufficient, as yet, to decide as to its nature in the
infinite or in the infinitesimal (cf. Essays, Vol. I. p. 320):
these writers are not concerned with any possibility of beings
different from ourselves, but simply with the everyday space we
know, which they investigate in the spirit of a chemist discussing
whether hydrogen is a metal, or an astronomer discussing the nebular
hypothesis.

But these are a minority: most, more cautious, admit that our space,
so far as observation extends, is Euclidean, and if not accurately
Euclidean, must be only slightly spherical or pseudo-spherical. Here
again, it is the space of daily life which is under discussion,
and here further the discussion is, I think, independent of any
philosophical assumption as to the nature of our space-intuition. For
even if this be purely subjective, the translation of an intuition
into a conception can only be accomplished approximately, within
the errors of observation incident to self-analysis; and until the
intuition of space has become a conception, we get no scientific
Geometry. The apodeictic certainty of the axiom of parallels shrinks
to an unmotived subjective conviction, and vanishes altogether in
those who entertain non-Euclidean doubts. To reinforce the Euclidean
faith, reason must now be brought to the aid of intuition; but
reason, unfortunately, abandons us, and we are left to the mercy of
approximate observations of stellar triangles--a meagre support,
indeed, for the cherished religion of our childhood.

=89.= But the possibility of an inaccuracy so slight, that our
finest instruments and our most distant parallaxes show no trace
of it, would trouble men's minds no more than the analogous chance
of inaccuracy in the law of gravitation, were it not for the
philosophical import of even the slenderest possibility in this
sphere. And it is the philosophical bearing of Metageometry alone,
I think, which constitutes its real importance. Even if, as we
will suppose for the moment, observation had established, beyond
the possibility of doubt, that our space might be safely regarded
as Euclidean, still Metageometry would have shown a philosophical
possibility, and on this ground alone it could claim, I think, very
nearly all the attention which it at present deserves.

But what is this possibility? A thing is possible, according to
Bradley (Logic, p. 187), when it would follow from a certain number
of conditions, some of which are known to be realized. Now the
conditions to which a form of externality must conform, in order to
be affirmed, are: first, of course, that it should be experienced,
or legitimately inferred from something experienced; but secondly,
that it should conform to certain logical conditions, detailed in
Chapter III., which may be summed up in the relativity of position.
Now what Metageometry has done, in any case, is to suggest the proof
that the second of these conditions is fulfilled by non-Euclidean
spaces. Euclid is affirmed, therefore, on the ground of immediate
experience alone, and his truth, as unmediated by logical necessity,
is merely assertorical, or, if we prefer it, empirical. This is
the most important sense, it seems to me, in which non-Euclidean
spaces are possible. They are, in short, a step in a philosophical
argument, rather than in the investigation of fact: they throw light
on the nature of the grounds for Euclid, rather than on the actual
conformation of space[102]. This import of Metageometry is denied by
Lotze, on the ground that non-Euclidean logic is faulty, a ground
which he endeavours, by much detail and through many pages, to make
good--with what success, we will now proceed to examine.

=90.= Lotze's attack on Metageometry--although it remains, so
far as I know, the best hostile criticism extant, and although
its arguments have become part of the regular stock-in-trade of
Euclidean philosophers--contains, if I am not mistaken, several
misunderstandings due to insufficient mathematical knowledge of the
subject. As these misunderstandings have been widely spread among
philosophers, and cannot be easily removed except by a critic who has
gone into non-Euclidean Geometry with some care, it seems desirable
to discuss Lotze's strictures point by point.

[Illustration]

=91.= The mathematical criticism begins (§ 131) with a somewhat
question-begging definition of parallel straight lines. Two straight
lines _aα_, _bβ_, according to this definition, are parallel
when--_a_ and _b_ being arbitrary points on the two lines--if _aα_
= _bβ_, then _ab_ = _αβ_, where _α_, _β_ are two other points on the
two straight lines respectively. This definition--which contains
Euclid's axiom and definition combined in a very convenient and
enticing form--is of course thoroughly suitable to Euclidean
Geometry, and leads immediately to all the Euclidean propositions
about parallels. But it is perhaps more honest to follow Euclid's
course; when an axiom is thus buried in a definition, it is apt to
seem, since definitions are supposed to be arbitrary, as though the
difficulty had been overcome, while in reality, the possibility of
parallels, as above defined, involves the very point in question,
namely, the disputed axiom of parallels. For what this axiom asserts
is simply the existence of lines conforming to Lotze's definition.
The deduction of the principal propositions on parallels, with
which Lotze follows up his definition, is of course a very simple
proceeding--a proceeding, however, in which the first step begs the
question.

=92.= The next argument for the apriority of Euclidean Geometry has,
oddly enough, an exactly opposite bearing, although it is a great
favourite with opponents of Metageometry. Measurements of stellar
triangles, and all similar attempts at an empirical determination
of the space-constant are, according to Lotze, beside the mark; for
any observed departure from two right angles, or any finite annual
parallax for distant stars, would be attributed to some new kind of
refraction, or, as in the case of aberration, to some other physical
cause, and never to the geometrical nature of space. This is a
strong argument for the empirical validity of Euclid, but as an
argument for the apodeictic certainty of the orthodox system, it has
an opposite tendency. For observations of the kind contemplated would
have to be due to departures from Euclidean straightness, hitherto
unknown, on the part of stellar light-rays. Such departure could,
in certain cases, be accounted for by a finite space-constant, but
it could also, probably, be accounted for by a change in Optics,
for example, by attributing refractive properties to the ether.
Such properties could only exist if ether were of varying density,
if (say) it were denser in the neighbourhood of any of the heavenly
bodies. But such an assumption would, I believe, destroy the utility
of ether for Physics; a slight alteration in our Geometry, so slight
as not appreciably to affect distances within the Solar System,
would probably be in the end, therefore, should such errors ever
be discovered, a simpler explanation than any that Physics could
offer. But this is not the point of my contention. The point is
that, if the physical explanation, as Lotze holds, be possible in
the above case, the converse must also hold: it must be possible to
explain the present phenomena by supposing ether refractive and space
non-Euclidean. From this conclusion there is no escape. If every
conceivable behaviour of light-rays can be explained, within Euclid,
by physical causes, it must also be possible, by a suitable choice
of hypothetical physical causes, to explain the actual phenomena
as belonging to a non-Euclidean space. Such a hypothesis would be
rightly rejected by Science, for the present, on account of its
unnecessary complexity. Nevertheless it would remain, for philosophy,
a possibility to be reckoned with, and the choice could only be
decided upon empirical grounds of simplicity. It may well be doubted
whether, in the world we know, the phenomena could be attributed
to a distinctly non-Euclidean space, but this conclusion follows
inevitably from the contention that no phenomena could force us to
assume such a space. Lotze's argument, therefore, if pushed home,
disproves his own view, and puts Euclidean space, as an empirical
explanation of phenomena, on a level with luminiferous ether[103].

=93.= Lotze now proceeds (§ 132) to a detailed criticism of
Helmholtz, whom he regards as a typical exponent of Metageometry.
It is possible that, at the time when he wrote, Helmholtz really
did occupy this position; but it is unfortunate that, in the minds
of philosophers, he should still continue to do so, after the very
material advances brought about by the projective treatment of the
subject. It is also unfortunate that his somewhat careless attempts
to popularise mathematical results have so often been disposed of,
without due attention to his more technical and solid contributions.
Thus his romances about Flatland and Sphereland--at best only
fairy-tale analogies of doubtful value--have been attacked as if they
formed an essential feature of Metageometry.

But to proceed to particulars: Lotze readily allows that the
Flatlanders would set up Plane Geometry, as we know it, but refuses
to admit that the Spherelanders could, without inferring the third
dimension, set up a two-dimensional spherical Geometry which should
be free from contradictions. I will endeavour to give a free
rendering of Lotze's argument on this point.

[Illustration]

Suppose, he says, a north and south pole, _N_ and _S_, arbitrarily
fixed, and an equator _EW_. Suppose a being, _B_, capable of
impressions only from things on the surface of the sphere, to move
in a meridian _NBS_. Let _B_ start from some point _a_, and finally,
after describing a great circle, return to the same point _a_. If
_a_ is known only by the quality of the impression it makes on _B_,
_B_ may imagine he has not reached the same point _a_, but another
similar point _a′_, bearing a relation to _a_ similar to that of
the octave in singing: he might even not arrange his impressions
spatially at all. In order that this may occur, we require the
further assumption, that every difference in the above-mentioned
feelings (as he describes the meridian) may be presented as a spatial
distance between two places. Even now, _B_ may think he is describing
a Euclidean straight line, containing similar points at certain
intervals. Allowing, however, that he realizes the identity of _a_
with his initial position, he will now seem, by motion in a straight
line, to have returned to the point from which he started, for his
motion cannot, without the third dimension, seem to him other than
rectilinear.

Up to this point, there seems little ground for objection, except,
perhaps, to the idea of a straight line with periodical similar
points--if _B_ were as philosophical as, in these discussions,
we usually suppose him to be, he would probably object to this
interpretation of his experiences, on the ground that it regards
empty space as something independent of the objects in it. It is
worth pointing out, also, that _B_ would not need to describe the
whole circle, in order suddenly to find himself home again with his
old friends. Accurate measurements of small triangles would suffice
to determine his space-constant, and show him the length of a great
circle (or straight line, as he would call it). We must admit, also,
that so hypothetical a being as _B_ might form no space-intuition at
all, but as he is introduced solely for the purposes of the analogy,
it is convenient to allow him all possible qualifications for his
post. But these points do not touch the kernel of the argument, which
lies in the statement that such a straight line, returning into
itself after a finite time, would appear to _B_ as an "unendurable
contradiction," and thus force him, for logical though not for
sensational purposes, into the assumption of a third dimension. This
assertion seems to me quite unwarranted: the whole of Metageometry
is a solid array in disproof of it. Helmholtz's argument is, it must
be remembered, only an analogy, and the contradiction would exist
_only_ for a Euclidean. A complete _three_-dimensional Geometry has,
we have seen in Chapter I., been developed on the assumption that
straight lines are of finite length. A _constant_ value for the
measure of curvature, as our discussion of Riemann showed, involves
neither reference to the fourth dimension, nor any kind of internal
contradiction. This fact disproves Lotze's contention, which arises
solely from inability to divest his imagination of Euclidean ideas.

Lotze next attacks Helmholtz for the assertion that _B_ would know
nothing of parallel lines--parallel _straight_ lines, as the context
shows, he meant to say[104]. Lotze, however, takes him as meaning,
apparently, mere curves of constant distance from a given straight
line, which are part of the regular stock-in-trade of Metageometry.
Parallels of latitude, in the geographical sense, would not--with
the exception of the equator--appear to _B_ as straight lines, but
as circles. _Great_ circles he _would_ call straight, and this fact
seems to have misled Lotze into thinking _all_ circles were to be
treated as straight lines. Parallels of latitude, therefore, though
_B_ might call them parallels, would not invalidate Helmholtz's
contention, which applies only to straight lines.

The argument that such small circles would be parallel, which we
have just disposed of, is only the preface to another proof that _B_
would need a third dimension. Let us call two of these parallels
of latitude _l{n}_ and _l{s}_, and let them be equidistant from
the equator, one in the northern, one in the southern hemisphere.
Consecutive tangent planes, along these parallels, converge, in
the one case northwards, in the other southwards. Either _B_ could
become aware of their difference, says Lotze, or he could not. In
the former case, which he regards as the more probable, he easily
proves that _B_ would infer a third dimension. But this alternative
is, I think, wholly inadmissible. Tangent planes, like Euclidean
planes in general, would have no meaning to _B_; unless, indeed,
he were a metageometrician, which, with all his metaphysical and
mathematical subtlety, the argument supposes him not to be--and to
such a supposition Lotze, surely, is the last person who has a right
to object. Lotze's attempted proof that this is the right alternative
rests, if I understand him aright, on a sheer error in ordinary
spherical Geometry. _B_ would observe, he says, that the meridians
made smaller angles with his path towards the nearer than towards the
further pole--as a matter of fact, they would be simply perpendicular
to his path in both directions. What Lotze means is, perhaps, that
all the meridians would meet sooner in one direction than in the
other, and this, of course, is true. But the poles, in which the
meridians meet, would appear to _B_ as the centres of the respective
parallels, while the parallels themselves would appear to be circles.
Now I am at a loss to see what difficulty would arise, to _B_, in
supposing two different circles to have different centres[105]. We
must, therefore, take the first alternative, that _B_ would have no
sort of knowledge as to the direction in which the tangent planes
converged. Here Lotze attempts, if I have not misunderstood him, to
prove a _reductio ad absurdum_: _B_ would think, he says, that he
was describing two paths wholly the same in direction, and then he
_might_ regard both paths as circles in a plane. It may be observed
that direction, when applied to a circle as a whole, is meaningless;
indeed direction, in all Metageometry, can only mean, even when
applied to straight lines, direction towards a point. To speak of
two lines, which do not meet, as having the same direction, is a
surreptitious introduction of the axiom of parallels. Apart from
this, I cannot conceive any objection, on _B_'s part, to such a
view--one should say _must_, not _might_. The whole argumentation,
therefore, unless its obscurity has led me astray, must be pronounced
fruitless and inconclusive.

=94.= After this preliminary discussion of Sphereland, Lotze proceeds
to the question of a fourth dimension, and thence to spherical and
pseudo-spherical space. As before, he appears to know only the
more careless and popular utterances of Helmholtz and Riemann, and
to have taken no trouble to understand even the foundations of
mathematical Metageometry. By this neglect, much of what he says
is rendered wholly worthless. To begin with, he regards, as the
purpose of Helmholtz's fairy tale, the suggestion of a possible
fourth dimension, whereas the real purpose was quite the opposite--to
make intelligible a purely three-dimensional non-Euclidean space.
Helmholtz introduced Flatland only because its relation to Sphereland
is analogous to the relation of ours to spherical space[106]. But
Lotze says: The Flatlanders would find no difficulty in a third
dimension, since it would in no way contradict their own Geometry,
while the people in Sphereland, from the contradictions in their
two-dimensional system, would already have been led to it. The
latter contention I have already tried to answer; the former has an
odd sound, in view of the attempt, a few pages later, to prove _à
priori_ that all forms of intuition, in any way analogous to space,
_must_ have three dimensions. One cannot help suspecting that the
Flatlanders, with two instead of three dimensions, would make a
similar attempt. But to return to Lotze's argument: Neither analogy
can be used, he says, to prove that we ought perhaps to set up a
fourth dimension, since, for us, no contradictions or otherwise
inexplicable phenomena exist. The only people, so far as I know, who
have used this analogy, are Dr Abbot and a few Spiritualists--the
former in joke, the latter to explain certain phenomena more simply
explained, perhaps, by Maskelyne and Cooke. But although Lotze's
conclusion in this matter is sound, and one with which Helmholtz
might have agreed, his arguments, to my mind, are irrelevant and
unconvincing. There is this difference, he says, between us and the
Spherelanders: the latter were logically forced to a new dimension,
and found it possible; we are not forced to it, and find it, in our
space, impossible. I have contended that, on the contrary, nothing
would force the Spherelanders to assume a third dimension, while they
would find it impossible exactly as we find a fourth impossible--not
logically, that is to say, but only as a presentable construction in
given space.

After a somewhat elephantine piece of humour, about socialistic
whales in a four-dimensional sea of Fourrier's _eau sucrée_, Lotze
proceeds to a proof, by logic, that every form of intuition, which
embraces the whole system of ordered relations of a coexisting
manifold, _must_ have three dimensions. One might object, on _à
priori_ grounds, to any such attempt: what belongs to pure intuition
could hardly, one would have thought, be determined by _à priori_
reasoning[107]. I will not, however, develop this argument here,
but endeavour to point out, as far as its obscurity will allow, the
particular fallacy of the proof in question.

[Illustration]

Lotze's argument is as follows. In this discussion, though our
terminology is necessarily taken from space, we are really concerned
with a much more general conception. We assume, in order to preserve
the homogeneity of dimensions, that the difference (distance) between
any two elements (points) of our manifold--to borrow Riemann's
word--is of the same kind as, and commensurable with, the difference
between any other two elements. Let us take a series of elements at
successive distances _x_ such that the distance between any two is
the sum of the distances between intermediate elements. Such a series
corresponds to a straight line, which is taken as the _x_-axis. Then
a series _OY_ is called perpendicular to the _x_-axis _OX_, when
the distances of any element _y_, on _OY_, from +_mx_ and -_mx_ are
equal. By our hypothesis, these distances are comparable with, and
qualitatively similar to, _x_ and _y_. So long as _OY_ is defined
only by relation to _OX_, it is conceptually unique. But now let
us suppose the same relation as that between _OX_ and _OY_, to be
possible between _OY_ and a new series _OZ_; we then get a third
series _OZ_ perpendicular to _OY_, and again conceptually unique, so
long as it is defined by relation to _OY_ alone. We might proceed,
in the same way, to a fourth line _OU_ perpendicular to _OZ_. But it
is necessary, for our purposes, that _OZ_ should be perpendicular to
_OX_ as well as _OY_. Without this condition, _OZ_ might extend into
another world, and have no corresponding relation to _OX_--this is a
possibility only excluded by our unavoidable spatial images. At this
point comes the crux of the argument. _That OZ_, says Lotze, which,
besides being perpendicular to _OY_, is also perpendicular to _OX_,
must be among the series of _OY_'s, for these were defined only by
perpendicularity to _OX_. _Hence_, he concludes, there can only be
even a third dimension if _OZ_ coincides with one, and--as soon as
_OX_ is considered fixed--with _only_ one, of the many members of the
_OY_ series.

In this argument it is difficult--to me at any rate--to see any
force at all. The only way I can account for it is, to suppose that
Lotze has neglected the possibility of any but single infinities.
On this interpretation, the argument might be stated thus: There is
an infinite series of continuously varying _OY_'s; to the common
property of these, we add another property, which will divide their
total number by infinity. The remaining _OZ_, therefore, must be
uniquely determined. The same form of argument, however, would prove
that two surfaces can only cut one another in a single point, and
numberless other absurdities. The fact is, that infinities may be of
different orders. For example, the number of points in a line may be
taken as a single infinity, and so may the number of lines in a plane
through any point; hence, by multiplication, the number of points in
a plane is a double infinity, ∞^{2}, and if we divide this number
by a single infinity, we get still an infinite number left. Thus
Lotze's argument assumes what he has to prove, that the number of
lines perpendicular to a given line, through any point, is a single
infinity, which is equivalent to the axiom of three dimensions. The
whole passage is so obscure, that its meaning may have escaped me. It
is obvious _à priori_, however, as I pointed out in the beginning,
that any proof of the axiom must be fallacious somewhere, and the
above interpretation of the argument is the only one I have been able
to find.

=95.= The rest of the Chapter is devoted to an attack on spherical
and pseudo-spherical space, on the ground that they interfere with
the homogeneity of the three dimensions, and with the similarity
of all parts of space. This is simply false. Such spaces, like the
surface of a sphere, _are_ exactly alike throughout. Lotze shows,
here and elsewhere, that he has not taken the pains to find out
what Metageometry really is. I hold myself, and have tried to prove
in this Essay, that Congruence is an _à priori_ axiom, without
which Geometry would be impossible; but the wish to uphold this
axiom is, as Lotze ought to have known, the precise motive which
led Metageometry to limit itself to spaces of constant measure of
curvature. We see here the importance of distinguishing between
Helmholtz the philosopher and Helmholtz the mathematician. Though the
philosopher wished to dispense with Congruence, the mathematician, as
we saw in Chapter I., retained and strongly emphasized it. A little
later Lotze shows, again, how he has been misled by the unfortunate
analogy of Sphereland. A spherical _surface_, he says, he can
understand; but how are we to pass from this to a spherical space?
Either this surface is the whole of our space, as in Sphereland, or
it generates space by a gradually growing radius. Such concentric
spheres, as Lotze triumphantly points out, of course generate
Euclidean space. His disjunction, however, is utterly and entirely
false, and could never have been suggested by any one with even a
superficial knowledge of Metageometry. This point is less laboured
than the former, which, in all its nakedness, is thus re-stated in
the last sentence of the Chapter: "I cannot persuade myself that one
could, without the elements of homogeneous space, even form or define
the presentation of heterogeneous spaces, or of such as had variable
measures of curvature." As though such spaces were ever set up by
non-Euclidean mathematics!

In conclusion, Lotze expresses a hope that Philosophy, on this
point, will not allow itself to be imposed upon by Mathematics. I
must, instead, rejoice that Mathematics has not been imposed upon by
Philosophy, but has developed freely an important and self-consistent
system, which deserves, for its subtle analysis into logical and
factual elements, the gratitude of all who seek for a philosophy of
space.

=96.= The objections to non-Euclidean Geometry which have just been
discussed fall under four heads:

I. Non-Euclidean spaces are not homogeneous; Metageometry therefore
unduly reifies space.

II. They involve a reference to a fourth dimension.

III. They cannot be set up without an implicit reference to Euclidean
space, or to the Euclidean straight line, on which they are therefore
dependent.

IV. They are self-contradictory in one or more ways.

The reader who has followed me in regarding these four objections as
fallacious, will have no difficulty in disposing of any other critic
of Metageometry, as these are the only mathematical arguments, so
far as I know, ever urged against non-Euclideans[108]. The logical
validity of Metageometry, and the mathematical possibility of
three-dimensional non-Euclidean spaces, will therefore be regarded,
throughout the remainder of the work, as sufficiently established.

=97.= Two other objections may, indeed, be urged against
Metageometry, but these are rather of a philosophical than of a
strictly mathematical import. The first of these, which has been
made the base of operations by Delbœuf, applies equally to all
non-Euclidean spaces. The second, which has not, so far as I know,
been much employed, but yet seems to me deserving of notice, bears
directly against spaces of positive curvature alone; but if it could
discredit these, it might throw doubt on the method by which all
alike are obtained. The two objections are:

I. Space must be such as to allow of similarity, _i.e._ of the
increase or diminution, in a constant ratio, of all the lines in a
figure, without change of angles; whereas in non-Euclid, lines, like
angles, have absolute magnitude.

II. Space must be infinite, whereas spherical and elliptic spaces are
finite.

I will discuss the first objection in connection with Delbœuf's
articles referred to above. The second, which has not, to my
knowledge, been widely used in criticism, will be better deferred to
Chapter III.


Delbœuf.

=98.= M. Delbœuf's four articles in the Revue Philosophique contain
much matter that has already been dealt with in the criticism of
Lotze, and much that is irrelevant for our present purpose. The only
point, which I wish to discuss here, is the question of absolute
magnitude, as it is called--the question, that is, whether the
possibility of similar but unequal geometrical figures can be known
_à priori_[109].

In discussing this question, it is important, to begin with, to
distinguish clearly the sense in which absolute magnitude _is_
required in non-Euclidean Geometry, from another sense, in which
it would be absurd to regard any magnitude as absolute. Judgments
of magnitude can only result from comparison, and if Metageometry
required magnitudes which could be determined without comparison, it
would certainly deserve condemnation. But this is not required. All
we require is, that it shall be impossible, while the rest of space
is unaffected, to alter the magnitude of any figure, as compared with
other figures, while leaving the relative internal magnitudes of its
parts unchanged. This construction, which is possible in Euclid,
is impossible in Metageometry. We have to discuss whether such an
impossibility renders non-Euclidean spaces logically faulty.

M. Delbœuf's position on this axiom--which he calls the postulate
of homogeneity[110]--is, that all Geometry must presuppose it, and
that Metageometry, consequently, though logically sound, is logically
subsequent to Euclid, and can only make its constructions within a
Euclidean "homogeneous" space (Rev. Phil. Vol. XXXVII., pp. 380-1).
He would appear to think, nevertheless, that homogeneity (in his
sense) is learnt from experience, though on this point he is not
very explicit. (See Vol. XXXVIII., p. 129.) No _à priori_ proof, at
any rate, is offered in his articles. As a result of experience,
every one would admit, similarity is known to be possible within the
limits of observation; but the fact that this possibility extends to
Ordnance maps, which deal with a spherical surface, should make us
chary of inferring, from such a datum, the certainty of Euclid for
large spaces. Moreover if homogeneity be empirical, Metageometry,
which dispenses with it, is not necessarily in _logical_ dependence
upon Euclid, since homogeneity and isogeneity are _logically_
separable. I shall assume, therefore, as the only contention which
can be interesting to our argument, that homogeneity is regarded as
_à priori_, and as logically essential to Geometry.

