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Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally
      printed in subscript. Letter subscripts are preceded by an
      underscore, like C_n.

(2) Characters following a carat (^) were printed in superscript.

(3) Side-notes were relocated to function as titles of their respective
      paragraphs.

(4) Macrons and breves above letters and dots below letters were not
      inserted.

(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek
      letters.

(6) The following typographical errors have been corrected:

    ARTICLE GEOFFREY: "... his history in chiefly one of quarrels, with
      the see of Canterbury, with the chancellor William Longchamp, with
      his half-brothers Richard and John, and especially with his canons
      at York." 'William' amended from 'Willian'.

    ARTICLE GEOLOGY: "... and at the same time greater appreciation has
      been shown of the signification and strength of the geological
      proofs of the high antiquity of our planet." 'strength' amended
      from 'stength'.

    ARTICLE GEOLOGY: "... it can be demonstrated that sometimes an inch
      or two of sediment might, on certain horizons, represent the
      deposit of an enormously longer period than a hundred or a thousand
      times the same amount of sediment on other horizons." 'might'
      amended from 'much'.

    ARTICLE GEOLOGY: "But no such extensive disturbance of the normal
      conditions of the distribution of life can take place without
      carrying with it many secondary effects, and setting in motion a
      wide cycle of change and of reaction in the animal and vegetable
      kingdoms." 'kingdoms' amended from 'kindgoms'.

    ARTICLE GEOMETRY: "The bases and altitudes of equal solid
      parallelepipeds are reciprocally proportional; and if the bases and
      altitudes be reciprocally proportional, the solid parallelepipeds
      are equal." 'are' amended from 'and'.

    ARTICLE GEOMETRY: "An alternative method of testing a relation is
      illustrated in the following example:--If A, B, C, D, E,
      F be six collinear points, then" 'following example:--' amended
      from 'example: following'.

    ARTICLE GEOMETRY: "3. In an hyperbolic involution any two conjugate
      points are harmonic conjugates with regard to the two foci." 'an'
      amended from 'a'.

    ARTICLE GEOMETRY: "If two lines, given by their projections,
      intersect, the intersection of their planes and the intersection of
      their elevations must lie in a line perpendicular to the axis,
      because they must be the projections of the point common to the two
      lines." 'planes' amended from 'plans'.

    ARTICLE GEOMETRY: "Where this is the case, if [alpha] be the measure
      of curvature, the linear element can be put into the form" 'if'
      amended from 'it'.

    ARTICLE GEOMETRY: "The development of the consequences of these
      metrical definitions is the subject of non-Euclidean geometry."
      'subject' amended from 'subjct'.




          ENCYCLOPAEDIA BRITANNICA

  A DICTIONARY OF ARTS, SCIENCES, LITERATURE
           AND GENERAL INFORMATION

              ELEVENTH EDITION


            VOLUME XI, SLICE VI

            GEODESY to GEOMETRY




ARTICLES IN THIS SLICE:


  GEODESY                          GEOFFROY, ÉTIENNE FRANÇOIS
  GEOFFREY (Martel)                GEOFFROY, JULIEN LOUIS
  GEOFFREY (Plantagenet)           GEOFFROY SAINT-HILAIRE, ÉTIENNE
  GEOFFREY (duke of Brittany)      GEOFFROY SAINT-HILAIRE, ISIDORE
  GEOFFREY (archbishop of York)    GEOGRAPHY
  GEOFFREY DE MONTBRAY             GEOID
  GEOFFREY OF MONMOUTH             GEOK-TEPE
  GEOFFREY OF PARIS                GEOLOGY
  GEOFFREY THE BAKER               GEOMETRICAL CONTINUITY
  GEOFFRIN, MARIE THÉRÈSE RODET    GEOMETRY




GEODESY (from the Gr. [Greek: gê], the earth, and [Greek: daiein], to
divide), the science of surveying (q.v.) extended to large tracts of
country, having in view not only the production of a system of maps of
very great accuracy, but the determination of the curvature of the
surface of the earth, and eventually of the figure and dimensions of the
earth. This last, indeed, may be the sole object in view, as was the
case in the operations conducted in Peru and in Lapland by the
celebrated French astronomers P. Bouguer, C.M. de la Condamine, P.L.M.
de Maupertuis, A.C. Clairault and others; and the measurement of the
meridian arc of France by P.F.A. Méchain and J.B.J. Delambre had for
its end the determination of the true length of the "metre" which was to
be the legal standard of length of France (see EARTH, FIGURE OF THE).

The basis of every extensive survey is an accurate triangulation, and
the operations of geodesy consist in the measurement, by theodolites, of
the angles of the triangles; the measurement of one or more sides of
these triangles on the ground; the determination by astronomical
observations of the azimuth of the whole network of triangles; the
determination of the actual position of the same on the surface of the
earth by observations, first for latitude at some of the stations, and
secondly for longitude; the determination of altitude for all stations.

For the computation, the points of the actual surface of the earth are
imagined as projected along their plumb lines on the mathematical
figure, which is given by the stationary sea-level, and the extension of
the sea through the continents by a system of imaginary canals. For many
purposes the mathematical surface is assumed to be a plane; in other
cases a sphere of radius 6371 kilometres (20,900,000 ft.). In the case
of extensive operations the surface must be considered as a compressed
ellipsoid of rotation, whose minor axis coincides with the earth's axis,
and whose compression, flattening, or ellipticity is about 1/298.


_Measurement of Base Lines._

  To determine by actual measurement on the ground the length of a side
  of one of the triangles ("base line"), wherefrom to infer the lengths
  of all the other sides in the triangulation, is not the least
  difficult operation of a trigonometrical survey. When the problem is
  stated thus--To determine the number of times that a certain standard
  or unit of length is contained between two finely marked points on the
  surface of the earth at a distance of some miles asunder, so that the
  error of the result may be pronounced to lie between certain very
  narrow limits,--then the question demands very serious consideration.
  The representation of the unit of length by means of the distance
  between two fine lines on the surface of a bar of metal at a certain
  temperature is never itself free from uncertainty and probable error,
  owing to the difficulty of knowing at any moment the precise
  temperature of the bar; and the transference of this unit, or a
  multiple of it, to a measuring bar will be affected not only with
  errors of observation, but with errors arising from uncertainty of
  temperature of both bars. If the measuring bar be not
  self-compensating for temperature, its expansion must be determined by
  very careful experiments. The thermometers required for this purpose
  must be very carefully studied, and their errors of division and index
  error determined.

  In order to avoid the difficulty in exactly determining the
  temperature of a bar by the mercury thermometer, F.W. Bessel
  introduced in 1834 near Königsberg a compound bar which constituted a
  metallic thermometer.[1] A zinc bar is laid on an iron bar two toises
  long, both bars being perfectly planed and in free contact, the zinc
  bar being slightly shorter and the two bars rigidly united at one end.
  As the temperature varies, the difference of the lengths of the bars,
  as perceived by the other end, also varies, and affords a quantitative
  correction for temperature variations, which is applied to reduce the
  length to standard temperature. During the measurement of the base
  line the bars were not allowed to come into contact, the interval
  being measured by the insertion of glass wedges. The results of the
  comparisons of four measuring rods with one another and with the
  standards were elaborately computed by the method of least-squares.
  The probable error of the measured length of 935 toises (about 6000
  ft.) has been estimated as 1/863500 or 1.2 µ (µ denoting a millionth).
  With this apparatus fourteen base lines were measured in Prussia and
  some neighbouring states; in these cases a somewhat higher degree of
  accuracy was obtained.

  The principal triangulation of Great Britain and Ireland has seven
  base lines: five have been measured by steel chains, and two, more
  exactly, by the compensation bars of General T.F. Colby, an apparatus
  introduced in 1827-1828 at Lough Foyle in Ireland. Ten base lines were
  measured in India in 1831-1869 by the same apparatus. This is a system
  of six compound-bars self-correcting for temperature. The bars may be
  thus described: Two bars, one of brass and the other of iron, are laid
  in parallelism side by side, firmly united at their centres, from
  which they may freely expand or contract; at the standard temperature
  they are of the same length. Let AB be one bar, A'B' the other; draw
  lines through the corresponding extremities AA' (to P) and BB' (to Q),
  and make A'P = B'Q, AA' being equal to BB'. If the ratio A'P/AP equals
  the ratio of the coefficients of expansion of the bars A'B' and AB,
  then, obviously, the distance PQ is constant (or nearly so). In the
  actual instrument P and Q are finely engraved dots 10 ft. apart. In
  practice the bars, when aligned, are not in contact, an interval of 6
  in. being allowed between each bar and its neighbour. This distance is
  accurately measured by an ingenious micrometrical arrangement
  constructed on exactly the same principle as the bars themselves.

  The last base line measured in India had a length of 8913 ft. In
  consequence of some suspicion as to the accuracy of the compensation
  apparatus, the measurement was repeated four times, the operations
  being conducted so as to determine the actual values of the probable
  errors of the apparatus. The direction of the line (which is at Cape
  Comorin) is north and south. In two of the measurements the brass
  component was to the west, in the others to the east; the differences
  between the individual measurements and the mean of the four were
  +0.0017, -0.0049, -0.0015, +0.0045 ft. These differences are very
  small; an elaborate investigation of all sources of error shows that
  the probable error of a base line in India is on the average ±2.8 µ.
  These compensation bars were also used by Sir Thomas Maclear in the
  measurement of the base line in his extension of Lacaille's arc at the
  Cape. The account of this operation will be found in a volume entitled
  _Verification and Extension of Lacaille's Arc of Meridian at the Cape
  of Good Hope_, by Sir Thomas Maclear, published in 1866. A
  rediscussion has been given by Sir David Gill in his _Report on the
  Geodetic Survey of South Africa, &c., 1896_.

  A very simple base apparatus was employed by W. Struve in his
  triangulations in Russia from 1817 to 1855. This consisted of four
  wrought-iron bars, each two toises (rather more than 13 ft.) long; one
  end of each bar is terminated in a small steel cylinder presenting a
  slightly convex surface for contact, the other end carries a contact
  lever rigidly connected with the bar. The shorter arm of the lever
  terminates below in a polished hemisphere, the upper and longer arm
  traversing a vertical divided arc. In measuring, the plane end of one
  bar is brought into contact with the short arm of the contact lever
  (pushed forward by a weak spring) of the next bar. Each bar has two
  thermometers, and a level for determining the inclination of the bar
  in measuring. The manner of transferring the end of a bar to the
  ground is simply this: under the end of the bar a stake is driven very
  firmly into the ground, carrying on its upper surface a disk, capable
  of movement in the direction of the measured line by means of
  slow-motion screws. A fine mark on this disk is brought vertically
  under the end of the bar by means of a theodolite which is planted at
  a distance of 25 ft. from the stake in a direction perpendicular to
  the base. Struve investigated for each base the probable errors of the
  measurement arising from each of these seven causes: Alignment,
  inclination, comparisons with standards, readings of index, personal
  errors, uncertainties of temperature, and the probable errors of
  adopted rates of expansion. He found that ±0.8 µ was the mean of the
  probable errors of the seven bases measured by him. The
  Austro-Hungarian apparatus is similar; the distance of the rods is
  measured by a slider, which rests on one of the ends of each rod.
  Twenty-two base lines were measured in 1840-1899.

  General Carlos Ibañez employed in 1858-1879, for the measurement of
  nine base lines in Spain, two apparatus similar to the apparatus
  previously employed by Porro in Italy; one is complicated, the other
  simplified. The first, an apparatus of the brothers Brunner of Paris,
  was a thermometric combination of two bars, one of platinum and one of
  brass, in length 4 metres, furnished with three levels and four
  thermometers. Suppose A, B, C three micrometer microscopes very firmly
  supported at intervals of 4 metres with their axes vertical, and
  aligned in the plane of the base line by means of a transit
  instrument, their micrometer screws being in the line of measurement.
  The measuring bar is brought under say A and B, and those micrometers
  read; the bar is then shifted and brought under B and C. By repetition
  of this process, the reading of a micrometer indicating the end of
  each position of the bar, the measurement is made.

  Quite similar apparatus (among others) has been employed by the French
  and Germans. Since, however, it only permitted a distance of about 300
  m. to be measured daily, Ibañez introduced a simplification; the
  measuring rod being made simply of steel, and provided with inlaid
  mercury thermometers. This apparatus was used in Switzerland for the
  measurement of three base lines. The accuracy is shown by the
  estimated probable errors: ±0.2 µ to ±0.8 µ. The distance measured
  daily amounts at least to 800 m.

  A greater daily distance can be measured with the same accuracy by
  means of Bessel's apparatus; this permits the ready measurement of
  2000 m. daily. For this, however, it is important to notice that a
  large staff and favourable ground are necessary. An important
  improvement was introduced by Edward Jäderin of Stockholm, who
  measures with stretched wires of about 24 metres long; these wires are
  about 1.65 mm. in diameter, and when in use are stretched by an
  accurate spring balance with a tension of 10 kg.[2] The nature of the
  ground has a very trifling effect on this method. The difficulty of
  temperature determinations is removed by employing wires made of
  invar, an alloy of steel (64%) and nickel (36%) which has practically
  no linear expansion for small thermal changes at ordinary
  temperatures; this alloy was discovered in 1896 by Benôit and
  Guillaume of the International Bureau of Weights and Measures at
  Breteuil. Apparently the future of base-line measurements rests with
  the invar wires of the Jäderin apparatus; next comes Porro's apparatus
  with invar bars 4 to 5 metres long.

  Results have been obtained in the United States, of great importance
  in view of their accuracy, rapidity of determination and economy. For
  the measurement of the arc of meridian in longitude 98° E., in 1900,
  nine base lines of a total length of 69.2 km. were measured in six
  months. The total cost of one base was $1231. At the beginning and at
  the end of the field-season a distance of exactly 100 m. was measured
  with R.S. Woodward's "5-m. ice-bar" (invented in 1891); by means of
  the remeasurement of this length the standardization of the apparatus
  was done under the same conditions as existed in the case of the base
  measurements. For the measurements there were employed two steel tapes
  of 100 m. long, provided with supports at distances of 25 m., two of
  50 m., and the duplex apparatus of Eimbeck, consisting of four 5-m.
  rods. Each base was divided into sections of about 1000 m.; one of
  these, the "test kilometre," was measured with all the five apparatus,
  the others only with two apparatus, mostly tapes. The probable error
  was about ±0.8 µ, and the day's work a distance of about 2000 m. Each
  of the four rods of the duplex apparatus consists of two bars of brass
  and steel. Mercury thermometers are inserted in both bars; these serve
  for the measurement of the length of the base lines by each of the
  bars, as they are brought into their consecutive positions, the
  contact being made by an elastic-sliding contact. The length of the
  base lines may be calculated for each bar only, and also by the
  supposition that both bars have the same temperature. The apparatus
  thus affords three sets of results, which mutually control themselves,
  and the contact adjustments permit rapid work. The same device has
  been applied to the older bimetallic-compensating apparatus of
  Bache-Würdemann (six bases, 1847-1857) and of Schott. There was also
  employed a single rod bimetallic apparatus on F. Porro's principle,
  constructed by the brothers Repsold for some base lines. Excellent
  results have been more recently obtained with invar tapes.

  The following results show the lengths of the same German base lines
  as measured by different apparatus:

                                                 metres.
    Base at Berlin   1864 Apparatus of Bessel   2336·3920
        "     "      1880     "        Brunner      ·3924
    Base at Strehlen 1854     "        Bessel   2762·5824
        "     "      1879     "        Brunner      ·5852
    Old base at Bonn 1847     "        Bessel   2133·9095
        "     "      1892     "          "          ·9097
    New base at Bonn 1892     "          "      2512·9612
        "     "      1892     "        Brunner      ·9696

  It is necessary that the altitude above the level of the sea of every
  part of a base line be ascertained by spirit levelling, in order that
  the measured length may be reduced to what it would have been had the
  measurement been made on the surface of the sea, produced in
  imagination. Thus if l be the length of a measuring bar, h its height
  at any given position in the measurement, r the radius of the earth,
  then the length radially projected on to the level of the sea is l(1 -
  h/r). In the Salisbury Plain base line the reduction to the level of
  the sea is -0.6294 ft.

  The total number of base lines measured in Europe up to the present
  time is about one hundred and ten, nineteen of which do not exceed in
  length 2500 metres, or about 1½ miles, and three--one in France, the
  others in Bavaria--exceed 19,000 metres. The question has been
  frequently discussed whether or not the advantage of a long base is
  sufficiently great to warrant the expenditure of time that it
  requires, or whether as much precision is not obtainable in the end by
  careful triangulation from a short base. But the answer cannot be
  given generally; it must depend on the circumstances of each
  particular case. With Jäderin's apparatus, provided with invar wires,
  bases of 20 to 30 km. long are obtained without difficulty.

  [Illustration: FIG. 1.]

  In working away from a base line ab, stations c, d, e, f are carefully
  selected so as to obtain from well-shaped triangles gradually
  increasing sides. Before, however, finally leaving the base line, it
  is usual to verify it by triangulation thus: during the measurement
  two or more points, as p, q (fig. 1), are marked in the base in
  positions such that the lengths of the different segments of the line
  are known; then, taking suitable external stations, as h, k, the
  angles of the triangles bhp, phq, hqk, kqa are measured. From these
  angles can be computed the ratios of the segments, which must agree,
  if all operations are correctly performed, with the ratios resulting
  from the measures. Leaving the base line, the sides increase up to
  10, 30 or 50 miles occasionally, but seldom reaching 100 miles. The
  triangulation points may either be natural objects presenting
  themselves in suitable positions, such as church towers; or they may
  be objects specially constructed in stone or wood on mountain tops or
  other prominent ground. In every case it is necessary that the precise
  centre of the station be marked by some permanent mark. In India no
  expense is spared in making permanent the principal trigonometrical
  stations--costly towers in masonry being erected. It is essential that
  every trigonometrical station shall present a fine object for
  observation from surrounding stations.


  _Horizontal Angles._

  In placing the theodolite over a station to be observed from, the
  first point to be attended to is that it shall rest upon a perfectly
  solid foundation. The method of obtaining this desideratum must depend
  entirely on the nature of the ground; the instrument must if possible
  be supported on rock, or if that be impossible a solid foundation must
  be obtained by digging. When the theodolite is required to be raised
  above the surface of the ground in order to command particular points,
  it is necessary to build two scaffolds,--the outer one to carry the
  observatory, the inner one to carry the instrument,--and these two
  edifices must have no point of contact. Many cases of high scaffolding
  have occurred on the English Ordnance Survey, as for instance at
  Thaxted church, where the tower, 80 ft. high, is surmounted by a spire
  of 90 ft. The scaffold for the observatory was carried from the base
  to the top of the spire; that for the instrument was raised from a
  point of the spire 140 ft. above the ground, having its bearing upon
  timbers passing through the spire at that height. Thus the instrument,
  at a height of 178 ft. above the ground, was insulated, and not
  affected by the action of the wind on the observatory.

  At every station it is necessary to examine and correct the
  adjustments of the theodolite, which are these: the line of
  collimation of the telescope must be perpendicular to its axis of
  rotation; this axis perpendicular to the vertical axis of the
  instrument; and the latter perpendicular to the plane of the horizon.
  The micrometer microscopes must also measure correct quantities on the
  divided circle or circles. The method of observing is this. Let A, B,
  C ... be the stations to be observed taken in order of azimuth; the
  telescope is first directed to A and the cross-hairs of the telescope
  made to bisect the object presented by A, then the microscopes or
  verniers of the horizontal circle (also of the vertical circle if
  necessary) are read and recorded. The telescope is then turned to B,
  which is observed in the same manner; then C and the other stations.
  Coming round by continuous motion to A, it is again observed, and the
  agreement of this second reading with the first is some test of the
  stability of the instrument. In taking this round of angles--or "arc,"
  as it is called on the Ordnance Survey--it is desirable that the
  interval of time between the first and second observations of A should
  be as small as may be consistent with due care. Before taking the next
  arc the horizontal circle is moved through 20° or 30°; thus a
  different set of divisions of the circle is used in each arc, which
  tends to eliminate the errors of division.

  It is very desirable that all arcs at a station should contain one
  point in common, to which all angular measurements are thus
  referred,--the observations on each arc commencing and ending with
  this point, which is on the Ordnance Survey called the "referring
  object." It is usual for this purpose to select, from among the points
  which have to be observed, that one which affords the best object for
  precise observation. For mountain tops a "referring object" is
  constructed of two rectangular plates of metal in the same vertical
  plane, their edges parallel and placed at such a distance apart that
  the light of the sky seen through appears as a vertical line about 10"
  in width. The best distance for this object is from 1 to 2 miles.

  This method seems at first sight very advantageous; but if, however,
  it be desired to attain the highest accuracy, it is better, as shown
  by General Schreiber of Berlin in 1878, to measure only single angles,
  and as many of these as possible between the directions to be
  determined. Division-errors are thus more perfectly eliminated, and
  errors due to the variation in the stability, &c., of the instruments
  are diminished. This method is rapidly gaining precedence.

  The theodolites used in geodesy vary in pattern and in size--the
  horizontal circles ranging from 10 in. to 36 in. in diameter. In
  Ramsden's 36-in. theodolite the telescope has a focal length of 36 in.
  and an aperture of 2.5 in., the ordinarily used magnifying power being
  54; this last, however, can of course be changed at the requirements
  of the observer or of the weather. The probable error of a single
  observation of a fine object with this theodolite is about 0".2. Fig.
  2 represents an altazimuth theodolite of an improved pattern used on
  the Ordnance Survey. The horizontal circle of 14-in. diameter is read
  by three micrometer microscopes; the vertical circle has a diameter of
  12 in., and is read by two microscopes. In the great trigonometrical
  survey of India the theodolites used in the more important parts of
  the work have been of 2 and 3 ft. diameter--the circle read by five
  equidistant microscopes. Every angle is measured twice in each
  position of the zero of the horizontal circle, of which there are
  generally ten; the entire number of measures of an angle is never
  less than 20. An examination of 1407 angles showed that the probable
  error of an observed angle is on the average ± 0".28.

  For the observations of very distant stations it is usual to employ a
  heliotrope (from the Gr. [Greek: hêlios], sun; [Greek: tropos], a
  turn), invented by Gauss at Göttingen in 1821. In its simplest form
  this is a plane mirror, 4, 6, or 8 in. in diameter, capable of
  rotation round a horizontal and a vertical axis. This mirror is placed
  at the station to be observed, and in fine weather it is kept so
  directed that the rays of the sun reflected by it strike the distant
  observing telescope. To the observer the heliotrope presents the
  appearance of a star of the first or second magnitude, and is
  generally a pleasant object for observing.

  Observations at night, with the aid of light-signals, have been
  repeatedly made, and with good results, particularly in France by
  General François Perrier, and more recently in the United States by
  the Coast and Geodetic Survey; the signal employed being an acetylene
  bicycle-lamp, with a lens 5 in. in diameter. Particularly noteworthy
  are the trigonometrical connexions of Spain and Algeria, which were
  carried out in 1879 by Generals Ibañez and Perrier (over a distance of
  270 km.), of Sicily and Malta in 1900, and of the islands of Elba and
  Sardinia in 1902 by Dr Guarducci (over distances up to 230 km.); in
  these cases artificial light was employed: in the first case electric
  light and in the two others acetylene lamps.

  [Illustration: FIG. 2.--Altazimuth Theodolite.]


  _Astronomical Observations._

  The direction of the meridian is determined either by a theodolite or
  a portable transit instrument. In the former case the operation
  consists in observing the angle between a terrestrial
  object--generally a mark specially erected and capable of illumination
  at night--and a close circumpolar star at its greatest eastern or
  western azimuth, or, at any rate, when very near that position. If the
  observation be made t minutes of time before or after the time of
  greatest azimuth, the azimuth then will differ from its maximum value
  by (450t)² sin 1" sin 2[delta]/ sin z, in seconds of angle, omitting
  smaller terms, [delta] being the star's declination and z its zenith
  distance. The collimation and level errors are very carefully
  determined before and after these observations, and it is usual to
  arrange the observations by the reversal of the telescope so that
  collimation error shall disappear. If b, c be the level and
  collimation errors, the correction to the circle reading is b cot z ±
  c cosec z, b being positive when the west end of the axis is high. It
  is clear that any uncertainty as to the real state of the level will
  produce a corresponding uncertainty in the resulting value of the
  azimuth,--an uncertainty which increases with the latitude and is very
  large in high latitudes. This may be partly remedied by observing in
  connexion with the star its reflection in mercury. In determining the
  value of "one division" of a level tube, it is necessary to bear in
  mind that in some the value varies considerably with the temperature.
  By experiments on the level of Ramsden's 3-foot theodolite, it was
  found that though at the ordinary temperature of 66° the value of a
  division was about one second, yet at 32° it was about five seconds.

  In a very excellent portable transit used on the Ordnance Survey, the
  uprights carrying the telescope are constructed of mahogany, each
  upright being built of several pieces glued and screwed together; the
  base, which is a solid and heavy plate of iron, carries a reversing
  apparatus for lifting the telescope out of its bearings, reversing it
  and letting it down again. Thus is avoided the change of temperature
  which the telescope would incur by being lifted by the hands of the
  observer. Another form of transit is the German diagonal form, in
  which the rays of light after passing through the object-glass are
  turned by a total reflection prism through one of the transverse arms
  of the telescope, at the extremity of which arm is the eye-piece. The
  unused half of the ordinary telescope being cut away is replaced by a
  counterpoise. In this instrument there is the advantage that the
  observer without moving the position of his eye commands the whole
  meridian, and that the level may remain on the pivots whatever be the
  elevation of the telescope. But there is the disadvantage that the
  flexure of the transverse axis causes a variable collimation error
  depending on the zenith distance of the star to which it is directed;
  and moreover it has been found that in some cases the personal error
  of an observer is not the same in the two positions of the telescope.

  To determine the direction of the meridian, it is well to erect two
  marks at nearly equal angular distances on either side of the north
  meridian line, so that the pole star crosses the vertical of each mark
  a short time before and after attaining its greatest eastern and
  western azimuths.

  If now the instrument, perfectly levelled, is adjusted to have its
  centre wire on one of the marks, then when elevated to the star, the
  star will traverse the wire, and its exact position in the field at
  any moment can be measured by the micrometer wire. Alternate
  observations of the star and the terrestrial mark, combined with
  careful level readings and reversals of the instrument, will enable
  one, even with only one mark, to determine the direction of the
  meridian in the course of an hour with a probable error of less than a
  second. The second mark enables one to complete the station more
  rapidly and gives a check upon the work. As an instance, at Findlay
  Seat, in latitude 57° 35', the resulting azimuths of the two marks
  were 177° 45' 37".29 ± 0".20 and 182° 17' 15".61 ± 0".13, while the
  angle between the two marks directly measured by a theodolite was
  found to be 4° 31' 37".43 ± 0".23.

  [Illustration: FIG. 3.]

  We now come to the consideration of the determination of time with the
  transit instrument. Let fig. 3 represent the sphere stereographically
  projected on the plane of the horizon,--ns being the meridian, we the
  prime vertical, Z, P the zenith and the pole. Let p be the point in
  which the production of the axis of the instrument meets the celestial
  sphere, S the position of a star when observed on a wire whose
  distance from the collimation centre is c. Let a be the azimuthal
  deviation, namely, the angle wZp, b the level error so that Zp = 90° -
  b. Let also the hour angle corresponding to p be 90° - n, and the
  declination of the same = m, the star's declination being [delta], and
  the latitude [phi]. Then to find the hour angle ZPS = [tau] of the
  star when observed, in the triangles pPS, pPZ we have, since pPS = 90
  + [tau] - n,

         -Sin c = sin m sin [delta] + cos m cos [delta] sin (n - [tau]),
          Sin m = sin b sin [phi]   - cos b cos [phi] sin a,
    Cos m sin n = sin b cos [phi]   + cos b sin [phi] sin a.

  And these equations solve the problem, however large be the errors of
  the instrument. Supposing, as usual, a, b, m, n to be small, we have
  at once [tau] = n + c sec [delta] + m tan [delta], which is the
  correction to the observed time of transit. Or, eliminating m and n by
  means of the second and third equations, and putting z for the zenith
  distance of the star, t for the observed time of transit, the
  corrected time is t + (a sin z + b cos z + c) / cos [delta]. Another
  very convenient form for stars near the zenith is [tau] = b sec [phi]
  + c sec [delta] + m (tan [delta] - tan [phi]).

  Suppose that in commencing to observe at a station the error of the
  chronometer is not known; then having secured for the instrument a
  very solid foundation, removed as far as possible level and
  collimation errors, and placed it by estimation nearly in the
  meridian, let two stars differing considerably in declination be
  observed--the instrument not being reversed between them. From these
  two stars, neither of which should be a close circumpolar star, a good
  approximation to the chronometer error can be obtained; thus let
  [epsilon]1, [epsilon]2, be the apparent clock errors given by these
  stars if [delta]1, [delta]2 be their declinations the real error is

    [epsilon] = [epsilon]1 + ([epsilon]1 - [epsilon]2)
      (tan [phi] - tan [delta]1) / (tan [delta]1 - tan [delta]2).

  Of course this is still only approximate, but it will enable the
  observer (who by the help of a table of natural tangents can compute
  [epsilon] in a few minutes) to find the meridian by placing at the
  proper time, which he now knows approximately, the centre wire of his
  instrument on the first star that passes--not near the zenith.

  The transit instrument is always reversed at least once in the course
  of an evening's observing, the level being frequently read and
  recorded. It is necessary in most instruments to add a correction for
  the difference in size of the pivots.

  The transit instrument is also used in the prime vertical for the
  determination of latitudes. In the preceding figure let q be the point
  in which the northern extremity of the axis of the instrument produced
  meets the celestial sphere. Let nZq be the azimuthal deviation = a,
  and b being the level error, Zq = 90° - b; let also nPq = [tau] and Pq
  = [psi]. Let S' be the position of a star when observed on a wire
  whose distance from the collimation centre is c, positive when to the
  south, and let h be the observed hour angle of the star, viz. ZPS'.
  Then the triangles qPS', gPZ give

    -Sin c = sin [delta] cos [psi] - cos [delta] sin [psi] cos (h + [tau]),
     Cos [psi] = sin b sin [phi] + cos b cos [phi] cos a,
     Sin [psi] sin [tau] = cos b sin a.

  Now when a and b are very small, we see from the last two equations
  that [psi] = [phi] - b, a = [tau] sin [psi], and if we calculate
  [phi]' by the formula cot [phi]' = cot [delta] cos h, the first
  equation leads us to this result--

    [phi] = [phi]' + (a sin z + b cos z + c)/cos z,

  the correction for instrumental error being very similar to that
  applied to the observed time of transit in the case of meridian
  observations. When a is not very small and z is small, the formulae
  required are more complicated.

  [Illustration: FIG. 4.--Zenith Telescope constructed for the
  International Stations at Mizusawa, Carloforte, Gaithersburg and
  Ukiah, by Hermann Wanschaff, Berlin.]

  The method of determining latitude by transits in the prime vertical
  has the disadvantage of being a somewhat slow process, and of
  requiring a very precise knowledge of the time, a disadvantage from
  which the zenith telescope is free. In principle this instrument is
  based on the proposition that when the meridian zenith distances of
  two stars at their upper culminations--one being to the north and the
  other to the south of the zenith--are equal, the latitude is the mean
  of their declinations; or, if the zenith distance of a star
  culminating to the south of the zenith be Z, its declination being
  [delta], and that of another culminating to the north with zenith
  distance Z' and declination [delta]', then clearly the latitude is
  ½([delta] + [delta]') + ½(Z - Z'). Now the zenith telescope does away
  with the divided circle, and substitutes the measurement
  micrometrically of the quantity Z' - Z.

  In fig. 4 is shown a zenith telescope by H. Wanschaff of Berlin, which
  is the type used (according to the Central Bureau at Potsdam) since
  about 1890 for the determination of the variations of latitude due to
  different, but as yet imperfectly understood, influences. The
  instrument is supported on a strong tripod, fitted with levelling
  screws; to this tripod is fixed the azimuth circle and a long vertical
  steel axis. Fitting on this axis is a hollow axis which carries on its
  upper end a short transverse horizontal axis with a level. This latter
  carries the telescope, which, supported at the centre of its length,
  is free to rotate in a vertical plane. The telescope is thus mounted
  eccentrically with respect to the vertical axis around which it
  revolves. Two extremely sensitive levels are attached to the
  telescope, which latter carries a micrometer in its eye-piece, with a
  screw of long range for measuring differences of zenith distance. Two
  levels are employed for controlling and increasing the accuracy. For
  this instrument stars are selected in pairs, passing north and south
  of the zenith, culminating within a few minutes of time and within
  about twenty minutes (angular) of zenith distance of each other. When
  a pair of stars is to be observed, the telescope is set to the mean of
  the zenith distances and in the plane of the meridian. The first star
  on passing the central meridional wire is bisected by the micrometer;
  then the telescope is rotated very carefully through 180° round the
  vertical axis, and the second star on passing through the field is
  bisected by the micrometer on the centre wire. The micrometer has thus
  measured the difference of the zenith distances, and the calculation
  to get the latitude is most simple. Of course it is necessary to read
  the level, and the observations are not necessarily confined to the
  centre wire. In fact if n, s be the north and south readings of the
  level for the south star, n', s' the same for the north star, l the
  value of one division of the level, m the value of one division of the
  micrometer, r, r' the refraction corrections, µ, µ' the micrometer
  readings of the south and north star, the micrometer being supposed to
  read from the zenith, then, supposing the observation made on the
  centre wire,--

    [phi] = ½([delta] + [delta]') + ½(µ - µ')m + ¼(n + n' - s - s')l +
      ½(r - r').

  It is of course of the highest importance that the value m of the
  screw be well determined. This is done most effectually by observing
  the vertical movement of a close circumpolar star when at its greatest
  azimuth.

  In a single night with this instrument a very accurate result, say
  with a probable error of about 0".2, could be obtained for latitude
  from, say, twenty pair of stars; but when the latitude is required to
  be obtained with the highest possible precision, two nights at least
  are necessary. The weak point of the zenith telescope lies in the
  circumstance that its requirements prevent the selection of stars
  whose positions are well fixed; very frequently it is necessary to
  have the declinations of the stars selected for this instrument
  specially observed at fixed observatories. The zenith telescope is
  made in various sizes from 30 to 54 in. in focal length; a 30-in.
  telescope is sufficient for the highest purposes and is very portable.
  The net observation probable-error for one pair of stars is only
  ±0".1.

  The zenith telescope is a particularly pleasant instrument to work
  with, and an observer has been known (a sergeant of Royal Engineers,
  on one occasion) to take every star in his list during eleven hours on
  a stretch, namely, from 6 o'clock P.M. until 5 A.M., and this on a
  very cold November night on one of the highest points of the
  Grampians. Observers accustomed to geodetic operations attain
  considerable powers of endurance. Shortly after the commencement of
  the observations on one of the hills in the Isle of Skye a storm
  carried away the wooden houses of the men and left the observatory
  roofless. Three observatory roofs were subsequently demolished, and
  for some time the observatory was used without a roof, being filled
  with snow every night and emptied every morning. Quite different,
  however, was the experience of the same party when on the top of Ben
  Nevis, 4406 ft. high. For about a fortnight the state of the
  atmosphere was unusually calm, so much so, that a lighted candle could
  often be carried between the tents of the men and the observatory,
  whilst at the foot of the hill the weather was wild and stormy.

  The determination of the difference of longitude between two stations
  A and B resolves itself into the determination of the local time at
  each of the stations, and the comparison by signals of the clocks at A
  and B. Whenever telegraphic lines are available these comparisons are
  made by telegraphy. A small and delicately-made apparatus introduced
  into the mechanism of an astronomical clock or chronometer breaks or
  closes by the action of the clock an electric circuit every second. In
  order to record the minutes as well as seconds, one second in each
  minute, namely that numbered 0 or 60, is omitted. The seconds are
  recorded on a chronograph, which consists of a cylinder revolving
  uniformly at the rate of one revolution per minute covered with white
  paper, on which a pen having a slow movement in the direction of the
  axis of the cylinder describes a continuous spiral. This pen is
  deflected through the agency of an electromagnet every second, and
  thus the seconds of the clock are recorded on the chronograph by
  offsets from the spiral curve. An observer having his hand on a
  contact key in the same circuit can record in the same manner his
  observed times of transits of stars. The method of determination of
  difference of longitude is, therefore, virtually as follows. After the
  necessary observations for instrumental corrections, which are
  recorded only at the station of observation, the clock at A is put in
  connexion with the circuit so as to write on both chronographs,
  namely, that at A and that at B. Then the clock at B is made to write
  on both chronographs. It is clear that by this double operation one
  can eliminate the effect of the small interval of time consumed in the
  transmission of signals, for the difference of longitude obtained from
  the one chronograph will be in excess by as much as that obtained from
  the other will be in defect. The determination of the personal errors
  of the observers in this delicate operation is a matter of the
  greatest importance, as therein lies probably the chief source of
  residual error.

  These errors can nevertheless be almost entirely avoided by using the
  impersonal micrometer of Dr Repsold (Hamburg, 1889). In this device
  there is a movable micrometer wire which is brought by hand into
  coincidence with the star and moved along with it; at fixed points
  there are electrical contacts, which replace the fixed wires.
  Experiments at the Geodetic Institute and Central Bureau at Potsdam in
  1891 gave the following personal equations in the case of four
  observers:--

          Older Procedure.   New Procedure.

    A-B      -0^s.108          -0^s.004
    A-G      -0^s.314          -0^s.035
    A-S      -0^s.184          -0^s.027
    B-G      -0^s.225          +0^s.013
    B-S      -0^s.086          -0^s.023
    G-S      +0^s.109          -0^s.006

  These results show that in the later method the personal equation is
  small and not so variable; and consequently the repetition of
  longitude determinations with exchanged observers and apparatus
  entirely eliminates the constant errors, the probable error of such
  determinations on ten nights being scarcely ±0^s.01.


  _Calculation of Triangulation._

  The surface of Great Britain and Ireland is uniformly covered by
  triangulation, of which the sides are of various lengths from 10 to
  111 miles. The largest triangle has one angle at Snowdon in Wales,
  another on Slieve Donard in Ireland, and a third at Scaw Fell in
  Cumberland; each side is over a hundred miles and the spherical excess
  is 64". The more ordinary method of triangulation is, however, that of
  chains of triangles, in the direction of the meridian and
  perpendicular thereto. The principal triangulations of France, Spain,
  Austria and India are so arranged. Oblique chains of triangles are
  formed in Italy, Sweden and Norway, also in Germany and Russia, and in
  the United States. Chains are composed sometimes merely of consecutive
  plain triangles; sometimes, and more frequently in India, of
  combinations of triangles forming consecutive polygonal figures. In
  this method of triangulating, the sides of the triangles are generally
  from 20 to 30 miles in length--seldom exceeding 40.

  The inevitable errors of observation, which are inseparable from all
  angular as well as other measurements, introduce a great difficulty
  into the calculation of the sides of a triangulation. Starting from a
  given base in order to get a required distance, it may generally be
  obtained in several different ways--that is, by using different sets
  of triangles. The results will certainly differ one from another, and
  probably no two will agree. The experience of the computer will then
  come to his aid, and enable him to say which is the most trustworthy
  result; but no experience or ability will carry him through a large
  network of triangles with anything like assurance. The only way to
  obtain trustworthy results is to employ the method of least squares.
  We cannot here give any illustration of this method as applied to
  general triangulation, for it is most laborious, even for the simplest
  cases.

  Three stations, projected on the surface of the sea, give a spherical
  or spheroidal triangle according to the adoption of the sphere or the
  ellipsoid as the form of the surface. A spheroidal triangle differs
  from a spherical triangle, not only in that the curvatures of the
  sides are different one from another, but more especially in this
  that, while in the spherical triangle the normals to the surface at
  the angular points meet at the centre of the sphere, in the spheroidal
  triangle the normals at the angles A, B, C meet the axis of revolution
  of the spheroid in three different points, which we may designate
  [alpha], ß, [gamma] respectively. Now the angle A of the triangle as
  measured by a theodolite is the inclination of the planes BA[alpha]
  and CA[alpha], and the angle at B is that contained by the planes ABß
  and CBß. But the planes AB[alpha] and ABß containing the line AB in
  common cut the surface in two distinct plane curves. In order,
  therefore, that a spheroidal triangle may be exactly defined, it is
  necessary that the nature of the lines joining the three vertices be
  stated. In a mathematical point of view the most natural definition is
  that the sides be geodetic or shortest lines. C.C.G. Andrae, of
  Copenhagen, has also shown that other lines give a less convenient
  computation.

  K.F. Gauss, in his treatise, _Disquisitiones generales circa
  superficies curvas_, entered fully into the subject of geodetic (or
  geodesic) triangles, and investigated expressions for the angles of a
  geodetic triangle whose sides are given, not certainly finite
  expressions, but approximations inclusive of small quantities of the
  fourth order, the side of the triangle or its ratio to the radius of
  the nearly spherical surface being a small quantity of the first
  order. The terms of the fourth order, as given by Gauss for any
  surface in general, are very complicated even when the surface is a
  spheroid. If we retain small quantities of the second order only, and
  put [A], [B], [C] for the angles of the geodetic triangle, while A, B,
  C are those of a plane triangle having sides equal respectively to
  those of the geodetic triangle, then, [sigma] being the area of the
  plane triangle and [a], [b], [c] the measures of curvature at the
  angular points,

    [A] = A + [sigma](2[a] + [b] + [c])/12,
    [B] = B + [sigma]([a] + 2[b] + [c])/12,
    [C] = C + [sigma]([a] + [b] + 2[c])/12.

  For the sphere [a] = [b] = [r], and making this simplification, we
  obtain the theorem previously given by A.M. Legendre. With the terms
  of the fourth order, we have (after Andrae):

              [epsilon]   [sigma]   /m² - a²    [a] - k \
    [A] - A = --------- + -------k ( -------k + -------  ),
                  3          3      \  20          4k   /

              [epsilon]   [sigma]   /m² - b²    [b] - k \
    [B] - B = --------- + -------k ( -------k + -------- ),
                  3          3      \  20          4k   /

              [epsilon]   [sigma]   /m² - c²    [c] - k \
    [C] - C = --------- + -------k ( -------k + -------- ),
                  3          3      \  20          4k   /

  in which [epsilon] = [sigma] k {1 + (m²k / 8)}, 3m² = a² + b² + c², 3k
  = [a] + [b] + [c]. For the ellipsoid of rotation the measure of
  curvature is equal to 1 / [rho]n, [rho] and n being the radii of
  curvature of the meridian and perpendicular.

  It is rarely that the terms of the fourth order are required. As a
  rule spheroidal triangles are calculated as spherical (after
  Legendre), i.e. like plane triangles with a decrease of each angle of
  about [epsilon] / 3; [epsilon] must, however, be calculated for each
  triangle separately with its mean measure of curvature k.

  The geodetic line being the shortest that can be drawn on any surface
  between two given points, we may be conducted to its most important
  characteristics by the following considerations: let p, q be adjacent
  points on a curved surface; through s the middle point of the chord pq
  imagine a plane drawn perpendicular to pq, and let S be any point in
  the intersection of this plane with the surface; then pS + Sq is
  evidently least when sS is a minimum, which is when sS is a normal to
  the surface; hence it follows that of all plane curves on the surface
  joining p, q, when those points are indefinitely near to one another,
  that is the shortest which is made by the normal plane. That is to
  say, the osculating plane at any point of a geodetic line contains the
  normal to the surface at that point. Imagine now three points in
  space, A, B, C, such that AB = BC = c; let the direction cosines of AB
  be l, m, n, those of BC l', m', n', then x, y, z being the
  co-ordinates of B, those of A and C will be respectively--

    x - cl : y - cm : z - cn
    x + cl': y + cm': z + cn'.

  Hence the co-ordinates of the middle point M of AC are x + ½c(l' - l),
  y + ½c(m' - m), z + ½c(n' - n), and the direction cosines of BM are
  therefore proportional to l' - l : m' - m : n' - n. If the angle made
  by BC with AB be indefinitely small, the direction cosines of BM are
  as [delta]l : [delta]m : [delta]n. Now if AB, BC be two contiguous
  elements of a geodetic, then BM must be a normal to the surface, and
  since [delta]l, [delta]m, [delta]n are in this case represented by
  [delta](dx/ds), [delta](dy/ds), [delta](dz/ds), and if the equation of
  the surface be u = 0, we have

    d²x   / du   d²y   / du   d²z   / du
    ---  /  -- = ---  /  -- = ---  /  --,
    ds² /   dx   ds² /   dy   ds² /   dz

  which, however, are equivalent to only one equation. In the case of
  the spheroid this equation becomes

      d²x     d²y
    y --- - x --- = 0,
      ds²     ds²

  which integrated gives ydx - xdy = Cds. This again may be put in the
  form r sin a = C, where a is the azimuth of the geodetic at any
  point--the angle between its direction and that of the meridian--and r
  the distance of the point from the axis of revolution.

  From this it may be shown that the azimuth at A of the geodetic
  joining AB is not the same as the astronomical azimuth at A of B or
  that determined by the vertical plane A[alpha]B. Generally speaking,
  the geodetic lies between the two plane section curves joining A and B
  which are formed by the two vertical planes, supposing these points
  not far apart. If, however, A and B are nearly in the same latitude,
  the geodetic may cross (between A and B) that plane curve which lies
  nearest the adjacent pole of the spheroid. The condition of crossing
  is this. Suppose that for a moment we drop the consideration of the
  earth's non-sphericity, and draw a perpendicular from the pole C on
  AB, meeting it in S between A and B. Then A being that point which is
  nearest the pole, the geodetic will cross the plane curve if AS be
  between ¼AB and 3/8 AB. If AS lie between this last value and ½AB, the
  geodetic will lie wholly to the north of both plane curves, that is,
  supposing both points to be in the northern hemisphere.