=99.= Now we saw, in discussing Erdmann's views of the judgment of
quantity, that in non-Euclidean space, as in Euclidean, a change
of all spatial magnitudes, in the same ratio, would be no change
at all; the ratios of all magnitudes to the space-constant would
be unchanged, and the space-constant, as the ultimate standard of
comparison, cannot, in any intelligible sense, be said to have any
particular magnitude. The absolute magnitudes of Metageometry,
therefore, are absolute only as against any other _particular_
magnitude, not as against other magnitudes in general. If this were
not the case, the comparative nature of the judgment of magnitude
would be contradicted, and metrical Metageometry would become absurd.
But as it is, the difference from Euclid consists only in this: that
in Metageometry we have, while in Euclid we have not, a standard of
comparison involved in the nature of our space as a whole, which we
call the space-constant. We have to discuss whether the assertion of
such a standard involves an undue reification of space.

I do not believe that this is the case. For an undue reification of
space would only arise, if we were no longer able to regard position
as wholly relative, and as geometrically definable only by departure
from other positions. But the relativity of position, as we have
abundantly seen, is preserved by all spaces of constant curvature--in
all of these, positions can only be defined, geometrically, by
relations to fresh positions[111]. This series of definitions may
lead to an infinite regress, but it may also, as in spherical space,
form a vicious circle, and return again to the position from which
it started. No reification of space, no independent existence of
mere relations, seems involved in such a procedure. The whole of
Metageometry, in short, is a proof that the relativity of position
is compatible with absolute magnitude, in the only sense required
by non-Euclidean spaces. We must conclude, therefore, that there
is nothing incompatible, in a denial of homogeneity (in Delbœuf's
sense), either with the relational nature of space, or with the
comparative nature of magnitude. This last _à priori_ objection to
Metageometry, therefore, cannot be maintained, and the issue must be
decided on empirical grounds alone.

=100.= The foundations of Geometry have been the subject of much
recent speculation in France, and this seems to demand some notice.
But in spite of the splendid work which the French have done on the
allied question of number and continuous quantity, I cannot persuade
myself that they have succeeded in greatly advancing the subject
of geometrical philosophy. The chief writers have been, from the
mathematical side, _Calinon_ and _Poincaré_, from the philosophical,
_Renouvier_ and _Delbœuf_; as a mediator between mathematics and
philosophy, _Lechalas_.

_Calinon_, in an interesting article on the geometrical
indeterminateness of the universe, maintains that any Geometry may be
applied to the actual world by a suitable hypothesis as to the course
of light-rays. For the earth only is known to us otherwise than by
Optics, and the earth is an infinitesimal part of the universe.
This line of argument has been already discussed in connection with
Lotze, but Calinon adds a new suggestion, that the space-constant may
perhaps vary with the time. This would involve a causal connection
between space and other things, which seems hardly conceivable, and
which, if regarded as possible, must surely destroy Geometry, since
Geometry depends throughout on the irrelevance of Causation[112].
Moreover, in all operations of measurement, some time is spent;
unless we knew that space was unchanging throughout the operation,
it is hard to see how our results could be trustworthy, and how,
consequently, a change in the parameter could be discovered. The
same difficulties would arise, in fact, as those which result from
supposing space not homogeneous.

_Poincaré_ maintains that the question, whether Euclid or
Metageometry should be accepted, is one of convenience and
convention, not of truth; axioms are definitions in disguise, and the
choice between definitions is arbitrary. This view has been discussed
in Chapter I., in connection with Cayley's theory of distance, on
which it depends.

_Lechalas_ is a philosophical disciple of Calinon. He is a
rationalist of the pre-Kantian type, but a believer in the validity
of Metageometry. He holds that Geometry can dispense with all purely
spatial postulates, and work with axioms of magnitude alone[113],
which, in his opinion, are purely analytic. The principle of
contradiction, to him, is the sole and only test of truth; we make
long chains of reasoning from our premisses to see if contradictions
will emerge. It might be objected that this view, though it saves
general Geometry from being logically empirical, leaves it only
empirically logical; this must, in fact, be the fate of every piece
of _à priori_ knowledge, if M. Lechalas's were the only test of
truth. However, he concludes that general Geometry is apodeictic,
while the space of our actual world, like all other phenomena, is
contingent.

_Delbœuf_ criticizes non-Euclidean space from an ultra-realist
standpoint: he holds that _real_ space is neither homogeneous nor
isogeneous, but that _conceived_ space, as abstracted from real
space, has both these properties. He offers no justification for
his real space, which seems to be maintained in the spirit of naïve
realism, nor does he show how he has acquired his intimate knowledge
of its constitution[114]. His arguments against Metageometry, in so
far as they are not repetitions of Lotze, have been discussed above.

_Renouvier_, finally, is a pure Kantian, of the most orthodox type.
His views as to the importance, for Geometry, of the distinction
between synthetic and analytic judgments, have been discussed, in
connection with Kant, at the beginning of the present Chapter[115].

=101.= Before beginning the constructive argument of the next
Chapter, let us endeavour briefly to sum up the theories which
have been polemically advocated throughout the criticisms we have
just concluded. We agreed to accept, with Kant, necessity for any
possible experience as the test of the _à priori_, but we refused,
for the present, to discuss the connection of the _à priori_ with
the subjective, regarding the purely logical test as sufficient for
our immediate purpose. We also refused to attach importance to the
distinction of analytic and synthetic, since it seemed to apply, not
to different judgments, but only to different aspects of any judgment.

We then discussed Riemann's attempt to identify the empirical element
in Geometry with the element not deducible from ideas of magnitude,
and we decided that this identification was due to a confusion as
to the nature of magnitude. For judgments of magnitude, we said,
require always some qualitative basis, which is not quantitatively
expressible.

In criticizing Helmholtz, we decided that Mechanics logically
presupposes Geometry, though space presupposes matter; but that the
matter which space presupposes, and to which Geometry indirectly
refers, is a more abstract matter than that of Mechanics, a matter
destitute of force and of causal attributes, and possessed only
of the purely spatial attributes required for the possibility
of spatial figures. But we conceded that Geometry, when applied
to mixed mathematics or to daily life, demands more than this,
demands, in fact, some means of discovering, in the more concrete
matter of Mechanics, either a rigid body, or a body whose departure
from rigidity follows some empirically discoverable law. _Actual_
measurement, therefore, we agreed to regard as empirical.

Our conclusions, as regards the empiricism of Riemann and Helmholtz,
were reinforced by a criticism of Erdmann. We then had an opposite
task to perform, in defending Metageometry against Lotze. Here we
saw that there are two senses in which Metageometry is possible. The
first concerns our actual space, and asserts that it may have a very
small space-constant; the second concerns philosophical theories
of space, and asserts a purely logical possibility, which leaves
the decision to experience. We saw also that Lotze's mathematical
strictures arose from insufficient knowledge of the subject, and
could all be refuted by a better acquaintance with Metageometry.

Finally, we discussed the question of absolute magnitude, and found
in it no logical obstacle to non-Euclidean spaces. Our conclusion,
then, in so far as we are as yet entitled to a conclusion, is
that all spaces with a space-constant are _à priori_ justifiable,
and that the decision between them must be the work of experience.
Spaces without a space-constant, on the other hand, spaces, that is,
which are not homogeneous throughout, we found logically unsound
and impossible to know, and therefore to be condemned _à priori_.
The constructive proof of this thesis will form the argument of the
following chapter.

FOOTNOTES:

[67] The Critical Philosophy of Kant, Vol. I. p. 287.

[68] For a discussion of Kant from a less purely mathematical
standpoint, see Chap. IV.

[69] Cf. Vaihinger's Commentar, II. pp. 202, 265. Also p. 336 ff.

[70] E.g. second edition, p. 39: "So werden auch alle geometrischen
Grundsätze, z. B. dass in einem Triangel zwei Seiten zusammen grösser
sind als die dritte, niemals aus allgemeinen Begriffen von Linie
und Triangel, sondern aus der Anschauung, und zwar _à priori_ mit
apodiktischer Gewissheit abgeleitet."

[71] Cf. Bradley's Logic, Bk. III. Pt. I. Chap. VI.; Bosanquet's
Logic, Bk. I. Chap. I. pp. 97-103.

[72] Philosophie de la Règle et du Compas, Année Philosophique, II.
pp. 1-66.

[73] I have stated this doctrine dogmatically, as a proof would
require a whole treatise on Logic. I accept the proofs offered by
Bradley and Bosanquet, to which the reader is referred.

[74] For a further discussion of this point, see Chaps. III. and IV.

[75] See Chap. IV. for a discussion of this argument.

[76] See Chap. IV. § 185.

[77] An Otherness of substance, rather than of attribute, is here
intended; an Otherness which may perhaps be called real as opposed to
logical diversity.

[78] This proposition will be argued at length in Chap. IV.

[79] See Psychologie als Wissenschaft, I. Section III. Chap. VII.;
II. Section I. Chap. III. and Section II. Chap. III. Compare also
Synechologie, Section I. Chaps. II. and III.

[80] On the influence of Herbart on Riemann, compare Erdmann, Die
Axiome der Geometrie, p. 30.

[81] I do not mean that measurement of colours is effected without
reference to their relations, since all measurement is essentially
comparison. But in colours, it is the elements which are compared,
while in space, it is the relations between elements.

[82] For a discussion of this point, see Chap. III. Sec. B, § 176.

[83] The works of Helmholtz on geometrical philosophy comprise, in
addition to the articles quoted in Chap. I., the following articles:
"Ursprung und Sinn der geometrischen Axiome, gegen Land," Wiss. Abh.
Vol. II. p. 640, 1878. (Also Mind, Vol. III.: an answer to Land in
Mind, Vol. II.) "Ursprung und Bedeutung der geometrischen Axiome,"
1870, Vorträge und Reden, Vol. II. p. 1. (Also Mind, Vol. I.) Two
Appendices to "Die Thatsachen in der Wahrnehmung," entitled: II. "Der
Raum kann transcendental sein, ohne dass es die Axiome sind"; and
III. "Die Anwendbarkeit der Axiome auf die physische Welt," 1878,
Vorträge und Reden, Vol. II. p. 256 ff.

The two Appendices last mentioned are popularizings and expansions of
the article in Mind, Vol. III. The most widely read, though also, to
my mind, the least valuable, of all Helmholtz's writings on Geometry,
is the article in Mind, Vol. I. This contains the famous and much
misunderstood analogies of Flatland and Sphereland, which will be
discussed, and as far as possible defended, in answering Lotze's
attack on Metageometry--an attack based, apparently, almost entirely
on this one popular article. The present discussion, therefore, may
be confined almost entirely to Mind, Vol. III., and the philosophical
portions of the two papers quoted in Chap. I., _i.e._ to the articles
in Wiss. Abh. Vol. II. pp. 610-660. His other works are popular, and
important only because of the large public to which they appeal.

[84] In the answer to Land, Mind, Vol. III. and Wiss. Abh. II. p. 640.

[85] See also Die Thatsachen in der Wahrnehmung, Zusatz II., Der Raum
kann transcendental sein, ohne dass es die Axiome sind. Vorträge und
Reden, Vol. II.

[86] See below, criticism of Erdmann, § 84.

[87] See Prof. Land, in Mind, Vol. II.

[88] See concluding paragraph of Helmholtz's article in Mind, Vol.
III.

[89] Cf. Veronese, Grundzüge der Geometrie (German translation), p.
ix. Also pp. xxxiv, 304, and Note II. pp. 692-4.

[90] See Chap. IV. § 197 ff.

[91] Cf. the opinion of Bolyai, quoted by Erdmann, Axiome, p. 26; cf.
also ib. p. 60.

[92] Die Axiome der Geometrie: Eine philosophische Untersuchung der
Riemann-Helmholtz'schen Raumtheorie, Leipzig, 1877.

[93] On the influence of Mill, cf. Stallo, Concepts of Modern
Physics, p. 216.

[94] This view seems to be derived, through Riemann, from Herbart.
See Psych. als Wiss. ed. Hart. Vol. V. p. 262.

[95] The same irreducibility of space to mere magnitude is proved by
Kant's hands and spherical triangles, in which a difference persists
in spite of complete quantitative equality.

[96] See §§ 146-7.

[97] "Jeder Versuch, Kant's Lehre von der Apriorität als
des subjectiven, von aller Erfahrung absolut unabhängigen
Erkenntnissfactors, trotzdem zu halten, ist deshalb von voruherein
aussichtslos."

[98] Ausdehnungslehre von 1844, 2nd edition, pp. xxii. xxiii.

[99] See § 129 ff.

[100] Metaphysik, Book II. Chap. II. My references are to the
original.

[101] See Lectures and Essays, Vol. I. p. 261.

[102] On the meaning of geometrical possibility, cf. Veronese,
Grundzüge der Geometrie (German translation), pp. xi.-xiii.

[103] Compare Calinon, "Sur l'Indétermination géométrique de
l'Univers," Revue Philosophique, 1893, Vol. XXXVI. pp. 595-607.

[104] Vorträge und Reden, Vol. II. p. 9: "Parallele Linien würden
die Bewohner der Kugel gar nicht kennen. Sie würden behaupten, dass
jede beliebige zwei _geradeste_ Linien, gehörig verlangert, sich
schliesslich nicht nur in einem, sondern in zwei Punkten schneiden
müssten." (The italics are mine.) The omission of _straight_ in such
phrases is a frequent laxity of mathematicians.

[105] It has been suggested to me that Lotze regards the meridians
as projected on to a plane, as in a map. If this be so, there is an
obviously illegitimate introduction of the third dimension.

[106] This is proved by Helmholtz's remark at the end of a detailed
attempt to make spherical and pseudo-spherical spaces imaginable
(l.c. p. 28): "Anders ist es mit den drei Dimensionen des Raumes. Da
alle unsere Mittel sinnlicher Anschauung sich nur auf einen Raum von
drei Dimensionen erstrecken, und die vierte Dimension nicht bloss
eine Abänderung von Vorhandenem, sondern etwas vollkommen Neues wäre,
so befinden wir uns schon wegen unserer körperlichen Organisation in
der absoluten Unmöglichkeit, uns eine Anschauungsweise einer vierten
Dimension vorzustellen."

[107] Cf. Grassmann, Ausdehnungslehre von 1844, 2nd Edition, p. xxiii.

[108] See especially Stallo, Concepts of Modern Physics,
International Science Series, Vol. XLII. Chaps. XIII. and
XIV.; Renouvier, "Philosophie de la règle et du compas," Année
Philosophique, II.; Delbœuf, "L'ancienne et les nouvelles
géométries," Revue Philosophique, Vols. XXXVI.-XXXIX.

[109] M. Delbœuf deserves credit for having based Euclid, already in
1860, in his "Prolégomènes Philosophiques de la Géométrie," on this
axiom--certainly a better basis, at first sight, than the axiom of
parallels.

[110] This meaning of homogeneity must not be confounded with the
sense in which I have used the word. In Delbœuf's sense, it means
that figures may be similar though of different sizes; in my sense it
means that figures may be similar though in different places. This
property of space is called by Delbœuf isogeneity.

[111] For a full proof of this proposition, see Chap. III.

[112] See Chap. III., especially § 133.

[113] For a criticism of this view, see the above discussions on
Riemann and Erdmann.

[114] Cf. Couturat, "De l'Infini Mathématique," Paris, Félix Alcan,
1896, p. 544.

[115] The following is a list of the most important recent French
philosophical writings on Geometry, so far as I am acquainted with
them.

  Andrade: "Les bases expérimentales de la géométrie euclidienne";
  Rev. Phil. 1890, II., and 1891, I.

  Bonnel: "Les hypothèses dans la géométrie"; Gauthier-Villars,
  1897.

  L'Abbé de Broglie: "La géométrie non-euclidienne," two articles;
  Annales de Phil. Chrét. 1890.

  Calinon: "Les espaces géométriques"; Rev. Phil. 1889, I., and
  1891, II. "Sur l'indétermination géométrique de l'univers"; ib.
  1893, II.

  Couturat: "L'Année Philosophique de F. Pillon," Rev. de Mét. et
  de Morale, Jan. 1893.

  "Note sur la géométrie non-euclidienne et la relativité de
  l'espace"; ib., May, 1893.

  "Études sur l'espace et le temps," ib. Sep. 1896.

  Delbœuf: "L'ancienne et les nouvelles géométries," four articles;
  Rev. Phil. 1893-5.

  Lechalas: "La géométrie générale"; Crit. Phil. 1889.

  "La géométrie générale et les jugements synthétiques à priori"
  and "Les bases expérimentales de la géométrie"; Rev. Phil. 1890,
  II.

  "M. Delbœuf et Le problème des mondes semblables"; ib. 1894, I.

  "Note sur la géométrie non-euclidienne et le principe de
  similitude"; Rev. de Mét. et de Morale, March, 1893.

  "La courbure et la distance en géométrie générale"; ib., March,
  1896.

  "La géométrie générale et l'intuition"; Annales de Phil. Chrét.,
  1890.

  "Etude sur l'espace et le temps"; Paris, Alcan, 1896.

  Liard: "Des définitions géométrie et des définitions empiriques,"
  2nd ed.; Paris, Alcan, 1888.

  Mansion: "Premiers principes de la métagéométrie"; two
  articles in Rev. Néo-Scholastique, 1896. Separately published,
  Gauthier-Villars, 1896.

  Milhaud: "La géométrie non-euclidienne et la théorie de la
  connaissance"; Rev. Phil. 1888, I.

  Poincaré: "Non-Euclidian Geometry"; Nature, Vol. XLV., 1891-2.

  "L'espace et la géométrie"; Rev. de Mét. et de Morale, Nov. 1895.

  "Résponse à quelques critiques," ib. Jan. 1897.

  Renouvier: "Philosophie de la règle et du compas"; Crit. Phil.,
  1889, and L'Année Phil., II^{me} année, 1891.

  Sorel: "Sur la géométrie non-euclidienne"; Rev. Phil., 1891, I.

  Tannery: "Théorie de la connaissance mathématique"; Rev. Phil.,
  1894, II.




CHAPTER III.


Section A.

THE AXIOMS OF PROJECTIVE GEOMETRY.

=102.= Projective Geometry proper, as we saw in Chapter I., does not
employ the conception of magnitude, and does not, therefore, require
those axioms which, in the systems of the second or metrical period,
were required solely to render possible the application of magnitude
to space. But we saw, also, that Cayley's reduction of metrical
to projective properties was purely technical and philosophically
irrelevant. Now it is in metrical properties alone--apart from
the exception to the axiom of the straight line, which itself,
however, presupposes metrical properties[116]--that non-Euclidean
and Euclidean spaces differ. The properties dealt with by projective
Geometry, therefore, in so far as these are obtained without the use
of imaginaries, are properties common to all spaces. Finally, the
differences which appear between the Geometries of different spaces
of the same curvature--_e.g._ between the Geometries of the plane
and the cylinder--are differences in projective properties[117].
Thus the necessity which arises, in metrical Geometry, for further
qualifications besides those of constant curvature, disappears when
our general space is defined by purely projective properties.

=103.= We have good ground for expecting, therefore, that the axioms
of projective Geometry will be the simplest and most complete
expression of the indispensable requisites of any geometrical
reasoning: and this expectation, I hope, will not be disappointed.
Projective Geometry, in so far as it deals only with the properties
common to all spaces, will be found, if I am not mistaken, to be
wholly _à priori_, to take nothing from experience, and to have, like
Arithmetic, a creature of the pure intellect for its object. If this
be so, it is that branch of pure mathematics which Grassmann, in his
_Ausdehnungslehre_ of 1844, felt to be possible, and endeavoured, in
a brilliant failure, to construct without any appeal to the space of
intuition.

=104.= But unfortunately, the task of discovering the axioms of
projective Geometry is far from easy. They have, as yet, found
no Riemann or Helmholtz to formulate them philosophically. Many
geometers have constructed systems, which they intended to be, and
which, with sufficient care in interpretation, really are, free from
metrical presuppositions. But these presuppositions are so rooted
in all the very elements of Geometry, that the task of eliminating
them demands a reconstruction of the whole geometrical edifice. Thus
Euclid, for example, deals, from the start, with spatial equality--he
employs the circle, which is necessarily defined by means of
equality, and he bases all his later propositions on the congruence
of triangles as discussed in Book I.[118] Before we can use any
elementary proposition of Euclid, therefore, even if this expresses a
projective property, we have to prove that the property in question
can be deduced by projective methods. This has not, in general,
been done by projective geometers, who have too often assumed, for
example, that the quadrilateral construction--by which, as we saw in
Chap. I., they introduce projective coordinates--or anharmonic ratio,
which is _primâ facie_ metrical, could be satisfactorily established
on their principles. Both these assumptions, however, can be
justified, and we may admit, therefore, that the claims of projective
Geometry to logical independence of measurement or congruence are
valid. Let us see, then, how it proceeds.

=105.= In the first place, it is important to realize that
when coordinates are used, in projective Geometry, they are not
coordinates in the ordinary metrical sense, _i.e._ the numerical
measures of certain spatial magnitudes. On the contrary, they are a
set of numbers, arbitrarily but systematically assigned to different
points, like the numbers of houses in a street, and serving only,
from a philosophical standpoint, as convenient designations for
points which the investigation wishes to distinguish. But for the
brevity of the alphabet, in fact, they might, as in Euclid, be
replaced by letters. How they are introduced, and what they mean, has
been discussed in Chapter I. Here we have only to repeat a caution,
whose neglect has led to much misunderstanding.

=106.= The distinction between various points, then, is not a result,
but a condition, of the projective coordinate system. The coordinate
system is a wholly extraneous, and merely convenient, set of marks,
which in no way touches the essence of projective Geometry. What we
must begin with, in this domain, is the possibility of distinguishing
various points from one another. This may be designated, with
Veronese, as the first axiom of Geometry[119]. How we are to define
a point, and how we distinguish it from other points, is for the
moment irrelevant; for here we only wish to discover the nature
of projective Geometry, and the kind of properties which it uses
and demonstrates. How, and with what justification, it uses and
demonstrates them, we will discuss later.

=107.= Now it is obvious that a mere collection of points,
distinguished one from another, cannot found a Geometry: we must
have some idea of the manner in which the points are interrelated,
in order to have an adequate subject-matter for discussion. But
since all ideas of quantity are excluded, the relations of points
cannot be relations of distance in the ordinary sense, nor even, in
the sense of ordinary Geometry, anharmonic ratios, for anharmonic
ratios are usually defined as the ratios of four distances, or of
four sines, and are thus quantitative. But since all quantitative
comparison presupposes an identity of quality, we may expect to find,
in projective Geometry, the qualitative substrata of the metrical
superstructure.

And this, we shall see, is actually the case. We have not distance,
but we _have_ the straight line; we have not quantitative anharmonic
ratio, but we _have_ the property, in any four points on a line,
of being the intersections with the rays of a given pencil. And
from this basis, we can build up a qualitative science of abstract
externality, which is projective Geometry. How this happens, I shall
now proceed to show.

=108.= All geometrical reasoning is, in the last resort, circular:
if we start by assuming points, they can only be defined by the
lines or planes which relate them; and if we start by assuming lines
or planes, they can only be defined by the points through which
they pass. This is an inevitable circle, whose ground of necessity
will appear as we proceed. It is, therefore, somewhat arbitrary to
start either with points or with lines, as the eminently projective
principle of duality mathematically illustrates; nevertheless we will
elect, with most geometers, to start with points[120]. We suppose,
therefore, as our datum, a set of discrete points, for the moment
without regard to their interconnections. But since connections are
essential to any reasoning about them as a system, we introduce, to
begin with, the axiom of the straight line. Any two of our points,
we say, lie on a line which those two points completely define.
This line, being determined by the two points, may be regarded as
a relation of the two points, or an adjective of the system formed
by both together. This is the only purely qualitative adjective--as
will be proved later--of a system of two points. Now projective
Geometry can only take account of qualitative adjectives, and can
distinguish between different points only by their relations to
other points, since all points, _per se_, are qualitatively similar.
Hence it comes that, for projective Geometry, when two points only
are given, they are qualitatively indistinguishable from any two
other points on the same straight line, since any two such other
points have the same qualitative relation. Reciprocally, since one
straight line is a figure determined by any two of its points, and
all points are qualitatively similar, it follows that all straight
lines are qualitatively similar. We may regard a point, therefore, as
determined by two straight lines which meet in it, and the point,
on this view, becomes the only qualitative relation between the two
straight lines. Hence, if the point only be regarded as given, the
two straight lines are qualitatively indistinguishable from any other
pair through the point.