  The difference of the azimuths of the vertical section AB and of the
  geodetic AB, i.e. the astronomical and geodetic azimuths, is very
  small for all observable distances, being approximately:--

  Geod. azimuth = Astr. azimuth -(1/12) [e²/(1 - e²)] (s²/[rho]n)
  (cos²[phi] sin 2[alpha] + (s/4a)|sin 2[phi] sin [alpha]), in which: e
  and a are the numerical eccentricity and semi-major axis respectively
  of the meridian ellipse, [phi] and [alpha] are the latitude and
  azimuth at A, s = AB, and [rho] and n are the radii of curvature of
  the meridian and perpendicular at A. For s = 100 kilometres, only the
  first term is of moment; its value is 0".028 cos² [phi] sin 2[alpha],
  and it lies well within the errors of observation. If we imagine the
  geodetic AB, it will generally trisect the angles between the vertical
  sections at A and B, so that the geodetic at A is near the vertical
  section AB, and at B near the section BA.[3] The greatest distance of
  the vertical sections one from another is e²s³ cos² [phi]0 sin
  2[alpha]0/16a², in which [phi]0 and [alpha]0 are the mean latitude and
  azimuth respectively of the middle point of AB. For the value s = 64
  kilometres, the maximum distance is 3 mm.

  An idea of the course of a longer geodetic line may be gathered from
  the following example. Let the line be that joining Cadiz and St
  Petersburg, whose approximate positions are--

        Cadiz.         St Petersburg.
    Lat. 36° 22' N.      59° 56' N.
    Long. 6° 18' W.      30° 17' E.

  If G be the point on the geodetic corresponding to F on that one of
  the plane curves which contains the normal at Cadiz (by
  "corresponding" we mean that F and G are on a meridian) then G is to
  the north of F; at a quarter of the whole distance from Cadiz GF is
  458 ft., at half the distance it is 637 ft., and at three-quarters it
  is 473 ft. The azimuth of the geodetic at Cadiz differs 20" from that
  of the vertical plane, which is the astronomical azimuth.

  The azimuth of a geodetic line cannot be observed, so that the line
  does not enter of necessity into practical geodesy, although many
  formulae connected with its use are of great simplicity and elegance.
  The geodetic line has always held a more important place in the
  science of geodesy among the mathematicians of France, Germany and
  Russia than has been assigned to it in the operations of the English
  and Indian triangulations. Although the observed angles of a
  triangulation are not geodetic angles, yet in the calculation of the
  distance and reciprocal bearings of two points which are far apart,
  and are connected by a long chain of triangles, we may fall upon the
  geodetic line in this manner:--

  If A, Z be the points, then to start the calculation from A, we obtain
  by some preliminary calculation the approximate azimuth of Z, or the
  angle made by the direction of Z with the side AB or AC of the first
  triangle. Let P1 be the point where this line intersects BC; then, to
  find P2, where the line cuts the next triangle side CD, we make the
  angle BP1P2 such that BP1P2 + BP1A = 180°. This fixes P2, and P3 is
  fixed by a repetition of the same process; so for P4, P5 .... Now it
  is clear that the points P1, P2, P3 so computed are those which would
  be actually fixed by an observer with a theodolite, proceeding in the
  following manner. Having set the instrument up at A, and turned the
  telescope in the direction of the computed bearing, an assistant
  places a mark P1 on the line BC, adjusting it till bisected by the
  cross-hairs of the telescope at A. The theodolite is then placed over
  P1, and the telescope turned to A; the horizontal circle is then moved
  through 180°. The assistant then places a mark P2 on the line CD, so
  as to be bisected by the telescope, which is then moved to P2, and in
  the same manner P3 is fixed. Now it is clear that the series of points
  P1, P2, P3 approaches to the geodetic line, for the plane of any two
  consecutive elements P_(n-1) P_n, P_n P_(n+1) contains the normal at
  P_n.

  If the objection be raised that not the geodetic azimuths but the
  astronomical azimuths are observed, it is necessary to consider that
  the observed vertical sections do not correspond to points on the
  sea-level but to elevated points. Since the normals of the ellipsoid
  of rotation do not in general intersect, there consequently arises an
  influence of the height on the azimuth. In the case of the measurement
  of the azimuth from A to B, the instrument is set to a point A' over
  the surface of the ellipsoid (the sea-level), and it is then adjusted
  to a point B', also over the surface, say at a height h'. The vertical
  plane containing A' and B' also contains A but not B: it must
  therefore be rotated through a small azimuth in order to contain B.
  The correction amounts approximately to -e²h' cos²[phi] sin
  2[alpha]/2a; in the case of h' = 1000 m., its value is 0".108
  cos²[phi] sin 2[alpha].

  This correction is therefore of greater importance in the case of
  observed azimuths and horizontal angles than in the previously
  considered case of the astronomical and the geodetic azimuths. The
  observed azimuths and horizontal angles must therefore also be
  corrected in the case, where it is required to dispense with geodetic
  lines.

  When the angles of a triangulation have been adjusted by the method of
  least squares, and the sides are calculated, the next process is to
  calculate the latitudes and longitudes of all the stations starting
  from one given point. The calculated latitudes, longitudes and
  azimuths, which are designated geodetic latitudes, longitudes and
  azimuths, are not to be confounded with the observed latitudes,
  longitudes and azimuths, for these last are subject to somewhat large
  errors. Supposing the latitudes of a number of stations in the
  triangulation to be observed, practically the mean of these determines
  the position in latitude of the network, taken as a whole. So the
  orientation or general azimuth of the whole is inferred from all the
  azimuth observations. The triangulation is then supposed to be
  projected on a spheroid of given elements, representing as nearly as
  one knows the real figure of the earth. Then, taking the latitude of
  one point and the direction of the meridian there as given--obtained,
  namely, from the astronomical observations there--one can compute the
  latitudes of all the other points with any degree of precision that
  may be considered desirable. It is necessary to employ for this
  purpose formulae which will give results true even for the longest
  distances to the second place of decimals of seconds, otherwise there
  will arise an accumulation of errors from imperfect calculation which
  should always be avoided. For very long distances, eight places of
  decimals should be employed in logarithmic calculations; if seven
  places only are available very great care will be required to keep the
  last place true. Now let [phi], [phi]' be the latitudes of two
  stations A and B; [alpha], [alpha]^* their mutual azimuths counted
  from north by east continuously from 0° to 360°; [omega] their
  difference of longitude measured from west to east; and s the distance
  AB.

  First compute a latitude [phi]1 by means of the formula [phi]1 = [phi]
  + (s cos [alpha]) / [rho], where [rho] is the radius of curvature of
  the meridian at the latitude [phi]; this will require but four places
  of logarithms. Then, in the first two of the following, five places
  are sufficient--

                   s²                                   s²
    [epsilon] = ------- sin [alpha] cos a,   [eta] = ------- sin²[alpha] tan[phi]1,
                2[rho]n                              2[rho]n

                      s
    [phi]' - [phi] = ---- cos ([alpha] - 2/3[epsilon]) - [eta],
                     rho0

              s sin (alpha - 1/3[epsilon])
    [omega] = ----------------------------,
               n cos ([phi]' + 1/3[eta])

    [alpha]^* - [alpha] = [omega] sin ([phi]' + 2/3[eta]) - [epsilon] + 180°.

  Here n is the normal or radius of curvature perpendicular to the
  meridian; both n and [rho] correspond to latitude [phi]1, and [rho]0
  to latitude ½([phi] + [phi]'). For calculations of latitude and
  longitude, tables of the logarithmic values of [rho] sin 1", n sin 1",
  and 2 n [rho] sin 1" are necessary. The following table contains these
  logarithms for every ten minutes of latitude from 52° to 53° computed
  with the elements a = 20926060 and a : b = 295 : 294 :--

    +------+------------------+--------------+--------------------+
    |      |          1       |        1     |           1        |
    | Lat. | Log.------------.| Log.--------.| Log.--------------.|
    |      |     [rho] sin 1" |     n sin 1" |     2[rho]n sin 1" |
    +------+------------------+--------------+--------------------+
    | °  ' |                  |              |                    |
    |52  0 |    7.9939434     |  7.9928231   |      0.37131       |
    |   10 |         9309     |       8190   |           29       |
    |   20 |         9185     |       8148   |           28       |
    |   30 |         9060     |       8107   |           26       |
    |   40 |         8936     |       8065   |           24       |
    |   50 |         8812     |       8024   |           23       |
    |53  0 |         8688     |       7982   |           22       |
    +------+------------------+--------------+--------------------+

  The logarithm in the last column is that required also for the
  calculation of spherical excesses, the spherical excess of a triangle
  being expressed by a b sin (C/2[rho]n) sin 1".

  It is frequently necessary to obtain the co-ordinates of one point
  with reference to another point; that is, let a perpendicular arc be
  drawn from B to the meridian of A meeting it in P, then, [alpha] being
  the azimuth of B at A, the co-ordinates of B with reference to A are

    AP = s cos ([alpha] - 2/3[epsilon]), BP = s sin ([alpha] -
      1/3[epsilon]),

  where [epsilon] is the spherical excess of APB, viz. s² sin [alpha]
  cos [alpha] multiplied by the quantity whose logarithm is in the
  fourth column of the above table.

  If it be necessary to determine the geographical latitude and
  longitude as well as the azimuths to a greater degree of accuracy than
  is given by the above formulae, we make use of the following formula:
  given the latitude [phi] of A, and the azimuth [alpha] and the
  distance s of B, to determine the latitude [phi]' and longitude
  [omega] of B, and the back azimuth [alpha]'. Here it is understood
  that [alpha]' is symmetrical to [alpha], so that [alpha]^* + [alpha]'
  = 360°.

  Let

    [theta] = s [Delta] / a, where [Delta] = (1 - e² sin² [phi])^½

  and

           e² [theta]²
    [xi] = -----------  cos² [phi] sin 2[alpha],
           (4 (1 - e²)

            e² [theta]³
    [xi]' = ----------- cos² [phi] cos² [alpha];
            (6 (1 - e²)

  [xi], [xi]' are always very minute quantities even for the longest
  distances; then, putting [kappa] = 90° - [phi],

       [alpha]' + [xi] - [omega]   sin ½([kappa] - [theta] - [xi]')     [alpha]
    tan------------------------- = -------------------------------- cot -------
                   2               sin ½([kappa] + [theta] + [xi]')        2

       [alpha]' + [xi] + [omega]   cos ½([kappa] - [theta] - [xi]')     [alpha]
    tan------------------------- = -------------------------------- cot -------
                   2               cos ½([kappa] + [theta] + [xi]')        2

                        s sin ½([alpha]' + [xi] - [alpha])    /    [theta]²     [alpha]' - [alpha]\
    [phi]' - [phi] = --------------------------------------- ( 1 + --------cos² ------------------ );
                     [rho]0 sin ½([alpha]' + [xi] + [alpha])  \       12                 2        /

  here [rho]0 is the radius of curvature of the meridian for the mean
  latitude ½([phi] + [phi]'). These formulae are approximate only, but
  they are sufficiently precise even for very long distances.

  For lines of any length the formulae of F.W. Bessel (_Astr. Nach._,
  1823, iv. 241) are suitable.

  If the two points A and B be defined by their geographical
  co-ordinates, we can accurately calculate the corresponding
  astronomical azimuths, i.e. those of the vertical section, and then
  proceed, in the case of not too great distances, to determine the
  length and the azimuth of the shortest lines. For _any_ distances
  recourse must again be made to Bessel's formula.[4]

  Let [alpha], [alpha]' be the mutual azimuths of two points A, B on a
  spheroid, k the chord line joining them, µ, µ' the angles made by the
  chord with the normals at A and B, [phi], [phi]', [omega] their
  latitudes and difference of longitude, and (x² + y²)/a² + z² b² = 1
  the equation of the surface; then if the plane xz passes through A the
  co-ordinates of A and B will be

    x = (a/[Delta]) cos [phi], x' = (a/[Delta]') cos [phi]' cos [omega],

    y = 0                      y' = (a/[Delta]') cos [phi]' sin [omega],

    z = (a/[Delta]) (1 - e²) sin [phi], z' = (a/[Delta]') (1 - e²) sin [phi]',

  where [Delta] = (1 - e² sin² [phi])^½, [Delta]' = (1 - e² sin²
  [phi]')^½, and e is the eccentricity. Let f, g, h be the direction
  cosines of the normal to that plane which contains the normal at A and
  the point B, and whose inclinations to the meridian plane of A is =
  [alpha]; let also l, m, n and l', m', n' be the direction cosines of
  the normal at A, and of the tangent to the surface at A which lies in
  the plane passing through B, then since the first line is
  perpendicular to each of the other two and to the chord k, whose
  direction cosines are proportional to x' - x, y' - y, z' - z, we have
  these three equations

    f(x' - x) + gy' + h(z' - z) = 0

                fl + gm + hn    = 0

                fl' + gm' + hn' = 0.

  Eliminate f, g, h from these equations, and substitute

    l = cos [phi]   l' = - sin [phi] cos [alpha]

    m = 0           m' = sin [alpha]

    n = sin [phi]   n' = cos [phi] cos [alpha],

  and we get

    (x' - x) sin [phi] + y' cot [alpha] - (z' - z) cos [phi] = 0.

  The substitution of the values of x, z, x', y', z' in this equation
  will give immediately the value of cot [alpha]; and if we put [zeta],
  [zeta]' for the corresponding azimuths on a sphere, or on the
  supposition e = 0, the following relations exist

                                     cos [phi] Q
    cot [alpha] - cot [zeta] = e² ------------------
                                  cos [phi]' [Delta]

                                       cos [phi]' Q
    cot [alpha]' - cot [zeta]' = e² ------------------
                                    cos [phi] [Delta]'

    [Delta]' sin [phi] - [Delta] sin [phi]' = Q sin [omega].

  If from B we let fall a perpendicular on the meridian plane of A, and
  from A let fall a perpendicular on the meridian plane of B, then the
  following equations become geometrically evident:

    k sin µ sin [alpha] = (a/[Delta]') cos [phi]' sin [omega]

    k sin µ' sin [alpha]' = (a/[Delta]) cos [phi] sin [omega].

  Now in any surface u = 0 we have

    k² = (x' - x)² + (y' - y)² + (z' - z)²
               _                                      _
              |         du            du            du |  /   / du²   du²   du² \ ½
    -cos µ =  |(x' - x) -- + (y' - y) -- + (z' - z) -- | / k (  --- + --- + ---  )
              |_        dx            dy            dz_|/     \ dx²   dy²   dz² /
               _                                         _
              |         du             du             du  |  /   / du²    du²    du²  \ ½
    -cos µ' = |(x' - x) --- + (y' - y) --- + (z' - z) --- | / k (  ---- + ---- + ----  ).
              |_        dx'            dy'            dz'_|/     \ dx'²   dy'²   dz'² /

  In the present case, if we put

        xx'   zz'
    1 - --- - --- = U,
        a²    b²

  then

    k²            /z' - z \ ²
    -- = 2U - e² ( ------  )
    a²            \   b   /

    cos µ = (a/k) [Delta]U; cos µ' = (a/k) [Delta]'U.

  Let u be such an angle that

    (1 - e²)^½ sin [phi] = [Delta] sin u

               cos [phi] = [Delta] cos u,

  then on expressing x, x', z, z' in terms of u and u',

    U = 1 - cos u cos u' cos [omega] - sin u sin u';

  also, if v be the third side of a spherical triangle, of which two
  sides are ½[pi] - u and ½[pi] - u' and the included angle [omega],
  using a subsidiary angle [psi] such that

    sin [psi] sin ½v = e sin ½(u' - u) cos ½(u' + u),

  we obtain finally the following equations:--

                      k = 2a cos [psi] sin ½v

                  cos µ = [Delta] sec [psi] sin ½v

                 cos µ' = [Delta]' sec [psi] sin ½v

      sin µ sin [alpha] = (a/k) cos u' sin [omega]

    sin µ' sin [alpha]' = (a/k) cos u sin [omega].

  These determine rigorously the distance, and the mutual zenith
  distances and azimuths, of any two points on a spheroid whose
  latitudes and difference of longitude are given.

  By a series of reductions from the equations containing [zeta],
  [zeta]' it may be shown that

    [alpha] + [alpha]' = [zeta] + [zeta]' + ¼e^4[omega]([phi]' - [phi])²
      cos^4 [phi]0 sin [phi]0 + ...,

  where [phi]0 is the mean of [phi] and [phi]', and the higher powers of
  e are neglected. A short computation will show that the small quantity
  on the right-hand side of this equation cannot amount even to the
  thousandth part of a second for k < 0.1a, which is, practically
  speaking, zero; consequently the sum of the azimuths [alpha] +
  [alpha]' on the spheroid is equal to the sum of the spherical
  azimuths, whence follows this very important theorem (known as Dalby's
  theorem). If [phi], [phi]' be the latitudes of two points on the
  surface of a spheroid, [omega] their difference of longitude, [alpha],
  [alpha]' their reciprocal azimuths,

    tan ½[omega] = cot ½([alpha] + [alpha]') {cos ½([phi]' - [phi])/
      sin ½([phi]' + [phi])}.

  The computation of the geodetic from the astronomical azimuths has
  been given above. From k we can now compute the length s of the
  vertical section, and from this the shortest length. The difference of
  length of the geodetic line and either of the plane curves is

    e^4 s^5 cos^4 [phi]0 sin² 2[alpha]0/360 a^4.

  At least this is an approximate expression. Supposing s = 0.1a, this
  quantity would be less than one-hundredth of a millimetre. The line s
  is now to be calculated as a circular arc with a mean radius r along
  AB. If [phi]0 = ½([phi] + [phi]'), [alpha]0 = ½(180° + [alpha] -
  [alpha]'), [Delta]0 = (1 - e² sin² [phi]0)^½, then 1/r = [Delta]0/a [1
  + e²/(1 - e²) (cos² [phi]0 cos² [alpha]0)], and approximately sin
  (s/2r) = k/2r. These formulae give, in the case of k = 0.1a, values
  certain to eight logarithmic decimal places. An excellent series of
  formulae for the solution of the problem, to determine the azimuths,
  chord and distance along the surface from the geographical
  co-ordinates, was given in 1882 by Ch. M. Schols (_Archives
  Néerlandaises_, vol. xvii.).


  _Irregularities of the Earth's Surface._

  In considering the effect of unequal distribution of matter in the
  earth's crust on the form of the surface, we may simplify the matter
  by disregarding the considerations of rotation and eccentricity. In
  the first place, supposing the earth a sphere covered with a film of
  water, let the density [rho] be a function of the distance from the
  centre so that surfaces of equal density are concentric spheres. Let
  now a disturbance of the arrangement of matter take place, so that the
  density is no longer to be expressed by [rho], a function of r only,
  but is expressed by [rho] + [rho]', where [rho]' is a function of
  three co-ordinates [theta], [phi], r. Then [rho]' is the density of
  what may be designated disturbing matter; it is positive in some
  places and negative in others, and the whole quantity of matter whose
  density is [rho]' is zero. The previously spherical surface of the sea
  of radius a now takes a new form. Let P be a point on the disturbed
  surface, P' the corresponding point vertically below it on the
  undisturbed surface, PP' = N. The knowledge of N over the whole
  surface gives us the form of the disturbed or actual surface of the
  sea; it is an equipotential surface, and if V be the potential at P of
  the disturbing matter [rho]', M the mass of the earth (the
  attraction-constant is assumed equal to unity)

      M             M    M
    ----- + V = C = -- - -- N + V.
    a + N           a    a²

  As far as we know, N is always a very small quantity, and we have with
  sufficient approximation N = 3V/4[pi][delta]a, where [delta] is the
  mean density of the earth. Thus we have the disturbance in elevation
  of the sea-level expressed in terms of the potential of the disturbing
  matter. If at any point P the value of N remain constant when we pass
  to any adjacent point, then the actual surface is there parallel to
  the ideal spherical surface; as a rule, however, the normal at P is
  inclined to that at P', and astronomical observations have shown that
  this inclination, the deflection or deviation, amounting ordinarily to
  one or two seconds, may in some cases exceed 10", or, as at the foot
  of the Himalayas, even 60". By the expression "mathematical figure of
  the earth" we mean the surface of the sea produced in imagination so
  as to percolate the continents. We see then that the effect of the
  uneven distribution of matter in the crust of the earth is to produce
  small elevations and depressions on the mathematical surface which
  would be otherwise spheroidal. No geodesist can proceed far in his
  work without encountering the irregularities of the mathematical
  surface, and it is necessary that he should know how they affect his
  astronomical observations. The whole of this subject is dealt with in
  his usual elegant manner by Bessel in the _Astronomische Nachrichten_,
  Nos. 329, 330, 331, in a paper entitled "Ueber den Einfluss der
  Unregelmässigkeiten der Figur der Erde auf geodätische Arbeiten, &c."
  But without entering into further details it is not difficult to see
  how local attraction at any station affects the determinations of
  latitude, longitude and azimuth there.

  Let there be at the station an attraction to the north-east throwing
  the zenith to the south-west, so that it takes in the celestial sphere
  a position Z', its undisturbed position being Z. Let the rectangular
  components of the displacement ZZ' be [xi] measured southwards and
  [eta] measured westwards. Now the great circle joining Z' with the
  pole of the heavens P makes there an angle with the meridian PZ =
  [eta] cosec PZ' = [eta] sec [phi], where [phi] is the latitude of the
  station. Also this great circle meets the horizon in a point whose
  distance from the great circle PZ is [eta] sec [phi] sin [phi] = [eta]
  tan [phi]. That is, a meridian mark, fixed by observations of the pole
  star, will be placed that amount to the east of north. Hence the
  observed latitude requires the correction [xi]; the observed longitude
  a correction [eta] sec [phi]; and any observed azimuth a correction
  [eta] tan [phi]. Here it is supposed that azimuths are measured from
  north by east, and longitudes eastwards. The horizontal angles are
  also influenced by the deflections of the plumb-line, in fact, just as
  if the direction of the vertical axis of the theodolite varied by the
  same amount. This influence, however, is slight, so long as the sights
  point almost horizontally at the objects, which is always the case in
  the observation of distant points.

  The expression given for N enables one to form an approximate estimate
  of the effect of a compact mountain in raising the sea-level. Take,
  for instance, Ben Nevis, which contains about a couple of cubic miles;
  a simple calculation shows that the elevation produced would only
  amount to about 3 in. In the case of a mountain mass like the
  Himalayas, stretching over some 1500 miles of country with a breadth
  of 300 and an average height of 3 miles, although it is difficult or
  impossible to find an expression for V, yet we may ascertain that an
  elevation amounting to several hundred feet may exist near their base.
  The geodetical operations, however, rather negative this idea, for it
  was shown by Colonel Clarke (_Phil. Mag._, 1878) that the form of the
  sea-level along the Indian arc departs but slightly from that of the
  mean figure of the earth. If this be so, the action of the Himalayas
  must be counteracted by subterranean tenuity.

  Suppose now that A, B, C, ... are the stations of a network of
  triangulation projected on or lying on a spheroid of semiaxis major
  and eccentricity a, e, this spheroid having its axis parallel to the
  axis of rotation of the earth, and its surface coinciding with the
  mathematical surface of the earth at A. Then basing the calculations
  on the observed elements at A, the calculated latitudes, longitudes
  and directions of the meridian at the other points will be the true
  latitudes, &c., of the points as projected on the spheroid. On
  comparing these geodetic elements with the corresponding astronomical
  determinations, there will appear a system of differences which
  represent the inclinations, at the various points, of the actual
  irregular surface to the surface of the spheroid of reference. These
  differences will suggest two things,--first, that we may improve the
  agreement of the two surfaces, by not restricting the spheroid of
  reference by the condition of making its surface coincide with the
  mathematical surface of the earth at A; and secondly, by altering the
  form and dimensions of the spheroid. With respect to the first
  circumstance, we may allow the spheroid two degrees of freedom, that
  is, the normals of the surfaces at A may be allowed to separate a
  small quantity, compounded of a meridional difference and a difference
  perpendicular to the same. Let the spheroid be so placed that its
  normal at A lies to the north of the normal to the earth's surface by
  the small quantity [xi] and to the east by the quantity [eta]. Then in
  starting the calculation of geodetic latitudes, longitudes and
  azimuths from A, we must take, not the observed elements [phi],
  [alpha], but for [phi], [phi] + [xi], and for [alpha], [alpha] + [eta]
  tan [phi], and zero longitude must be replaced by [eta] sec [phi]. At
  the same time suppose the elements of the spheroid to be altered from
  a, e to a + da, e + de. Confining our attention at first to the two
  points A, B, let ([phi]'), ([alpha]'), ([omega]) be the numerical
  elements at B as obtained in the first calculation, viz. before the
  shifting and alteration of the spheroid; they will now take the form

    ([phi]') + f[xi] + g[eta] + hda + kde,

    ([alpha]') + f'[xi] + g'[eta] + h'da + k'de,

    [omega] + f"[xi] + g"[eta] + h"da + k"de,

  where the coefficients f, g, ... &c. can be numerically calculated.
  Now these elements, corresponding to the projection of B on the
  spheroid of reference, must be equal severally to the astronomically
  determined elements at B, corrected for the inclination of the
  surfaces there. If [xi]', [eta]' be the components of the inclination
  at that point, then we have

    [xi]' = ([phi]') - [phi]' + f[xi] + g[eta] + hda + kde,

    [eta]' tan [phi]' = ([alpha]') - [alpha]' + f'[xi] + g'[eta] + h'da + k'de,

    [eta]' sec [phi]' = ([omega]) - [omega] + f"[xi] + g"[eta] + h"da + k"de,

  where [phi]', [alpha]', [omega] are the observed elements at B. Here
  it appears that the observation of longitude gives no additional
  information, but is available as a check upon the azimuthal
  observations.

  If now there be a number of astronomical stations in the
  triangulation, and we form equations such as the above for each point,
  then we can from them determine those values of [xi], [eta], da, de,
  which make the quantity [xi]² + [eta]² + [xi]'² + [eta]'² + ... a
  minimum. Thus we obtain that spheroid which best represents the
  surface covered by the triangulation.

  In the _Account of the Principal Triangulation of Great Britain and
  Ireland_ will be found the determination, from 75 equations, of the
  spheroid best representing the surface of the British Isles. Its
  elements are a = 20927005 ± 295 ft., b : a - b = 280 ± 8; and it is so
  placed that at Greenwich Observatory [xi] = 1".864, [eta] = -0".546.

  Taking Durham Observatory as the origin, and the tangent plane to the
  surface (determined by [xi] = -0".664, [eta] = -4".117) as the plane
  of x and y, the former measured northwards, and z measured vertically
  downwards, the equation to the surface is

    .99524953 x² + .99288005 y² + .99763052 z² - 0.00671003xz - 41655070z = 0.


  _Altitudes._

  The precise determination of the altitude of his station is a matter
  of secondary importance to the geodesist; nevertheless it is usual to
  observe the zenith distances of all trigonometrical points. Of great
  importance is a knowledge of the height of the base for its reduction
  to the sea-level. Again the height of a station does influence a
  little the observation of terrestrial angles, for a vertical line at B
  does not lie generally in the vertical plane of A (see above). The
  height above the sea-level also influences the geographical latitude,
  inasmuch as the centrifugal force is increased and the magnitude and
  direction of the attraction of the earth are altered, and the effect
  upon the latitude is a very small term expressed by the formula h (g'-
  g) sin 2 [phi] / ag, where g, g' are the values of gravity at the
  equator and at the pole. This is h sin 2 [phi] / 5820 seconds, h being
  in metres, a quantity which may be neglected, since for ordinary
  mountain heights it amounts to only a few hundredths of a second. We
  can assume this amount as joined with the northern component of the
  plumb-line perturbations.

  The uncertainties of terrestrial refraction render it impossible to
  determine accurately by vertical angles the heights of distant points.
  Generally speaking, refraction is greatest at about daybreak; from
  that time it diminishes, being at a minimum for a couple of hours
  before and after mid-day; later in the afternoon it again increases.
  This at least is the general march of the phenomenon, but it is by no
  means regular. The vertical angles measured at the station on Hart
  Fell showed on one occasion in the month of September a refraction of
  double the average amount, lasting from 1 P.M. to 5 P.M. The mean
  value of the coefficient of refraction k determined from a very large
  number of observations of terrestrial zenith distances in Great
  Britain is .0792 ± .0047; and if we separate those rays which for a
  considerable portion of their length cross the sea from those which do
  not, the former give k = .0813 and the latter k = .0753. These values
  are determined from high stations and long distances; when the
  distance is short, and the rays graze the ground, the amount of
  refraction is extremely uncertain and variable. A case is noted in the
  Indian survey where the zenith distance of a station 10.5 miles off
  varied from a depression of 4' 52".6 at 4.30 P.M. to an elevation of
  2' 24".0 at 10.50 P.M.

  If h, h' be the heights above the level of the sea of two stations,
  90° + [delta], 90° + [delta]' their mutual zenith distances ([delta]
  being that observed at h), s their distance apart, the earth being
  regarded as a sphere of radius = a, then, with sufficient precision,

                    /   1 - 2k           \                   / 1 - 2k            \
    h' - h = s tan ( s -------- - [delta] ), h - h' = s tan ( -------- - [delta]' ).
                    \     2a             /                   \   2a              /

  If from a station whose height is h the horizon of the sea be observed
  to have a zenith distance 90° + [delta], then the above formula gives
  for h the value

        a  tan² [delta]
    h = -- ------------.
        2     1 - 2k

  Suppose the depression [delta] to be n minutes, then h = 1.054n² if
  the ray be for the greater part of its length crossing the sea; if
  otherwise, h = 1.040n². To take an example: the mean of eight
  observations of the zenith distance of the sea horizon at the top of
  Ben Nevis is 91° 4' 48", or [delta] = 64.8; the ray is pretty equally
  disposed over land and water, and hence h = 1.047n² = 4396 ft. The
  actual height of the hill by spirit-levelling is 4406 ft., so that the
  error of the height thus obtained is only 10 ft.

  The determination of altitudes by means of spirit-levelling is
  undoubtedly the most exact method, particularly in its present
  development as precise-levelling, by which there have been determined
  in all civilized countries close-meshed nets of elevated points
  covering the entire land.     (A. R. C; F. R. H.)


FOOTNOTES:

  [1] An arrangement acting similarly had been previously introduced by
    Borda.

  [2] _Geodetic Survey of South Africa_, vol. iii. (1905), p. viii;
    _Les Nouveaux Appareils pour la mesure rapide des bases géod._, par
    J. René Benoît et Ch. Éd. Guillaume (1906).

  [3] See a paper "On the Course of Geodetic Lines on the Earth's
    Surface" in the _Phil. Mag._ 1870; Helmert, _Theorien der höheren
    Geodäsie_, 1. 321.

  [4] Helmert, Theorien der höheren Geodäsie, 1. 232, 247.




GEOFFREY, surnamed MARTEL (1006-1060), count of Anjou, son of the count
Fulk Nerra (q.v.) and of the countess Hildegarde or Audegarde, was born
on the 14th of October 1006. During his father's lifetime he was
recognized as suzerain by Fulk l'Oison ("the Gosling"), count of
Vendôme, the son of his half-sister Adela. Fulk having revolted, he
confiscated the countship, which he did not restore till 1050. On the
1st of January 1032 he married Agnes, widow of William the Great, duke
of Aquitaine, and taking arms against William the Fat, eldest son and
successor of William the Great, defeated him and took him prisoner at
Mont-Couër near Saint-Jouin-de-Marnes on the 20th of September 1033. He
then tried to win recognition as dukes of Aquitaine for the sons of his
wife Agnes by William the Great, who were still minors, but Fulk Nerra
promptly took up arms to defend his suzerain William the Fat, from whom
he held the Loudunois and Saintonge in fief against his son. In 1036
Geoffrey Martel had to liberate William the Fat, on payment of a heavy
ransom, but the latter having died in 1038, and the second son of
William the Great, Odo, duke of Gascony, having fallen in his turn at
the siege of Mauzé (10th of March 1039) Geoffrey made peace with his
father in the autumn of 1039, and had his wife's two sons recognized as
dukes. About this time, also, he had interfered in the affairs of Maine,
though without much result, for having sided against Gervais, bishop of
Le Mans, who was trying to make himself guardian of the young count of
Maine, Hugh, he had been beaten and forced to make terms with Gervais in
1038. In 1040 he succeeded his father in Anjou and was able to conquer
Touraine (1044) and assert his authority over Maine (see ANJOU). About
1050 he repudiated Agnes, his first wife, and married Grécie, the widow
of Bellay, lord of Montreuil-Bellay (before August 1052), whom he
subsequently left in order to marry Adela, daughter of a certain Count
Odo. Later he returned to Grécie, but again left her to marry Adelaide
the German. When, however, he died on the 14th of November 1060, at the
monastery of St Nicholas at Angers, he left no children, and transmitted
the countship to Geoffrey the Bearded, the eldest of his nephews (see
ANJOU).

  See Louis Halphen, _Le Comté d'Anjou au XI^e siècle_ (Paris, 1906). A
  summary biography is given by Célestin Port, _Dictionnaire historique,
  géographique et biographique de Maine-et-Loire_ (3 vols.,
  Paris-Angers, 1874-1878), vol. ii. pp. 252-253, and a sketch of the
  wars by Kate Norgate, _England under the Angevin Kings_ (2 vols.,
  London, 1887), vol. i. chs. iii. iv.     (L. H.*)




GEOFFREY, surnamed PLANTAGENET [or PLANTEGENET] (1113-1151), count of
Anjou, was the son of Count Fulk the Young and of Eremburge (or
Arembourg of La Flèche); he was born on the 24th of August 1113. He is
also called "le bel" or "the handsome," and received the surname of
Plantagenet from the habit which he is said to have had of wearing in
his cap a sprig of broom (_genêt_). In 1127 he was made a knight, and on
the 2nd of June 1129 married Matilda, daughter of Henry I. of England,
and widow of the emperor Henry V. Some months afterwards he succeeded to
his father, who gave up the countship when he definitively went to the
kingdom of Jerusalem. The years of his government were spent in subduing
the Angevin barons and in conquering Normandy (see ANJOU). In 1151,
while returning from the siege of Montreuil-Bellay, he took cold, in
consequence of bathing in the Loir at Château-du-Loir, and died on the
7th of September. He was buried in the cathedral of Le Mans. By his wife
Matilda he had three sons: Henry Plantagenet, born at Le Mans on Sunday,
the 5th of March 1133; Geoffrey, born at Argentan on the 1st of June
1134; and William Long-Sword, born on the 22nd of July 1136.

  See Kate Norgate, _England under the Angevin Kings_ (2 vols., London,
  1887), vol. i. chs. v.-viii.; Célestin Port, _Dictionnaire historique,
  géographique et biographique de Maine-et-Loire_ (3 vols.,
  Paris-Angers, 1874-1878), vol. ii. pp. 254-256. A history of Geoffrey
  le Bel has yet to be written; there is a biography of him written in
  the 12th century by Jean, a monk of Marmoutier, _Historia Gaufredi,
  ducis Normannorum et comitis Andegavorum_, published by Marchegay et
  Salmon; "Chroniques des comtes d'Anjou" (_Société de l'histoire de
  France_, Paris, 1856), pp. 229-310.     (L. H.*)




GEOFFREY (1158-1186), duke of Brittany, fourth son of the English king
Henry II. and his wife Eleanor of Aquitaine, was born on the 23rd of
September 1158. In 1167 Henry suggested a marriage between Geoffrey and
Constance (d. 1201), daughter and heiress of Conan IV., duke of Brittany
(d. 1171); and Conan not only assented, perhaps under compulsion, to
this proposal, but surrendered the greater part of his unruly duchy to
the English king. Having received the homage of the Breton nobles,
Geoffrey joined his brothers, Henry and Richard, who, in alliance with
Louis VII. of France, were in revolt against their father; but he made
his peace in 1174, afterwards helping to restore order in Brittany and
Normandy, and aiding the new French king, Philip Augustus, to crush some
rebellious vassals. In July 1181 his marriage with Constance was
celebrated, and practically the whole of his subsequent life was spent
in warfare with his brother Richard. In 1183 he made peace with his
father, who had come to Richard's assistance; but a fresh struggle soon
broke out for the possession of Anjou, and Geoffrey was in Paris
treating for aid with Philip Augustus, when he died on the 19th of
August 1186. He left a daughter, Eleanor, and his wife bore a
posthumous son, the unfortunate Arthur.




GEOFFREY (c. 1152-1212), archbishop of York, was a bastard son of Henry
II., king of England. He was distinguished from his legitimate
half-brothers by his consistent attachment and fidelity to his father.
He was made bishop of Lincoln at the age of twenty-one (1173); but
though he enjoyed the temporalities he was never consecrated and
resigned the see in 1183. He then became his father's chancellor,
holding a large number of lucrative benefices in plurality. Richard
nominated him archbishop of York in 1189, but he was not consecrated
till 1191, or enthroned till 1194. Geoffrey, though of high character,
was a man of uneven temper; his history in chiefly one of quarrels, with
the see of Canterbury, with the chancellor William Longchamp, with his
half-brothers Richard and John, and especially with his canons at York.
This last dispute kept him in litigation before Richard and the pope for
many years. He led the clergy in their refusal to be taxed by John and
was forced to fly the kingdom in 1207. He died in Normandy on the 12th
of December 1212.

  See Giraldus Cambrensis, _Vita Galfridi_; Stubbs's prefaces to _Roger
  de Hoveden_, vols. iii. and iv. (Rolls Series).     (H. W. C. D.)




GEOFFREY DE MONTBRAY (d. 1093), bishop of Coutances (_Constantiensis_),
a right-hand man of William the Conqueror, was a type of the great
feudal prelate, warrior and administrator at need. He knew, says
Orderic, more about marshalling mailed knights than edifying
psalm-singing clerks. Obtaining, as a young man, in 1048, the see of
Coutances, by his brother's influence (see MOWBRAY), he raised from his
fellow nobles and from their Sicilian spoils funds for completing his
cathedral, which was consecrated in 1056. With bishop Odo, a warrior
like himself, he was on the battle-field of Hastings, exhorting the
Normans to victory; and at William's coronation it was he who called on
them to acclaim their duke as king. His reward in England was a mighty
fief scattered over twelve counties. He accompanied William on his visit
to Normandy (1067), but, returning, led a royal force to the relief of
Montacute in September 1069. In 1075 he again took the field, leading
with Bishop Odo a vast host against the rebel earl of Norfolk, whose
stronghold at Norwich they besieged and captured.

Meanwhile the Conqueror had invested him with important judicial
functions. In 1072 he had presided over the great Kentish suit between
the primate and Bishop Odo, and about the same time over those between
the abbot of Ely and his despoilers, and between the bishop of Worcester
and the abbot of Ely, and there is some reason to think that he acted as
a Domesday commissioner (1086), and was placed about the same time in
charge of Northumberland. The bishop, who attended the Conqueror's
funeral, joined in the great rising against William Rufus next year
(1088), making Bristol, with which (as Domesday shows) he was closely
connected and where he had built a strong castle, his base of
operations. He burned Bath and ravaged Somerset, but had submitted to
the king before the end of the year. He appears to have been at Dover
with William in January 1090, but, withdrawing to Normandy, died at
Coutances three years later. In his fidelity to Duke Robert he seems to
have there held out for him against his brother Henry, when the latter
obtained the Cotentin.

  See E.A. Freeman, _Norman Conquest_ and _William Rufus_; J.H. Round,
  _Feudal England_; and, for original authorities, the works of Orderic
  Vitalis and William of Poitiers, and of Florence of Worcester; the
  Anglo-Saxon Chronicle; William of Malmesbury's _Gesta pontificum_, and
  Lanfranc's works, ed. Giles; Domesday Book.     (J. H. R.)




GEOFFREY OF MONMOUTH (d. 1154), bishop of St Asaph and writer on early
British history, was born about the year 1100. Of his early life little
is known, except that he received a liberal education under the eye of
his paternal uncle, Uchtryd, who was at that time archdeacon, and
subsequently bishop, of Llandaff. In 1129 Geoffrey appears at Oxford
among the witnesses of an Oseney charter. He subscribes himself Geoffrey
Arturus; from this we may perhaps infer that he had already begun his
experiments in the manufacture of Celtic mythology. A first edition of
his _Historia Britonum_ was in circulation by the year 1139, although
the text which we possess appears to date from 1147. This famous work,
which the author has the audacity to place on the same level with the
histories of William of Malmesbury and Henry of Huntingdon, professes to
be a translation from a Celtic source; "a very old book in the British
tongue" which Walter, archdeacon of Oxford, had brought from Brittany.
Walter the archdeacon is a historical personage; whether his book has
any real existence may be fairly questioned. There is nothing in the
matter or the style of the _Historia_ to preclude us from supposing that
Geoffrey drew partly upon confused traditions, partly on his own powers
of invention, and to a very slight degree upon the accepted authorities
for early British history. His chronology is fantastic and incredible;
William of Newburgh justly remarks that, if we accepted the events which
Geoffrey relates, we should have to suppose that they had happened in
another world. William of Newburgh wrote, however, in the reign of
Richard I. when the reputation of Geoffrey's work was too well
established to be shaken by such criticisms. The fearless romancer had
achieved an immediate success. He was patronized by Robert, earl of
Gloucester, and by two bishops of Lincoln; he obtained, about 1140, the
archdeaconry of Llandaff "on account of his learning"; and in 1151 was
promoted to the see of St Asaph.

Before his death the _Historia Britonum_ had already become a model and
a quarry for poets and chroniclers. The list of imitators begins with
Geoffrey Gaimar, the author of the _Estorie des Engles_ (c. 1147), and
Wace, whose _Roman de Brut_ (1155) is partly a translation and partly a
free paraphrase of the _Historia_. In the next century the influence of
Geoffrey is unmistakably attested by the _Brut_ of Layamon, and the
rhyming English chronicle of Robert of Gloucester. Among later
historians who were deceived by the _Historia Britonum_ it is only
needful to mention Higdon, Hardyng, Fabyan (1512), Holinshed (1580) and
John Milton. Still greater was the influence of Geoffrey upon those
writers who, like Warner in _Albion's England_ (1586), and Drayton in
_Polyolbion_ (1613), deliberately made their accounts of English history
as poetical as possible. The stories which Geoffrey preserved or
invented were not infrequently a source of inspiration to literary
artists. The earliest English tragedy, _Gorboduc_ (1565), the _Mirror
for Magistrates_ (1587), and Shakespeare's Lear, are instances in point.
It was, however, the Arthurian legend which of all his fabrications
attained the greatest vogue. In the work of expanding and elaborating
this theme the successors of Geoffrey went as far beyond him as he had
gone beyond Nennius; but he retains the credit due to the founder of a
great school. Marie de France, who wrote at the court of Henry II., and
Chrétien de Troyes, her French contemporary, were the earliest of the
avowed romancers to take up the theme. The succeeding age saw the
Arthurian story popularized, through translations of the French
romances, as far afield as Germany and Scandinavia. It produced in
England the _Roman du Saint Graal_ and the _Roman de Merlin_, both from
the pen of Robert de Borron; the _Roman de Lancelot_; the _Roman de
Tristan_, which is attributed to a fictitious Lucas de Gast. In the
reign of Edward IV. Sir Thomas Malory paraphrased and arranged the best
episodes of these romances in English prose. His _Morte d'Arthur_,
printed by Caxton in 1485, epitomizes the rich mythology which
Geoffrey's work had first called into life, and gave the Arthurian story
a lasting place in the English imagination. The influence of the
_Historia Britonum_ may be illustrated in another way, by enumerating
the more familiar of the legends to which it first gave popularity. Of
the twelve books into which it is divided only three (Bks. IX., X., XI.)
are concerned with Arthur. Earlier in the work, however, we have the
adventures of Brutus; of his follower Corineus, the vanquisher of the
Cornish giant Goemagol (Gogmagog); of Locrinus and his daughter Sabre
(immortalized in Milton's _Comus_); of Bladud the builder of Bath; of
Lear and his daughters; of the three pairs of brothers, Ferrex and
Porrex, Brennius and Belinus, Elidure and Peridure. The story of
Vortigern and Rowena takes its final form in the _Historia Britonum_;
and Merlin makes his first appearance in the prelude to the Arthur
legend. Besides the _Historia Britonum_ Geoffrey is also credited with
a _Life of Merlin_ composed in Latin verse. The authorship of this work
has, however, been disputed, on the ground that the style is distinctly
superior to that of the _Historia_. A minor composition, the _Prophecies
of Merlin_, was written before 1136, and afterwards incorporated with
the _Historia_, of which it forms the seventh book.

  For a discussion of the manuscripts of Geoffrey's work, see Sir T.D.
  Hardy's _Descriptive Catalogue_ (Rolls Series), i. pp. 341 ff. The
  _Historia Britonum_ has been critically edited by San Marte (Halle,
  1854). There is an English translation by J.A. Giles (London, 1842).
  The _Vita Merlini_ has been edited by F. Michel and T. Wright (Paris,
  1837). See also the _Dublin Univ. Magazine_ for April 1876, for an
  article by T. Gilray on the literary influence of Geoffrey; G.
  Heeger's _Trojanersage der Britten_ (1889); and La Borderie's _Études
  historiques bretonnes_ (1883).     (H. W. C. D.)




GEOFFREY OF PARIS (d. c. 1320), French chronicler, was probably the
author of the _Chronique métrique de Philippe le Bel, or Chronique rimée
de Geoffroi de Paris_. This work, which deals with the history of France
from 1300 to 1316, contains 7918 verses, and is valuable as that of a
writer who had a personal knowledge of many of the events which he
relates. Various short historical poems have also been attributed to
Geoffrey, but there is no certain information about either his life or
his writings.

  The _Chronique_ was published by J.A. Buchon in his _Collection des
  chroniques_, tome ix. (Paris, 1827), and it has also been printed in
  tome xxii. of the _Recueil des historiens des Gaules et de la France_
  (Paris, 1865). See G. Paris, _Histoire de la littérature française au
  moyen âge_ (Paris, 1890); and A. Molinier, _Les Sources de l'histoire
  de France_, tome iii. (Paris, 1903).