=109.= The extension of these two reciprocal principles is the
essence of all projective transformations, and indeed of all
projective Geometry. The fundamental operations, by which figures
are projectively transformed, are called projection and section. The
various forms of projection and section are defined in Cremona's
"Projective Geometry," Chapter I., from which I quote the following
account.

"_To project from a fixed point S_ (the _centre of projection_) a
figure (_ABCD_ ... _abcd_ ...) composed of points and straight lines,
is to construct the straight lines or _projecting rays SA_, _SB_,
_SC_, _SD_, ... and the planes (_projecting planes_) _Sa_, _Sb_,
_Sc_, _Sd_, ... We thus obtain a new figure composed of straight
lines and planes which all pass through the centre _S_.

"_To cut by a fixed plane σ (transversal plane_) a figure (_αβγδ_ ...
_abcd_ ...) made up of planes and straight lines, is to construct the
straight lines or _traces σα, σβ, σγ_ ... and the points or _traces
σa, σb, σc_....[121] By this means we obtain a new figure
composed of straight lines and points lying in the plane _σ_.

"_To project from a fixed straight line s_ (the _axis_) a figure
_ABCD_ composed of points, is to construct the planes _sA_, _sB_,
_sC_.... The figure thus obtained is composed of planes which all
pass through the axis _s_.

"_To cut by a fixed straight line s_ (a _transversal_) a figure
_αβγδ_ ... composed of planes, is to construct the points _sα_, _sβ_,
_sγ_.... In this way a new figure is obtained, composed of points all
lying on the fixed transversal _s_.

"If a figure is composed of straight lines _a_, _b_, _c_ ... which
all pass through a fixed point or _centre S_, it can be _projected_
from a straight line or _axis s_ passing through _S_; the result is a
figure composed of planes _sa_, _sb_, _sc_....

"If a figure is composed of straight lines _a_, _b_, _c_ ...
all lying in a fixed plane, it may be cut by a straight line
(transversal) _s_ lying in the same plane; the figure which results
is formed by the points _sa_, _sb_, _sc_...."

=110.= The successive application, to any figure, of two reciprocal
operations of projection and section, is regarded as producing a
figure protectively indistinguishable from the first, provided only
that the dimensions of the original figure were the same as those
of the resulting figure, that, for example, if the second operation
be section by a plane, the original figure shall have been a plane
figure. The figures obtained from a given figure, by projection
or section alone, are related to that figure by the principle of
duality, of which we shall have to speak later on.

I shall endeavour to show, in what follows, first, in what sense
figures obtained from each other by projective transformation are
qualitatively alike; secondly, what axioms, or adjectives of space,
are involved in the principle of projective transformation; and
thirdly, that these adjectives must belong to any form of externality
with more than one dimension, and are, therefore, _à priori_
properties of any possible space.

For the sake of simplicity, I shall in general confine myself to two
dimensions. In so doing, I shall introduce no important difference of
principle, and shall greatly simplify the mathematics involved.

=111.= The two mathematically fundamental things in projective
Geometry are anharmonic ratio, and the quadrilateral construction.
Everything else follows mathematically from these two. Now what is
meant, in projective Geometry, by anharmonic ratio?

[Illustration]

If we start from anharmonic ratio as ordinarily defined, we are
met by the difficulty of its quantitative nature[122]. But among
the properties deduced from this definition, many, if not most,
are purely qualitative. The most fundamental of these is that, if
through any four points in a straight line we draw four straight
lines which meet in a point, and if we then draw a new straight line
meeting these four, the four new points of intersection have the
same anharmonic ratio as the four points we started with. Thus, in
the figure, _abcd_, _a′b′c′d′_, _a″b″c″d″_, all have the same
anharmonic ratio. The reciprocal relation holds for the anharmonic
ratio of four straight lines. Here we have, plainly, the required
basis for a qualitative definition. The definition must be as follows:

Two sets of four points each are defined as having the same
anharmonic ratio, when (1) each set of four lies in one straight
line, and (2) corresponding points of different sets lie two by two
on four straight lines through a single point, or when both sets have
this relation to any third set[123]. And reciprocally: Two sets of
four straight lines are defined as having the same anharmonic ratio
when (1) each set of four passes through a single point, and (2)
corresponding lines of different sets pass, two by two, through four
points in one straight line, or when both sets have this relation to
any third set.

Two sets of points or of lines, which have the same anharmonic ratio,
are treated by projective Geometry as equivalent: this qualitative
equivalence replaces the quantitative equality of metrical Geometry,
and is obviously included, by its definition, in the above account of
projective transformations in general.

=112.= We have next to consider the quadrilateral construction[124].
This has a double purpose: first, to define the important special
case known as a harmonic range; and secondly, to afford an
unambiguous and exhaustive method of assigning different numbers to
different points. This last method has, again, a double purpose:
first, the purpose of giving a convenient symbolism for describing
and distinguishing different points, and of thus affording a means
for the introduction of analysis; and secondly, of so assigning
these numbers that, if they had the ordinary metrical significance,
as distances from some point on the numbered straight line, they
would yield -1 as the anharmonic ratio of a harmonic range, and
that, if four points have the same anharmonic ratio as four
others, so have the corresponding numbers. This last purpose is
due to purely technical motives: it avoids the confusion with our
preconceptions which would result from any other value for a harmonic
range; it allows us, when metrical interpretations of projective
results are desired, to make these interpretations without tedious
numerical transformations, and it enables us to perform projective
transformations by algebraical methods. At the same time, from the
strictly projective point of view, as observed above, the numbers
introduced have a purely conventional meaning; and until we pass to
metrical Geometry, no reason can be shown for assigning the value -1
to a harmonic range. With this preliminary, let us see in what the
quadrilateral construction consists.

[Illustration]

=113.= A harmonic range, in elementary Geometry, is one whose
anharmonic ratio is -1, or one in which the three segments formed by
the four points are in harmonic progression, or again, one in which
the ratio of the two internal segments is equal to the ratio of the
two external segments. If _a_, _b_, _c_, _d_ be the four points, it
is easily seen that these definitions are equivalent to one another:
they give respectively:

    (ab/bc)/(ad/dc) = -1, (1/ab) - (1/ac) = (1/ac) - (1/ad),

          and (ab/bc) = (ad/cd).

But as they are all quantitative, they cannot be used for our present
purpose. Nor are any definitions which involve bisection of lines or
angles available. We must have a definition which proceeds entirely
by the help of straight lines and points, without measurement of
distances or angles. Now from the above definitions of a harmonic
range, we see that _a_, _b_, _c_, _d_ have the same anharmonic ratio
as _c_, _b_, _a_, _d_. This gives us the property we require for our
definition. For it shows that, in a harmonic range, we can find a
projective transformation which will interchange _a_ and _c_. This is
a necessary and sufficient condition for a harmonic range, and the
quadrilateral construction is the general method for giving effect to
it.

[Illustration]

Given any three points _A_, _B_, _D_ in one straight line, the
quadrilateral construction finds the point _C_ harmonic to _A_ with
respect to _B_, _D_ by the following method: Take any point _O_
outside the straight line _ABD_, and join it to _B_ and _D_. Through
_A_ draw any straight line cutting _OD_, _OB_ in _P_ and _Q_. Join
_DQ_, _BP_, and let them intersect in _R_. Join _OR_, and let _OR_
meet _ABD_ in _C_. Then _C_ is the point required.

To prove this, let _DRQ_ meet _OA_ in _T_, and draw _AR_, meeting
_OD_ in _S_. Then a projective transformation of _A_, _B_, _C_, _D_
from _R_ on to _OD_ gives the points _S_, _P_, _O_, _D_, which,
projected from _A_ on to _DQ_, give _R_, _Q_, _T_, _D_. But these
again, projected from _O_ on to _ABD_, give _C_, _B_, _A_, _D_.
Hence _A_, _B_, _C_, _D_ can be projectively transformed into _C_,
_B_, _A_, _D_, and therefore form a harmonic range. From this point,
the proof that the construction is unique and general follows
simply[125].

The introduction of numbers, by this construction, offers no
difficulties of principle--except, indeed, those which always
attend the application of number to continua--and may be studied
satisfactorily in Klein's Nicht-Euklid (I. p. 337 ff.). The principle
of it is, to assign the numbers 0, 1, ∞ to _A_, _B_, _D_ and
therefore the number 2 to _C_, in order that the differences _AB_,
_AC_, _AD_ may be in harmonic progression. By taking _B_, _C_, _D_ as
a new triad corresponding to _A_, _B_, _D_, we find a point harmonic
to _B_ with respect to _C_, _D_ and assign to it the number 3, and so
on. In this way, we can obtain any number of points, and we are sure
of having no number and no point twice over, so that our coordinates
have the essential property of a unique correspondence with the
points they denote, and _vice versa_.

=114.= The point of importance in the above construction,
however, and the reason why I have reproduced it in detail, is
that it proceeds entirely by means of the general principles of
transformation enunciated above. From this stage onwards, everything
is effected by means of the two fundamental ideas we have just
discussed, and everything, therefore, depends on our general
principle of projective equivalence. This principle, as regards two
dimensions, may be stated more simply than in the passage quoted from
Cremona. It starts, in two dimensions, from the following definitions:

To project the points _A_, _B_, _C_, _D_ ... from a centre _O_, is to
construct the straight lines _OA_, _OB_, _OC_, _OD_....

To cut a number of straight lines _a_, _b_, _c_, _d_ ... by a
transversal _s_, is to construct the points _sa_, _sb_, _sc_,
_sd_....[126]

The successive application of these two operations, provided the
original figure consisted of points on one straight line or of
straight lines through one point, gives a figure projectively
indistinguishable from the former figure; and hence, by extension,
if any points in one straight line in the original figure lie in one
straight line in the derived figure, and reciprocally for straight
lines through points, the two operations have given projectively
similar figures. This general principle may be regarded as consisting
of two parts, according to the order of the operations: if we begin
with projection and end with section, we transform a figure of
points into another figure of points; by the converse order, we
transform a figure of lines into another figure of lines.

=115.= Before we can be clear as to the meaning of our principle, we
must have some notion as to our definition of points and straight
lines. But this definition, in projective Geometry, cannot be given
without some discussion of the principle of duality, the mathematical
form of the philosophical circle involved in geometrical definitions.

Confining ourselves for the moment to two dimensions, the principle
asserts, roughly speaking, that any theorem, dealing with lines
through a point and points on a line, remains true if these two
terms, wherever they occur, are interchanged. Thus: two points
lie on one straight line which they completely determine; and two
straight lines meet in one point, which they completely determine.
The four points of intersection of a transversal with four lines
through a point have an anharmonic ratio independent of the
particular transversal; and the four lines joining four points on one
straight line to a fifth point have an anharmonic ratio independent
of that fifth point. So also our general principle of projective
transformation has two sides: one in which points move along fixed
lines, and one in which lines turn about fixed points.

This duality suggests that any definition of points must be effected
by means of the straight line, and any definition of the straight
line must be effected by means of points. When we take the third
dimension into account, it is true, the duality is no longer so
simple; we have now to take account also of the plane, but this only
introduces a circle of three terms, which is scarcely preferable to
a circle of two terms. We now say: Three points, or a line and a
point, determine a plane: but conversely, three planes, or a line
and plane, determine a point. We may regard the straight line as a
relation between two of its points, but we may also regard the point
as a relation between two straight lines through it. We may regard
the plane as a relation between three points, or between a point and
a line, but we may also regard the point as a relation between three
planes, or between a line and a plane, which meet in it.

=116.= How are we to get outside this circle? The fact is that, in
pure Geometry, we cannot get outside it. For space, as we shall see
more fully hereafter, is nothing but relations; if, therefore, we
take any spatial figure, and seek for the terms between which it is a
relation, we are compelled, in Geometry, to seek these terms within
space, since we have nowhere else to seek them, but we are doomed,
since anything purely spatial is a mere relation, to find our terms
melting away as we grasp them.

Thus the relativity of space, while it is the essence of the
principle of duality, at the same time renders impossible the
expression of that principle, or of any other principle of pure
Geometry, in a manner which shall be free from contradictions.
Nevertheless, if we are to advance at all with our analysis of
geometrical reasoning and with our definitions of lines and points,
we must, for a while, ignore this contradiction; we must argue
as though it did not exist, so as to free our science from any
contradictions which are not inevitable.

=117.= In accordance with this procedure, then, let us define our
points as the terms of spatial relations, regarding whatever is not
a point as a relation between points. What, on this view, must our
points be taken to be? Obviously, if extension is mere relativity,
they must be taken to contain no extension; but if they are to
supply the terms for spatial relations, _e.g._ for straight lines,
these relations must exhibit them as the terms of the figures they
relate. In other words, since what can really be taken, without
contradiction, as the term of a spatial relation, is unextended, we
must take, as the term to be used in Geometry, where we cannot go
outside space, the least spatial thing which Geometry can deal with,
the thing which, though _in_ space, _contains_ no space; and this
thing we define as the point[127].

Neglecting, then, the fundamental contradiction in this definition,
the rest of our definitions follow without difficulty. The straight
line is the relation between two points, and the plane is the
relation between three. These definitions will be argued and defended
at length in section B of this Chapter[128], where we can discuss at
the same time the alternative metrical definitions; for our present
purpose, it is sufficient to observe that projective Geometry, from
the first, regards the straight line as determined by two points, and
the plane as determined by three, from which it follows, if we take
points as possible terms for spatial relations, that the straight
line and the plane may be regarded as relations between two and three
points respectively. If we agree on these definitions, we can proceed
to discuss the fundamental principle of projective Geometry, and to
analyse the axioms implicated in its truth.

=118.= Projective Geometry, we have seen, does not deal with
quantity, and therefore recognizes no difference where the
difference is purely quantitative. Now quantitative comparison
depends on a recognized identity of quality; the recognition of
qualitative identity, therefore, is logically prior to quantity, and
presupposed by every judgment of quantity. Hence all figures, whose
differences can be exhaustively described by quantity, _i.e._ by
pure measurement, must have an identity of quality, and this must be
recognizable without appeal to quantity. It follows that, by defining
the word quality in geometrical matters, we shall discover what
sets of figures are projectively indiscernible. If our definition
is correct, it ought to yield the general projective principle with
which we set out.

=119.= We agreed to regard points as the terms of spatial relations,
and we agreed that different points could be distinguished. But
we postponed the discussion of the conditions under which this
distinction could be effected. This discussion will yield us the
definition of quality and the proof of our general projective
principle.

Points, to begin with, have been defined as nothing but the terms for
spatial relations. They have, therefore, no intrinsic properties;
but are distinguished solely by means of their relations. Now the
relation between two points, we said, is the straight line on which
they lie. This gives that identity of quality for all pairs of points
on the same straight line, which is required both by our projective
principle and by metrical Geometry. (For only where there is identity
of quality can quantity be properly applied.) If only two points are
given, they cannot, without the use of quantity, be distinguished
from any two other points on the same straight line; for the
qualitative relation between any two such points is the same as for
the original pair, and only by a difference of relation can points be
distinguished from one another.

But conversely, one straight line is nothing but the relation
between two of its points, and all points are qualitatively alike.
Hence there can be nothing to distinguish one straight line from
another except the points through which it passes, and these are
distinguished from other points only by the fact that it passes
through them. Thus we get the reciprocal transformation: if we are
given only one point, any pair of straight lines through that point
is qualitatively indistinguishable from any other. This again is, on
the one hand, the basis of the second part of our general projective
principle, and on the other hand the condition of applying quantity,
in the measurement of angles, to the departure of two intersecting
straight lines.

=120.= We can now see the reason for what may have hitherto seemed a
somewhat arbitrary fact, namely, the necessity of _four_ collinear
points for anharmonic ratio. Recurring to the quadrilateral
construction and the consequent introduction of number, we see
that anharmonic ratio is an intrinsic projective relation of four
collinear points or concurrent straight lines, such that given three
terms and the relation, the fourth term can be uniquely determined
by projective methods. Now consider first a pair of points. Since
all straight lines are projectively equivalent, the relation between
one pair of points is precisely equivalent to that between another
pair. Given one point only, therefore, no projective relation, to
any second point, can be assigned, which shall in any way limit our
choice of the second point. Given two points, however, there is such
a relation--the third point may be given collinear with the first
two. This limits its position to one straight line, but since two
points determine nothing but one straight line, the third point
cannot be further limited. Thus we see why no intrinsic projective
relation can be found between three points, which shall enable us,
from two, uniquely to determine the third. With three given collinear
points, however, we have more given than a mere straight line, and
the quadrilateral construction enables us uniquely to determine any
number of fresh collinear points. This shows why anharmonic ratio
must be a relation between four points, rather than between three.

=121.= We can now prove, I think, that two figures, which are
projectively related, are qualitatively similar. Let us begin with
a collection of points on a straight line. So long as these are
considered without reference to other points or figures, they are
all qualitatively similar. They can be distinguished by immediate
intuition, but when we endeavour, without quantity, to distinguish
them conceptually, we find the task impossible, since the only
qualitative relation of any two of them, the straight line, is
the same for any other two. But now let us choose, at hap-hazard,
some point outside the straight line. The points of our line now
acquire new adjectives, namely their relations to the new point,
_i.e._ the straight lines joining them to this new point. But these
straight lines, reciprocally, alone define our external point,
and all straight lines are qualitatively similar. If we take some
other external point, therefore, and join it to the same points
of our original straight line, we obtain a figure in which, so
long as quantity is excluded, there is no conceptual difference
from the former figure. Immediate intuition can distinguish the
two figures, but qualitative discrimination cannot do so. Thus we
obtain a projective transformation of four lines into four other
lines, as giving a figure qualitatively indistinguishable from the
original figure. A similar argument applies to the other projective
transformations. Thus the only reason, within projective Geometry,
for not regarding projective figures as actually identical, is the
intuitive perception of difference of position. This is fundamental,
and must be accepted as a _datum_. It is presupposed in the
distinction of various points, and forms the very life of Geometry.
It is, in fact, the essence of the notion of a form of externality,
which notion forms the subject-matter of projective Geometry.

=122.= We may now sum up the results of our analysis of projective
Geometry, and state the axioms on which its reasoning is based. We
shall then have to prove that these axioms are necessary to any form
of externality, with which we shall pass, from mere analysis, to a
transcendental argument.

The axioms which have been assumed in the above analysis, and which,
it would seem, suffice to found projective Geometry, may be roughly
stated as follows:

I. We can distinguish different parts of space, but all parts are
qualitatively similar, and are distinguished only by the immediate
fact that they lie outside one another.

II. Space is continuous and infinitely divisible; the result of
infinite division, the zero of extension, is called a _point_[129].

III. Any two points determine a unique figure, called a straight
line, any three in general determine a unique figure, the plane.
Any four determine a corresponding figure of three dimensions, and
for aught that appears to the contrary, the same may be true of any
number of points. But this process comes to an end, sooner or later,
with some number of points which determine the whole of space. For
if this were not the case, no number of relations of a point to a
collection of given points could ever determine its relation to fresh
points, and Geometry would become impossible[130].

This statement of the axioms is not intended to have any exclusive
precision: other statements equally valid could easily be made. For
all these axioms, as we shall see hereafter, are philosophically
interdependent, and may, therefore, be enunciated in many ways. The
above statement, however, includes, if I am not mistaken, everything
essential to projective Geometry, and everything required to prove
the principle of projective transformation. Before discussing the
apriority of these axioms, let us once more briefly recapitulate the
ends which they are intended to attain.

=123.= From the exclusively mathematical standpoint, as we have
seen, projective Geometry discusses only what figures can be obtained
from each other by projective transformations, _i.e._ by the
operations of projection and section. These operations, in all their
forms, presuppose the point, straight line, and plane[131], whose
necessity for projective Geometry, from the purely mathematical point
of view, is thus self-evident from the start. But philosophically,
projective Geometry has, as we saw, a wider aim. This wider aim,
which gives, to the investigation of projectively equivalent figures,
its chief importance, consists in the determination of qualitative
spatial similarity, in the determination, that is, of all the figures
which, when any one figure is given, can be distinguished from the
given figure, so long as quantity is excluded, only by the mere fact
that they are external to it.

=124.= Now when we consider what is involved in such absolute
qualitative equivalence, we find at once, as its most obvious
prerequisite, the perfect homogeneity of space. For it is assumed
that a figure can be completely defined by its internal relations,
and that the external relations, which constitute its position,
though they suffice to distinguish it from other figures, in
no way affect its internal properties, which are regarded as
qualitatively identical with those of figures with quite different
external relations. If this were not the case, anything analogous
to projective transformation would be impossible. For such
transformation always alters the position, _i.e._ the external
relations, of a figure, and could not, therefore, if figures were
dependent on their relations to other figures or to empty space,
be studied without reference to other figures, or to the absolute
position of the original figure. We require for our principle,
in short, what may be called the mutual passivity and reciprocal
independence of two parts or figures of space.

This passivity and this independence involve the homogeneity of
space, or its equivalent, the relativity of position. For if
the internal properties of a figure are the same, whatever its
external relations may be, it follows that all parts of space are
qualitatively similar, since a change of external relation is a
change in the part of space occupied. It follows, also, that all
position is relative and extrinsic, _i.e._, that the position of a
point, or the part of space occupied by a figure, is not, and has
no effect upon, any intrinsic property of the point or figure, but
is exclusively a relation to other points or figures in space, and
remains without effect except where such relations are considered.

=125.= The homogeneity of space and the relativity of position,
therefore, are presupposed in the qualitative spatial comparison
with which projective Geometry deals. The latter, as we saw, is
also the basis of the principle of duality. But these properties,
as I shall now endeavour to prove, belong of necessity to any form
of externality, and are thus _à priori_ properties of all possible
spaces. To prove this, however, we must first define the notion of a
form of externality in general.

Let us observe, to begin with, that the distinction between
Euclidean and non-Euclidean Geometries, so important in metrical
investigations, disappears in projective Geometry proper. This
suggests that projective Geometry, though originally invented as
the science of Euclidean space, and subsequently of non-Euclidean
spaces also, deals really with a wider conception, a conception which
includes both, and neglects the attributes in which they differ. This
conception I shall speak of as a form of externality.

=126.= In Grassmann's profound philosophical introduction to his
_Ausdehnungslehre_ of 1844, he suggested that Geometry, though
improperly regarded as pure, was really a branch of applied
mathematics, since it dealt with a subject-matter not created,
like number, by the intellect, but given to it, and therefore not
wholly subject to its laws alone. But it must be possible--so he
contended--to construct a branch of pure mathematics, a science, that
is, in which our object should be wholly a creature of the intellect,
which should yet deal, as Geometry does, with extension--extension
as conceived, however, not as empirically perceived in sensation or
intuition.

From this point of view, the controversy between Kantians and
anti-Kantians becomes wholly irrelevant, since the distinction
between pure and mixed mathematics does not lie in the distinction
between the subjective and the objective, but between the purely
intellectual on the one hand, and everything else on the other.
Now Kant had contended, with great emphasis, that space was not an
intellectual construction, but a subjective intuition. Geometry,
therefore, with Grassmann's distinction, belongs to mixed mathematics
as much on Kant's view as on that of his opponents. And Grassmann's
distinction, I contend, is the more important for Epistemology,
and the one to be adopted in distinguishing the _à priori_ from
the empirical. For what is merely intuitional can change, without
upsetting the laws of thought, without making knowledge formally
impossible: but what is purely intellectual cannot change, unless the
laws of thought should change, and all our knowledge simultaneously
collapse. I shall therefore follow Grassmann's distinction in
constructing an _à priori_ and purely conceptual form of externality.