GEOFFREY THE BAKER (d. c. 1360), English chronicler, is also called
Walter of Swinbroke, and was probably a secular clerk at Swinbrook in
Oxfordshire. He wrote a _Chronicon Angliae temporibus Edwardi II. et
Edwardi III._, which deals with the history of England from 1303 to
1356. From the beginning until about 1324 this work is based upon Adam
Murimuth's _Continuatio chronicarum_, but after this date it is valuable
and interesting, containing information not found elsewhere, and closing
with a good account of the battle of Poitiers. The author obtained his
knowledge about the last days of Edward II. from William Bisschop, a
companion of the king's murderers, Thomas Gurney and John Maltravers.
Geoffrey also wrote a _Chroniculum_ from the creation of the world until
1336, the value of which is very slight. His writings have been edited
with notes by Sir E.M. Thompson as the _Chronicon Galfridi le Baker de
Swynebroke_ (Oxford, 1889). Some doubt exists concerning Geoffrey's
share in the compilation of the _Vita et mors Edwardi II._, usually
attributed to Sir Thomas de la More, or Moor, and printed by Camden in
his _Anglica scripta_. It has been maintained by Camden and others that
More wrote an account of Edward's reign in French, and that this was
translated into Latin by Geoffrey and used by him in compiling his
_Chronicon_. Recent scholarship, however, asserts that More was no
writer, and that the _Vita et mors_ is an extract from Geoffrey's
_Chronicon_, and was attributed to More, who was the author's patron. In
the main this conclusion substantiates the verdict of Stubbs, who has
published the _Vita et mors_ in his _Chronicles of the reigns of Edward
I. and Edward II._ (London, 1883). The manuscripts of Geoffrey's works
are in the Bodleian library at Oxford.




GEOFFRIN, MARIE THÉRÈSE RODET (1699-1777), a Frenchwoman who played an
interesting part in French literary and artistic life, was born in Paris
in 1699. She married, on the 19th of July 1713, Pierre François
Geoffrin, a rich manufacturer and lieutenant-colonel of the National
Guard, who died in 1750. It was not till Mme Geoffrin was nearly fifty
years of age that we begin to hear of her as a power in Parisian
society. She had learned much from Mme de Tencin, and about 1748 began
to gather round her a literary and artistic circle. She had every week
two dinners, on Monday for artists, and on Wednesday for her friends the
Encyclopaedists and other men of letters. She received many foreigners
of distinction, Hume and Horace Walpole among others. Walpole spent much
time in her society before he was finally attached to Mme du Deffand,
and speaks of her in his letters as a model of common sense. She was
indeed somewhat of a small tyrant in her circle. She had adopted the
pose of an old woman earlier than necessary, and her coquetry, if such
it can be called, took the form of being mother and mentor to her
guests, many of whom were indebted to her generosity for substantial
help. Although her aim appears to have been to have the _Encyclopédie_
in conversation and action around her, she was extremely displeased with
any of her friends who were so rash as to incur open disgrace. Marmontel
lost her favour after the official censure of _Bélisaire_, and her
advanced views did not prevent her from observing the forms of religion.
A devoted Parisian, Mme Geoffrin rarely left the city, so that her
journey to Poland in 1766 to visit the king, Stanislas Poniatowski, whom
she had known in his early days in Paris, was a great event in her life.
Her experiences induced a sensible gratitude that she had been born
"_Française_" and "_particulière_." In her last illness her daughter,
Thérèse, marquise de la Ferté Imbault, excluded her mother's old friends
so that she might die as a good Christian, a proceeding wittily
described by the old lady: "My daughter is like Godfrey de Bouillon, she
wished to defend my tomb from the infidels." Mme Geoffrin died in Paris
on the 6th of October 1777.

  See _Correspondance inédite du roi Stanislas Auguste Poniatowski et de
  Madame Geoffrin_, edited by the comte de Mouÿ (1875); P. de Ségur, _Le
  Royaume de la rue Saint-Honoré, Madame Geoffrin et sa fille_ (1897);
  A. Tornezy, _Un Bureau d'esprit au XVIII^e siècle: le salon de Madame
  Geoffrin_ (1895); and Janet Aldis, _Madame Geoffrin, her Salon and her
  Times, 1750-1777_ (1905).




GEOFFROY, ÉTIENNE FRANÇOIS (1672-1731), French chemist, born in Paris on
the 13th of February 1672, was first an apothecary and then practised
medicine. After studying at Montpellier he accompanied Marshal Tallard
on his embassy to London in 1698 and thence travelled to Holland and
Italy. Returning to Paris he became professor of chemistry at the Jardin
du Roi and of pharmacy and medicine at the Collège de France, and dean
of the faculty of medicine. He died in Paris on the 6th of January 1731.
His name is best known in connexion with his tables of affinities
(_tables des rapports_), which he presented to the French Academy in
1718 and 1720. These were lists, prepared by collating observations on
the actions of substances one upon another, showing the varying degrees
of affinity exhibited by analogous bodies for different reagents, and
they retained their vogue for the rest of the century, until displaced
by the profounder conceptions introduced by C.L. Berthollet. Another of
his papers dealt with the delusions of the philosopher's stone, but
nevertheless he believed that iron could be artificially formed in the
combustion of vegetable matter. His _Tractatus de materia medica_,
published posthumously in 1741, was long celebrated.

His brother CLAUDE JOSEPH, known as Geoffroy the younger (1685-1752),
was also an apothecary and chemist who, having a considerable knowledge
of botany, devoted himself especially to the study of the essential oils
in plants.




GEOFFROY, JULIEN LOUIS (1743-1814), French critic, was born at Rennes in
1743. He studied in the school of his native town and at the Collège
Louis le Grand in Paris. He took orders and fulfilled for some time the
humble functions of an usher, eventually becoming professor of rhetoric
at the _Collège Mazarin_. A bad tragedy, Caton, was accepted at the
_Théâtre Français_, but was never acted. On the death of Élie Fréron in
1776 the other collaborators in the _Année littéraire_ asked Geoffroy to
succeed him, and he conducted the journal until in 1792 it ceased to
appear. Geoffroy was a bitter critic of Voltaire and his followers, and
made for himself many enemies. An enthusiastic royalist, he published
with Fréron's brother-in-law, the abbé Thomas Royou (1741-1792), a
journal, _L'Ami du roi_ (1790-1792), which possibly did more harm than
good to the king's cause by its ill-advised partisanship. During the
Terror Geoffroy hid in the neighbourhood of Paris, only returning in
1799. An attempt to revive the _Année littéraire_ failed, and Geoffroy
undertook the dramatic feuilleton of the _Journal des débats_. His
scathing criticisms had a success of notoriety, but their popularity was
ephemeral, and the publication of them (5 vols., 1819-1820) as _Cours de
littérature dramatique_ proved a failure. He was also the author of a
perfunctory _Commentaire_ on the works of Racine prefixed to Lenormant's
edition (1808). He died in Paris on the 27th of February 1814.




GEOFFROY SAINT-HILAIRE, ÉTIENNE (1772-1844), French naturalist, was the
son of Jean Gèrard Geoffroy, procurator and magistrate of Étampes,
Seine-et-Oise, where he was born on the 15th of April 1772. Destined for
the church he entered the college of Navarre, in Paris, where he studied
natural philosophy under M.J. Brisson; and in 1788 he obtained one of
the canonicates of the chapter of Sainte Croix at Étampes, and also a
benefice. Science, however, offered him a more congenial career, and he
gained from his father permission to remain in Paris, and to attend the
lectures at the Collège de France and the Jardin des Plantes, on the
condition that he should also read law. He accordingly took up his
residence at Cardinal Lemoine's college, and there became the pupil and
soon the esteemed associate of Brisson's friend, the abbé Haüy, the
mineralogist. Having, before the close of the year 1790, taken the
degree of bachelor in law, he became a student of medicine, and attended
the lectures of A.F. de Fourcroy at the Jardin des Plantes, and of
L.J.M. Daubenton at the Collège de France. His studies at Paris were at
length suddenly interrupted, for, in August 1792, Haüy and the other
professors of Lemoine's college, as also those of the college of
Navarre, were arrested by the revolutionists as priests, and confined in
the prison of St Firmin. Through the influence of Daubenton and others
Geoffroy on the 14th of August obtained an order for the release of Haüy
in the name of the Academy; still the other professors of the two
colleges, save C.F. Lhomond, who had been rescued by his pupil J.L.
Tallien, remained in confinement. Geoffroy, foreseeing their certain
destruction if they remained in the hands of the revolutionists,
determined if possible to secure their liberty by stratagem. By bribing
one of the officials at St Firmin, and disguising himself as a
commissioner of prisons, he gained admission to his friends, and
entreated them to effect their escape by following him. All, however,
dreading lest their deliverance should render the doom of their
fellow-captives the more certain, refused the offer, and one priest
only, who was unknown to Geoffroy, left the prison. Already on the night
of the 2nd of September the massacre of the proscribed had begun, when
Geoffroy, yet intent on saving the life of his friends and teachers,
repaired to St Firmin. At 4 o'clock on the morning of the 3rd of
September, after eight hours' waiting, he by means of a ladder assisted
the escape of twelve ecclesiastics, not of the number of his
acquaintance, and then the approach of dawn and the discharge of a gun
directed at him warned him, his chief purpose unaccomplished, to return
to his lodgings. Leaving Paris he retired to Étampes, where, in
consequence of the anxieties of which he had lately been the prey, and
the horrors which he had witnessed, he was for some time seriously ill.
At the beginning of the winter of 1792 he returned to his studies in
Paris, and in March of the following year Daubenton, through the
interest of Bernardin de Saint Pierre, procured him the office of
sub-keeper and assistant demonstrator of the cabinet of natural history,
vacant by the resignation of B.G.E. Lacépède. By a law passed in June
1793, Geoffroy was appointed one of the twelve professors of the newly
constituted museum of natural history, being assigned the chair of
zoology. In the same year he busied himself with the formation of a
menagerie at that institution.

In 1794 through the introduction of A.H. Tessier he entered into
correspondence with Georges Cuvier, to whom, after the perusal of some
of his manuscripts, he wrote: "Venez jouer parmi nous le rôle de Linné,
d'un autre législateur de l'histoire naturelle." Shortly after the
appointment of Cuvier as assistant at the Muséum d'Histoire Naturelle,
Geoffroy received him into his house. The two friends wrote together
five memoirs on natural history, one of which, on the classification of
mammals, puts forward the idea of the subordination of characters upon
which Cuvier based his zoological system. It was in a paper entitled
"Histoire des Makis, ou singes de Madagascar," written in 1795, that
Geoffroy first gave expression to his views on "the unity of organic
composition," the influence of which is perceptible in all his
subsequent writings; nature, he observes, presents us with only one plan
of construction, the same in principle, but varied in its accessory
parts.

In 1798 Geoffroy was chosen a member of the great scientific expedition
to Egypt, and on the capitulation of Alexandria in August 1801, he took
part in resisting the claim made by the British general to the
collections of the expedition, declaring that, were that demand
persisted in, history would have to record that he also had burnt a
library in Alexandria. Early in January 1802 Geoffroy returned to his
accustomed labours in Paris. He was elected a member of the academy of
sciences of that city in September 1807. In March of the following year
the emperor, who had already recognized his national services by the
award of the cross of the legion of honour, selected him to visit the
museums of Portugal, for the purpose of procuring collections from them,
and in the face of considerable opposition from the British he
eventually was successful in retaining them as a permanent possession
for his country. In 1809, the year after his return to France, he was
made professor of zoology at the faculty of sciences at Paris, and from
that period he devoted himself more exclusively than before to
anatomical study. In 1818 he gave to the world the first part of his
celebrated _Philosophie anatomique_, the second volume of which,
published in 1822, and subsequent memoirs account for the formation of
monstrosities on the principle of arrest of development, and of the
attraction of similar parts. When, in 1830, Geoffroy proceeded to apply
to the invertebrata his views as to the unity of animal composition, he
found a vigorous opponent in Georges Cuvier, and the discussion between
them, continued up to the time of the death of the latter, soon
attracted the attention of the scientific throughout Europe. Geoffroy, a
synthesist, contended, in accordance with his theory of unity of plan in
organic composition, that all animals are formed of the same elements,
in the same number, and with the same connexions: homologous parts,
however they differ in form and size, must remain associated in the same
invariable order. With Goethe he held that there is in nature a law of
compensation or balancing of growth, so that if one organ take on an
excess of development, it is at the expense of some other part; and he
maintained that, since nature takes no sudden leaps, even organs which
are superfluous in any given species, if they have played an important
part in other species of the same family, are retained as rudiments,
which testify to the permanence of the general plan of creation. It was
his conviction that, owing to the conditions of life, the same forms had
not been perpetuated since the origin of all things, although it was not
his belief that existing species are becoming modified. Cuvier, who was
an analytical observer of facts, admitted only the prevalence of "laws
of co-existence" or "harmony" in animal organs, and maintained the
absolute invariability of species, which he declared had been created
with a regard to the circumstances in which they were placed, each organ
contrived with a view to the function it had to fulfil, thus putting, in
Geoffroy's considerations, the effect for the cause.

In July 1840 Geoffroy became blind, and some months later he had a
paralytic attack. From that time his strength gradually failed him. He
resigned his chair at the museum in 1841, and died at Paris on the 19th
of June 1844.

  Geoffroy wrote: _Catalogue des mammifères du Muséum National
  d'Histoire Naturelle_ (1813), not quite completed; _Philosophie
  anatomique_--t. i., _Des organes respiratoires_ (1818), and t. ii.,
  _Des monstruosités humaines_ (1822); _Système dentaire des mammifères
  et des oiseaux_ (1st pt., 1824); _Sur le principe de l'unité de
  composition organique_ (1828); _Cours de l'histoire naturelle des
  mammifères_ (1829); _Principes de philosophie zoologique_ (1830);
  _Études progressives d'un naturaliste_ (1835); _Fragments
  biographiques_ (1832); _Notions synthétiques, historiques et
  physiologiques de philosophie naturelle_ (1838), and other works; also
  part of the _Description de l'Égypte par la commission des sciences_
  (1821-1830); and, with Frédéric Cuvier (1773-1838), a younger brother
  of G. Cuvier, _Histoire naturelle des mammifères_ (4 vols.,
  1820-1842); besides numerous papers on such subjects as the anatomy of
  marsupials, ruminants and electrical fishes, the vertebrate theory of
  the skull, the opercula of fishes, teratology, palaeontology and the
  influence of surrounding conditions in modifying animal forms.

  See _Vie, travaux, et doctrine scientifique d'Étienne Geoffroy
  Saint-Hilaire, par son fils M. Isidore Geoffroy Saint-Hilaire_ (Paris
  and Strasburg, 1847), to which is appended a list of Geoffroy's works;
  and Joly, in _Biog. universelle_, t. xvi. (1856).





GEOFFROY SAINT-HILAIRE, ISIDORE (1805-1861), French zoologist, son of
the preceding, was born at Paris on the 16th of December 1805. In his
earlier years he showed an aptitude for mathematics, but eventually he
devoted himself to the study of natural history and of medicine, and in
1824 he was appointed assistant naturalist to his father. On the
occasion of his taking the degree of doctor of medicine in September
1829, he read a thesis entitled _Propositions sur la monstruosité,
considérée chez l'homme et les animaux_; and in 1832-1837 was published
his great teratological work, _Histoire générale et particulière des
anomalies de l'organisation chez l'homme et les animaux_, 3 vols. 8vo.
with 20 plates. In 1829 he delivered for his father the second part of a
course of lectures on ornithology, and during the three following years
he taught zoology at the Athénée, and teratology at the École pratique.
He was elected a member of the academy of sciences at Paris in 1833, was
in 1837 appointed to act as deputy for his father at the faculty of
sciences in Paris, and in the following year was sent to Bordeaux to
organize a similar faculty there. He became successively inspector of
the academy of Paris (1840), professor of the museum on the retirement
of his father (1841), inspector-general of the university (1844), a
member of the royal council for public instruction (1845), and on the
death of H.M.D. de Blainville, professor of zoology at the faculty of
sciences (1850). In 1854 he founded the Acclimatization Society of
Paris, of which he was president. He died at Paris on the 10th of
November 1861.

  Besides the above-mentioned works, he wrote: _Essais de zoologie
  générale_ (1841); _Vie ... d'Étienne Geoffroy Saint-Hilaire_ (1847);
  _Acclimatation et domestication des animaux utiles_ (1849; 4th ed.,
  1861); _Lettres sur les substances alimentaires et particulièrement
  sur la viande de cheval_ (1856); and _Histoire naturelle générale des
  règnes organiques_ (3 vols., 1854-1862), which was not quite
  completed. He was the author also of various papers on zoology,
  comparative anatomy and palaeontology.




GEOGRAPHY (Gr. [Greek: gê], earth, and [Greek: graphein], to write), the
exact and organized knowledge of the distribution of phenomena on the
surface of the earth. The fundamental basis of geography is the vertical
relief of the earth's crust, which controls all mobile distributions.
The grander features of the relief of the lithosphere or stony crust of
the earth control the distribution of the hydrosphere or collected
waters which gather into the hollows, filling them up to a height
corresponding to the volume, and thus producing the important practical
division of the surface into land and water. The distribution of the
mass of the atmosphere over the surface of the earth is also controlled
by the relief of the crust, its greater or lesser density at the surface
corresponding to the lesser or greater elevation of the surface. The
simplicity of the zonal distribution of solar energy on the earth's
surface, which would characterize a uniform globe, is entirely destroyed
by the dissimilar action of land and water with regard to radiant heat,
and by the influence of crust-forms on the direction of the resulting
circulation. The influence of physical environment becomes clearer and
stronger when the distribution of plant and animal life is considered,
and if it is less distinct in the case of man, the reason is found in
the modifications of environment consciously produced by human effort.
Geography is a synthetic science, dependent for the data with which it
deals on the results of specialized sciences such as astronomy, geology,
oceanography, meteorology, biology and anthropology, as well as on
topographical description. The physical and natural sciences are
concerned in geography only so far as they deal with the forms of the
earth's surface, or as regards the distribution of phenomena. The
distinctive task of geography as a science is to investigate the control
exercised by the crust-forms directly or indirectly upon the various
mobile distributions. This gives to it unity and definiteness, and
renders superfluous the attempts that have been made from time to time
to define the limits which divide geography from geology on the one hand
and from history on the other. It is essential to classify the
subject-matter of geography in such a manner as to give prominence not
only to facts, but to their mutual relations and their natural and
inevitable order.

The fundamental conception of geography is form, including the figure of
the earth and the varieties of crustal relief. Hence mathematical
geography (see MAP), including cartography as a practical application,
comes first. It merges into physical geography, which takes account of the
forms of the lithosphere (geomorphology), and also of the distribution of
the hydrosphere and the rearrangements resulting from the workings of
solar energy throughout the hydrosphere and atmosphere (oceanography and
climatology). Next follows the distribution of plants and animals
(biogeography), and finally the distribution of mankind and the various
artificial boundaries and redistributions (anthropogeography). The
applications of anthropogeography to human uses give rise to political and
commercial geography, in the elucidation of which all the earlier
departments or stages have to be considered, together with historical and
other purely human conditions. The evolutionary idea has revolutionized
and unified geography as it did biology, breaking down the old
hard-and-fast partitions between the various departments, and substituting
the study of the nature and influence of actual terrestrial environments
for the earlier motive, the discovery and exploration of new lands.


  HISTORY OF GEOGRAPHICAL THEORY

  The earliest conceptions of the earth, like those held by the
  primitive peoples of the present day, are difficult to discover and
  almost impossible fully to grasp. Early generalizations, as far as
  they were made from known facts, were usually expressed in symbolic
  language, and for our present purpose it is not profitable to
  speculate on the underlying truths which may sometimes be suspected in
  the old mythological cosmogonies.


    Early Greek ideas.

    Flat earth of Homer.

    Hecataeus.

    Herodotus.

    The idea of symmetry.

  The first definite geographical theories to affect the western world
  were those evolved, or at least first expressed, by the Greeks.[1] The
  earliest theoretical problem of geography was the form of the earth.
  The natural supposition that the earth is a flat disk, circular or
  elliptical in outline, had in the time of Homer acquired a special
  definiteness by the introduction of the idea of the ocean river
  bounding the whole, an application of imperfectly understood
  observations. Thales of Miletus is claimed as the first exponent of
  the idea of a spherical earth; but, although this does not appear to
  be warranted, his disciple Anaximander (c. 580 B.C.) put forward the
  theory that the earth had the figure of a solid body hanging freely in
  the centre of the hollow sphere of the starry heavens. The Pythagorean
  school of philosophers adopted the theory of a spherical earth, but
  from metaphysical rather than scientific reasons; their convincing
  argument was that a sphere being the most perfect solid figure was the
  only one worthy to circumscribe the dwelling-place of man. The
  division of the sphere into parallel zones and some of the
  consequences of this generalization seem to have presented themselves
  to Parmenides (c. 450 B.C.); but these ideas did not influence the
  Ionian school of philosophers, who in their treatment of geography
  preferred to deal with facts demonstrable by travel rather than with
  speculations. Thus Hecataeus, claimed by H.F. Tozer[2] as the father
  of geography on account of his _Periodos_, or general treatise on the
  earth, did not advance beyond the primitive conception of a circular
  disk. He systematized the form of the land within the ring of
  ocean--the [Greek: oikoumenê], or habitable world--by recognizing two
  continents: Europe to the north, and Asia to the south of the midland
  sea. Herodotus, equally oblivious of the sphere, criticized and
  ridiculed the circular outline of the _oekumene_, which he knew to be
  longer from east to west than it was broad from north to south. He
  also pointed out reasons for accepting a division of the land into
  three continents--Europe, Asia and Africa. Beyond the limits of his
  personal travels Herodotus applied the characteristically Greek theory
  of symmetry to complete, in the unknown, outlines of lands and rivers
  analogous to those which had been explored. Symmetry was in fact the
  first geographical theory, and the effect of Herodotus's hypothesis
  that the Nile must flow from west to east before turning north in
  order to balance the Danube running from west to east before turning
  south lingered in the maps of Africa down to the time of Mungo
  Park.[3]


    Aristotle and the sphere.

  To Aristotle (384-322 B.C.) must be given the distinction of founding
  scientific geography. He demonstrated the sphericity of the earth by
  three arguments, two of which could be tested by observation. These
  were: (1) that the earth must be spherical, because of the tendency of
  matter to fall together towards a common centre; (2) that only a
  sphere could always throw a circular shadow on the moon during an
  eclipse; and (3) that the shifting of the horizon and the appearance
  of new constellations, or the disappearance of familiar stars, as one
  travelled from north to south, could only be explained on the
  hypothesis that the earth was a sphere. Aristotle, too, gave greater
  definiteness to the idea of zones conceived by Parmenides, who had
  pictured a torrid zone uninhabitable by reason of heat, two frigid
  zones uninhabitable by reason of cold, and two intermediate temperate
  zones fit for human occupation. Aristotle defined the temperate zone
  as extending from the tropic to the arctic circle, but there is some
  uncertainty as to the precise meaning he gave to the term "arctic
  circle." Soon after his time, however, this conception was clearly
  established, and with so large a generalization the mental horizon was
  widened to conceive of a geography which was a science. Aristotle had
  himself shown that in the southern temperate zone winds similar to
  those of the northern temperate zone should blow, but from the
  opposite direction.


    Fitting the oekumene to the sphere.

  While the theory of the sphere was being elaborated the efforts of
  practical geographers were steadily directed towards ascertaining the
  outline and configuration of the _oekumene_, or habitable world, the
  only portion of the terrestrial surface known to the ancients and to
  the medieval peoples, and still retaining a shadow of its old monopoly
  of geographical attention in its modern name of the "Old World." The
  fitting of the _oekumene_ to the sphere was the second theoretical
  problem. The circular outline had given way in geographical opinion to
  the elliptical with the long axis lying east and west, and Aristotle
  was inclined to view it as a very long and relatively narrow band
  almost encircling the globe in the temperate zone. His argument as to
  the narrowness of the sea between West Africa and East Asia, from the
  occurrence of elephants at both extremities, is difficult to
  understand, although it shows that he looked on the distribution of
  animals as a problem of geography.


    Problem of the Antipodes.

  Pythagoras had speculated as to the existence of antipodes, but it was
  not until the first approximately accurate measurements of the globe
  and estimates of the length and breadth of the _oekumene_ were made by
  Eratosthenes (c. 250 B.C.) that the fact that, as then known, it
  occupied less than a quarter of the surface of the sphere was clearly
  recognized. It was natural, if not strictly logical, that the ocean
  river should be extended from a narrow stream to a world-embracing
  sea, and here again Greek theory, or rather fancy, gave its modern
  name to the greatest feature of the globe. The old instinctive idea of
  symmetry must often have suggested other _oekumene_ balancing the
  known world in the other quarters of the globe. The Stoic
  philosophers, especially Crates of Mallus, arguing from the love of
  nature for life, placed an _oekumene_ in each quarter of the sphere,
  the three unknown world-islands being those of the Antoeci, Perioeci
  and Antipodes. This was a theory not only attractive to the
  philosophical mind, but eminently adapted to promote exploration. It
  had its opponents, however, for Herodotus showed that sea-basins
  existed cut off from the ocean, and it is still a matter of
  controversy how far the pre-Ptolemaic geographers believed in a
  water-connexion between the Atlantic and Indian oceans. It is quite
  clear that Pomponius Mela (c. A.D. 40), following Strabo, held that
  the southern temperate zone contained a habitable land, which he
  designated by the name _Antichthones_.


    Aristotle's geographical views.

  Aristotle left no work on geography, so that it is impossible to know
  what facts he associated with the science of the earth's surface. The
  word geography did not appear before Aristotle, the first use of it
  being in the [Greek: Peri kosmôn], which is one of the writings
  doubtfully ascribed to him, and H. Berger considers that the
  expression was introduced by Eratosthenes.[4] Aristotle was certainly
  conversant with many facts, such as the formation of deltas,
  coast-erosion, and to a certain extent the dependence of plants and
  animals on their physical surroundings. He formed a comprehensive
  theory of the variations of climate with latitude and season, and was
  convinced of the necessity of a circulation of water between the sea
  and rivers, though, like Plato, he held that this took place by water
  rising from the sea through crevices in the rocks, losing its
  dissolved salts in the process. He speculated on the differences in
  the character of races of mankind living in different climates, and
  correlated the political forms of communities with their situation on
  a seashore, or in the neighbourhood of natural strongholds.


    Strabo.

  Strabo (c. 50 B.C.-A.D. 24) followed Eratosthenes rather than
  Aristotle, but with sympathies which went out more to the human
  interests than the mathematical basis of geography. He compiled a very
  remarkable work dealing, in large measure from personal travel, with
  the countries surrounding the Mediterranean. He may be said to have
  set the pattern which was followed in succeeding ages by the compilers
  of "political geographies" dealing less with theories than with
  facts, and illustrating rather than formulating the principles of the
  science.


    Ptolemy.

  Claudius Ptolemaeus (c. A.D. 150) concentrated in his writings the
  final outcome of all Greek geographical learning, and passed it across
  the gulf of the middle ages by the hands of the Arabs, to form the
  starting-point of the science in modern times. His geography was based
  more immediately on the work of his predecessor, Marinus of Tyre, and
  on that of Hipparchus, the follower and critic of Eratosthenes. It was
  the ambition of Ptolemy to describe and represent accurately the
  surface of the _oekumene_, for which purpose he took immense trouble
  to collect all existing determinations of the latitude of places, all
  estimates of longitude, and to make every possible rectification in
  the estimates of distances by land or sea. His work was mainly
  cartographical in its aim, and theory was as far as possible excluded.
  The symmetrically placed hypothetical islands in the great continuous
  ocean disappeared, and the _oekumene_ acquired a new form by the
  representation of the Indian Ocean as a larger Mediterranean
  completely cut off by land from the Atlantic. The _terra incognita_
  uniting Africa and Farther Asia was an unfortunate hypothesis which
  helped to retard exploration. Ptolemy used the word _geography_ to
  signify the description of the whole _oekumene_ on mathematical
  principles, while _chorography_ signified the fuller description of a
  particular region, and _topography_ the very detailed description of a
  smaller locality. He introduced the simile that geography represented
  an artist's sketch of a whole portrait, while chorography corresponded
  to the careful and detailed drawing of an eye or an ear.[5]

  The Caliph al-Mam[ = u]n (c. A.D. 815), the son and successor of H[ =
  a]r[ = u]n al-Rash[ = i]d, caused an Arabic version of Ptolemy's great
  astronomical work ([Greek: Suntaxis megistê]) to be made, which is
  known as the _Almagest_, the word being nothing more than the Gr.
  [Greek: megistê] with the Arabic article _al_ prefixed. The geography
  of Ptolemy was also known and is constantly referred to by Arab
  writers. The Arab astronomers measured a degree on the plains of
  Mesopotamia, thereby deducing a fair approximation to the size of the
  earth. The caliph's librarian, Abu Jafar Muhammad Ben Musa, wrote a
  geographical work, now unfortunately lost, entitled _Rasm el Arsi_ ("A
  Description of the World"), which is often referred to by subsequent
  writers as having been composed on the model of that of Ptolemy.


    Geography in the middle ages.

  The middle ages saw geographical knowledge die out in Christendom,
  although it retained, through the Arabic translations of Ptolemy, a
  certain vitality in Islam. The verbal interpretation of Scripture led
  Lactantius (c. A.D. 320) and other ecclesiastics to denounce the
  spherical theory of the earth as heretical. The wretched subterfuge of
  Cosmas (c. A.D. 550) to explain the phenomena of the apparent
  movements of the sun by means of an earth modelled on the plan of the
  Jewish Tabernacle gave place ultimately to the wheel-maps--the T in an
  O--which reverted to the primitive ignorance of the times of Homer and
  Hecataeus.[6]

  The journey of Marco Polo, the increasing trade to the East and the
  voyages of the Arabs in the Indian Ocean prepared the way for the
  reacceptance of Ptolemy's ideas when the sealed books of the Greek
  original were translated into Latin by Angelus in 1410.


    Revival of geography.

  The old arguments of Aristotle and the old measurements of Ptolemy
  were used by Toscanelli and Columbus in urging a westward voyage to
  India; and mainly on this account did the crossing of the Atlantic
  rank higher in the history of scientific geography than the laborious
  feeling out of the coast-line of Africa. But not until the voyage of
  Magellan shook the scales from the eyes of Europe did modern geography
  begin to advance. Discovery had outrun theory; the rush of new facts
  made Ptolemy practically obsolete in a generation, after having been
  the fount and origin of all geography for a millennium.


    Apianus.

  The earliest evidence of the reincarnation of a sound theoretical
  geography is to be found in the text-books by Peter Apian and
  Sebastian Münster. Apian in his _Cosmographicus liber_, published in
  1524, and subsequently edited and added to by Gemma Frisius under the
  title of _Cosmographia_, based the whole science on mathematics and
  measurement. He followed Ptolemy closely, enlarging on his distinction
  between geography and chorography, and expressing the artistic analogy
  in a rough diagram. This slender distinction was made much of by most
  subsequent writers until Nathanael Carpenter in 1625 pointed out that
  the difference between geography and chorography was simply one of
  degree, not of kind.


    Münster.

  Sebastian Münster, on the other hand, in his _Cosmographia
  universalis_ of 1544, paid no regard to the mathematical basis of
  geography, but, following the model of Strabo, described the world
  according to its different political divisions, and entered with great
  zest into the question of the productions of countries, and into the
  manners and costumes of the various peoples. Thus early commenced the
  separation between what were long called mathematical and political
  geography, the one subject appealing mainly to mathematicians, the
  other to historians.

  Throughout the 16th and 17th centuries the rapidly accumulating store
  of facts as to the extent, outline and mountain and river systems of
  the lands of the earth were put in order by the generation of
  cartographers of which Mercator was the chief; but the writings of
  Apian and Münster held the field for a hundred years without a serious
  rival, unless the many annotated editions of Ptolemy might be so
  considered. Meanwhile the new facts were the subject of original study
  by philosophers and by practical men without reference to classical
  traditions. Bacon argued keenly on geographical matters and was a
  lover of maps, in which he observed and reasoned upon such
  resemblances as that between the outlines of South America and Africa.


    Cluverius.

  Philip Cluver's _Introductio in geographiam universam tam veterem quam
  novam_ was published in 1624. Geography he defined as "the description
  of the whole earth, so far as it is known to us." It is distinguished
  from cosmography by dealing with the earth alone, not with the
  universe, and from chorography and topography by dealing with the
  whole earth, not with a country or a place. The first book, of
  fourteen short chapters, is concerned with the general properties of
  the globe; the remaining six books treat in considerable detail of the
  countries of Europe and of the other continents. Each country is
  described with particular regard to its people as well as to its
  surface, and the prominence given to the human element is of special
  interest.


    Carpenter.

  A little-known book which appears to have escaped the attention of
  most writers on the history of modern geography was published at
  Oxford in 1625 by Nathanael Carpenter, fellow of Exeter College, with
  the title _Geographie delineated forth in Two Bookes, containing the
  Sphericall and Topicall parts thereof_. It is discursive in its style
  and verbose; but, considering the period at which it appeared, it is
  remarkable for the strong common sense displayed by the author, his
  comparative freedom from prejudice, and his firm application of the
  methods of scientific reasoning to the interpretation of phenomena.
  Basing his work on the principles of Ptolemy, he brings together
  illustrations from the most recent travellers, and does not hesitate
  to take as illustrative examples the familiar city of Oxford and his
  native county of Devon. He divides geography into _The Spherical
  Part_, or that for the study of which mathematics alone is required,
  and _The Topical Part_, or the description of the physical relations
  of parts of the earth's surface, preferring this division to that
  favoured by the ancient geographers--into general and special. It is
  distinguished from other English geographical books of the period by
  confining attention to the principles of geography, and not describing
  the countries of the world.


    Varenius.

  A much more important work in the history of geographical method is
  the _Geographia generalis_ of Bernhard Varenius, a German medical
  doctor of Leiden, who died at the age of twenty-eight in 1650, the
  year of the publication of his book. Although for a time it was lost
  sight of on the continent, Sir Isaac Newton thought so highly of this
  book that he prepared an annotated edition which was published in
  Cambridge in 1672, with the addition of the plates which had been
  planned by Varenius, but not produced by the original publishers. "The
  reason why this great man took so much care in correcting and
  publishing our author was, because he thought him necessary to be read
  by his audience, the young gentlemen of Cambridge, while he was
  delivering lectures on the same subject from the Lucasian Chair."[7]
  The treatise of Varenius is a model of logical arrangement and terse
  expression; it is a work of science and of genius; one of the few of
  that age which can still be studied with profit. The English
  translation renders the definition thus: "Geography is that part of
  _mixed mathematics_ which explains the state of the earth and of its
  parts, depending on quantity, viz. its figure, place, magnitude and
  motion, with the celestial appearances, &c. By some it is taken in too
  limited a sense, for a bare description of the several countries; and
  by others too extensively, who along with such a description would
  have their political constitution."

  Varenius was reluctant to include the human side of geography in his
  system, and only allowed it as a concession to custom, and in order to
  attract readers by imparting interest to the sterner details of the
  science. His division of geography was into two parts--(i.) General or
  universal, dealing with the earth in general, and explaining its
  properties without regard to particular countries; and (ii.) Special
  or particular, dealing with each country in turn from the
  chorographical or topographical point of view. General geography was
  divided into--(1) the _Absolute_ part, dealing with the form,
  dimensions, position and substance of the earth, the distribution of
  land and water, mountains, woods and deserts, hydrography (including
  all the waters of the earth) and the atmosphere; (2) the _Relative_
  part, including the celestial properties, i.e. latitude, climate
  zones, longitude, &c.; and (3) the _Comparative_ part, which
  "considers the particulars arising from comparing one part with
  another"; but under this head the questions discussed were longitude,
  the situation and distances of places, and navigation. Varenius does
  not treat of special geography, but gives a scheme for it under three
  heads--(1) _Terrestrial_, including position, outline, boundaries,
  mountains, mines, woods and deserts, waters, fertility and fruits, and
  living creatures; (2) _Celestial_, including appearance of the heavens
  and the climate; (3) _Human_, but this was added out of deference to
  popular usage.

  This system of geography founded a new epoch, and the book--translated
  into English, Dutch and French--was the unchallenged standard for more
  than a century. The framework was capable of accommodating itself to
  new facts, and was indeed far in advance of the knowledge of the
  period. The method included a recognition of the causes and effects of
  phenomena as well as the mere fact of their occurrence, and for the
  first time the importance of the vertical relief of the land was
  fairly recognized.

  The physical side of geography continued to be elaborated after
  Varenius's methods, while the historical side was developed
  separately. Both branches, although enriched by new facts, remained
  stationary so far as method is concerned until nearly the end of the
  18th century. The compilation of "geography books" by uninstructed
  writers led to the pernicious habit, which is not yet wholly overcome,
  of reducing the general or "physical" part to a few pages of
  concentrated information, and expanding the particular or "political"
  part by including unrevised travellers' stories and uncritical
  descriptions of the various countries of the world. Such books were in
  fact not geography, but merely compressed travel.


    Bergman.

  The next marked advance in the theory of geography may be taken as the
  nearly simultaneous studies of the physical earth carried out by the
  Swedish chemist, Torbern Bergman, acting under the impulse of
  Linnaeus, and by the German philosopher, Immanuel Kant. Bergman's
  _Physical Description of the Earth_ was published in Swedish in 1766,
  and translated into English in 1772 and into German in 1774. It is a
  plain, straightforward description of the globe, and of the various
  phenomena of the surface, dealing only with definitely ascertained
  facts in the natural order of their relationships, but avoiding any
  systematic classification or even definitions of terms.


    Kant.

  The problems of geography had been lightened by the destructive
  criticism of the French cartographer D'Anville (who had purged the map
  of the world of the last remnants of traditional fact unverified by
  modern observations) and rendered richer by the dawn of the new era of
  scientific travel, when Kant brought his logical powers to bear upon
  them. Kant's lectures on physical geography were delivered in the
  university of Königsberg from 1765 onwards.[8] Geography appealed to
  him as a valuable educational discipline, the joint foundation with
  anthropology of that "knowledge of the world" which was the result of
  reason and experience. In this connexion he divided the communication
  of experience from one person to another into two categories--the
  narrative or historical and the descriptive or geographical; both
  history and geography being viewed as descriptions, the former a
  description in order of time, the latter a description in order of
  space.

  Physical geography he viewed as a summary of nature, the basis not
  only of history but also of "all the other possible geographies," of
  which he enumerates five, viz. (1) _Mathematical geography_, which
  deals with the form, size and movements of the earth and its place in
  the solar system; (2) _Moral geography_, or an account of the
  different customs and characters of mankind according to the region
  they inhabit; (3) _Political geography_, the divisions according to
  their organized governments; (4) _Mercantile geography_, dealing with
  the trade in the surplus products of countries; (5) _Theological
  geography_, or the distribution of religions. Here there is a clear
  and formal statement of the interaction and causal relation of all the
  phenomena of distribution on the earth's surface, including the
  influence of physical geography upon the various activities of mankind
  from the lowest to the highest. Notwithstanding the form of this
  classification, Kant himself treats mathematical geography as
  preliminary to, and therefore not dependent on, physical geography.
  Physical geography itself is divided into two parts: a general, which
  has to do with the earth and all that belongs to it--water, air and
  land; and a particular, which deals with special products of the
  earth--mankind, animals, plants and minerals. Particular importance is
  given to the vertical relief of the land, on which the various
  branches of human geography are shown to depend.


    Humboldt.

  Alexander von Humboldt (1769-1859) was the first modern geographer to
  become a great traveller, and thus to acquire an extensive stock of
  first-hand information on which an improved system of geography might
  be founded. The impulse given to the study of natural history by the
  example of Linnaeus; the results brought back by Sir Joseph Banks, Dr
  Solander and the two Forsters, who accompanied Cook in his voyages of
  discovery; the studies of De Saussure in the Alps, and the lists of
  desiderata in physical geography drawn up by that investigator,
  combined to prepare the way for Humboldt. The theory of geography was
  advanced by Humboldt mainly by his insistence on the great principle
  of the unity of nature. He brought all the "observable things," which
  the eager collectors of the previous century had been heaping together
  regardless of order or system, into relation with the vertical relief
  and the horizontal forms of the earth's surface. Thus he demonstrated
  that the forms of the land exercise a directive and determining
  influence on climate, plant life, animal life and on man himself. This
  was no new idea; it had been familiar for centuries in a less definite
  form, deduced from a priori considerations, and so far as regards the
  influence of surrounding circumstances upon man, Kant had already
  given it full expression. Humboldt's concrete illustrations and the
  remarkable power of his personality enabled him to enforce these
  principles in a way that produced an immediate and lasting effect. The
  treatises on physical geography by Mrs Mary Somerville and Sir John
  Herschel (the latter written for the eighth edition of the
  _Encyclopaedia Britannica_) showed the effect produced in Great
  Britain by the stimulus of Humboldt's work.


    Ritter.

  Humboldt's contemporary, Carl Ritter (1779-1859), extended and
  disseminated the same views, and in his interpretation of "Comparative
  Geography" he laid stress on the importance of forming conclusions,
  not from the study of one region by itself, but from the comparison of
  the phenomena of many places. Impressed by the influence of
  terrestrial relief and climate on human movements, Ritter was led
  deeper and deeper into the study of history and archaeology. His
  monumental _Vergleichende Geographie_, which was to have made the
  whole world its theme, died out in a wilderness of detail in
  twenty-one volumes before it had covered more of the earth's surface
  than Asia and a portion of Africa. Some of his followers showed a
  tendency to look on geography rather as an auxiliary to history than
  as a study of intrinsic worth.


    Geography as a natural science.

  During the rapid development of physical geography many branches of
  the study of nature, which had been included in the cosmography of the
  early writers, the physiography of Linnaeus and even the _Erdkunde_ of
  Ritter, had been so much advanced by the labours of specialists that
  their connexion was apt to be forgotten. Thus geology, meteorology,
  oceanography and anthropology developed into distinct sciences. The
  absurd attempt was, and sometimes is still, made by geographers to
  include all natural science in geography; but it is more common for
  specialists in the various detailed sciences to think, and sometimes
  to assert, that the ground of physical geography is now fully occupied
  by these sciences. Political geography has been too often looked on
  from both sides as a mere summary of guide-book knowledge, useful in
  the schoolroom, a poor relation of physical geography that it was
  rarely necessary to recognize.

  The science of geography, passed on from antiquity by Ptolemy,
  re-established by Varenius and Newton, and systematized by Kant,
  included within itself definite aspects of all those terrestrial
  phenomena which are now treated exhaustively under the heads of
  geology, meteorology, oceanography and anthropology; and the inclusion
  of the requisite portions of the perfected results of these sciences
  in geography is simply the gathering in of fruit matured from the seed
  scattered by geography itself.

  The study of geography was advanced by improvements in cartography
  (see MAP), not only in the methods of survey and projection, but in
  the representation of the third dimension by means of contour lines
  introduced by Philippe Buache in 1737, and the more remarkable because
  less obvious invention of isotherms introduced by Humboldt in 1817.


    The teleological argument in geography.

  The "argument from design" had been a favourite form of reasoning
  amongst Christian theologians, and, as worked out by Paley in his
  _Natural Theology_, it served the useful purpose of emphasizing the
  fitness which exists between all the inhabitants of the earth and
  their physical environment. It was held that the earth had been
  created so as to fit the wants of man in every particular. This
  argument was tacitly accepted or explicitly avowed by almost every
  writer on the theory of geography, and Carl Ritter distinctly
  recognized and adopted it as the unifying principle of his system. As
  a student of nature, however, he did not fail to see, and as professor
  of geography he always taught, that man was in very large measure
  conditioned by his physical environment. The apparent opposition of
  the observed fact to the assigned theory he overcame by looking upon
  the forms of the land and the arrangement of land and sea as
  instruments of Divine Providence for guiding the destiny as well as
  for supplying the requirements of man. This was the central theme of
  Ritter's philosophy; his religion and his geography were one, and the
  consequent fervour with which he pursued his mission goes far to
  account for the immense influence he acquired in Germany.


    The theory of evolution in geography.

  The evolutionary theory, more than hinted at in Kant's "Physical
  Geography," has, since the writings of Charles Darwin, become the
  unifying principle in geography. The conception of the development of
  the plan of the earth from the first cooling of the surface of the
  planet throughout the long geological periods, the guiding power of
  environment on the circulation of water and of air, on the
  distribution of plants and animals, and finally on the movements of
  man, give to geography a philosophical dignity and a scientific
  completeness which it never previously possessed. The influence of
  environment on the organism may not be quite so potent as it was once
  believed to be, in the writings of Buckle, for instance,[9] and
  certainly man, the ultimate term in the series, reacts upon and
  greatly modifies his environment; yet the fact that environment does
  influence all distributions is established beyond the possibility of
  doubt. In this way also the position of geography, at the point where
  physical science meets and mingles with mental science, is explained
  and justified. The change which took place during the 19th century in
  the substance and style of geography may be well seen by comparing the
  eight volumes of Malte-Brun's _Géographie universelle_ (Paris,
  1812-1829) with the twenty-one volumes of Reclus's _Géographie
  universelle_ (Paris, 1876-1895).

  In estimating the influence of recent writers on geography it is usual
  to assign to Oscar Peschel (1826-1875) the credit of having corrected
  the preponderance which Ritter gave to the historical element, and of
  restoring physical geography to its old pre-eminence.[10] As a matter
  of fact, each of the leading modern exponents of theoretical
  geography--such as Ferdinand von Richthofen, Hermann Wagner, Friedrich
  Ratzel, William M. Davis, A. Penck, A. de Lapparent and Elisée
  Reclus--has his individual point of view, one devoting more attention
  to the results of geological processes, another to anthropological
  conditions, and the rest viewing the subject in various blendings of
  the extreme lights.