=127.= The pure doctrine of extension, as constructed by Grassmann,
need not be discussed--it included much empirical material, and was
philosophically a failure. But his principles, I think, will enable
us to prove that projective Geometry, abstractly interpreted, is
the science which he foresaw, and deals with a matter which can be
constructed by the pure intellect alone. If this be so, however,
it must be observed that projective Geometry, for the moment, is
rendered purely hypothetical[132]. All necessary truth, as Bradley
has shown, is hypothetical[133], and asserts, _primâ facie_, only
the ground on which rests the necessary connection of premisses
and conclusion. If we construct a mere conception of externality,
and thus abandon our actually given space, the result of our
construction, until we return to something actually given, remains
without existential import--if there _be_ experienced externality, it
asserts, then there must be a form of externality with such and such
properties. That there must be experienced externality, Kant's first
argument about space proves, I think, to those who admit experience
of a world of diverse but interrelated things. But this is a question
which belongs to the next Chapter.

What we have to do here is, not to discuss whether there is a form of
externality, but whether, if there be such a form, it must possess
the properties embodied in the axioms of projective Geometry. Now
first of all, what do we mean by such a form?

=128.= In any world in which perception presents us with various
things, with discriminated and differentiated contents, there must
be, in perception, at least one "principle of differentiation[134],"
an element, that is, by which the things presented are distinguished
as various. This element, taken in isolation, and abstracted from the
content which it differentiates, we may call a form of externality.
That it must, when taken in isolation, appear as a form, and not as
a mere diversity of material content, is, I think, fairly obvious.
For a diversity of material content cannot be studied apart from
that material content; what we wish to study here, on the contrary,
is the bare possibility of such diversity, which forms the residuum,
as I shall try to prove hereafter[135], when we abstract from any
sense-perception all that is distinctive of its particular matter.
This possibility, then, this principle of bare diversity, is our form
of externality. How far it is necessary to assume such a form, as
distinct from interrelated things, I shall consider later on[136].
For the present, since space, as dealt with by Geometry, is certainly
a form of this kind, we have only to ask: What properties must such a
form, when studied in abstraction, necessarily possess?

=129.= In the first place, externality is an essentially relative
conception--nothing can be external to itself. To be external to
something is to be another with some relation to that thing. Hence,
when we abstract a form of externality from all material content,
and study it in isolation, position will appear, of necessity,
as purely relative--a position can have no intrinsic quality, for
our form consists of pure externality, and externality contains
no shadow or trace of an intrinsic quality. Thus we obtain our
fundamental postulate, the relativity of position, or, as we may put
it, the complete absence, on the part of our form, of any vestige of
thinghood.

The same argument may also be stated as follows: If we abstract the
conception of externality, and endeavour to deal with it _per se_, it
is evident that we must obtain an object alike destitute of elements
and of totality. For we have abstracted from the diverse matter
which filled our form, while any element, or any whole, would retain
some of the qualities of a matter. Either an element or a whole, in
fact, would have to be a thing not external to itself, and would
thus contain something not pure externality. Hence arise infinite
divisibility, with the self-contradictory notion of the point, in the
search for elements, and unbounded extension, with the contradiction
of an infinite regress or a vicious circle, in the search for a
completed whole. Thus again, our form contains neither elements nor
totality, but only endless relations--the terms of these relations
being excluded by our abstraction from the matter which fills our
form.

=130.= In like manner we can deduce the homogeneity of our form. The
diversity of content, which was possible only within the form of
externality, has been abstracted from, leaving nothing but the bare
possibility of diversity, the bare principle of differentiation,
itself uniform and undifferentiated. For if diversity presupposes
such a form, the form cannot, unless it were contained in a fresh
form, be itself diverse or differentiated.

Or we may deduce the same property from the relativity of position.
For any quality in one position, by which it was marked out from
another, would be necessarily more or less intrinsic, and would
contradict the pure relativity. Hence all positions are qualitatively
alike, _i.e._ the form is homogeneous throughout.

=131.= From what has been said of homogeneity and relativity,
follows one of the strangest properties of a form of externality.
This property is, that the relation of externality between any two
things is infinitely divisible, and may be regarded, consequently,
as made up of an infinite number of the would-be elements of our
form, or again as the sum of two relations of externality[137]. To
speak of dividing or adding relations may well sound absurd--indeed
it reveals the impropriety of the word relation in this connexion.
It is difficult, however, to find an expression which shall be less
improper. The fact seems to be, that externality is not so much a
relation as bare relativity, or the bare possibility of a relation.
On this subject, I shall enlarge in Chapter IV.[138] At this point
it is only important to realize, what the subsequent argument will
assume, that the relation--if we may so call it--of externality
between two or more things must, since our form is homogeneous, be
capable of continuous alteration, and must, since our infinitely
divisible form is constituted by such relations, be capable of
infinite division. But the result of infinite division is defined
as the element of our form. (Our form has no elements, but we have
to imagine elements in order to reason about it, as will be shown
more fully in Chapter IV.) Hence it follows, that every relation
of externality may be regarded, for scientific purposes, as an
infinite congeries of elements, though philosophically, the relations
alone are valid, and the elements are a self-contradictory result
of hypostatizing the form of externality. This way of regarding
relations of externality is important in understanding the meaning of
such ideas as three or four collinear points.

As this point is difficult and important, I will repeat, in somewhat
greater detail, the explanation of the manner in which straight
lines and planes come to be regarded as congeries of points. From
the strictly projective standpoint, though all other figures _are_
merely a collection of any required number of points, lines or
planes, given by some projective construction, straight lines and
planes themselves are given integrally, and are not to be considered
as divisible or composed of parts. To say that a point lies on
a straight line means, for projective Geometry proper, that the
straight line is a relation between this and some other point.
Here the points concerned, if our statement is to be freed from
contradictions, must be regarded, if I may use such an expression, as
_real_ points--_i.e._ as unextended material centres[139]. Straight
lines and planes are then relations between these material atoms.
They are relations, however, which may undergo a metrical alteration
while remaining projectively unchanged. When the projective relation
between the two points _A_, _B_ is the same as that between the two
points _A_, _C_, while the metrical relation (distance) is different,
the three points _A_, _B_, _C_ are said to be collinear. Now the
metrical manner of regarding spatial figures demands that they should
be hypostatized, and no longer regarded as mere relations. For when
we regard a quantity as extensive, _i.e._ as divisible into parts,
we necessarily regard it as more than a mere relation or adjective,
since no mere relation or adjective can be divided. For quantitative
treatment, therefore, spatial relations must be hypostatized[140].
When this is done, we obtain, as we saw above, a homogeneous
and infinitely divisible form of externality. We find now that
distance, for example, may be continuously altered without changing
the straight line on which it is measured. We thus obtain, on the
straight line in question, a continuous series of points, which,
since it is continuous, we regard as constituting our straight line.
It is thus solely from the hypostatizing of relations, which metrical
Geometry requires, that the view of straight lines and planes as
_composed_ of points arises, and it is from this hypostatizing that
the difficulties of metrical Geometry spring.

=132.= The next step, in defining a form of externality, is obtained
from the idea of _dimensions_. Positions, we have seen, are defined
solely by their relations to other positions. But in order that
such definition may be possible, a finite number of relations must
suffice, since infinite numbers are philosophically inadmissible. A
position must be definable, therefore, if knowledge of our form is
to be possible at all, by some finite integral number of relations
to other positions. Every relation thus necessary for definition
we call a dimension. Hence we obtain the proposition: _Any form of
externality must have a finite integral number of dimensions_.

=133.= The above argument, it may be urged, has overlooked a
possibility. It has used a transcendental argument, so an opponent
may contend, without sufficiently proving that knowledge about
externality must be possible without reference to the matters
external to each other. The definition of a position may be
impossible, so long as we neglect the matter which fills the form,
but may become possible when this matter is taken into account.
Such an objection can, I think, be successfully met, by a reference
to the passivity and homogeneity of our form. For any dependence
of the definition of a position on the particular matter filling
that position, would involve some kind of interaction between the
matter and its position, some effect of the diverse content on
the homogeneous form. But since the form is totally destitute of
thinghood, perfectly impassive, and perfectly void of differences
between its parts, any such effect is inconceivable. An effect on
a position would have to alter it in some way, but how could it be
altered? It has no qualities except those which make it the position
it is, as opposed to other positions; it cannot change, therefore,
without becoming a different position. But such a change contradicts
the law of identity. Hence it is not the position which has changed,
but the content which has moved in the form. Thus it must be
possible, if knowledge of our form can be obtained at all, to obtain
this knowledge in logical independence of the particular matter which
fills it. The above argument, therefore, granted the possibility of
knowledge in the department in question, shows the necessity of a
finite integral number of dimensions.

=134.= Let us repeat our original argument in the light of this
elucidation. A position is completely defined when, and only when,
enough relations are known to enable us to determine its relation
to any fresh known position. Only by relations within the form of
externality, as we have just seen, and never by relations which
involve a reference to the particular matter filling the form,
can such a definition be effected. But the possibility of such a
definition follows from the Law of Excluded Middle, when this law is
interpreted to mean, as Bosanquet makes it mean, that "Reality ... is
a system of reciprocally determinate parts[141]." For this implies
that, given the relations of a part _A_ to other parts _B_, _C_ ...,
a sufficient wealth of such relations throws light on the relations
of _B_ to _C_, etc. If this were not the case, the parts _A_, _B_,
_C_ ... could not be said to form such a system; for in such a
system, to define _A_ is to define, at the same time, all the other
members, and to give an adjective to _A_, is to give an adjective to
_B_ and _C_. But the relations between positions are, when we restore
the matter from which the positions were abstracted, relations
between the things occupying those positions, and these relations, we
have seen, can be studied without reference to the particular nature,
in other respects, of the related things. It follows that, when we
apply the general principle of systematic unity to these relations in
particular, we find these relations to be dependent on each other,
since they are not dependent, for their definition, on anything else.
This gives the axiom of dimensions, in the above general form, as
the result, on our abstract geometrical level, of the relativity of
position and the law of excluded middle.

=135.= Before proceeding further, it is necessary to discuss the
important special case where a form of externality has only one
dimension. Of the two such forms, given in experience, one, namely
time, presents an instance of this special case. But it may be shown,
I think, that the function, in constituting the possibility of
experience, which we demand of such forms, could not be accomplished
by a one-dimensional form alone. For in a one-dimensional form, the
various contents may be arranged in a series, and cannot, without
interpenetration, change the order of contents in the series.
But interpenetration is impossible, since a form of externality
is the mere expression of diversity among things, from which it
follows that things cannot occupy the same position in a form,
unless there is another form by which to differentiate them. For
without externality, there is no diversity[142]. Thus two bodies
may occupy the same space, but only at different times: two things
may exist simultaneously, but only at different places. A form of
one dimension, therefore, could not, by itself, allow that change
of the relations of externality, by which alone a varied world
of interrelated things can be brought into consciousness. In a
one-dimensional space, for example, only a single object, which must
appear as a point, or two objects at most, one in front and one
behind, could ever be perceived. Thus two or more dimensions seem
an essential condition of anything worth calling an experience of
interrelated things.

=136.= It may be objected, to this argument, that its validity
depends upon the assumption that the change of a relation of
externality must be continuous. Both to make and to meet this
objection, in a manner which shall not imply time, seems almost
impossible. For we cannot speak of change, whether continuous or
discrete, without imagining time. Let us, therefore, allow time to
be known, and discuss whether the temporal change, in any other form
of externality, is necessarily continuous[143]. We must reply, I
think, that continuity is necessary. The change of relation, in our
non-temporal form, may be safely described as motion, and the law
of Causality--since we have already assumed time--may be applied to
this motion. It then follows that discrete motion would involve a
finite effect from an infinitesimal cause, for a cause acting only
for a moment of time would be infinitesimal. It involves, also, a
validity in the point of time, whereas what is valid in any form of
externality is not, as we have already seen, the infinitesimal and
self-contradictory element resulting from infinite division, but the
finite relation which mathematics analyzes into vanishing elements.
Hence change must be continuous, and the possibility of serial
arrangement holds good.

In a one-dimensional form other than time, the same argument must
hold. For something analogous to Causality would be necessary to
experience, and the relativity of the form would still necessarily
hold. Hence, since only these two properties of time have been
assumed, the above contention would remain valid of any second form
whose relations were correlated with those of the first, as the
analogue of Causality would require them to be.

=137.= The next step in the argument, which assumes two or more
dimensions, is concerned with the general analogues of straight
lines and planes, _i.e._ with figures--which may be regarded either
as relations between positions or as series of positions--uniquely
determined by two or by three positions. If this step can be
successfully taken, our deduction of the above projective axioms will
be complete, and descriptive Geometry will be established as the
abstract _à priori_ doctrine of forms of externality.

To prove this contention, consider of what nature the relations
can be by which positions are defined. We have seen already that
our form is purely relational and infinitely divisible, and that
positions (points) are the self-contradictory outcome of the search
for something other than relations. What we really mean, therefore,
by the relations defining a position, is, when we undo our previous
abstraction, the relations of externality by which some thing is
related to other things. But how, when we remain in the abstract
form, must such relations appear?

=138.= We have to prove that two positions must have a relation
independent of any reference to other positions. To prove this, let
us recur to what was said, in connection with dimensions, as to the
passivity and homogeneity of our form. Since positions are defined
only by relations, there must be relations, within the form, between
positions. But if there are such relations, there must be a relation
which is intrinsic to two positions. For to suppose the contrary,
is to attribute an interaction or causal connection, of some kind,
between those two positions and other positions--a supposition
which the perfect homogeneity of our form renders absurd, since all
positions are qualitatively similar, and cannot be changed without
losing their identity. We may put this argument thus: since positions
are only defined by their relations, such definition could never
begin, unless it began with a relation between only two positions.
For suppose three positions _A_, _B_, _C_ were necessary, and gave
rise to the relation _abc_ between the three. Then there would remain
no means of defining the different pairs _BC_, _CA_, _AB_, since the
only relation defining them would be one common to all three pairs.
Nothing would be gained, in this case, by reference to fresh points,
for it follows, from the homogeneity and passivity of the form, that
these fresh points could not affect the internal relations of our
triad, which relations, if they can give definiteness at all, must
give it without the aid of external reference. Two positions must,
therefore, if definition is to be possible, have some relation which
they by themselves suffice to define. Precisely the same argument
applies to three positions, or to four; the argument loses its scope
only when we have exhausted the dimensions of the form considered.
Thus, in three dimensions, five positions have no fresh relation,
not deducible from those already known, for by the definition of
dimensions, all the relations involved can be deduced from those of
the fourth point to the first three, together with those of the fifth
to the first three.

We may give the argument a more concrete, and perhaps a more
convincing shape, by considering the matter arranged in our form. If
two things are mutually external, they must since they belong to the
same world, have some relation of externality; there is, therefore,
a relation of externality between two things. But since our form is
homogeneous, the same relation of externality may subsist in other
parts of the form, _i.e._ while the two things considered alter their
relations of externality to other things. The relation of externality
between two things is, therefore, independent of other things. Hence,
when we return to the abstract language of the form, two positions
have a relation determined by those two positions alone, and
independent of other positions.

Precisely the same argument applies to the relations of three
positions, and in each case the relation must appear in the form as
not a mere inference from the positions it relates. For relations,
as we have seen, actually constitute a form of externality, and are
not mere inferences from terms, which are nowhere to be found in the
form[144].

To sum up: Since position is relative, two positions must have
_some_ relation to each other; and since our form of externality
is homogeneous, this relation can be kept unchanged while the two
positions change their relations to other positions. Hence their
relation is intrinsic, and independent of other positions. Since
the form is a mere complex of relations, the relation in question
must, if the form is sensuous or intuitive, be itself sensuous or
intuitive, and not a mere inference. In this case, a unique relation
must be a unique figure--in spatial terms, the straight line joining
the two points.

=139.= With this, our deduction of projective Geometry from the
_à priori_ conceptual properties of a form of externality is
completed. That such a form, when regarded as an independent thing,
is self-contradictory, has been abundantly evident throughout the
discussion. But the science of the form has been founded on the
opposite way of regarding it: we have held it throughout to be a mere
complex of relations, and have deduced its properties exclusively
from this view of it. The many difficulties, in applying such an _à
priori_ deduction to intuitive space, and in explaining, as logical
necessities, properties which appear as sensuous or intuitional data,
must be postponed to Chapter IV. For the present, I wish to point out
that projective Geometry is wholly _à priori_; that it deals with an
object whose properties are logically deduced from its definition,
not empirically discovered from data; that its definition, again, is
founded on the possibility of experiencing diversity in relation,
or multiplicity in unity; and that our whole science, therefore, is
logically implied in, and deducible from, the possibility of such
experience.

=140.= In metrical Geometry, on the contrary, we shall find a very
different result. Although the geometrical conditions which render
spatial measurement possible, will be found identical, except for
slight differences in the form of statement, with the _à priori_
axioms discussed above, yet the actual measurement--which deals
with actually given space, not the mere intellectual construction
we have been just discussing--gives results which can only be known
empirically and approximately, and can be deduced by no necessity
of thought. The Euclidean and non-Euclidean spaces give the various
results which are _à priori_ possible; the axioms peculiar to
Euclid--which are properly not axioms, but empirical results of
measurement--determine, within the errors of observation, which of
these _à priori_ possibilities is realized in our actual space. Thus
measurement deals throughout with an empirically given matter, not
with a creature of the intellect, and its _à priori_ elements are
only the conditions presupposed in the possibility of measurement.
What these conditions are, we shall see in the second section of this
chapter.


Section B.

THE AXIOMS OF METRICAL GEOMETRY.

=141.= We have now reviewed the axioms of projective Geometry, and
have seen that they are _à priori_ deductions from the fact that
we can experience externality, _i.e._ a coexistent multiplicity of
different but interrelated things. But projective Geometry, in spite
of its claims, is not the whole science of space, as is sufficiently
proved by the fact that it cannot discriminate between Euclidean and
non-Euclidean spaces[145]. For this purpose, spatial measurement
is required: metrical Geometry, with its quantitative tests, can
alone effect the discrimination. For all application of Geometry to
physics, also, measurement is required; the law of gravitation, for
example, requires the determination of actual distances. For many
purposes, in short, projective Geometry is wholly insufficient: thus
it is unable to distinguish between different kinds of conics, though
their distinction is of fundamental importance in many departments of
knowledge.

Metrical Geometry is, then, a necessary part of the science of
space, and a part not included in descriptive Geometry. Its _à
priori_ element, nevertheless, so far as this is spatial and not
arithmetical, is the same as the postulate of projective Geometry,
namely, the homogeneity of space, or its equivalent, the relativity
of position. We can see, in fact, that the _à priori_ element in
both is likely to be the same. For the _à priori_ in metrical
Geometry will be whatever is presupposed in the possibility of
spatial measurement, _i.e._ of quantitative spatial comparison. But
such comparison presupposes simply a known identity of quality,
the determination of which is precisely the problem of projective
Geometry. Hence the conditions for the possibility of measurement, in
so far as they are not arithmetical, will be precisely the same as
those for projective Geometry.

=142.= Metrical Geometry, therefore, though distinct from projective
Geometry, is not independent of it, but presupposes it, and arises
from its combination with the extraneous idea of _quantity_.
Nevertheless the mathematical form of the axioms, in metrical
Geometry, is slightly different from their form in projective
Geometry. The homogeneity of space is replaced by its equivalent, the
axiom of Free Mobility. The axiom of the straight line is replaced
by the axiom of distance: Two points determine a unique quantity,
distance, which is unaltered in any motion of the two points as a
single figure. This axiom, indeed, will be found to involve the axiom
of the straight line--such a quantity could not exist unless the
two points determined a unique curve--but its mathematical form is
changed. Another important change is the collapse of the principle
of duality: quantity can be applied to the straight line, because
it is divisible into similar parts, but cannot be applied to the
indivisible point. We thus obtain a reason, which was wanting in
descriptive Geometry, for preferring points, as spatial elements,
to straight lines or planes[146]. Finally, an entirely new idea is
introduced with quantity, namely, the idea of _Motion_. Not that we
study motion, or that any of our results have reference to motion,
but that they cannot, though in projective Geometry they could, be
obtained without at least an ideal motion of our figures through
space.

Let us now examine in detail the prerequisites of spatial
measurement. We shall find three axioms, without which such
measurement would be impossible, but with which it is adequate to
decide, empirically and approximately, the Euclidean or non-Euclidean
nature of our actual space. We shall find, further, that these three
axioms can be deduced from the conception of a form of externality,
and owe nothing to the evidence of intuition. They are, therefore,
like their equivalents the axioms of projective Geometry, _à priori_,
and deducible from the conditions of spatial experience. This
experience, accordingly, can never disprove them, since its very
existence presupposes them.


I. _The Axiom of Free Mobility._

=143.= Metrical Geometry, to begin with, may be defined as the
science which deals with the comparison and relations of spatial
magnitudes. The conception of magnitude, therefore, is necessary from
the start. Some of Euclid's axioms, accordingly, have been classed as
arithmetical, and have been supposed to have nothing particular to do
with space. Such are the axioms that equals added to or subtracted
from equals give equals, and that things which are equal to the same
thing are equal to one another. These axioms, it is said, are purely
arithmetical, and do not, like the others, ascribe an adjective to
space. As regards their use in arithmetic, this is of course true.
But if an arithmetical axiom is to be applied to spatial magnitudes,
it must have some spatial import[147], and thus even this class is
not, in Geometry, _merely_ arithmetical. Fortunately, the geometrical
element is the same in all the axioms of this class--we can see at
once, in fact, that it can amount to no more than a definition of
spatial magnitude[148]. Again, since the space with which Geometry
deals is infinitely divisible, a definition of spatial magnitude
reduces itself to a definition of spatial equality, for, as soon as
we have this last, we can compare two spatial magnitudes by dividing
each into a number of equal units, and counting the number of such
units in each[149]. The ratio of the number of units is, of course,
the ratio of the two magnitudes.

=144.= We require, then, at the very outset, some criterion of
spatial equality: without such a criterion metrical Geometry would
become wholly impossible. It might appear, at first sight, as though
this need not be an axiom, but might be a mere definition. In part
this is true, but not wholly. The part which is merely a definition
is given in Euclid's eighth axiom: "Magnitudes which exactly coincide
are equal." But this gives a sufficient criterion only when the
magnitudes to be compared already occupy the same position. When, as
will normally be the case, the two spatial magnitudes are external to
one another--as, indeed, must be the case, if they are distinct, and
not whole and part--the two magnitudes can only be made to coincide
by a motion of one or both of them. In order, therefore, that our
definition of spatial magnitude may give unambiguous results,
coincidence when superposed, if it can ever occur, must occur always,
whatever path be pursued in bringing it about. Hence, if mere motion
could alter shapes, our criterion of equality would break down.
It follows that the application of the conception of magnitude
to figures in space involves the following axiom[150]: _Spatial
magnitudes can be moved from place to place without distortion_; or,
as it may be put, _Shapes do not in any way depend upon absolute
position in space_.

The above axiom is the axiom of Free Mobility[151]. I propose to
prove (1) that the denial of this axiom would involve logical and
philosophical absurdities, so that it must be classed as wholly _à
priori_; (2) that metrical Geometry, if it refused this axiom, would
be unable, without a logical absurdity, to establish the notion of
spatial magnitude at all. The conclusion will be, that the axiom
cannot be proved or disproved by experience, but is an _à priori_
condition of metrical Geometry. As I shall thus be maintaining a
position which has been much controverted, especially by Helmholtz
and Erdmann, I shall have to enter into the arguments at some length.