  The two conceptions which may now be said to animate the theory of
  geography are the genetic, which depends upon processes of origin, and
  the morphological, which depends on facts of form and distribution.


  PROGRESS OF GEOGRAPHICAL DISCOVERY

  Exploration and geographical discovery must have started from more
  than one centre, and to deal justly with the matter one ought to treat
  of these separately in the early ages before the whole civilized world
  was bound together by the bonds of modern intercommunication. At the
  least there should be some consideration of four separate systems of
  discovery--the Eastern, in which Chinese and Japanese explorers
  acquired knowledge of the geography of Asia, and felt their way
  towards Europe and America; the Western, in which the dominant races
  of the Mexican and South American plateaus extended their knowledge of
  the American continent before Columbus; the Polynesian, in which the
  conquering races of the Pacific Islands found their way from group to
  group; and the Mediterranean. For some of these we have no certain
  information, and regarding others the tales narrated in the early
  records are so hard to reconcile with present knowledge that they are
  better fitted to be the battle-ground of scholars championing rival
  theories than the basis of definite history. So it has come about that
  the only practicable history of geographical exploration starts from
  the Mediterranean centre, the first home of that civilization which
  has come to be known as European, though its field of activity has
  long since overspread the habitable land of both temperate zones,
  eastern Asia alone in part excepted.

  From all centres the leading motives of exploration were probably the
  same--commercial intercourse, warlike operations, whether resulting in
  conquest or in flight, religious zeal expressed in pilgrimages or
  missionary journeys, or, from the other side, the avoidance of
  persecution, and, more particularly in later years, the advancement of
  knowledge for its own sake. At different times one or the other motive
  predominated.

  Before the 14th century B.C. the warrior kings of Egypt had carried
  the power of their arms southward from the delta of the Nile well-nigh
  to its source, and eastward to the confines of Assyria. The
  hieroglyphic inscriptions of Egypt and the cuneiform inscriptions of
  Assyria are rich in records of the movements and achievements of
  armies, the conquest of towns and the subjugation of peoples; but
  though many of the recorded sites have been identified, their
  discovery by wandering armies was isolated from their subsequent
  history and need not concern us here.


    The Phoenicians.

  The Phoenicians are the earliest Mediterranean people in the
  consecutive chain of geographical discovery which joins pre-historic
  time with the present. From Sidon, and later from its more famous
  rival Tyre, the merchant adventurers of Phoenicia explored and
  colonized the coasts of the Mediterranean and fared forth into the
  ocean beyond. They traded also on the Red sea, and opened up regular
  traffic with India as well as with the ports of the south and west, so
  that it was natural for Solomon to employ the merchant navies of Tyre
  in his oversea trade. The western emporium known in the scriptures as
  Tarshish was probably situated in the south of Spain, possibly at
  Cadiz, although some writers contend that it was Carthage in North
  Africa. Still more diversity of opinion prevails as to the southern
  gold-exporting port of Ophir, which some scholars place in Arabia,
  others at one or another point on the east coast of Africa. Whether
  associated with the exploitation of Ophir (q.v.) or not the first
  great voyage of African discovery appears to have been accomplished by
  the Phoenicians sailing the Red Sea. Herodotus (himself a notable
  traveller in the 5th century B.C.) relates that the Egyptian king
  Necho of the XXVIth Dynasty (c. 600 B.C.) built a fleet on the Red
  Sea, and confided it to Phoenician sailors with the orders to sail
  southward and return to Egypt by the Pillars of Hercules and the
  Mediterranean sea. According to the tradition, which Herodotus quotes
  sceptically, this was accomplished; but the story is too vague to be
  accepted as more than a possibility.

  The great Phoenician colony of Carthage, founded before 800 B.C.,
  perpetuated the commercial enterprise of the parent state, and
  extended the sphere of practical trade to the ocean shores of Africa
  and Europe. The most celebrated voyage of antiquity undertaken for the
  express purpose of discovery was that fitted out by the senate of
  Carthage under the command of Hanno, with the intention of founding
  new colonies along the west coast of Africa. According to Pliny, the
  only authority on this point, the period of the voyage was that of the
  greatest prosperity of Carthage, which may be taken as somewhere
  between 570 and 480 B.C. The extent of this voyage is doubtful, but it
  seems probable that the farthest point reached was on the east-running
  coast which bounds the Gulf of Guinea on the north. Himilco, a
  contemporary of Hanno, was charged with an expedition along the west
  coast of Iberia northward, and as far as the uncertain references to
  this voyage can be understood, he seems to have passed the Bay of
  Biscay and possibly sighted the coast of England.


    The Greeks.

  The sea power of the Greek communities on the coast of Asia Minor and
  in the Archipelago began to be a formidable rival to the Phoenician
  soon after the time of Hanno and Himilco, and peculiar interest
  attaches to the first recorded Greek voyage beyond the Pillars of
  Hercules. Pytheas, a navigator of the Phocean colony of Massilia
  (Marseilles), determined the latitude of that port with considerable
  precision by the somewhat clumsy method of ascertaining the length of
  the longest day, and when, about 330 B.C., he set out on exploration
  to the northward in search of the lands whence came gold, tin and
  amber, he followed this system of ascertaining his position from time
  to time. If on each occasion he himself made the observations his
  voyage must have extended over six years; but it is not impossible
  that he ascertained the approximate length of the longest day in some
  cases by questioning the natives. Pytheas, whose own narrative is not
  preserved, coasted the Bay of Biscay, sailed up the English Channel
  and followed the coast of Britain to its most northerly point. Beyond
  this he spoke of a land called _Thule_, which, if his estimate of the
  length of the longest day is correct, may have been Shetland, but was
  possibly Iceland; and from some confused statements as to a sea which
  could not be sailed through, it has been assumed that Pytheas was the
  first of the Greeks to obtain direct knowledge of the Arctic regions.
  During this or a second voyage Pytheas entered the Baltic, discovered
  the coasts where amber is obtained and returned to the Mediterranean.
  It does not seem that any maritime trade followed these discoveries,
  and indeed it is doubtful whether his contemporaries accepted the
  truth of Pytheas's narrative; Strabo four hundred years later
  certainly did not, but the critical studies of modern scholars have
  rehabilitated the Massilian explorer.


    Alexander the Great.

  The Greco-Persian wars had made the remoter parts of Asia Minor more
  than a name to the Greek geographers before the time of Alexander the
  Great, but the campaigns of that conqueror from 329 to 325 B.C. opened
  up the greater Asia to the knowledge of Europe. His armies crossed the
  plains beyond the Caspian, penetrated the wild mountain passes
  north-west of India, and did not turn back until they had entered on
  the Indo-Gangetic plain. This was one of the few great epochs of
  geographical discovery.

  The world was henceforth viewed as a very large place stretching far
  on every side beyond the Midland or Mediterranean Sea, and the land
  journey of Alexander resulted in a voyage of discovery in the outer
  ocean from the mouth of the Indus to that of the Tigris, thus opening
  direct intercourse between Grecian and Hindu civilization. The Greeks
  who accompanied Alexander described with care the towns and villages,
  the products and the aspect of the country. The conqueror also
  intended to open up trade by sea between Europe and India, and the
  narrative of his general Nearchus records this famous voyage of
  discovery, the detailed accounts of the chief pilot Onesicritus being
  lost. At the beginning of October 326 B.C. Nearchus left the Indus
  with his fleet, and the anchorages sought for each night are carefully
  recorded. He entered the Persian Gulf, and rejoined Alexander at Susa,
  when he was ordered to prepare another expedition for the
  circumnavigation of Arabia. Alexander died at Babylon in 323 B.C., and
  the fleet was dispersed without making the voyage.

  The dynasties founded by Alexander's generals, Seleucus, Antiochus and
  Ptolemy, encouraged the same spirit of enterprise which their master
  had fostered, and extended geographical knowledge in several
  directions. Seleucus Nicator established the Greco-Bactrian empire and
  continued the intercourse with India. Authentic information respecting
  the great valley of the Ganges was supplied by Megasthenes, an
  ambassador sent by Seleucus, who reached the remote city of
  Patali-putra, the modern Patna.


    The Ptolemies.

  The Ptolemies in Egypt showed equal anxiety to extend the bounds of
  geographical knowledge. Ptolemy Euergetes (247-222 B.C.) rendered the
  greatest service to geography by the protection and encouragement of
  Eratosthenes, whose labours gave the first approximate knowledge of
  the true size of the spherical earth. The second Euergetes and his
  successor Ptolemy Lathyrus (118-115 B.C.) furnished Eudoxus with a
  fleet to explore the Arabian sea. After two successful voyages,
  Eudoxus, impressed with the idea that Africa was surrounded by ocean
  on the south, left the Egyptian service, and proceeded to Cadiz and
  other Mediterranean centres of trade seeking a patron who would
  finance an expedition for the purpose of African discovery; and we
  learn from Strabo that the veteran explorer made at least two voyages
  southward along the coast of Africa. The Ptolemies continued to send
  fleets annually from their Red Sea ports of Berenice and Myos Hormus
  to Arabia, as well as to ports on the coasts of Africa and India.


    The Romans.

  The Romans did not encourage navigation and commerce with the same
  ardour as their predecessors; still the luxury of Rome, which gave
  rise to demands for the varied products of all the countries of the
  known world, led to an active trade both by ships and caravans. But it
  was the military genius of Rome, and the ambition for universal
  empire, which led, not only to the discovery, but also to the survey
  of nearly all Europe, and of large tracts in Asia and Africa. Every
  new war produced a new survey and itinerary of the countries which
  were conquered, and added one more to the imperishable roads that led
  from every quarter of the known world to Rome. In the height of their
  power the Romans had surveyed and explored all the coasts of the
  Mediterranean, Italy, Greece, the Balkan Peninsula, Spain, Gaul,
  western Germany and southern Britain. In Africa their empire included
  Egypt, Carthage, Numidia and Mauritania. In Asia they held Asia Minor
  and Syria, had sent expeditions into Arabia, and were acquainted with
  the more distant countries formerly invaded by Alexander, including
  Persia, Scythia, Bactria and India. Roman intercourse with India
  especially led to the extension of geographical knowledge.

  Before the Roman legions were sent into a new region to extend the
  limits of the empire, it was usual to send out exploring expeditions
  to report as to the nature of the country. It is narrated by Pliny and
  Seneca that the emperor Nero sent out two centurions on such a mission
  towards the source of the Nile (probably about A.D. 60), and that the
  travellers pushed southwards until they reached vast marshes through
  which they could not make their way either on foot or in boats. This
  seems to indicate that they had penetrated to about 9° N. Shortly
  before A.D. 79 Hippalus took advantage of the regular alternation of
  the monsoons to make the voyage from the Red Sea to India across the
  open ocean out of sight of land. Even though this sea-route was known,
  the author of the _Periplus of the Erythraean Sea_, published after
  the time of Pliny, recites the old itinerary around the coast of the
  Arabian Gulf. It was, however, in the reigns of Severus and his
  immediate successors that Roman intercourse with India was at its
  height, and from the writings of Pausanias (c. 174) it appears that
  direct communication between Rome and China had already taken place.

  After the division of the Roman empire, Constantinople became the last
  refuge of learning, arts and taste; while Alexandria continued to be
  the emporium whence were imported the commodities of the East. The
  emperor Justinian (483-565), in whose reign the greatness of the
  Eastern empire culminated, sent two Nestorian monks to China, who
  returned with eggs of the silkworm concealed in a hollow cane, and
  thus silk manufactures were established in the Peloponnesus and the
  Greek islands. It was also in the reign of Justinian that Cosmas
  Indicopleustes, an Egyptian merchant, made several voyages, and
  afterwards composed his [Greek: Christianikê topographia] (Christian
  Topography), containing, in addition to his absurd cosmogony, a
  tolerable description of India.


    The Arabs.

  The great outburst of Mahommedan conquest in the 7th century was
  followed by the Arab civilization, having its centres at Bagdad and
  Cordova, in connexion with which geography again received a share of
  attention. The works of the ancient Greek geographers were translated
  into Arabic, and starting with a sound basis of theoretical knowledge,
  exploration once more made progress. From the 9th to the 13th century
  intelligent Arab travellers wrote accounts of what they had seen and
  heard in distant lands. The earliest Arabian traveller whose
  observations have come down to us is the merchant Sulaiman, who
  embarked in the Persian Gulf and made several voyages to India and
  China, in the middle of the 9th century. Abu Zaid also wrote on India,
  and his work is the most important that we possess before the
  epoch-making discoveries of Marco Polo. Masudi, a great traveller who
  knew from personal experience all the countries between Spain and
  China, described the plains, mountains and seas, the dynasties and
  peoples, in his _Meadows of Gold_, an abstract made by himself of his
  larger work _News of the Time_. He died in 956, and was known, from
  the comprehensiveness of his survey, as the Pliny of the East. Amongst
  his contemporaries were Istakhri, who travelled through all the
  Mahommedan countries and wrote his _Book of Climates_ in 950, and Ibn
  Haukal, whose _Book of Roads and Kingdoms_, based on the work of
  Istakhri, was written in 976. Idrisi, the best known of the Arabian
  geographical authors, after travelling far and wide in the first half
  of the 12th century, settled in Sicily, where he wrote a treatise
  descriptive of an armillary sphere which he had constructed for Roger
  II., the Norman king, and in this work he incorporated all accessible
  results of contemporary travel.


    The Northmen.

  The Northmen of Denmark and Norway, whose piratical adventures were
  the terror of all the coasts of Europe, and who established themselves
  in Great Britain and Ireland, in France and Sicily, were also
  geographical explorers in their rough but practical way during the
  darkest period of the middle ages. All Northmen were not bent on
  rapine and plunder; many were peaceful merchants. Alfred the Great,
  king of the Saxons in England, not only educated his people in the
  learning of the past ages; he inserted in the geographical works he
  translated many narratives of the travel of his own time. Thus he
  placed on record the voyages of the merchant Ulfsten in the Baltic,
  including particulars of the geography of Germany. And in particular
  he told of the remarkable voyage of Other, a Norwegian of Helgeland,
  who was the first authentic Arctic explorer, the first to tell of the
  rounding of the North Cape and the sight of the midnight sun. This
  voyage of the middle of the 9th century deserves to be held in happy
  memory, for it unites the first Norwegian polar explorer with the
  first English collector of travels. Scandinavian merchants brought the
  products of India to England and Ireland. From the 8th to the 11th
  century a commercial route from India passed through Novgorod to the
  Baltic, and Arabian coins found in Sweden, and particularly in the
  island of Gotland, prove how closely the enterprise of the Northmen
  and of the Arabs intertwined. Five-sixths of these coins preserved at
  Stockholm were from the mints of the Samanian dynasty, which reigned
  in Khorasan and Transoxiana from about A.D. 900 to 1000. It was the
  trade with the East that originally gave importance to the city of
  Visby in Gotland.

  In the end of the 9th century Iceland was colonized from Norway; and
  about 985 the intrepid viking, Eric the Red, discovered Greenland, and
  induced some of his Icelandic countrymen to settle on its inhospitable
  shores. His son, Leif Ericsson, and others of his followers were
  concerned in the discovery of the North American coast (see VINLAND),
  which, but for the isolation of Iceland from the centres of European
  awakening, would have had momentous consequences. As things were, the
  importance of this discovery passed unrecognized. The story of two
  Venetians, Nicolo and Antonio Zeno, who gave a vague account of
  voyages in the northern seas in the end of the 13th century, is no
  longer to be accepted as history.


    Close of the dark ages.

  At length the long period of barbarism which accompanied and followed
  the fall of the Roman empire drew to a close in Europe. The Crusades
  had a favourable influence on the intellectual state of the Western
  nations. Interesting regions, known only by the scant reports of
  pilgrims, were made the objects of attention and study; while
  religious zeal, and the hope of gain, combined with motives of mere
  curiosity, induced several persons to travel by land into remote
  regions of the East, far beyond the countries to which the operations
  of the crusaders extended. Among these was Benjamin of Tudela, who set
  out from Spain in 1160, travelled by land to Constantinople, and
  having visited India and some of the eastern islands, returned to
  Europe by way of Egypt after an absence of thirteen years.


    Asiatic journeys.

  Joannes de Plano Carpini, a Franciscan monk, was the head of one of
  the missions despatched by Pope Innocent to call the chief and people
  of the Tatars to a better mind. He reached the headquarters of Batu,
  on the Volga, in February 1246; and, after some stay, went on to the
  camp of the great khan near Karakorum in central Asia, and returned
  safely in the autumn of 1247. A few years afterwards, a Fleming named
  Rubruquis was sent on a similar mission, and had the merit of being
  the first traveller of this era who gave a correct account of the
  Caspian Sea. He ascertained that it had no outlet. At nearly the same
  time Hayton, king of Armenia, made a journey to Karakorum in 1254, by
  a route far to the north of that followed by Carpini and Rubruquis. He
  was treated with honour and hospitality, and returned by way of
  Samarkand and Tabriz, to his own territory. The curious narrative of
  King Hayton was translated by Klaproth.

  While the republics of Italy, and above all the state of Venice, were
  engaged in distributing the rich products of India and the Far East
  over the Western world, it was impossible that motives of curiosity,
  as well as a desire of commercial advantage, should not be awakened to
  such a degree as to impel some of the merchants to visit those remote
  lands. Among these were the brothers Polo, who traded with the East
  and themselves visited Tatary. The recital of their travels fired the
  youthful imagination of young Marco Polo, son of Nicolo, and he set
  out for the court of Kublai Khan, with his father and uncle, in 1265.
  Marco remained for seventeen years in the service of the Great Khan,
  and was employed on many important missions. Besides what he learnt
  from his own observation, he collected much information from others
  concerning countries which he did not visit. He returned to Europe
  possessed of a vast store of knowledge respecting the eastern parts of
  the world, and, being afterwards made a prisoner by the Genoese, he
  dictated the narrative of his travels during his captivity. The work
  of Marco Polo is the most valuable narrative of travels that appeared
  during the middle ages, and despite a cold reception and many denials
  of the accuracy of the record, its substantial truthfulness has been
  abundantly proved.

  Missionaries continued to do useful geographical work. Among them were
  John of Monte Corvino, a Franciscan monk, Andrew of Perugia, John
  Marignioli and Friar Jordanus, who visited the west coast of India,
  and above all Friar Odoric of Pordenone. Odoric set out on his travels
  about 1318, and his journeys embraced parts of India, the Malay
  Archipelago, China and even Tibet, where he was the first European to
  enter Lhasa, not yet a forbidden city.

  Ibn Batuta, the great Arab traveller, is separated by a wide space of
  time from his countrymen already mentioned, and he finds his proper
  place in a chronological notice after the days of Marco Polo, for he
  did not begin his wanderings until 1325, his career thus coinciding in
  time with the fabled journeyings of Sir John Mandeville. While Arab
  learning flourished during the darkest ages of European ignorance, the
  last of the Arab geographers lived to see the dawn of the great period
  of the European awakening. Ibn Batuta went by land from Tangier to
  Cairo, then visited Syria, and performed the pilgrimages to Medina and
  Mecca. After exploring Persia, and again residing for some time at
  Mecca, he made a voyage down the Red sea to Yemen, and travelled
  through that country to Aden. Thence he visited the African coast,
  touching at Mombasa and Quiloa, and then sailed across to Ormuz and
  the Persian Gulf. He crossed Arabia from Bahrein to Jidda, traversed
  the Red sea and the desert to Syene, and descended the Nile to Cairo.
  After this he revisited Syria and Asia Minor, and crossed the Black
  sea, the desert from Astrakhan to Bokhara, and the Hindu Kush. He was
  in the service of Muhammad Tughluk, ruler of Delhi, about eight years,
  and was sent on an embassy to China, in the course of which the
  ambassadors sailed down the west coast of India to Calicut, and then
  visited the Maldive Islands and Ceylon. Ibn Batuta made the voyage
  through the Malay Archipelago to China, and on his return he proceeded
  from Malabar to Bagdad and Damascus, ultimately reaching Fez, the
  capital of his native country, in November 1349. After a journey into
  Spain he set out once more for Central Africa in 1352, and reached
  Timbuktu and the Niger, returning to Fez in 1353. His narrative was
  committed to writing from his dictation.


    Spanish exploration.

  The European country which had come the most completely under the
  influence of Arab culture now began to send forth explorers to distant
  lands, though the impulse came not from the Moors but from Italian
  merchant navigators in Spanish service. The peaceful reign of Henry
  III. of Castile is famous for the attempts of that prince to extend
  the diplomatic relations of Spain to the remotest parts of the earth.
  He sent embassies to all the princes of Christendom and to the Moors.
  In 1403 the Spanish king sent a knight of Madrid, Ruy Gonzalez de
  Clavijo, to the distant court of Timur, at Samarkand. He returned in
  1406, and wrote a valuable narrative of his travels.

  Italians continued to make important journeys in the East during the
  15th century. Among them was Nicolo Conti, who passed through Persia,
  sailed along the coast of Malabar, visited Sumatra, Java and the south
  of China, returned by the Red sea, and got home to Venice in 1444
  after an absence of twenty-five years. He related his adventures to
  Poggio Bracciolini, secretary to Pope Eugenius IV.; and the narrative
  contains much interesting information. One of the most remarkable of
  the Italian travellers was Ludovico di Varthema, who left his native
  land in 1502. He went to Egypt and Syria, and for the sake of visiting
  the holy cities became a Mahommedan. He was the first European who
  gave an account of the interior of Yemen. He afterwards visited and
  described many places in Persia, India and the Malay Archipelago,
  returning to Europe in a Portuguese ship after an absence of five
  years.


    Portuguese exploration--Prince Henry the Navigator.

  In the 15th century the time was approaching when the discovery of the
  Cape of Good Hope was to widen the scope of geographical enterprise.
  This great event was preceded by the general utilization in Europe of
  the polarity of the magnetic needle in the construction of the
  mariner's compass. Portugal took the lead along this new path, and
  foremost among her pioneers stands Prince Henry the Navigator
  (1394-1460), who was a patron both of exploration and of the study of
  geographical theory. The great westward projection of the coast of
  Africa, and the islands to the north-west of that continent, were the
  principal scene of the work of the mariners sent out at his expense;
  but his object was to push onward and reach India from the Atlantic.
  The progress of discovery received a check on his death, but only for
  a time. In 1462 Pedro de Cintra extended Portuguese exploration along
  the African coast and discovered Sierra Leone. Fernan Gomez followed
  in 1469, and opened trade with the Gold Coast; and in 1484 Diogo Cão
  discovered the mouth of the Congo. The king of Portugal next
  despatched Bartolomeu Diaz in 1486 to continue discoveries southwards;
  while, in the following year, he sent Pedro de Covilhão and Affonso de
  Payva to discover the country of Prester John. Diaz succeeded in
  rounding the southern point of Africa, which he named Cabo
  Tormentoso--the Cape of Storms--but King João II., foreseeing the
  realization of the long-sought passage to India, gave it the
  stimulating and enduring name of the Cape of Good Hope. Payva died at
  Cairo; but Covilhão, having heard that a Christian ruler reigned in
  the mountains of Ethiopia, penetrated into Abyssinia in 1490. He
  delivered the letter which João II. had addressed to Prester John to
  the Negus Alexander of Abyssinia, but he was detained by that prince
  and never allowed to leave the country.