=145.= A. _Philosophical Argument._ The denial of the axiom involves
absolute position, and an action of mere space, _per se_, on things.
For the axiom does not assert that real bodies, as a matter of
empirical fact, never change their shape in any way during their
passage from place to place: on the contrary, we know that such
changes do occur, sometimes in a very noticeable degree, and always
to some extent. But such changes are attributed, not to the change
of place as such, but to physical causes: changes of temperature,
pressure, etc. What our axiom has to deal with is not actual material
bodies, but geometrical figures[152], and it asserts that a figure
which is possible in any one position in space is possible in every
other. Its meaning will become clearer by reference to a case where
it does not hold, say the space formed by the surface of an egg.
Here, a triangle drawn near the equator cannot be moved without
distortion to the point, as it would no longer fit the greater
curvature of the new position: a triangle drawn near the point
cannot be fitted on to the flatter end, and so on. Thus the method
of superposition, such as Euclid employs in Book I. Prop. IV.,
becomes impossible; figures cannot be freely moved about, indeed,
given any figure, we can determine a certain series of possible
positions for it on the egg, outside which it becomes impossible.
What I assert is, then, that there is a philosophic absurdity in
supposing space in general to be of this nature. On the egg we have
marked points, such as the two ends; the space formed by its surface
is not homogeneous, and if things are moved about in it, it must of
itself exercise a distorting effect upon them, quite independently of
physical causes; if it did not exercise such an effect, the things
could not be moved. Thus such a space would not be homogeneous, but
would have marked points, by reference to which bodies would have
absolute position, quite independently of any other bodies. Space
would no longer be passive, but would exercise a definite effect upon
things, and we should have to accommodate ourselves to the notion
of marked points in empty space; these points being marked, not by
the bodies which occupied them, but by their effects on any bodies
which might from time to time occupy them. This want of homogeneity
and passivity is, however, absurd; space must, since it is a form of
externality, allow only of relative, not of absolute, position, and
must be completely homogeneous throughout. To suppose it otherwise,
is to give it a thinghood which no form of externality can possibly
possess. We must, then, on purely philosophical grounds, admit that a
geometrical figure which is possible anywhere is possible everywhere,
which is the axiom of Free Mobility.

=146.= B. _Geometrical Argument._ Let us see next what sort of
Geometry we could construct without this axiom. The ultimate standard
of comparison of spatial magnitudes must, as we saw in introducing
the axiom, be equality when superposed; but need we, from this
equality, infer equality when separated? It has been urged by Erdmann
that, for the more immediate purposes of Geometry, this would be
unnecessary[153]. We might construct a new Geometry, he thinks, in
which sizes varied with motion on any definite law. Such a view,
as I shall show below, involves a logical error as to the nature
of magnitude. But before pointing this out, let us discuss the
geometrical consequences of assuming its truth. Suppose the length
of an infinitesimal arc in some standard position were _ds_; then
in any other position _p_ its length would be _ds.f(p)_, where the
form of the function _f(p)_ must be supposed known. But how are we
to determine the position _p_? For this purpose, we require _p_'s
coordinates, _i.e._, some measurement of distance from the origin.
But the distance from the origin could only be measured if we assumed
our law _f(p)_ to measure it by. For suppose the origin to be _O_,
and _Op_ to be a straight line whose length is required. If we have
a measuring rod with which we travel along the line and measure
successive infinitesimal arcs, the measuring rod will change its
size as we move, so that an arc which appears by the measure to be
_ds_ will really be _f(s).ds_, where _s_ is the previously traversed
distance. If, on the other hand, we move our line _Op_ slowly through
the origin, and measure each piece as it passes through, our measure,
it is true, will not alter, but now we have no means of discovering
the law by which any element has changed its length in coming to
the origin. Hence, until we assume our function _f(p)_, we have no
means of determining _p_, for we have just seen that distances from
the origin can only be estimated by means of the law _f(p)_. It
follows that experience can neither prove nor disprove the constancy
of shapes throughout motion, since, if shapes were not constant, we
should have to _assume_ a law of their variation before measurement
became possible, and therefore measurement could not itself reveal
that variation to us[154].

Nevertheless, such an arbitrarily assumed law _does_, at first sight,
give a mathematically possible Geometry. The fundamental proposition,
that two magnitudes which can be superposed in any one position can
be superposed in any other, still holds. For two infinitesimal arcs,
whose lengths in the standard position are _ds{1}_ and _ds{2}_,
would, in any other position _p_, have lengths _f(p).ds{1}_ and
_f(p).ds{2}_, so that their ratio would be unaltered. From this
constancy of ratio, as we know through Riemann and Helmholtz, the
above proposition follows. Hence all that Geometry requires, it would
seem, as a basis for measurement, is an axiom that the alteration
of shapes during motion follows a definite known law, such as that
assumed above.

=147.= There is, however, in such a view, as I remarked above, a
logical error as to the nature of magnitude. This error has been
already pointed out in dealing with Erdmann[155], and need only be
briefly repeated here. A judgment of magnitude is essentially a
judgment of comparison: in unmeasured quantity, comparison as to
the mere more or less, but in measured magnitude, comparison as to
the precise how many times. To speak of differences of magnitude,
therefore, in a case where comparison cannot reveal them, is
logically absurd. Now in the case contemplated above, two magnitudes,
which appear equal in one position, appear equal also when compared
in another position. There is no sense, therefore, in supposing
the two magnitudes unequal when separated, nor in supposing,
consequently, that they have changed their magnitudes in motion.
This senselessness of our hypothesis is the logical ground of the
mathematical indeterminateness as to the law of variation. Since,
then, there is no means of comparing two spatial figures, as regards
magnitude, except superposition, the only logically possible axiom,
if spatial magnitude is to be self-consistent, is the axiom of Free
Mobility in the form first given above.

=148.= Although this axiom is _à priori_, its application to the
measurement of actual bodies, as we found in discussing Helmholtz's
views, always involves an empirical element[156]. Our axiom, then,
only supplies the _à priori_ condition for carrying out an operation
which, in the concrete, is empirical--just as arithmetic supplies the
_à priori_ condition for a census. As this topic has been discussed
at length in Chapter II., I shall say no more about it here.

=149.= There remain, however, a few objections and difficulties to be
discussed. First, how do we obtain equality in solids, and in Kant's
cases of right and left hands, or of right and left-handed screws,
where actual superposition is impossible? Secondly, how can we take
congruence as the only possible basis of spatial measurement, when we
have before us the case of time, where no such thing as congruence
is conceivable? Thirdly, it might be urged that we can immediately
estimate spatial equality by the eye, with more or less accuracy,
and thus have a measure independent of congruence. Fourthly, how is
metrical Geometry possible on non-congruent surfaces, if congruence
be the basis of spatial measurement? I will discuss these objections
successively.

=150.= (1) How do we measure the equality of solids? These could only
be brought into actual congruence if we had a fourth dimension to
operate in[157], and from what I have said before of the absolute
necessity of this test, it might seem as though we should be left
here in utter ignorance. Euclid is silent on the subject, and in
all works on Geometry it is assumed as self-evident that two cubes
of equal side are equal. This assumption suggests that we are not
so badly off as we should have been without congruence, as a test
of equality in one or two dimensions; for now we can at least be
sure that two cubes have all their sides and all their faces equal.
Two such cubes differ, then, in no sensible spatial quality save
position, for volume, in this case at any rate, is not a sensible
quality. They are, therefore, as far as such qualities are concerned,
indiscernible. If their places were interchanged, we might know the
change by their colour, or by some other non-geometrical property;
but so far as any property of which Geometry can take cognisance is
concerned, everything would seem as before. To suppose a difference
of volume, then, would be to ascribe an effect to mere position,
which we saw to be inadmissible while discussing Free Mobility.
Except as regards position, they are geometrically indiscernible, and
we may call to our aid the Identity of Indiscernibles to establish
their agreement in the one remaining geometrical property of volume.
This may seem rather a strange principle to use in Mathematics,
and for Geometry their equality is, perhaps, best regarded as
a definition; but if we demand a philosophical ground for this
definition, it is, I believe, only to be found in the Identity of
Indiscernibles. We can, without error, make our _definition_ of
three-dimensional equality rest on two-dimensional congruence. For
since direct comparison as to volume is impossible, we are at liberty
to _define_ two volumes as equal, when all their various lines,
surfaces, angles and solid angles are congruent, since there remains,
in such a case, no _measurable_ difference between the figures
composing the two volumes. Of course, as soon as we have established
this one case of equality of volumes, the rest of the theory follows;
as appears from the ordinary method of integrating volumes, by
dividing them into small cubes.

Thus congruence _helps_ to establish three-dimensional equality,
though it cannot directly _prove_ such equality; and the same
philosophical principle, of the homogeneity of space, by which
congruence was proved, comes to our rescue here. But how about
right-handed and left-handed screws? Here we can no longer apply the
Identity of Indiscernibles, for the two are very well discernible.
But as with solids, so here, Free Mobility can help us much. It
can enable us, by ordinary measurement, to show that the internal
relations of both screws are the same, and that the difference lies
only in their relation to other things in space. Knowing these
internal relations, we can calculate, by the Geometry which Free
Mobility has rendered possible, all the geometrical properties of
these screws--radius, pitch, etc.--and can show them to be severally
equal in both. But this is all we require. Mediate comparison is
possible, though immediate comparison is not. Both can, for instance,
be compared with the cylinder on which both would fit, and thus their
equality can be proved. A precisely similar proof holds, of course,
for the other cases, right and left hands, spherical triangles, etc.
On the whole, these cases confirm my argument; for they show, as Kant
intended them to show[158], the essential relativity of space.

=151.= (2) As regards time, no congruence is here conceivable,
for to effect congruence requires always--as we saw in the case
of solids--one more dimension than belongs to the magnitudes
compared. No day can be brought into temporal coincidence with any
other day, to show that the two exactly cover each other; we are
therefore reduced to the arbitrary assumption that some motion or
set of motions, given us in experience, is uniform. Fortunately,
we have a large set of motions which all roughly agree; the swing
of the pendulum, the rotation and revolution of the earth and the
planets, etc. These do not exactly agree, but they lead us to the
laws of motion, by which we are able, on our arbitrary hypothesis,
to estimate their small departures from uniformity; just as the
assumption of Free Mobility enabled us to measure the departures
of actual bodies from rigidity. But here, as there, another
possibility is mathematically open to us, and can only be excluded
by its philosophic absurdity; we might have assumed that the above
set of approximately agreeing motions all had velocities which
varied approximately as some arbitrarily assumed function of the
time, _f(t)_ say, measured from some arbitrary origin. Such an
assumption would still keep them as nearly synchronous as before,
and would give an equally possible, though more complex, system
of Mechanics; instead of the first law of motion, we should have
the following: A particle perseveres in its state of rest, or of
rectilinear motion with velocity varying as _f(t)_, except in so far
as it is compelled to alter that state by the action of external
forces. Such a hypothesis _is_ mathematically possible, but, like the
similar one for space, it is excluded logically by the comparative
nature of the judgment of quantity, and philosophically by the fact
that it involves absolute time, as a determining agent in change,
whereas time can never, philosophically, be anything but a passive
form, abstracted from change. I have introduced this parallel from
time, not as directly bearing on the argument, but as a simpler case
which may serve to illustrate my reasoning in the more complex case
of space. For since time, in mathematics, is one-dimensional, the
mathematical difficulties are simpler than in Geometry; and although
nothing accurately corresponds to congruence, there is a very similar
mixture of mathematical and philosophical necessity, giving, finally,
a thoroughly definite axiom as the basis of time-measurement,
corresponding to congruence as the basis of space-measurement[159].

=152.= (3) The case of time-measurement suggests the third of the
above objections to the absolute necessity of the axiom of Free
Mobility. Psycho-physics has shown that we have an approximate
power, by means of what may be called the sense of duration, of
immediately estimating equal short times. This establishes a rough
measure independent of any assumed uniform motion, and in space also,
it may be said, we have a similar power of immediate comparison.
We can see, by immediate inspection, that the sub-divisions on
a foot rule are not grossly inaccurate; and so, it may be said,
we both have a measure independent of congruence, and also could
discover, by experience, any gross departure from Free Mobility.
Against this view, however, there is at the outset a very fundamental
psychological objection. It has been urged that all our comparison
of spatial magnitudes proceeds by ideal superposition. Thus James
says (Psychology, Vol. II. p. 152): "Even where we only feel one
sub-division to be vaguely larger or less, the mind must pass rapidly
between it and the other sub-division, and receive the immediate
sensible shock of the more," and "so far as the sub-divisions of
a sense-space are to be _measured_ exactly against each other,
objective forms occupying one sub-division must be directly or
indirectly superposed upon the other[160]."

Even if we waive this fundamental objection, however, others remain.
To begin with, such judgments of equality are only very rough
approximations, and cannot be applied to lines of more than a certain
length, if only for the reason that such lines cannot well be seen
together. Thus this method can only give us any security in our own
immediate neighbourhood, and could in no wise warrant such operations
as would be required for the construction of maps &c., much less the
measurement of astronomical distances. They might just enable us to
say that some lines were longer than others, but they would leave
Geometry in a position no better than that of the Hedonical Calculus,
in which we depend on a purely subjective measure. So inaccurate, in
fact, is such a method acknowledged to be, that the foot-rule is as
much a need of daily life as of science. Besides, no one would trust
such immediate judgments, but for the fact that the stricter test of
congruence to some extent confirms them; if we could not apply this
test, we should have no ground for trusting them even as much as
we do. Thus we should have, here, no real escape from our absolute
dependence upon the axiom of Free Mobility.

=153.= (4) One last elucidatory remark is necessary before our proof
of this axiom can be considered complete. We spoke above of the
Geometry on an egg, where Free Mobility does not hold. What, I may
be asked, is there about a thoroughly non-congruent Geometry, more
impossible than this Geometry on the egg? The answer is obvious. The
Geometry of non-congruent surfaces is _only_ possible by the use of
infinitesimals, and in the infinitesimal all surfaces become plane.
The fundamental formula, that for the length of an infinitesimal
arc, is only obtained on the assumption that such an arc may be
treated as a straight line, and that Euclidean Plane Geometry may be
applied in the immediate neighbourhood of any point. If we had not
our Euclidean measure, which could be moved without distortion, we
should have no method of comparing small arcs in different places,
and the Geometry of non-congruent surfaces would break down. Thus
the axiom of Free Mobility, as regards three-dimensional space, is
necessarily implied and presupposed in the Geometry of non-congruent
surfaces; the possibility of the latter, therefore, is a dependent
and derivative possibility, and can form no argument against the _à
priori_ necessity of congruence as the test of equality.

=154.= It is to be observed that the axiom of Free Mobility, as I
have enunciated it, includes also the axiom to which Helmholtz gives
the name of Monodromy. This asserts that a body does not alter its
dimensions in consequence of a complete revolution through four
right angles, but occupies at the end the same position as at the
beginning. The supposed mathematical necessity of making a separate
axiom of this property of space has been disproved by Sophus Lie (v.
Chap. I. § 45); philosophically, it is plainly a particular case of
Free Mobility[161], and indeed a particularly obvious case, for a
translation really does make some change in a body, namely, a change
in position, but a rotation through four right angles may be supposed
to have been performed any number of times without appearing in the
result, and the absurdity of ascribing to space the power of making
bodies grow in the process is palpable; everything that was said
above on congruence in general applies with even greater evidence to
this special case.

=155.= The axiom of Free Mobility involves, if it is to be true, the
homogeneity of space, or the complete relativity of position. For if
any shape, which is possible in one part of space, be always possible
in another, it follows that all parts of space are qualitatively
similar, and cannot, therefore, be distinguished by any intrinsic
property. Hence positions in space, if our axiom be true, must be
wholly defined by external relations, _i.e._ _Position is not an
intrinsic, but a purely relative, property of things in space_. If
there could be such a thing as absolute position, in short, metrical
Geometry would be impossible. This relativity of position is the
fundamental postulate of all Geometry, to which each of the necessary
metrical axioms leads, and from which, conversely, each of these
axioms can be deduced.

=156.= This converse deduction, as regards Free Mobility, is not very
difficult, and follows from the argument of Section A[162], which
I will briefly recapitulate. In the first place, externality is an
essentially relative conception--nothing can be external to itself.
To be external to something is to be an other with some relation to
that thing. Hence, when we abstract a form of externality from all
material content, and study it in isolation, position will appear
of necessity as purely relative--it can have no intrinsic quality,
for our form consists of pure externality, and externality contains
no shadow or trace of an intrinsic quality. Hence we derive our
fundamental postulate, the relativity of position. From this follows
the homogeneity of our form, for any quality in one position, which
marked out that position from another, would be necessarily more or
less intrinsic, and would contradict the pure relativity. Finally
Free Mobility follows from homogeneity, for our form would not be
homogeneous unless it allowed, in every part, shapes or systems
of relations, which it allowed in any other part. Free Mobility,
therefore, is a necessary property of every possible form of
externality.

=157.= In summing up the argument we have just concluded, we may
exhibit it, in consequence of the two preceding paragraphs, in the
form of a completed circle. Starting from the conditions of spatial
measurement, we found that the comparison, required for measurement,
could only be effected by superposition. But we found, further, that
the result of such comparison will only be unambiguous, if spatial
magnitudes and shapes are unaltered by motion in space, if, in other
words, shapes do not depend upon absolute position in space. But this
axiom can only be true if space is homogeneous and position merely
relative. Conversely, if position is assumed to be merely relative,
a change of magnitude in motion--involving as it does, the assertion
of absolute position--is impossible, and our test of spatial equality
is therefore adequate. But position in any form of externality must
be purely relative, since externality cannot be an intrinsic property
of anything. Our axiom, therefore, is _à priori_ in a double sense.
It is presupposed in all spatial measurement, and it is a necessary
property of any form of externality. A similar double apriority, we
shall see, appears in our other necessary axioms.


II. _The Axiom of Dimensions[163]._

=158.= We have seen, in discussing the axiom of Free Mobility, that
all position is relative, that is, a position exists only by virtue
of relations[164]. It follows that, if positions can be defined at
all, they must be uniquely and exhaustively defined by some finite
number of such relations. If Geometry is to be possible, it must
happen that, after enough relations have been given to determine a
point uniquely, its relations to any fresh known point are deducible
from the relations already given. Hence we obtain, as an _à priori_
condition of Geometry, logically indispensable to its existence, the
axiom that _Space must have a finite integral number of Dimensions_.
For every relation required in the definition of a point constitutes
a dimension, and a fraction of a relation is meaningless. The number
of relations required must be finite, since an infinite number of
dimensions would be practically impossible to determine. If we
remember our axiom of Free Mobility, and remember also that space
is a continuum, we may state our axiom, for metrical Geometry, in
the form given by Helmholtz (v. Chap. I. § 25): "In a space of n
dimensions, the position of every point is uniquely determined by the
measurement of n continuous independent variables (coordinates).[165]"

=159.= So much, then, is _à priori_ necessary to metrical Geometry.
The restriction of the dimensions to three seems, on the contrary,
to be wholly the work of experience[166]. This restriction cannot
be logically necessary, for as soon as we have formulated any
analytical system, it appears wholly arbitrary. Why, we are driven
to ask, cannot we add a fourth coordinate to our _x_, _y_, _z_, or
give a geometrical meaning to _x^{4}_? In this more special form, we
are tempted to regard the axiom of dimensions, like the number of
inhabitants of a town, as a purely statistical fact, with no greater
necessity than such facts have.

Geometry affords intrinsic evidence of the truth of my division of
the axiom of dimensions into an _à priori_ and empirical portion.
For while the extension of the number of dimensions to four, or to
_n_, alters nothing in plane and solid Geometry, but only adds a
new branch which interferes in no way with the old, _some_ definite
number of dimensions is assumed in all Geometries, nor is it
possible to conceive of a Geometry which should be free from this
assumption[167].

=160.= Let us, since the point seems of some interest, repeat our
proof of the apriority of this axiom from a slightly different point
of view. We will begin, this time, from the most abstract conception
of space, such as we find in Riemann's dissertation, or in Erdmann's
extents. We have here, an ordered manifold, infinitely divisible and
allowing of Free Mobility[168]. Free Mobility involves, as we saw,
the power of passing continuously from any one point to any other, by
any course which may seem pleasant to us; it involves, also, that,
in such a course, no changes occur except changes of mere position,
_i.e._, positions do not differ from one another in any qualitative
way. (This absence of qualitative difference is the distinguishing
mark of space as opposed to other manifolds, such as the colour- and
tone-systems: in these, every element has a definite qualitative
sensational value, whereas in space, the sensational value of a
position depends wholly on its spatial relation to our own body, and
is thus not intrinsic, but relative.) From the absence of qualitative
differences among positions, it follows logically that positions
exist only by virtue of other positions; one position differs from
another just because they are two, not because of anything intrinsic
in either. Position is thus defined simply and solely by relation to
other positions. Any position, therefore, is completely defined when,
and only when, enough such relations have been given to enable us to
determine its relation to any new position, this new position being
defined by the same number of relations. Now, in order that such
definition may be at all possible, a finite number of relations must
suffice. But every such relation constitutes a dimension. Therefore,
if Geometry is to be possible, it is _à priori_ necessary that space
should have a finite integral number of dimensions.

=161.= The limitation of the dimensions to three is, as we have
seen, empirical; nevertheless, it is not liable to the inaccuracy
and uncertainty which usually belong to empirical knowledge. For
the alternatives which logic leaves to sense are discrete--if the
dimensions are not three, they must be two or four or some other
number--so that _small_ errors are out of the question[169]. Hence
the final certainty of the axiom of three dimensions, though in part
due to experience, is of quite a different order from that of (say)
the law of Gravitation. In the latter, a small inaccuracy might exist
and remain undetected; in the former, an error would have to be so
considerable as to be utterly impossible to overlook. It follows that
the certainty of our whole axiom, that the number of dimensions is
three, is almost as great as that of the _à priori_ element, since
this element leaves to sense a definite disjunction of discrete
possibilities.


III. _The Axiom of Distance._

=162.= We have already seen, in discussing projective Geometry, that
two points must determine a unique curve, the straight line. In
metrical Geometry, the corresponding axiom is, that two points must
determine a unique spatial quantity, distance. I propose to prove,
in what follows, (1) that if distance, as a quantity completely
determined by two points, did not exist, spatial magnitude would
not be measurable; (2) that distance can only be determined by two
points, if there is an actual curve in space determined by those two
points; (3) that the existence of such a curve can be deduced from
the conception of a form of externality, and (4) that the application
of quantity to such a curve necessarily leads to a certain magnitude,
namely distance, uniquely determined by any two points which
determine the curve. The conclusion will be, if these propositions
can be successfully maintained, that the axiom of distance is _à
priori_ in the same double sense as the axiom of Free Mobility,
_i.e._ it is presupposed in the possibility of measurement, and it is
necessarily true of any possible form of externality.

=163.= (1) The possibility of spatial measurement allows us to
infer the existence of a magnitude uniquely determined by any two
points. The proof of this depends on the axiom of Free Mobility, or
its equivalent, the homogeneity of space. We have seen that these
are involved in the possibility of spatial measurement; we may
employ them, therefore, in any argument as to the conditions of this
possibility.

Now to begin with, two points must, if Geometry is to be possible,
have _some_ relation to each other, for we have seen that such
relations alone constitute position or localization. But if two
points have a relation to each other, this must be an intrinsic
relation. For it follows, from the axiom of Free Mobility, that
two points, forming a figure congruent with the given pair, can be
constructed in any part of space. If this were not possible, we have
seen that metrical Geometry could not exist. But both the figures may
be regarded as composed of two points and their relation; if the
two figures are congruent, therefore, it follows that the relation
is quantitatively the same for both figures, since congruence is the
test of spatial equality. Hence the two points have a quantitative
relation, which is such that they can traverse all space in a
combined motion without in any way altering that relation. But in
such a general motion, any external relation of the two points,
any relation involving other points or figures in space, must be
altered[170]. Hence the relation between the two points, being
unaltered, must be an intrinsic relation, a relation involving no
other point or figure in space; and this intrinsic relation we call
distance[171].

=164.= It might be objected, to the above argument, that it involves
a _petitio principii_. For it has been assumed that the two points
and their relation form a figure, to which other figures can be
congruent. Now if two points have no intrinsic relation, it would
seem that they cannot form such a figure. The argument, therefore,
apparently assumes what it had to prove. Why, it may be asked, should
not three points be required, before we obtain any relation, which
Free Mobility allows us to construct afresh in other parts of space?

The answer to this, as to the corresponding question in the first
section of this chapter, lies, I think, in the passivity of space,
or the mutual independence of its parts. For it follows, from this
independence, that any figure, or any assemblage of points, may
be discussed without reference to other figures or points. This
principle is the basis of infinite divisibility, of the use of
quantity in Geometry, and of all possibility of isolating particular
figures for discussion. It follows that two points cannot be
dependent, as to their relation, on any other points or figures, for
if they were so dependent, we should have to suppose some action
of such points or figures on the two points considered, which
would contradict the mutual independence of different positions.
To illustrate by an example: the relation of two given points does
not depend on the other points of the straight line on which the
given points lie. For only through their relation, _i.e._ through
the straight line which they determine, can the other points of the
straight line be known to have any peculiar connection with the given
pair.

=165.= But why, it may be asked, should there be only one such
relation between two points? Why not several? The answer to this lies
in the fact that points are wholly constituted by relations, and have
no intrinsic nature of their own[172]. A point is defined by its
relations to other points, and when once the relations necessary for
definition have been given, no fresh relations to the points used in
definition are possible, since the point defined has no qualities
from which such relations could flow. Now one relation to any one
other point is as good for definition as more would be, since however
many we had, they would all remain unaltered in a combined motion of
both points. Hence there can only be one relation determined by any
two points.

=166.= (2) We have thus established our first proposition--two points
have one and only one relation uniquely determined by those two
points. This relation we call their distance apart. It remains to
consider the conditions of the measurement of distance, _i.e._, how
far a unique value for distance involves a curve uniquely determined
by the two points.

In the first place, some curve joining the two points is involved
in the above notion of a combined motion of the two points, or of
two other points forming a figure congruent with the first two.
For without some such curve, the two point-pairs cannot be known
as congruent, nor can we have any test by which to discover when
a point-pair is moving as a single figure[173]. Distance must be
measured, therefore, by some line which joins the two points. But
need this be a line which the two points completely determine?

=167.= We are accustomed to the definition of the straight line
as the _shortest_ distance between two points, which implies that
distance might equally well be measured by curved lines. This
implication I believe to be false, for the following reasons. When we
speak of the length of a curve, we can give a meaning to our words
only by supposing the curve divided into infinitesimal rectilinear
arcs, whose sum gives the length of an equivalent straight line;
thus unless we presuppose the straight line, we have no means of
comparing the lengths of different curves, and can therefore never
discover the applicability of our definition. It might be thought,
perhaps, that some other line, say a circle, might be used as the
basis of measurement. But in order to estimate in this way the length
of any curve other than a circle, we should have to divide the curve
into infinitesimal circular arcs. Now two successive points do not
determine a circle, so that an arc of two points would have an
indeterminate length. It is true that, if we exclude infinitesimal
radii for the measuring circles, the lengths of the infinitesimal
arcs would be determinate, even if the circles varied, but that is
only because all the small circular arcs through two consecutive
points coincide with the straight line through those two points.
Thus, even with the help of the arbitrary restriction to a finite
radius, all that happens is that we are brought back to the straight
line. If, to mend matters, we take three consecutive points of our
curve, and reckon distance by the arc of the circle of curvature,
the notion of distance loses its fundamental property of being a
relation between _two_ points. For two consecutive points of the
arc could not then be said to have any corresponding distance
apart--three points would be necessary before the notion of distance
became applicable. Thus the circle is not a possible basis for
measurement, and similar objections apply, of course, with increased
force, to any other curve. All this argument is designed to show,
in detail, the logical impossibility of measuring distance by any
curve not completely defined by the two points whose distance apart
is required. If in the above we had taken distance as measured
by circles of _given radius_, we should have introduced into its
definition a relation to other points besides the two whose distance
was to be measured, which we saw to be a logical fallacy. Moreover,
how are we to know that all the circles have equal radii, until we
have an independent measure of distance?

=168.= A straight line, then, is not the _shortest_ distance, but is
simply _the_ distance between two points--so far, this conclusion
has stood firm. But suppose we had two or more curves through
two points, and that all these curves were congruent _inter se_.
We should then say, in accordance with the definition of spatial
equality, that the lengths of all these curves were equal. Now
it might happen that, although no one of the curves was uniquely
determined by the two end-points, yet the common length of all the
curves was so determined. In this case, what would hinder us from
calling this common length the distance apart, although no unique
figure in space corresponded to it? This is the case contemplated by
spherical Geometry, where, as on a sphere, antipodes can be joined by
an infinite number of geodesics, all of which are of equal length.
The difficulty supposed is, therefore, not a purely imaginary one,
but one which modern Geometry forces us to face. I shall consequently
discuss it at some length.

=169.= To begin with, I must point out that my axiom is not quite
equivalent to Euclid's. Euclid's axiom states that two straight lines
cannot enclose a space, _i.e._, cannot have more than one common
point. Now if every two points, without exception, determine a unique
straight line, it follows, of course, that two different straight
lines can have only one point in common--so far, the two axioms are
equivalent. But it may happen, as in spherical space, that two points
_in general_ determine a unique straight line, but fail to do so
when they have to each other the special relation of being antipodes.
In such a system every pair of straight lines in the same plane meet
in two points, which are each other's antipodes; but two points, _in
general_, still determine a unique straight line. We are still able,
therefore, to obtain distances from unique straight lines, except in
limiting cases; and in such cases, we can take any point intermediate
between the two antipodes, join it by the _same_ straight line to
both antipodes, and measure its distance from those antipodes in the
usual way. The sum of these distances then gives a unique value for
the distance between the antipodes.

Thus even in spherical space, we are greatly assisted by the axiom
of the straight line; all linear measurement is effected by it, and
exceptional cases can be treated, through its help, by the usual
methods for limits. Spherical space, therefore, is not so adverse
as it at first appeared to be to the _à priori_ necessity of the
axiom. Nevertheless we have, so far, not attacked the kernel of the
objection which spherical space suggested. To this attack it is now
our duty to proceed.

=170.= It will be remembered that, in our _à priori_ proof that
two points must have one definite relation, we held it impossible
for those two points to have, to the rest of space, any relation
which would be unaltered by motion. Now in spherical space, in the
particular case where the two points are antipodes, they _have_ a
relation, unaltered by motion, to the rest of space--the relation,
namely, that their distance is half the circumference of the
universe. In our former discussion, we assumed that any relation
to outside space must be a relation of position--and a relation of
position must be altered by motion. But with a finite space, in
which we have absolute magnitude, another relation becomes possible,
namely, a relation of magnitude. Antipodal points, accordingly,
like coincident points, no longer determine a unique straight line.
And it is instructive to observe that there is, in consequence, an
ambiguity in the expression for distance, like the ordinary ambiguity
in angular measurement. If 1/k^{2} be the space constant, and _d_
be one value for the distance between two points, 2πkn ± d, where
_n_ is any integer, is an equally good value. Distance is, in short,
a periodic function like angle. Thus such a state of things rather
confirms than destroys my contention, that distance depends on a
curve uniquely determined by two points. For as soon as we drop this
unique determination, we see ambiguities creeping into our expression
for distance. Distance still has a set of _discrete_ values,
corresponding to the fact that, given one point, the straight line
is uniquely determined for all other points but one, the antipodal
point. It is tempting to go on, and say: If through _every_ pair of
points there were an infinite number of the curves used in measuring
distance, distance would be able, for the same pair of points, to
take, not only a discrete series, but an infinite _continuous_ series
of values.

=171.= This, however, is mere speculation. I come now to the _pièce
de résistance_ of my argument. The ambiguity in spherical space
arose, as we saw, from a relation of _magnitude_ to the rest of
space--such a relation being unaltered by a motion of the two points,
and therefore falling outside our introductory reasoning. But what
is this relation of magnitude? Simply a relation of the _distance_
between the two points to a _distance_ given in the nature of the
space in question. It follows that such a relation _presupposes_ a
measure of distance, and need not, therefore, be contemplated in
any argument which deals with the _à priori_ requisites for the
possibility of definite distances[174].

=172.= I have now shown, I hope conclusively, that spherical space
affords no objection to the apriority of my axiom. Any two points
have one relation, their distance, which is independent of the
rest of space, and this relation requires, as its measure, a curve
uniquely determined by those two points. I might have taken the bull
by the horns, and said: Two points _can_ have no relation but what
is given by lines which join them, and therefore, if they have a
relation independent of the rest of space, there must be one line
joining them which they completely determine. Thus James says[175]:

"Just as, in the field of quantity, the relation between two numbers
is another number, so in the field of space the relations are facts
of the same order with the facts they relate.... When we speak of
the relation of direction of two points towards each other, we mean
simply the sensation of the line that joins the two points together.
_The line is the relation...._ The relation of position between the
top and bottom points of a vertical line is that line, and nothing
else."

If I had been willing to use this doctrine at the beginning, I
might have avoided all discussion. A unique relation between two
points _must_ in this case, involve a unique line between them. But
it seemed better to avoid a doctrine not universally accepted, the
more so as I was approaching the question from the logical, not the
psychological, side. After disposing of the objections, however, it
is interesting to find this confirmation of the above theory from so
different a standpoint. Indeed, I believe James's doctrine could be
proved to be a logical necessity, as well as a psychological fact.
For what sort of thing can a spatial relation between two distinct
points be? It must be something spatial, and it must, since points
are wholly constituted by their relations, be something at least
as real and tangible as the points it relates. There seems nothing
which can satisfy these requirements, except a line joining them.
Hence, once more, a unique relation must involve a unique line. That
is, linear magnitude is logically impossible, unless space allows of
curves uniquely determined by any two of their points.

=173.= (3) But farther, the existence of curves uniquely determined
by two points can be deduced from the nature of any form of
externality[176]. For we saw, in discussing Free Mobility, that this
axiom, together with homogeneity and the relativity of position,
can be so deduced, and we saw in the beginning of our discussion on
distance, that the existence of a unique relation between two points
could be deduced from the homogeneity of space. Since position is
relative, we may say, any two points must have _some_ relation to
each other: since our form of externality is homogeneous, this
relation can be kept unchanged while the two points move in the
form, _i.e._, change their relations to other points; hence their
relation to each other is an intrinsic relation, independent of their
relations to other points. But since our form _is_ merely a complex
of relations, a relation of externality must appear in the form, with
the same evidence as anything else in the form; thus if the form be
intuitive or sensational, the relation must be immediately presented,
and not a mere inference. Hence the intrinsic relation between two
points must be a unique figure in our form, _i.e._ in spatial terms,
the straight line joining the two points.

=174.= (4) Finally, we have to prove that the existence of such a
curve necessarily leads, when quantity is applied to the relation
between two points, to a unique magnitude, which those two points
completely determine. With this, we shall be brought back to
distance, from which we started, and shall complete the circle of our
argument.

We saw, in section A § 119, that the figure formed by two points is
projectively indistinguishable from that formed by any two other
points in the same straight line; the figure, in both cases, is,
from the projective standpoint, simply the straight line on which
the two points lie. The difference of relation, in the two cases,
is not qualitative, since projective Geometry cannot deal with it;
nevertheless, there is some difference of relation. For instance, if
one point be kept fixed, while the other moves, there is obviously
some change of relation. This change, since all parts of the straight
line are qualitatively alike, must be a change of quantity. If two
points, therefore, determine a unique figure, there must exist, for
the distinction between the various other points of this figure, a
unique quantitative relation between the two determining points, and
therefore, since these points are arbitrary, between only two points.
This relation is _distance_, with which our argument began, and to
which it at least returns.

=175.= To sum up: If points are defined simply by relations to other
points, _i.e._, if all position is relative, _every point must have
to every other point one, and only one, relation independent of
the rest of space. This relation is the distance between the two
points._ Now a relation between two points can only be defined by a
line joining them--nay further, it may be contended that a relation
can only _be_ a line joining them. Hence a unique relation involves
a unique line, _i.e._, a line determined by any two of its points.
Only in a space which admits of such a line is linear magnitude a
logically possible conception. But when once we have established the
possibility, _in general_, of drawing such lines, and therefore of
measuring linear magnitudes, we may find that a certain magnitude has
a peculiar relation to the constitution of space. The straight line
may turn out to be of finite length, and in this case its length will
give a certain peculiar magnitude, the space-constant. Two antipodal
points, that is, points which bisect the entire straight line, will
then have a relation of magnitude which, though unaltered by motion,
is rendered peculiar by a certain constant relation to the rest of
space. This peculiarity presupposes a measure of linear magnitude in
general, and cannot, therefore, upset the apriority of the axiom of
the straight line. But it destroys, for points having the peculiar
antipodal relation to each other, the argument which proved that the
relation between two points could not, since it was unchanged by
motion, have reference to the rest of space. Thus it is intelligible
that, for such special points, the axiom breaks down, and an infinite
number of straight lines are possible between them; but unless we had
started with assuming the general validity of the axiom, we could
never have reached a position in which antipodal points could have
been known to be peculiar, or, indeed, a position which would have
enabled us to give any quantitative definition whatever of particular
points.

Distance and the straight line, as relations uniquely determined
by two points, are thus _à priori_ necessary to metrical Geometry.
But further, they are properties which must belong to any form of
externality. Since their necessity for Geometry was deduced from
homogeneity and the relativity of position, and since these are
necessary properties of any form of externality, the same argument
proves both conclusions. We thus obtain, as in the case of Free
Mobility, a double apriority: The axiom of Distance, and its
implication, the axiom of the Straight Line, are, on the one hand,
presupposed in the possibility of spatial magnitude, and cannot,
therefore, be contradicted by any experience resulting from the
measurement of space; while they are consequences, on the other hand,
of the necessary properties of any form of externality which is to
render possible experience of an external world.

=176.= In connection with the straight line, it will be convenient
to discuss the conditions of a metrical coordinate system. The
projective coordinate system, as we have seen, aims only at a
convenient nomenclature for different points, and can be set up
without introducing the notion of spatial quantity. But a metrical
coordinate system does much more than this. It defines every point
quantitatively, by its quantitative spatial relations to a certain
coordinate figure. Only when the system of coordinates is thus
metrical, _i.e._, when every coordinate represents some spatial
magnitude, which is itself a relation of the point defined to some
other point or figure--can operations with coordinates lead to a
metrical result. When, as in projective Geometry, the coordinates
are not spatial magnitudes, no amount of transformation can give a
metrical result. I wish to prove, here, that a metrical coordinate
system necessarily involves the straight line, and cannot, without a
logical fallacy, be set up on any other basis. The projective system
of coordinates, as we saw, is entirely based on the straight line;
but the metrical system is more important, since its quantities
embody actual information as to spatial magnitudes, which, in
projective Geometry, is not the case.

In the first place, a point's metrical coordinates constitute a
complete quantitative definition of it; now a point can only be
defined, as we have seen, by its relations to other points, and
these relations can only be defined by means of the straight line.
Consequently, any metrical system of coordinates must involve the
straight line, as the basis of its definitions of points.

This _à priori_ argument, however, though I believe it to be quite
sound, is not likely to carry conviction to any one persuaded of the
opposite. Let us, therefore, examine metrical coordinate systems in
detail, and show, in each case, their dependence on the straight line.

We have already seen that the notion of distance is impossible
without the straight line. We cannot, therefore, define our
coordinates in any of the ordinary ways, as the distances from three
planes, lines, points, spheres, or what not. Polar coordinates
are impossible, since,--waiving the straightness of the radius
vector--the length of the radius vector becomes unmeaning. Triangular
coordinates involve not only angles, which must in the limit be
rectilinear, but straight lines, or at any rate some well-defined
curves. Now curves can only be metrically defined in two ways:
_Either_ by relation to the straight line, as, _e.g._, by the
curvature at any point, _or_ by purely analytical equations, which
presuppose an intelligible system of metrical coordinates. What
methods remain for assigning these arbitrary values to different
points? Nay, how are we to get any estimate of the difference--to
avoid the more special notion of distance--between two points?
The very notion of a point has become illusory. When we have a
coordinate system, we may define a point by its three coordinates;
in the absence of such a system, we may define the notion of point
_in general_ as the intersection of three surfaces or of two curves.
Here we take surfaces and curves as notions which intuition makes
plain, but if we wish them to give us a precise numerical definition
of _particular_ points, we must specify the kind of surface or
curve to be used. Now this, as we have seen, is only possible when
we presuppose either the straight line, or a coordinate system. It
follows that every coordinate system presupposes the straight line,
and is logically impossible without it.

=177.= The above three axioms, we have seen, are _à priori_ necessary
to metrical Geometry. No others can be necessary, since metrical
systems, logically as unassailable as Euclid's, and dealing with
spaces equally homogeneous and equally relational, have been
constructed by the metageometers, without the help of any other
axioms. The remaining axioms of Euclidean Geometry--the axiom of
parallels, the axiom that the number of dimensions is three, and
Euclid's form of the axiom of the straight line (two straight lines
cannot enclose a space)--are not essential to the possibility of
metrical Geometry, _i.e._, are not deducible from the fact that
a science of spatial magnitudes is possible. They are rather
to be regarded as empirical laws, obtained, like the empirical
laws of other sciences, by actual investigation of the given
subject-matter--in this instance, experienced space.

=178.= In summing up the distinctive argument of this Section,
we may give it a more general form, and discuss the conditions
of measurement in any continuous manifold, _i.e._, the qualities
necessary to the manifold, in order that quantities in it may be
determinable, not only as to the more or less, but as to the precise
_how much_.

Measurement, we may say, is the application of number to continua,
or, if we prefer it, the transformation of mere quantity into number
of units. Using _quantity_ to denote the vague more or less, and
_magnitude_ to denote the precise number of units, the problem of
measurement may be defined as the transformation of quantity into
magnitude.

Now a number, to begin with, is a whole consisting of smaller units,
all of these units being qualitatively alike. In order, therefore,
that a continuous quantity may be expressible as a number, it must,
on the one hand, be itself a whole, and must, on the other hand,
be divisible into qualitatively similar parts. In the aspect of a
whole, the quantity is _intensive_; in the aspect of an aggregate of
parts, it is _extensive_. A purely intensive quantity, therefore,
is not numerable--a purely extensive quantity, if any such could
be imagined, would not be a single quantity at all, since it would
have to consist of wholly unsynthesized particulars. A measurable
quantity, therefore, is a whole divisible into similar parts. But
a continuous quantity, if divisible at all, must be _infinitely_
divisible. For otherwise the points at which it could be divided
would form natural barriers, and so destroy its continuity. But
further, it is not sufficient that there should be a possibility
of division into mutually external parts; while the parts, to be
perceptible as parts, must be mutually external, they must also, to
be knowable as _equal_ parts, be capable of overcoming their mutual
externality. For this, as we have seen, we require superposition,
which involves Free Mobility and homogeneity--the absence of Free
Mobility in time, where all other requisites of measurement are
fulfilled, renders direct measurement of time impossible. Hence
infinite divisibility, free mobility, and homogeneity are necessary
for the possibility of measurement in _any_ continuous manifold, and
these, as we have seen, are equivalent to our three axioms. These
axioms are necessary, therefore, not only for spatial measurement,
but for all measurement. The only manifold given in experience, in
which these conditions are satisfied, is space. All other exact
measurement--as could be proved, I believe, for every separate
case--is effected, as we saw in the case of time, by reduction to
a spatial correlative. This explains the paramount importance, to
exact science, of the mechanical view of nature, which reduces
all phenomena to motions in time and space. For number is, of
all conceptions, the easiest to operate with, and science seeks
everywhere for an opportunity to apply it, but finds this opportunity
only by means of spatial equivalents to phenomena[177].

=179.= We have now seen in what the _à priori_ element of Geometry
consists. This _à priori_ element may be defined as the axioms common
to Euclidean and non-Euclidean spaces, as the axioms deducible from
the conception of a form of externality, or--in metrical Geometry--as
the axioms required for the possibility of measurement. It remains
to discuss, in a final chapter, some questions of a more general
philosophic nature, in which we shall have to desert the firm ground
of mathematics and enter on speculations which I put forward very
tentatively, and with little faith in their ultimate validity. The
chief questions for this final chapter will be two: (1) How is such
_à priori_ and purely logical necessity possible, as applied to an
actually given subject-matter like space? (2) How can we remove the
contradictions which have haunted us in this chapter, arising out of
the relativity, infinite divisibility, and unbounded extension of
space? These two questions are forced upon us by the present chapter,
but as they open some of the fundamental problems of philosophy, it
would be rash to expect a conclusive or wholly satisfactory answer. A
few hints and suggestions may be hoped for, but a complete solution
could only be obtained from a complete philosophy, of which the
prospects are far too slender to encourage a confident frame of mind.


FOOTNOTES:

[116] See infra, Axiom of Distance, in Sec. B. of this Chapter.

[117] Thus on a cylinder, two geodesics, _e.g._ a generator and
a helix, may have any number of intersections--a very important
difference from the plane.

[118] Cf. Cremona, Projective Geometry (Clarendon Press, 2nd ed.
1893) p. 50: "Most of the propositions in Euclid's Elements are
metrical, and it is not easy to find among them an example of a
purely descriptive theorem."

[119] Op. cit. p. 226.

[120] Some ground for this choice will appear when we come to
metrical Geometry.

[121] The straight line _σa_ denotes the straight line common to the
planes _σ_ and _a_, the point _σa_ denotes the point common to the
plane _σ_ and the straight line _a_, and similarly for the rest of the
notation.

[122] Cremona (op. cit. Chap. IX. p. 50) defines anharmonic ratio as
a metrical property which is unaltered by projection. This, however,
destroys the logical independence of projective Geometry, which can
only be maintained by a purely descriptive definition.

[123] There is no corresponding property of _three_ points on a line,
because they can be projectively transformed into any other three
points on the same line. See § 120.

[124] Due to v. Staudt's "Geometrie der Lage."

[125] See Cremona, op. cit. Chapter VIII.

[126] The corresponding definitions, for the two-dimensional manifold
of lines through a point, follow by the principle of duality.

[127] It is important to observe that this definition of the Point
introduces metrical ideas. Without metrical ideas, we saw, nothing
appears to give the Point precedence of the straight line, or indeed
to distinguish it conceptually from the straight line. A reference
to quantity is therefore inevitable in defining the Point, if the
definition is to be geometrical. A non-metrical definition would have
to be also non-geometrical. See Chap. IV. §§ 196-199.

[128] §§ 163-175.

[129] On this axiom, however, compare § 131.

[130] For the proof of this proposition, see Chap. III. Sec. B, Axiom
of Dimensions.

[131] The straight line and plane, in all discussions of general
Geometry, are not necessarily Euclidean. They are simply figures
determined, in general, by two and by three points respectively;
whether they conform to the axiom of parallels and to Euclid's form
of the axiom of the straight line, is not to be considered in the
general definition.

[132] That projective Geometry must have existential import, I shall
attempt to prove in Chapter IV.

[133] Logic, Book I. Chapter II.

[134] Cf. Bradley's Logic, p. 63. It will be seen that the sense in
which I have spoken of space as a principle of differentiation is not
the sense of a "principle of individuation" which Bradley objects to.

[135] Chap. IV. §§ 186-191.

[136] Chap. IV. § 201 ff.

[137] It is important to observe, however, that this way of regarding
spatial relations is metrical; from the projective standpoint, the
relation between two points is the whole unbounded straight line on
which they lie, and need not be regarded as divisible into parts or
as built up of points.

[138] §§ 207, 208. Cf. Hegel, Naturphilosophie, § 254.

[139] See Chap. IV. §§ 196-199.

[140] See a forthcoming article on "The relations of number and
quantity" by the present writer in _Mind_, July, 1897.

[141] Logic, Vol. II. Chap. VII. p. 211.

[142] Real, as opposed to logical, diversity is throughout intended.
Diverse aspects may coexist in a thing at one time and place, but two
diverse real things cannot so coexist.

[143] On the insufficiency of time alone, see Chapter IV. § 191.

[144] Geometrically, the axiom of the plane is, not that three
points determine a figure at all, which follows from the axiom of
the straight line, but that the straight line joining two casual
points of the plane lies wholly in the plane. This axiom requires a
projective method of constructing the plane, _i.e._ of finding all
the triads of points which determine the same projective figure as
the given triad. The required construction will be obtained if we
can find any projective figure determined by three points, and any
projective method of reaching other points which determine the same
figure.

[Illustration]

Let _O_, _P_, _Q_ be the three points whose projective relation
is required. Then we have given us the three straight lines _PQ_,
_QO_, _OP_. Metrically, the relation between these points is made
up of the area, and the magnitude of the sides and angles, of the
triangle _OPQ_, just as the relation between two points is distance.
But projectively, the figure is unchanged when _P_ and _Q_ travel
along _OP_ and _OQ_, or when _OP_ and _OQ_ turn about _O_ in such a
way as still to meet _PQ_. This is a result of the general principle
of projective equivalence enunciated above (§§ 108, 109). Hence the
projective relation between _O_, _P_, _Q_ is the same as that between
_O_, _p_, _q_ or _O_, _P′_, _Q′_; that is, _p_, _q_ and _P′_, _Q′_
lie in the plane _OPQ_. In this way, any number of points on the
plane may be obtained, and by repeating the construction with fresh
triads, every point of the plane can be reached. We have to prove
that, when the plane is so constructed, the straight line joining any
two points of the plane lies wholly in the plane.

It is evident, from the manner of construction, that any point of
_PQ_, _OP_, _OQ_, _OP′_ or _OQ′_ lies in the plane. If we can prove
that any point of _pq_ lies in the plane, we shall have proved all
that is required, since _pq_ may be transformed, by successive
repetitions of the same construction, into any straight line joining
two points of the plane. But we have seen that the same plane is
determined by _O_, _p_, _q_ and by _O_, _P_, _Q_. The straight lines
_PQ_, _pq_ have, therefore, the same relation to the plane. But _PQ_
lies wholly in the plane; therefore _pq_ also lies wholly in the
plane. Hence our axiom is proved.

[145] A detailed proof has been given above, Chap. I. 3rd period. It
is to be observed that any reference to infinitely distant elements
involves metrical ideas.

[146] Cf. Section A, §§ 115-117.

[147] Contrast Erdmann, op. cit. p. 138.

[148] Cf. Erdmann, op. cit. p. 164.

[149] Strictly speaking, this method is only applicable where the two
magnitudes are commensurable. But if we take infinite divisibility
rigidly, the units can theoretically be taken so small as to obtain
any required degree of approximation. The difficulty is the universal
one of applying to continua the essentially discrete conception of
number.

[150] Cf. Erdmann, op. cit. p. 50.

[151] Also called the axiom of congruence. I have taken congruence to
be the _definition_ of spatial equality by superposition, and shall
therefore generally speak of the _axiom_ as Free Mobility.

[152] For the sense in which these figures are to be regarded as
material, see criticism of Helmholtz, Chapter II. §§ 69 ff.

[153] Op. cit. p. 60.

[154] The view of Helmholtz and Erdmann, that mechanical experience
suffices here, though geometrical experience fails us, has been
discussed above, Chapter II. §§ 73, 82.

[155] Chapter II. § 81.

[156] Chapter II. § 72.

[157] Contrast Delbœuf, L'ancienne et les nouvelles géométries, II.
Rev. Phil. 1894, Vol. xxxvii. p. 354.

[158] Prolegomena, § 13. See Vaihinger's Commentar, II. pp. 518-532
esp. pp. 521-2. The above was Kant's whole purpose in 1768, but only
part of his purpose in the Prolegomena, where the intuitive nature of
space was also to be proved.

[159] On the subject of time measurement, cf. Bosanquet's Logic, Vol.
i. pp. 178-183. Since time, in the above account, is measured by
motion, its measurement presupposes that of spatial magnitudes.

[160] Cf. Stumpf. Ursprung der Raumvorstellung, p. 68.

[161] As is Helmholtz's other axiom, that the possibility of
superposition is independent of the course pursued in bringing it
about.

[162] Cf. §§ 129, 130.

[163] This deduction is practically the same as that in Sec. A, but
I have stated it here with more special reference to space and to
metrical Geometry.

[164] The question: "Relations to what?" is a question involving
many difficulties. It will be touched on later in this chapter, and
answered, as far as possible, in the fourth chapter. For the present,
in spite of the glaring circle involved, I shall take the relations
as relations to other positions.

[165] Wiss. Abh. Vol. II. p. 614.

[166] Cp. Grassmann, Ausdehnungslehre von 1844, 2nd ed. p. XXIII.

[167] Delbœuf, it is true, speaks of Geometries with _m_/_n_
dimensions, but gives no reference (Rev. Phil. T. xxxvi. p. 450).

[168] In criticizing Erdmann, it will be remembered, we saw that Free
Mobility is a necessary property of his extents, though he does not
regard it as such.

[169] Cf. Riemann, Hypothesen welche der Geometrie zu Grunde liegen,
Gesammelte Werke, p. 266; also Erdmann, op. cit. p. 154.

[170] This is subject, in spherical space, to the modification
pointed out below, in dealing with the exception to the axiom of the
straight line. See §§ 168-171.

[171] In speaking of distance at once as a quantity and as an
intrinsic relation, I am anxious to guard against an apparent
inconsistency. I have spoken of the judgment of quantity, throughout,
as one of comparison; how, then, can a quantity be intrinsic? The
reply is that, although measurement and the judgment of quantity
express the result of comparison, yet the terms compared must exist
before the comparison; in this case, the terms compared in measuring
distances, _i.e._ in comparing them _inter se_, are intrinsic
relations between points. Thus, although the _measurement_ of
distance involves a reference to other distances, and its expression
as a magnitude requires such a reference, yet its existence does not
depend on any external reference, but exclusively on the two points
whose distance it is.

[172] See the end of the argument on Free Mobility, § 155 ff.

[173] In Frischauf's "Absolute Geometrie nach Johann Bolyai," Anhang,
there is a series of definitions, starting from the sphere, as the
locus of congruent point-pairs when one point of the pair is fixed,
and hence obtaining the circle and the straight line. From the above
it follows, that the sphere so defined already involves a curve
between the points of the point-pair, by which various point-pairs
can be known as congruent; and it will appear, as we proceed, that
this curve must be a straight line. Frischauf's definition by means
of the sphere involves, therefore, a vicious circle, since the sphere
presupposes the straight line, as the test of congruent point-pairs.

[174] Nor in any argument which, like those of projective Geometry,
avoids the notion of magnitude or distance altogether. It follows
that the propositions of projective Geometry apply, without reserve,
to spherical space, since the exception to the axiom of the straight
line arises only on metrical ground.

[175] Psychology, Vol. II. pp. 149-150.

[176] This step in the argument has been put very briefly, since it
is a mere repetition of the corresponding argument in Section A, and
is inserted here only for the sake of logical completeness. See § 137
ff.

[177] Cf. Hannequin, Essai critique sur l'hypothèse des atomes,
Paris, 1895, passim.




CHAPTER IV.

PHILOSOPHICAL CONSEQUENCES.


=180.= In the present chapter, we have to discuss two questions
which, though scarcely geometrical, are of fundamental importance
to the theory of Geometry propounded above. The first of these
questions is this: What relation can a purely logical and deductive
proof, like that from the nature of a form of externality, bear to an
experienced subject-matter such as space? You have merely framed a
general conception, I may be told, containing space as a particular
species, and you have then shown, what should have been obvious from
the beginning, that this general conception contained some of the
attributes of space. But what ground does this give for regarding
these attributes as _à priori_? The conception Mammal has some of the
attributes of a horse; but are these attributes therefore _à priori_
adjectives of the horse? The answer to this obvious objection is so
difficult, and involves so much general philosophy, that I have kept
it for a final chapter, in order not to interrupt the argument on
specially geometrical topics.

=181.= I have already indicated, in general terms, the ground for
regarding as _à priori_ the properties of any form of externality.
This ground is transcendental, _i.e._ it is to be found in the
conditions required for the possibility of experience. The form of
externality, like Riemann's manifolds, is a general class-conception,
including time as well as Euclidean and non-Euclidean spaces. It
is not motived, however, like the manifolds, by a _quantitative_
resemblance to space, but by the fact that it fulfils, if it has
more than one dimension, all those functions which, in our actual
world, are fulfilled by space. But a form of externality, in order
to accomplish this, must be, not a mere conception, but an actually
experienced intuition. Hence the conception of such a form is the
general _conception_, containing under it every logically possible
_intuition_ which can fulfil the function actually fulfilled by
space. And this function is, to render possible experience of
diverse but interrelated things. Some form in sense-perception,
then, whose conception is included under our form of externality,
is _à priori_ necessary to experience of diversity in relation, and
without experience of this, we should, as modern logic shows, have no
experience at all. This still leaves untouched the relation of the
_à priori_ to the subjective: the form of externality is necessary
to experience, but is not, _on that account_, to be declared purely
subjective. Of course, necessity for experience can only arise from
the nature of the mind which experiences; but it does not follow that
the necessary conditions could be fulfilled, unless the objective
world had certain properties. The _ground_ of necessity, we may
safely say, arises from the mind; but it by no means follows that
the _truth_ of what is necessary depends only on the constitution of
the mind. Where this is not the case, our conclusion, when a piece
of knowledge has been declared _à priori_, can only be: Owing to the
constitution of the _mind_, experience will be impossible unless the
_world_ accepts certain adjectives.

Such, in outline, will be the argument of the first half of this
chapter, and such will be the justification for regarding as _à
priori_ those axioms of Geometry, which were deduced above from the
conception of a form of externality. For these axioms, and these
only, are necessarily true of any world in which experience is
possible.

=182[178].= The view suggested has, obviously, much in common with
that of the Transcendental Aesthetic. Indeed the whole of it, I
believe, can be obtained by a certain limitation and interpretation
of Kant's classic arguments. But as it differs, in many important
points, from the conclusions aimed at by Kant, and as the agreement
may easily seem greater than it is, I will begin by a brief
comparison, and endeavour, by reference to authoritative criticisms,
to establish the legitimacy of my divergence from him.

=183.= In the first place, the psychological element is much larger
in Kant's thesis than in mine. I shall contend, it is true, that a
form of externality, if it is to do its work, must not be a mere
conception or a mere inference, but must be a given element in
sense-perception--not, of course, originally given in isolation,
but discoverable, through analysis, by attention to the object of
sense-perception[179]. But Kant contended, not only that this element
is given, but also that it is subjective. Space, for him, is, on the
one hand, not conceptual, but on the other hand, not sensational.
It forms, for him, no part of the data of sense, but is added by a
subjective intuition, which he regards as not only logically, but
psychologically, prior to objects in space[180].

This part of Kant's argument is wholly irrelevant for us. Whether a
form of externality be given in sense, or in a pure intuition, is for
us unimportant, since we neglect the question as to the connection
of the _à priori_ and the subjective; while the temporal priority of
space to objects in it has been generally recognized as irrelevant
to Epistemology, and has often been regarded as forming no part of
Kant's thesis[181]. If we call intuitional whatever is given in
sense-perception, then we may contend that a form of externality must
be intuitional; but whether it is a pure intuition, in Kant's sense,
or not, is irrelevant to us, as is its priority to the objects in it.

That the non-sensational nature of space is no essential part of
Kant's _logical_ teaching, appears from an examination of his
argument. He has made, in the introduction, the purely logical
distinction of matter and form, but has given to this distinction,
in the very moment of suggesting it, a psychological implication.
This he does by the assertion that the form, in which the matter
of sensations is ordered, cannot itself be sensational. From this
assumption it follows, of course, that space cannot be sensational.
But the assumption is totally unsupported by argument, being set
forth, apparently, as a self-evident axiom; it has been severely
criticized by Stumpf[182] and others[183], and has been described by
Vaihinger as a fatal _petitio principii_[184]; it is irrelevant to
the logical argument, when this argument is separated, as we have
separated it, from all connection with psychological subjectivity;
and finally, it leaves us a prey to psychological theories of space,
which have seemed, of late, but little favourable to the pure Kantian
doctrine.

=184.= We have a right, therefore, in an epistemological inquiry, to
neglect Kant's psychological teaching--in so far, at any rate, as it
distinguishes spatial intuition from sensation--and attend rather to
the logical aspect alone. That part of his psychological teaching,
which maintains that space is not a mere conception, is, with certain
limitations, sufficiently evident as applied to actual space; but for
us, it must be transformed into a much more difficult thesis, namely,
that _no_ form of externality, which renders experience of diversity
in relation possible, can be merely conceptual. This question, to
which we must return later, is no longer psychological, but belongs
wholly to Epistemology.

=185.= What, then, remains the kernel, for our purposes, of Kant's
first argument for the apriority of space? His argument, in the form
in which he gave it, is concerned with the eccentric projection
of sensations. In order that I may refer sensations, he says,
to something outside myself, I must already have the subjective
space-form in the mind. In this shape, as Vaihinger points out
(Commentar, II. pp. 69, 165), the argument rests on a _petitio
principii_, for only if sensations are necessarily non-spatial does
their projection demand a subjective space-form. But, further, is the
logical apriority of space concerned with the externality of things
to ourselves?

Space _seems_ to perform two functions: on the one hand, it reveals
things, by the eccentric projection of sensations, as external
to the self, while, on the other hand, it reveals simultaneously
presented things as mutually external. These two functions, though
often treated as coordinate and almost equivalent[185], seem to me
widely different. Before we discuss the apriority of space, we must
carefully distinguish, I think, between these two functions, and
decide which of them we are to argue about.

Now externality to the Self, it would seem, must necessarily raise
the whole question of the nature and limits of the Ego, and what is
more, it cannot be derived from spatial presentation, unless we give
the Self a definite position in space. But things acquire a position
in space only when they can appear in sense-perception; we are
forced, therefore, if we adopt this view of the function of space, to
regard the Self as a phenomenon presented to sense-perception. But
this reduces externality to the Self to externality to the body. The
body, however, is a presented object like any other, and externality
of objects to it is, therefore, a special case of the mutual
externality of presented things. Hence we cannot regard space as
giving, primarily at any rate, externality to the Self, but only the
mutual externality of the things presented to sense-perception[186].

=186.= This, then, is the kind of externality we are to expect
from space, and our question must be: Would the existence of
diverse but interrelated things be unknowable, if there were not,
in sense-perception, some form of externality? This is the crucial
question, on which turns the apriority of our form, and hence of the
necessary axioms of Geometry.

=187.= The converse argument to mine, the argument from the
spatio-temporal element in perception to a world of interrelated
but diverse things, is developed at length in Bradley's Logic. It
is put briefly in the following sentence (p. 44, note): "If space
and time are continuous, and if all appearance must occupy some time
or space--and it is not hard to support both these _theses_--we can
at once proceed to the conclusion, no mere particular exists. Every
phenomenon will exist in more times or spaces than one; and against
that diversity will be itself an universal[187]." The importance of
this fact appears, when we consider that, if any _mere_ particular
existed, all judgment and inference as to that particular would be
impossible, since all judgment and inference necessarily operate
by means of universal. But all reality is constructed from the
_This_ of immediate presentation, from which judgment and inference
necessarily spring. Owing, however, to the continuity and relativity
of space and time, no _This_ can be regarded either as simple or
as self-subsistent. Every _This_, on the one hand, can be analyzed
into _Thises_, and on the other hand, is found to be necessarily
related to other things, outside the limits of the given object of
sense-perception. This function of space and time is presupposed in
the following statement from Bosanquet's Logic (Vol. I. pp. 77-78):
"Reality is given for me _in_ present sensuous perception, and _in_
the immediate feeling of my own sentient existence that goes with
it. The real world, as a definite organized system, is _for me_ an
extension of this present sensation and self feeling by means of
judgment, and it is the essence of judgment to effect and sustain
such an extension.... The subject in every judgment of Perception
is some given spot or point in sensuous contact with the percipient
self. But, as all reality is continuous, the subject is not _merely_
this given spot or point."

=188.= This doctrine of Bradley and Bosanquet is the converse of
the epistemological doctrine I have to advocate. Owing to the
continuity and relativity of space and time, they say, we are able
to construct a systematic world, by judgment and inference, out of
that fragmentary and yet necessarily complex existence which is
given in sense-perception. My contention is, conversely, that since
all knowledge is necessarily derived by an extension of the _This_
of sense-perception, and since such extension is only possible if
the _This_ has that fragmentary and yet complex character conferred
by a form of externality, therefore some form of externality,
given with the _This_, is essential to all knowledge, and is thus
logically _à priori_. Bradley's argument, if sound, already proves
this contention; for while, on the one hand, he uses no properties of
space and time but those which belong to every form of externality,
he proves, on the other hand, that judgment and inference require
the _This_ to be neither single nor self-subsistent. But I will
endeavour, since the point is of fundamental importance, to
reproduce the proof, in a form more suited than Bradley's to the
epistemological question.

=189.= The essence of my contention is that, if experience is to be
possible, every sensational _This_ must, when attended to, be found,
on the one hand, resolvable into _Thises_, and on the other hand
dependent, for some of its adjectives, on external reference. The
second of these theses follows from the first, for if we take one
of the _Thises_ contained in the first _This_, we get a new _This_
necessarily related to the other _Thises_ which make up the original
_This_. I may, therefore, confine myself to the first proposition,
which affirms that the object of perception must contain a diversity,
not only of conceptual content, but of existence, and that this can
only be known if sense-perception contains, as an element, some form
of externality.

My premiss, in this argument, is that all knowledge involves a
recognition of diversity in relation, or, if we prefer it, of
identity in difference. This premiss I accept from Logic, as
resulting from the analysis of judgment and inference. To prove
such a premiss, would require a treatise on Logic; I must refer the
reader, therefore, to the works of Bradley and Bosanquet on the
subject. It follows at once, from my premiss, that knowledge would be
impossible, unless the object of attention could be complex, _i.e._
not a _mere_ particular. Now could the mental object--_i.e._, in this
connection, the object of a cognition--be complex, if the object of
immediate perception were always simple?

=190.= We might be inclined, at first sight, to answer this question
affirmatively. But several difficulties, I think, would prevent
such an answer. In the first place, knowledge must start from
perception. Hence, either we could have no knowledge except of our
present perception, or else we must be able to contrast and compare
it with some other perception. Now in the first case, since the
present perception, by hypothesis, is a mere particular, knowledge
of it is impossible, according to our premiss. But in the second
case, the other perception, with which we compare our first, must
have occurred at some other time, and with time, we have at once a
form of externality. But what is more, our present perception is
no longer a mere particular. For the power of comparing it with
another perception involves a point of identity between the two, and
thus renders both complex. Moreover, time must be continuous, and
the present, as Bradley points out, is no mere point of time[188].
Thus our present perception contains the complexity involved in
duration throughout the specious present: its mere particularity
and its simplicity are lost. Its self-subsistence is also lost, for
beyond the specious present, lie the past and the future, to which
our present perception thus unavoidably refers us. Time at least,
therefore, is essential to that identity in difference, which all
knowledge postulates.

=191.= But we have derived, from all this, no ground for affirming a
multiplicity of real things, or a form of externality of more than
one dimension, which, we saw, was necessary for the truth of two out
of our three axioms. This brings us to the question: Have we enough,
with time alone as a form of externality, for the possibility of
knowledge?

This question we must, I think, answer in the negative. With time
alone, we have seen, our presented object must be complex, but its
complexity must, if I may use such a phrase, be merely adjectival.
Without a second form of externality, only one thing can be given at
one moment[189], and this one thing, therefore, must constitute the
whole of our world. The object of past perception must--since our one
thing has nothing external to it, by which it could be created or
destroyed--be regarded as the same thing in a different state. The
complexity, therefore, will lie only in the changing states of our
one thing--it will be adjectival, not substantival. Moreover we have
the following dilemma: Either the one thing must be ourselves, or
else self-consciousness could never arise. But the chief difficulty
of such a world would lie in the changes of the thing. What could
cause these changes, since we should know of nothing external to our
thing? It would be like a Leibnitzian monad, without any God outside
it to prearrange its changes. Causality, in such a world, could not
be applied, and change would be wholly inexplicable.

Hence we require also the possibility of a diversity of
simultaneously existing things, not merely of successive adjectives;
and this, we have seen, cannot be given by time alone, but only by
a form of externality for simultaneous parts of one presentation.
We could never, in other words, infer the existence of diverse but
interrelated things, unless the object of sense-perception could have
substantival complexity, and for such complexity we require a form
of externality other than time. Such a form, moreover, as was shown
in Chapter III., Section A (§ 135), can only fulfil its functions
if it has more than one dimension. In our actual world, this form
is given by space; in any world, knowable to beings with our laws
of thought, some such form, as we have now seen, must be given in
sense-perception.

This argument may be briefly summed up, by assuming the doctrine
of Bradley, that all knowledge is obtained by inference from the
_This_ of sense-perception. For, if this be so, the _This_--in
order that inference, which depends on identity in difference, may
be possible at all--must itself be complex, and must, on analysis,
reveal adjectives having a reference beyond itself. But this, as was
shown above, can only happen by means of a form of externality. This
establishes the à priori axioms of Geometry, as necessarily having
existential import and validity in any intelligible world.

=192.= The above argument, I hope, has explained why I hold it
possible to deduce, from a mere conception like that of a form
of externality, the logical apriority of certain axioms as to
experienced space. The Kantian argument--which was correct, if our
reasoning has been sound, in asserting that real diversity, in our
actual world, could only be known by the help of space--was only
mistaken, so far as its purely logical scope extends, in overlooking
the possibility of other forms of externality, which could, if they
existed, perform the same task with equal efficiency. In so far as
space differs, therefore, from these other conceptions of possible
intuitional forms, it is a mere experienced fact, while in so far as
its properties are those which all such forms must have, it is _à
priori_ necessary to the possibility of experience.

I cannot hope, however, that no difficulty will remain, for the
reader, in such a deduction, from abstract conceptions, of the
properties of an actual _datum_ in sense-perception. Let us consider,
for example, such a property as impenetrability. To suppose two
things simultaneously in the same position in a form of externality,
is a logical contradiction; but can we say as much of actual space
and time? Is not the impossibility, here, a matter of experience
rather than of logic? Not if the above argument has been sound,
I reply. For in that case, we infer real diversity, _i.e._ the
existence of different things, only from difference of position in
space or time. It follows, that to suppose two things in the same
point of space and time, is still a logical contradiction: not
because we have constructed the data of sense out of logic, but
because logic is dependent, as regards its application, on the nature
of these data. This instance illustrates, what I am anxious to make
plain, that my argument has not attempted to construct the living
wealth of sense-perception out of "bloodless categories," but only to
point out that, unless sense-perception contained a certain element,
these categories would be powerless to grapple with it.

=193.= How we are to account for the fortunate realization of these
requirements--whether by a pre-established harmony, by Darwinian
adaptation to our environment, by the subjectivity of the necessary
element in sense-perception, or by a fundamental identity and unity
between ourselves and the rest of reality--is a further question,
belonging rather to metaphysics than to our present line of argument.
The _à priori_, we have said throughout, is that which is necessary
for the possibility of experience, and in this we have a purely
logical criterion, giving results which only Logic and Epistemology
can prove or disprove. What is subjective in experience, on the
contrary, is primarily a question for psychology, and should be
decided on psychological grounds alone. When these two questions have
been separately answered, but not till then, we may frame theories as
to the connection of the _à priori_ and the subjective; to allow such
theories to influence our decision, on either of the two previous
questions, is liable, surely, to confuse the issue, and prevent a
clear discrimination between fundamentally different points of view.

=194.= I come now to the second question with which this chapter
has to deal, the question, namely: What are we to do with the
contradictions which obtruded themselves in Chapter III., whenever we
came to a point which seemed fundamental? I shall treat this question
briefly, as I have little to add to answers with which we are all
familiar. I have only to prove, first, that the contradictions are
inevitable, and therefore form no objection to my argument; secondly,
that the first step in removing them is to restore the notion of
matter, as that which, in the data of sense-perception, is localized
and interrelated in space.

=195.= The contradictions in space are an ancient theme--as ancient,
in fact, as Zeno's refutation of motion. They are, roughly, of two
kinds, though the two kinds cannot be sharply divided. There are
the contradictions inherent in the notion of the continuum, and the
contradictions which spring from the fact that space, while it must,
to be knowable, be pure relativity, must also, it would seem, since
it is immediately experienced, be something more than mere relations.
The first class of contradictions has been encountered more
frequently in this essay, and is also, I think, the more definite,
and the more important for our present purpose. I doubt, however,
whether the two classes are really distinct; for any continuum,
I believe, in which the elements are not data, but intellectual
constructions resulting from analysis, can be shown to have the same
relational and yet not wholly relational character as belongs to
space.

The three following contradictions, which I shall discuss
successively, seem to me the most prominent in a theory of Geometry.

(1) Though the parts of space are intuitively distinguished, no
conception is adequate to differentiate them. Hence arises a
vain search for elements, by which the differentiation could be
accomplished, and for a whole, of which the parts of space are to be
components. Thus we get the point, or zero extension, as the spatial
element, and an infinite regress or a vicious circle in the search
for a whole.

(2) All positions being relative, positions can only be defined by
their relations, _i.e._ by the straight lines or planes through
them; but straight lines and planes, being all qualitatively similar,
can only be defined by the positions they relate. Hence, again, we
get a vicious circle.

(3) Spatial figures must be regarded as relations. But a relation
is necessarily indivisible, while spatial figures are necessarily
divisible _ad infinitum_.

=196.= (1) _Points._ The antinomy of the point--which arises wherever
a continuum is given, and elements have to be sought in it--is
fundamental to Geometry. It has been given, perhaps unintentionally,
by Veronese as the first axiom, in the form: "There are different
points. All points are identical" (_op. cit._ p. 226). We saw, in
discussing projective Geometry, that straight lines and planes must
be regarded, on the one hand as relations between points, and on the
other hand as made up of points[190]. We saw again, in dealing with
measurement, how space must be regarded as infinitely divisible,
and yet as mere relativity. But what is divisible and consists of
parts, as space does, must lead at last, by continued analysis, to
a simple and unanalyzable part, as the unit of differentiation. For
whatever can be divided, and has parts, possesses some thinghood,
and must, therefore, contain two ultimate units, the whole namely,
and the smallest element possessing thinghood. But in space this is
notoriously not the case. After hypostatizing space, as Geometry
is compelled to do, the mind imperatively demands elements, and
insists on having them, whether possible or not. Of this demand,
all the geometrical applications of the infinitesimal calculus are
evidence[191]. But what sort of elements do we thus obtain? Analysis,
being unable to find any earlier halting-place, finds its elements
in points, that is, in zero quanta of space. Such a conception is
a palpable contradiction, only rendered tolerable by its necessity
and familiarity. A point must be spatial, otherwise it would not
fulfil the function of a spatial element; but again it must contain
no space, for any finite extension is capable of further analysis.
Points can never be given in intuition, which has no concern with the
infinitesimal: they are a purely conceptual construction, arising
out of the need of terms between which spatial relations can hold.
If space be more than relativity, spatial relations must involve
spatial relata; but no relata appear, until we have analyzed our
spatial data down to nothing. The contradictory notion of the point,
as a thing in space without spatial magnitude, is the only outcome
of our search for spatial relata. This _reductio ad absurdum_ surely
suffices, by itself, to prove the essential relativity of space.

=197.= Thus Geometry is forced, since it wishes to regard space
as independent, to hypostatize its abstractions, and therefore to
invent a self-contradictory notion as the spatial element. A similar
absurdity appears, even more obviously, in the notion of a whole of
space. The antinomy may, therefore, be stated thus: Space, as we have
seen throughout, must, if knowledge of it is to be possible, be mere
relativity; but it must also, if _independent_ knowledge of it, such
as Geometry seeks, is to be possible, be something more than mere
relativity, since it is divisible and has parts. But we saw, in Chap.
III., Section A (§ 133) that knowledge of a form of externality must
be logically independent of the particular matter filling the form.
How then are we to extricate ourselves from this dilemma?

The only way, I think, is, not to make Geometry dependent on
Physics, which we have seen to be erroneous[192], but to give every
geometrical proposition a certain reference to matter in general. And
at this point an important distinction must be made. We have hitherto
spoken of space as relational, and of spatial figures as relations.
But space, it would seem, is rather relativity than relations--itself
not a relation, it gives the bare possibility of relations between
diverse things[193]. As applied to a spatial figure, which can only
arise by a differentiation of space, and hence by the introduction
of some differentiating matter, the word relation is, perhaps, less
misleading than any other; as applied to empty undifferentiated
space, it seems by no means an accurate description.

But a bare possibility cannot exist, or be given in sense-perception!
What becomes, then, of the arguments of the first part of this
chapter? I reply, it is not empty space, but spatial figures, which
sense-perception reveals, and spatial figures, as we have just seen,
involve a differentiation of space, and therefore a reference to the
matter which is in space. It is spatial figures, also, and not empty
space, with which Geometry has to deal. The antinomy discussed above
arises then--so it would seem--from the attempt to deal with empty
space, rather than with spatial figures and the matter to which they
necessarily refer.

=198.= Let us see whether, by this change, we can overcome the
antinomy of the point. Spatial figures, we shall now say, are
relations between the matter which differentiates empty space. Their
divisibility, which seemed to contradict their relational character,
may be explained in two ways: first, as holding of the figures
considered as parts of empty space, which is itself not a relation;
second, as denoting the possibility of continuous change in the
relation expressed by the spatial figure. These two ways are, at
bottom, the same; for empty space is a possibility of relations, and
the figure, when viewed in connection with empty space, thus becomes
a _possible_ relation, with which other possible relations may be
contrasted or compared. But the second way of regarding divisibility
is the better way, since it introduces a reference to the matter
which differentiates empty space, without which, spatial figures,
and therefore Geometry, could not exist. It is empty space, then--so
we must conclude--which gives rise to the antinomy in question; for
empty space is a bare possibility of relations, undifferentiated and
homogeneous, and thus wholly destitute of parts or of thinghood.
To speak of parts of a possibility is nonsense; the parts and
differentiations arise only through a reference to the matter which
is differentiated in space.

=199.= But what nature must we ascribe to this matter, which is
to be involved in all geometrical propositions? In criticizing
Helmholtz (Chap. II. § 73), it may be remembered, we decided that
Geometry refers to a peculiar and abstract kind of matter, which is
not regarded as possessing any causal qualities, as exerting or as
subject to the action of forces. And this is the matter, I think,
which we require for the needs of the moment. Not that we affirm,
of course, that actual matter can be destitute of the properties
with which Physics is cognizant, but that we abstract from these
properties, as being irrelevant to Geometry. All that we require, for
our immediate purpose, is a subject of that diversity which space
renders possible, or terms for those relations by which empty space,
if space is to be studied at all, must be differentiated. But how
must a matter, which is to fulfil this function, be regarded?

Empty space, we have said, is a possibility of diversity in
relation, but spatial figures, with which Geometry necessarily
deals, are the actual relations rendered possible by empty space.
Our matter, therefore, must supply the terms for these relations.
It must be differentiated, since such differentiation, as we have
seen, is the special work of space. We must find, therefore, in
our matter, that unit of differentiation, or atom[194], which in
space we could not find. This atom must be simple, _i.e._ it must
contain no real diversity; it must be a _This_ not resolvable into
_Thises_. Being simple, it can contain no relations within itself,
and consequently, since spatial figures are mere relations, it cannot
appear as a spatial figure; for every spatial figure involves some
diversity of matter. But our atom must have spatial relations with
other atoms, since to supply terms for these relations is its only
function. It is also capable of having these relations, since it is
differentiated from other atoms. Hence we obtain an unextended term
for spatial relations, precisely of the kind we require. So long as
we sought this term without reference to anything more than space,
the self-contradictory notion of the point was the only outcome
of our search; but now that we allow a reference to the matter
differentiated by space, we find at once the term which was needed,
namely, a non-spatial simple element, with spatial relations to other
elements. To Geometry such a term will appear, owing to its spatial
relations, as a point; but the contradiction of the point, as we now
see, is a result only of the undue abstraction with which Geometry
deals.

=200.= (2) _The circle in the definition of straight lines and
planes._ This difficulty need not long detain us, since we have
already, with the material atom, broken through the relativity
which caused our circle. Straight lines, in the purely geometrical
procedure, are defined only by points, and points only by straight
lines. But points, now, are replaced by material atoms: the duality
of points and lines, therefore, has disappeared, and the straight
line may be defined as the spatial relation between two unextended
atoms. These atoms have spatial adjectives, derived from their
relations to other atoms; but they have no _intrinsic_ spatial
adjectives, such as could belong to them if they had extension or
figure. Thus straight lines and planes are the true spatial units,
and points result only from the attempt to find, within space, those
terms for spatial relations which exist only in a more than spatial
matter. Straight lines, planes and volumes are the spatial relations
between two, three or four unextended atoms, and points are a merely
convenient geometrical fiction, by which possible atoms are replaced.
For, since space, as we saw, is a possibility, Geometry deals not
with actually realized spatial relations, but with the whole scheme
of possible relations.

=201.= (3) _Space is at once relational and more than relational._
We have already touched on the question how far space is other than
relations, but as this question is quite fundamental, as _relation_
is an ambiguous and dangerous word, as I have made constant use of
the relativity of space without attempting to define a relation, it
will be necessary to discuss this antinomy at length.

=202.= Now for this discussion it is essential to distinguish clearly
between empty space and spatial figures. Empty space, as a form
of externality, is not actual relations, but the possibility of
relations: if we ascribe existential import to it, as the ground,
in reality, of all diversity in relation, we at once have space as
something not itself relations, though giving the possibility of all
relations. In this sense, space is to be distinguished from spatial
order. Spatial order, it may be said, presupposes space, as that in
which this order is possible. Thus Stumpf says[195]: "There is no
order or relation without a positive absolute content, underlying
it, and making it possible to order anything in this manner. Why and
how should we otherwise distinguish one order from another?... To
distinguish different orders from one another, we must everywhere
recognize a particular absolute content, in relation to which the
order takes place. And so space, too, is not a mere order, but just
that by which the spatial order, side-by-sideness (_Nebeneinander_)
distinguishes itself from the rest."

May we not, then, resolve the antinomy very simply, by a reference to
this ambiguity of space? Bradley contends (Appearance and Reality,
pp. 36-7) that, on the one hand, space has parts, and is therefore
not mere relations, while on the other hand, when we try to say what
these parts are, we find them after all to be mere relations. But
cannot the space which has parts be regarded as empty space, Stumpf's
absolute underlying content, which is not mere relations, while the
parts, in so far as they turn out to be mere relations, are those
relations which constitute spatial order, not empty space? If this
can be maintained, the antinomy no longer exists.

But such an explanation, though I believe it to be a first step
towards a solution, will, I fear, itself demand almost as much
explanation as the original difficulty. For the connection of empty
space with spatial order is itself a question full of difficulty, to
be answered only after much labour.

=203.= Let us consider what this empty space is. (I speak of "empty"
space without necessarily implying the absence of matter, but only
to denote a space which is not a mere order of material things.)
Stumpf regards it as given in sense; Kant, in the last two arguments
of his metaphysical deduction, argues that it is an intuition, not
a concept, and must be known before spatial order becomes possible.
I wish to maintain, on the contrary, that it is wholly conceptual;
that space is given only as spatial order; that spatial relations,
being given, appear as more than mere relations, and so become
hypostatized; that when hypostatized, the whole collection of
them is regarded as contained in empty space; but that this empty
space itself, if it means more than the logical possibility of
space-relations, is an unnecessary and self-contradictory assumption.
Let us begin by considering Kant's arguments on this point.

Leibnitz had affirmed that space was only relations, while Newton
had maintained the objective reality of absolute space. Kant adopted
a middle course: he asserted absolute space, but regarded it as
purely subjective. The assertion of absolute space is the object of
his second argument; for if space were mere relations between things,
it would necessarily disappear with the disappearance of the things
in it; but this the second argument denies[196]. Now spatial order
obviously does disappear with matter, but absolute or empty space may
be supposed to remain. It is this, then, which Kant is arguing about,
and it is this which he affirms to be a pure intuition, necessarily
presupposed by spatial order[197].

=204.= But can we agree in regarding empty space, the "infinite
given whole," as really given? Must we not, in spite of Kant's
argument, regard it as wholly conceptual? It is not required, in
the first place, by the argument of the first half of this chapter,
which required only that every _This_ of sense-perception should
be resolvable into _Thises_, and thus involved only an order among
_Thises_, not anything given originally without reference to them
at all. In the second place, Kant's two arguments[198] designed to
prove that empty space is not conceptual, are inadequate to their
purpose. The argument that the parts of space are not contained
_under_ it, but _in_ it, proves certainly that space is not a general
conception, of which spatial figures are the instances; but it by
no means follows that empty space is not a conception. Empty space
is undifferentiated and homogeneous; parts of space, or spatial
figures, arise only by reference to some differentiating matter, and
thus belong rather to spatial order than to empty space. If empty
space be the pre-condition of spatial order, we cannot expect it
to be connected with spatial relations as genus with species. But
empty space may nevertheless be a universal conception; it may be
related to spatial order as the state to the citizens. These are not
instances of the state, but are contained in it; they also, in a
sense, presuppose it, for a man can only become a citizen by being
related to other citizens in a state[199].

The uniqueness of space, again, seems hardly a valid argument for
its intuitional nature; to regard it as an argument implies, indeed,
that all conceptions are abstracted from a series of instances--a
view which has been criticized in Chapter II. (§ 77), and need not be
further discussed here[200]. There is no ground, therefore, in Kant's
two arguments for the intuitional nature of empty space, which can be
maintained against criticism.

=205.= Another ground for condemning empty space is to be found in
the mathematical antinomies. For it is no solution, as Lotze points
out (Metaphysik, Bk. II. Chap. I., § 106), to regard empty space as
purely subjective: contradictions in a necessary subjective intuition
form as great a difficulty as in anything else. But these antinomies
arise only in connection with empty space, not with spatial order
as an aggregate of relations. For only when space is regarded as
possessed of some thinghood, can a whole or a true element be
demanded. This we have seen already in connection with the Point.
When space is regarded, so far as it is valid, as only spatial order,
unbounded extension and infinite divisibility both disappear. What
is divided is not spatial relations, but matter; and if matter, as
we have seen that Geometry requires, consists of unextended atoms
with spatial relations, there is no reason to regard matter either as
infinitely divisible, or as consisting of atoms of finite extension.

=206.= But whence arises, on this view, the paradox that we cannot
but regard space as having more or less thinghood, and as divisible
_ad infinitum_? This must be explained, I think, as a psychological
illusion, unavoidably arising from the fact that spatial relations
are immediately presented. They thus have a peculiar psychical
quality, as immediate experiences, by which quality they can be
distinguished from time-relations or any other order in which things
may be arranged. To Stumpf, whose problem is psychological, such a
psychical quality would constitute an absolute underlying content,
and would fully justify his thesis; to us, however, whose problem is
epistemological, it would not do so, but would leave the _meaning_ of
the spatial element in sense-perception free from any implication
of an absolute or empty space[201]. May we not, then, abandon empty
space, and say: Spatial order consists of _felt_ relations, and _quâ_
felt has, for Psychology, an existence not wholly resolvable into
relations, and unavoidably _seeming_ to be more than mere relations.
But when we examine the information, as to space, which we derive
from sense-perception, we find ourselves plunged in contradictions,
as soon as we allow this information to consist of more than
relations. This leaves spatial order alone in the field, and reduces
empty space to a mere name for the logical possibility of spatial
relations.

=207.= The apparent divisibility of the relations which constitute
spatial order, then, may be explained in two ways, though these are
at bottom equivalent. We may take the relation as considered in
connection with empty space, in which case it becomes more than a
relation; but being falsely hypostatized, it appears as a complex
thing, necessarily composed of elements, which elements, however,
nowhere emerge until we analyze the pseudo-thing down to nothing,
and arrive at the point. In this sense, the divisibility of spatial
relations is an unavoidable illusion. Or again, we may take the
relation in connection with the material atoms it relates. In this
case, other atoms may be imagined, differently localized by different
spatial relations. If they are localized on the straight line joining
two of the original atoms, this straight line appears as divided by
them. But the original relation is not really divided: all that has
happened is, that two or more equivalent relations have replaced
it, as two compounded relations of father and son may replace the
equivalent relation of grandfather and grandson. These two ways of
viewing the apparent divisibility are equivalent: for empty space, in
so far as it is not illusion, is a name for the aggregate of possible
space-relations. To regard a figure in empty space as divided,
therefore, means, if it means anything, to regard two or more other
possible relations as substituted for it, which gives the second way
of viewing the question.

The same reference to matter, then, by which the antinomy of the
Point was solved, solves also the antinomy as to the relational
nature of space. Space, if it is to be freed from contradictions,
must be regarded exclusively as spatial order, as relations between
unextended material atoms. Empty space, which arises, by an
inevitable illusion, out of the spatial element in sense-perception,
may be regarded, if we wish to retain it, as the bare principle of
relativity, the bare logical possibility of relations between diverse
things. In this sense, empty space is wholly conceptual; spatial
order alone is immediately experienced.

=208.= But in what sense does spatial order consist of relations?
We have hitherto spoken of externality as a relation, and in a
sense such a manner of speaking is justified. Externality, when
predicated of anything, is an adjective of that thing, and implies
a reference to some other thing. To this extent, then, externality
is analogous to other relations; and only to this extent, in our
previous arguments, has it been regarded as a relation. But when we
take account of further qualities of relations, externality begins to
appear, not so much as a relation, but rather as a necessary aspect
or element in every relation. And this is borne out by the necessity,
for the existence of relations, of some given form of externality.

Every relation, we may say, involves a diversity between the
related terms, but also some unity. Mere diversity does not give
a ground for that interaction, and that interdependence, which a
relation requires. Mere unity leaves the terms identical, and thus
destroys the reference of one to another required for a relation.
Mere externality, taken in abstraction, gives only the element of
diversity required for a relation, and is thus more abstract than any
actual relation. But mere diversity does not give that indivisible
whole of which any actual relation must consist, and is thus, when
regarded abstractly, not subject to the restrictions of ordinary
relations.

But with mere diversity, we seem to have returned to empty space, and
abandoned spatial order. Mere diversity, surely, is either complete
or non-existent; degrees of diversity, or a quantitative measure of
it, are nonsense. We cannot, therefore, reduce spatial order to mere
diversity. Two things, if they occupy different positions in space,
are necessarily diverse, but are as necessarily something more;
otherwise spatial order becomes unmeaning.

Empty space, then, in the above sense of the possibility of spatial
relations, contains only one aspect of a relation, namely the aspect
of diversity; but spatial order, by its reference to matter, becomes
more concrete, and contains also the element of unity, arising out
of the connection of the different material atoms. Spatial order,
then, consists of relations in the ordinary sense; its merely
spatial element, however--if one may make such a distinction--the
element, that is, which can be abstracted from matter and regarded
as constituting empty space, is only one aspect of a relation, but
an aspect which, in the concrete, must be inseparably bound up
with the other aspect. Here, once more, we see the ground of the
contradictions in empty space, and the reason why spatial order is
free from these contradictions.


_Conclusion._

=209.= We have now completed our review of the foundations of
Geometry. It will be well, before we take leave of the subject,
briefly to review and recapitulate the results we have won.

In the first chapter, we watched the development of a branch of
Mathematics designed, at first, only to establish the logical
independence of Euclid's axiom of parallels, and the possibility
of a self-consistent Geometry which dispensed with it. We found
the further development of the subject entangled, for a while, in
philosophical controversy; having shown one axiom to be superfluous,
the geometers of the second period hoped to prove the same conclusion
of all the others, but failed to construct any system free from three
fundamental axioms. Being concerned with analytical and metrical
Geometry, they tended to regard Algebra as _à priori_, but held that
those properties of spatial magnitudes, which were not deducible from
the laws of Algebra, must be empirical. In all this, they aimed as
much at discrediting Kant as at advancing Mathematics. But with the
third period, the interest in Philosophy diminishes, the opposition
to Euclid becomes less marked, and most important of all, measurement
is no longer regarded as fundamental, and space is dealt with by
descriptive rather than quantitative methods. But nevertheless,
three axioms, substantially the same as those retained in the second
period, are still retained by all geometers.

In the second chapter, we endeavoured, by a criticism of some
geometrical philosophies, to prepare the ground for a constructive
theory of Geometry. We saw that Kant, in applying the argument of
the Transcendental Aesthetic to space, had gone too far, since
its logical scope extended only to some form of externality in
general. We saw that Riemann, Helmholtz and Erdmann, misled by the
quantitative bias, overlooked the qualitative substratum required
by all judgments of quantity, and thus mistook the direction in
which the necessary axioms of Geometry are to be found. We rejected,
also, Helmholtz's view that Geometry depends on Physics, because we
found that Physics must assume a knowledge of Geometry before it
can become possible. But we admitted, in Geometry, a reference to
matter--not, however, to matter as empirically known in Physics, but
to a more abstract matter, whose sole function is to appear in space,
and supply the terms for spatial relations. We admitted, however,
besides this, that all _actual_ measurement must be effected by means
of _actual_ matter, and is only empirically possible, through the
empirical knowledge of approximately rigid bodies. In criticizing
Lotze, we saw that the most important sense, in which non-Euclidean
spaces are possible, is a philosophical sense, namely, that they are
not condemned by any _à priori_ argument as to the necessity of space
for experience, and that consequently, if they are not affirmed,
this must be on empirical grounds alone. Lotze's strictures on the
mathematical procedure of Metageometry we found to be wholly due to
ignorance of the subject.

Proceeding, in the third chapter, to a constructive theory of
Geometry, we saw that projective Geometry, which has no reference to
quantity, is necessarily true of any form of externality. Its three
axioms--homogeneity, dimensions, and the straight line--were all
deduced from the conception of a form of externality, and, since some
such form is necessary to experience, were all declared _à priori_.
In metrical Geometry, on the contrary, we found an empirical element,
arising out of the alternatives of Euclidean and non-Euclidean space.
Three _à priori_ axioms, common to these spaces, and necessary
conditions of the possibility of measurement, still remained; these
were the axiom of Free Mobility, the axiom that space has a finite
integral number of dimensions, and the axiom of distance. Except for
the new idea of motion, these were found equivalent to the projective
triad, and thus necessarily true of any form of externality. But
the remaining axioms of Euclid--the axiom of three dimensions, the
axiom that two straight lines can never enclose a space, and the
axiom of parallels--were regarded as empirical laws, derived from the
investigation and measurement of our actual space, and true only, as
far as the last two are concerned, within the limits set by errors of
observation.

In the present chapter, we completed our proof of the apriority
of the projective and equivalent metrical axioms, by showing the
necessity, for experience, of some form of externality, given by
sensation or intuition, and not merely inferred from other data.
Without this, we said, a knowledge of diverse but interrelated
things, the corner-stone of all experience, would be impossible.
Finally, we discussed the contradictions arising out of the
relativity and continuity of space, and endeavoured to overcome
them by a reference to matter. This matter, we found, must consist
of unextended atoms, localized by their spatial relations, and
appearing, in Geometry, as points. But the non-spatial adjectives
of matter, we contended, are irrelevant to Geometry, and its
causal properties may be left out of account. To deal with the new
contradictions, involved in such a notion of matter, would demand
a fresh treatise, leading us, through Kinematics, into the domains
of Dynamics and Physics. But to discuss the special difficulties
of space is all that is possible in an essay on the Foundations of
Geometry.

FOOTNOTES:

[178] Compare, with the following paragraphs, the admirable
discussion in Mr Hobhouse's Theory of Knowledge (Methuen 1896), Part
I. Chapter II.

[179] I speak of sense-perception instead of sensation, so as not to
prejudge the issue as to the sensational nature of space.

[180] See Vaihinger's Commentar, II. pp. 86-7, 168-171.

[181] See Caird, Critical Philosophy of Kant, Vol. I. p. 287.

[182] Ursprung der Raumvorstellung, pp. 12-30.

[183] See the references in Vaihinger's Commentar, II. p. 76 ff.

[184] Commentar, II. p. 71 ff.

[185] _E.g._ by Caird, _op. cit._ Vol. I. p. 286.

[186] I have no wish to deny, however, that space is essential in the
subsequent distinction of Self and not-Self.

[187] See also Book I. Chap II. passim; especially p. 51 ff. and pp.
70-1.

[188] Logic, p. 51 ff.

[189] For the _This_, on such a hypothesis, has a purely temporal
complexity, and is not resolvable into coexisting _Thises_.

[190] Chapter III. Section A, (§ 131).

[191] Cf. Hannequin, Essai critique sur l'hypothèse des atomes, Paris
1895, Chap. I. Section III.; especially p. 43.

[192] See Chapter II. § 69 ff.

[193] See third antinomy below, § 201 ff.

[194] This atom, of course, must not be confounded with the atom of
Chemistry.

[195] Ursprung der Raumvorstellung, p. 15.

[196] See Vaihinger's Commentar, II. pp. 189-190.

[197] See ibid. p. 224 ff. for Kant's inconsistencies on this point.

[198] The fourth and fifth in the first edition, the third and fourth
in the second.

[199] Cf. Vaihinger's Commentar, II. p. 218.

[200] Cf. Vaihinger's Commentar, II. p. 207.

[201] Cf. James, Psychology, Vol. II., p. 148 ff.




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