    Columbus.

  The Portuguese, following the lead of Prince Henry, continued to look
  for the road to India by the Cape of Good Hope. The same end was
  sought by Christopher Columbus, following the suggestion of
  Toscanelli, and under-estimating the diameter of the globe, by sailing
  due west. The voyages of Columbus (1492-1498) resulted in the
  discovery of the West Indies and North America which barred the way to
  the Far East. In 1493 the pope, Alexander VI., issued a bull
  instituting the famous "line of demarcation" running from N. to S. 100
  leagues W. of the Azores, to the west of which the Spaniards were
  authorized to explore and to the east of which the Portuguese received
  the monopoly of discovery. The direct line of Portuguese exploration
  resulted in the discovery of the Cape route to India by Vasco da Gama
  (1498), and in 1500 to the independent discovery of South America by
  Pedro Alvarez Cabral. The voyages of Columbus and of Vasco da Gama
  were so important that it is unnecessary to detail their results in
  this place. See COLUMBUS, CHRISTOPHER; GAMA, VASCO DA.


    Vasco da Gama.

  The three voyages of Vasco da Gama (who died on the scene of his
  labours, at Cochin, in 1524) revolutionized the commerce of the East.
  Until then the Venetians held the carrying trade of India, which was
  brought by the Persian Gulf and Red sea into Syria and Egypt, the
  Venetians receiving the products of the East at Alexandria and Beirut
  and distributing them over Europe. This commerce was a great source of
  wealth to Venice; but after the discovery of the new passage round the
  Cape, and the conquests of the Portuguese, the trade of the East
  passed into other hands.


    Spaniards in America.

  The discoveries of Columbus awakened a spirit of enterprise in Spain
  which continued in full force for a century; adventurers flocked
  eagerly across the Atlantic, and discovery followed discovery in rapid
  succession. Many of the companions of Columbus continued his work.
  Vicente Yañez Pinzon in 1500 reached the mouth of the Amazon. In the
  same year Alonso de Ojeda, accompanied by Juan de la Cosa, from whose
  maps we learn much of the discoveries of the 16th century navigators,
  and by a Florentine named Amerigo Vespucci, touched the coast of South
  America somewhere near Surinam, following the shore as far as the Gulf
  of Maracaibo. Vespucci afterwards made three voyages to the Brazilian
  coast; and in 1504 he wrote an account of his four voyages, which was
  widely circulated, and became the means of procuring for its author at
  the hands of the cartographer Waldseemüller in 1507 the
  disproportionate distinction of giving his name to the whole
  continent. In 1508 Alonso de Ojeda obtained the government of the
  coast of South America from Cabo de la Vela to the Gulf of Darien;
  Ojeda landed at Cartagena in 1510, and sustained a defeat from the
  natives, in which his lieutenant, Juan de la Cosa, was killed. After
  another reverse on the east side of the Gulf of Darien Ojeda returned
  to Hispaniola and died there. The Spaniards in the Gulf of Darien were
  left by Ojeda under the command of Francisco Pizarro, the future
  conqueror of Peru. After suffering much from famine and disease,
  Pizarro resolved to leave, and embarked the survivors in small
  vessels, but outside the harbour they met a ship which proved to be
  that of Martin Fernandez Enciso, Ojeda's partner, coming with
  provisions and reinforcements. One of the crew of Enciso's ship, Vasco
  Nuñez de Balboa, the future discoverer of the Pacific Ocean, induced
  his commander to form a settlement on the other side of the Gulf of
  Darien. The soldiers became discontented and deposed Enciso, who was a
  man of learning and an accomplished cosmographer. His work _Suma de
  Geografia_, which was printed in 1519, is the first Spanish book which
  gives an account of America. Vasco Nuñez, the new commander, entered
  upon a career of conquest in the neighbourhood of Darien, which ended
  in the discovery of the Pacific Ocean on the 25th of September 1513.
  Vasco Nuñez was beheaded in 1517 by Pedrarias de Avila, who was sent
  out to supersede him. This was one of the greatest calamities that
  could have happened to South America; for the discoverer of the South
  sea was on the point of sailing with a little fleet into his unknown
  ocean, and a humane and judicious man would probably have been the
  conqueror of Peru, instead of the cruel and ignorant Pizarro. In the
  year 1519 Panama was founded by Pedrarias; and the conquest of Peru by
  Pizarro followed a few years afterwards. Hernan Cortes overran and
  conquered Mexico from 1518 to 1521, and the discovery and conquest of
  Guatemala by Alvarado, the invasion of Florida by De Soto, and of
  Nueva Granada by Quesada, followed in rapid succession. The first
  detailed account of the west coast of South America was written by a
  keenly observant old soldier, Pedro de Cieza de Leon, who was
  travelling in South America from 1533 to 1550, and published his story
  at Seville in 1553.


    Pacific Ocean.

  The great desire of the Spanish government at that time was to find a
  westward route to the Moluccas. For this purpose Juan Diaz de Solis
  was despatched in October 1515, and in January 1516 he discovered the
  mouth of the Rio de la Plata. He was, however, killed by the natives,
  and his ships returned. In the following year the Portuguese
  Ferdinando Magalhães, familiarly known as Magellan, laid before
  Charles V., at Valladolid, a scheme for reaching the Spice Islands by
  sailing westward. He started on the 21st of September 1519, entered
  the strait which now bears his name in October 1520, worked his way
  through between Patagonia and Tierra del Fuego, and entered on the
  vast Pacific which he crossed without sighting any of its innumerable
  island groups. This was unquestionably the greatest of the voyages
  which followed from the impulse of Prince Henry, and it was rendered
  possible only by the magnificent courage of the commander in spite of
  rebellion, mutiny and starvation. It was the 6th of March 1521 when he
  reached the Ladrone Islands. Thence Magellan proceeded to the
  Philippines, and there his career ended in an unimportant encounter
  with hostile natives. Eventually a Biscayan named Sebastian del Cano,
  sailing home by way of the Cape of Good Hope, reached San Lucar in
  command of the "Victoria" on the 6th of September 1522, with eighteen
  survivors; this one ship of the squadron which sailed on the quest
  succeeded in accomplishing the first circumnavigation of the globe.
  Del Cano was received with great distinction by the emperor, who
  granted him a globe for his crest, and the motto _Primus circumdedisti
  me_.


    Portuguese in Africa and the East.

  While the Spaniards were circumnavigating the world and completing
  their knowledge of the coasts of Central and South America, the
  Portuguese were actively engaged on similar work as regards Africa and
  the East Indies.

  With Abyssinia the mission of Covilhão led to further intercourse. In
  April 1520 Vasco da Gama, as viceroy of the Indies, took a fleet into
  the Red sea, and landed an embassy consisting of Dom Rodriguez de Lima
  and Father Francisco Alvarez, a priest whose detailed narrative is the
  earliest and not the least interesting account we possess of
  Abyssinia. It was not until 1526 that the embassy was dismissed; and
  not many years afterwards the negus entreated the help of the
  Portuguese against Mahommedan invaders, and the viceroy sent an
  expeditionary force, commanded by his brother Cristoforo da Gama, with
  450 musketeers. Da Gama was taken prisoner and killed, but his
  followers enabled the Christians of Abyssinia to regain their power,
  and a Jesuit mission remained in the country. The Portuguese also
  established a close connexion with the kingdom of Congo on the west
  side of Africa, and obtained much information respecting the interior
  of the continent. Duarte Lopez, a Portuguese settled in the country,
  was sent on a mission to Rome by the king of Congo, and Pope Sixtus V.
  caused him to recount to his chamberlain, Felipe Pigafetta, all he had
  learned during the nine years he had been in Africa, from 1578 to
  1587. This narrative, under the title of _Description of the Kingdom
  of Congo_, was published at Rome by Pigafetta in 1591. A map was
  attached on which several great equatorial lakes are shown, and the
  empire of Monomwezi or Unyamwezi is laid down. The most valuable work
  on Africa about this time is, however, that written by the Moor Leo
  Africanus in the early part of the 16th century. Leo travelled
  extensively in the north and west of Africa, and was eventually taken
  by pirates and sold to a master who presented him to Pope Leo X. At
  the pope's desire he translated his work on Africa into Italian.

  In Further India and the Malay Archipelago the Portuguese acquired
  predominating influence at sea, establishing factories on the Malabar
  coast, in the Persian Gulf, at Malacca, and in the Spice Islands, and
  extending their commercial enterprises from the Red sea to China.
  Their missionaries were received at the court of Akbar, and Benedict
  Goes, a native of the Azores, was despatched on a journey overland
  from Agra to China. He started in 1603, and, after traversing the
  least-known parts of Central Asia, he reached the confines of China.
  He appears to have ascended from Kabul to the plateau of the Pamir,
  and thence onwards by Yarkand, Khotan and Aksu. He died on the journey
  in March 1607; and thus, as one of the brethren pronounced his
  epitaph, "seeking Cathay he found heaven."


    English, Dutch and French.

  The activity and love of adventure, which became a passion for two or
  three generations in Spain and Portugal, spread to other countries. It
  was the spirit of the age; and England, Holland and France were fired
  by it. English enterprise was first aroused by John and Sebastian
  Cabot, father and son, who came from Venice and settled at Bristol in
  the time of Henry VII. The Cabots received a patent in 1496,
  empowering them to seek unknown lands; and John Cabot discovered
  Newfoundland and part of the coast of America. Sebastian afterwards
  made a voyage to Rio de la Plata in the service of Spain, but he
  returned to England in 1548 and received a pension from Edward VI. At
  his suggestion a voyage was undertaken for the discovery of a
  north-east passage to Cathay, with Sir Hugh Willoughby as
  captain-general of the fleet and Richard Chancellor as pilot-major.
  They sailed in May 1553, but Willoughby and all his crew perished on
  the Lapland coast. Chancellor, however, was more fortunate. He reached
  the White Sea, performed the journey overland to Moscow, where he was
  well received, and may be said to have been the founder of the trade
  between Russia and England. He returned to Archangel and brought his
  ship back in safety to England. On a second voyage, in 1556,
  Chancellor was drowned; and three subsequent voyages, led by Stephen
  Burrough, Arthur Pet and Charles Jackman, in small craft of 50 tons
  and under, carried on an examination of the straits which lead into
  the Kara sea.

  The French followed closely on the track of John Cabot, and Norman and
  Breton fishermen frequented the banks of Newfoundland at the beginning
  of the 16th century. In 1524 Francis I. sent Giovanni da Verazzano of
  Florence on an expedition of discovery to the coast of North America;
  and the details of his voyage were embodied in a letter addressed by
  him to the king of France from Dieppe, in July 1524. In 1534 Jacques
  Cartier set out to continue the discoveries of Verazzano, and visited
  Newfoundland and the Gulf of St Lawrence. In the following year he
  made another voyage, discovered the island of Anticosti, and ascended
  the St Lawrence to Hochelaga, now Montreal. He returned, after passing
  two winters in Canada; and on another occasion he also failed to
  establish a colony. Admiral de Coligny made several unsuccessful
  endeavours to form a colony in Florida under Jean Ribault of Dieppe,
  René de Laudonnière and others, but the settlers were furiously
  assailed by the Spaniards and the attempt was abandoned.


    The Elizabethan era.

  The reign of Elizabeth is famous for the gallant enterprises that were
  undertaken by sea and land to discover and bring to light the unknown
  parts of the earth. The great promoter of geographical discovery in
  the Elizabethan period was Richard Hakluyt (1553-1616), who was active
  in the formation of the two companies for colonizing Virginia in 1606;
  and devoted his life to encouraging and recording similar
  undertakings. He published much, and left many valuable papers at his
  death, most of which, together with many other narratives, were
  published in 1622 in the great work of the Rev. Samuel Purchas,
  entitled _Hakluytus Posthumus, or Purchas his Pilgrimes_.

  It is from these works that our knowledge of the gallant deeds of the
  English and other explorers of the Elizabethan age is mainly derived.
  The great and splendidly illustrated collections of voyages and
  travels of Theodorus de Bry and Hulsius served a similar useful
  purpose on the continent of Europe. One important object of English
  maritime adventurers of those days was to discover a route to Cathay
  by the north-west, a second was to settle Virginia, and a third was to
  raid the Spanish settlements in the West Indies. Nor was the trade to
  Muscovy and Turkey neglected; while latterly a resolute and successful
  attempt was made to establish direct commercial relations with India.

  The conception of the north-western route to Cathay now leads the
  story of exploration, for the first time as far as important and
  sustained efforts are concerned, towards the Arctic seas. This part of
  the story is fully told under the heading of POLAR REGIONS, and only
  the names of Martin Frobisher (1576), John Davis (1585), Henry Hudson
  (1607) and William Baffin (1616) need be mentioned here in order to
  preserve the complete conspectus of the history of discovery. The
  Dutch emulated the British in the Arctic seas during this period,
  directing their efforts mainly towards the discovery of a north-east
  passage round the northern end of Novaya Zemlya; and William Barents
  or Barendsz (1594-1597) is the most famous name in this connexion, his
  boat voyage along the coast of Novaya Zemlya after losing his ship and
  wintering in a high latitude, being one of the most remarkable
  achievements in polar annals.

  Many English voyages were also made to Guinea and the West Indies, and
  twice English vessels followed in the track of Magellan, and
  circumnavigated the globe. In 1577 Francis Drake, who had previously
  served with Hawkins in the West Indies, undertook his celebrated
  voyage round the world. Reaching the Pacific through the Strait of
  Magellan, Drake proceeded northward along the west coast of America,
  resolved to attempt the discovery of a northern passage from the
  Pacific to the Atlantic. The coast from the southern extremity of the
  Californian peninsula to Cape Mendocino had been discovered by Juan
  Rodriguez Cabrillo and Francisco de Ulloa in 1539. Drake's discoveries
  extended from Cape Mendocino to 48° N., in which latitude he gave up
  his quest, sailed across the Pacific and reached the Philippine
  Islands, returning home round the Cape of Good Hope in 1580.

  Thomas Cavendish, emulous of Drake's example, fitted out three vessels
  for an expedition to the South sea in 1586. He took the same route as
  Drake along the west coast of America. From Cape San Lucas Cavendish
  steered across the Pacific, seeing no land until he reached the
  Ladrone Islands. He returned to England in 1588. The third English
  voyage into the Pacific was not so fortunate. Sir Richard Hawkins
  (1593) on reaching the bay of Atacames, in 1°N. in 1594, was attacked
  by a Spanish fleet, and, after a desperate naval engagement, was
  forced to surrender. Hawkins declared his object to be discovery and
  the survey of unknown lands, and his voyage, though terminating in
  disaster, bore good fruit. _The Observations of Sir Richard Hawkins in
  his Voyage into the South Sea_, published in 1622, are very valuable.
  It was long before another British ship entered the Pacific Ocean. Sir
  John Narborough took two ships through the Strait of Magellan in 1670
  and touched on the coast of Chile, but it was not until 1685 that
  Dampier sailed over the part of the Pacific where Hawkins met his
  defeat.

  The exploring enterprise of the Spanish nation did not wane after the
  conquest of Peru and Mexico, and the acquisition of the vast empire of
  the Indies. It was spurred into renewed activity by the audacity of
  Sir John Hawkins in the West Indies, and by the appearance of Drake,
  Cavendish and Richard Hawkins in the Pacific.

  In the interior of South America the Spanish conquerors had explored
  the region of the Andes from the isthmus of Panama to Chile. Pedro de
  Valdivia in 1540 made an expedition into the country of the Araucanian
  Indians of Chile, and was the first to explore the eastern base of
  the Andes in what is now Argentine Patagonia. In 1541 Francisco de
  Orellana discovered the whole course of the Amazon from its source in
  the Andes to the Atlantic. A second voyage on the Amazon was made in
  1561 by the mad pirate Lope de Aguirre; but it was not until 1639 that
  a full account was written of the great river by Father Cristoval de
  Acuña, who ascended it from its mouth and reached the city of Quito.


    Spaniards in the Pacific.

  The voyage of Drake across the Pacific was preceded by that of Alvaro
  de Mendaña, who was despatched from Peru in 1567 to discover the great
  Antarctic continent which was believed to extend far northward into
  the South sea, the search for which now became one of the leading
  motives of exploration. After a voyage of eighty days across the
  Pacific, Mendaña discovered the Solomon Islands; and the expedition
  returned in safety to Callao. The appearance of Drake on the Peruvian
  coast led to an expedition being fitted out at Callao, to go in chase
  of him, under the command of Pedro Sarmiento. He sailed from Callao in
  October 1579, and made a careful survey of the Strait of Magellan,
  with the object of fortifying that entrance to the South sea. The
  colony which he afterwards took out from Spain was a complete failure,
  and is only remembered now from the name of "Port Famine," which
  Cavendish gave to the site at which he found the starving remnant of
  Sarmiento's settlers. In June 1595 Mendaña sailed from the coast of
  Peru in command of a second expedition to colonize the Solomon
  Islands. After discovering the Marquesas, he reached the island of
  Santa Cruz of evil memory, where he and many of the settlers died. His
  young widow took command of the survivors and brought them safely to
  Manila. The viceroys of Peru still persevered in their attempts to
  plant a colony in the hypothetical southern continent. Pedro Fernandez
  de Quiros, who was pilot under Mendaña and Luis Vaez de Torres, were
  sent in command of two ships to continue the work of exploration. They
  sailed from Callao in December 1605, and discovered several islands of
  the New Hebrides group. They anchored in a bay of a large island which
  Quiros named "Australia del Espiritu Santo." From this place Quiros
  returned to America, but Torres continued the voyage, passed through
  the strait between Australia and New Guinea which bears his name, and
  explored and mapped the southern and eastern coasts of New Guinea.

  The Portuguese, in the early part of the 17th century (1578-1640),
  were under the dominion of Spain, and their enterprise was to some
  extent damped; but their missionaries extended geographical knowledge
  in Africa. Father Francisco Paez acquired great influence in
  Abyssinia, and explored its highlands from 1600 to 1622. Fathers
  Mendez and Lobo traversed the deserts between the coast of the Red sea
  and the mountains, became acquainted with Lake Tsana, and discovered
  the sources of the Blue Nile in 1624-1633.


    Rivalry in the East.

  But the attention of the Portuguese was mainly devoted to vain
  attempts to maintain their monopoly of the trade of India against the
  powerful rivalry of the English and Dutch. The English enterprises
  were persevering, continuous and successful. James Lancaster made a
  voyage to the Indian Ocean from 1591 to 1594; and in 1599 the
  merchants and adventurers of London resolved to form a company, with
  the object of establishing a trade with the East Indies. On the 31st
  of December 1599 Queen Elizabeth granted the charter of incorporation
  to the East India Company, and Sir James Lancaster, one of the
  directors, was appointed general of their first fleet. He was
  accompanied by John Davis, the great Arctic navigator, as pilot-major.
  This voyage was eminently successful. The ships touched at Achin in
  Sumatra and at Java, returning with full ladings of pepper in 1603.
  The second voyage was commanded by Sir Henry Middleton; but it was in
  the third voyage, under Keelinge and Hawkins, that the mainland of
  India was first reached in 1607. Captain Hawkins landed at Surat and
  travelled overland to Agra, passing some time at the court of the
  Great Mogul. In the voyage of Sir Edward Michelborne in 1605, John
  Davis lost his life in a fight with a Japanese junk. The eighth
  voyage, led by Captain Saris, extended the operations of the company
  to Japan; and in 1613 the Japanese government granted privileges to
  the company; but the British retired in 1623, giving up their factory.
  The chief result of this early intercourse between Great Britain and
  Japan was the interesting series of letters written by William Adams
  from 1611 to 1617. From the tenth voyage of the East India Company,
  commanded by Captain Best, who left England in 1612, dates the
  establishment of permanent British factories on the coast of India. It
  was Captain Best who secured a regular _firman_ for trade from the
  Great Mogul. From that time a fleet was despatched every year, and the
  company's operations greatly increased geographical knowledge of India
  and the Eastern Archipelago. British visits to Eastern countries, at
  this time, were not confined to the voyages of the company. Journeys
  were also made by land, and, among others, the entertaining author of
  the _Crudities_, Thomas Coryate, of Odcombe in Somersetshire, wandered
  on foot from France to India, and died (1617) in the company's factory
  at Surat. In 1561 Anthony Jenkinson arrived in Persia with a letter
  from Queen Elizabeth to the shah. He travelled through Russia to
  Bokhara, and returned by the Caspian and Volga. In 1579 Christopher
  Burroughs built a ship at Nizhniy Novgorod and traded across the
  Caspian to Baku; and in 1598 Sir Anthony and Robert Shirley arrived in
  Persia, and Robert was afterwards sent by the shah to Europe as his
  ambassador. He was followed by a Spanish mission under Garcia de
  Silva, who wrote an interesting account of his travels; and to Sir
  Dormer Cotton's mission, in 1628, we are indebted for Sir Thomas
  Herbert's charming narrative. In like manner Sir Thomas Roe's mission
  to India resulted not only in a large collection of valuable reports
  and letters of his own, but also in the detailed account of his
  chaplain Terry. But the most learned and intelligent traveller in the
  East, during the 17th century, was the German, Engelbrecht Kaempfer,
  who accompanied an embassy to Persia, in 1684, and was afterwards a
  surgeon in the service of the Dutch East India Company. He was in the
  Persian Gulf, India and Java, and resided for more than two years in
  Japan, of which he wrote a history.


    Dutch exploration, 16th-17th centuries.

  The Dutch nation, as soon as it was emancipated from Spanish tyranny,
  displayed an amount of enterprise, which, for a long time, was fully
  equal to that of the British. The Arctic voyages of Barents were
  quickly followed by the establishment of a Dutch East India Company;
  and the Dutch, ousting the Portuguese, not only established factories
  on the mainland of India and in Japan, but acquired a preponderating
  influence throughout the Malay Archipelago. In 1583 Jan Hugen van
  Linschoten made a voyage to India with a Portuguese fleet, and his
  full and graphic descriptions of India, Africa, China and the Malay
  Archipelago must have been of no small use to his countrymen in their
  distant voyages. The first of the Dutch Indian voyages was performed
  by ships which sailed in April 1595, and rounded the Cape of Good
  Hope. A second large Dutch fleet sailed in 1598; and, so eager was the
  republic to extend her commerce over the world that another fleet,
  consisting of five ships of Rotterdam, was sent in the same year by
  way of Magellan's Strait, under Jacob Mahu as admiral, with William
  Adams as pilot. Mahu died on the passage out, and was succeeded by
  Simon de Cordes, who was killed on the coast of Chile. In September
  1599 the fleet had entered the Pacific. The ships were then steered
  direct for Japan, and anchored off Bungo in April 1600. In the same
  year, 1598, a third expedition was despatched under Oliver van Noort,
  a native of Utrecht, but the voyage contributed nothing to geography.
  The Dutch Company in 1614 again resolved to send a fleet to the
  Moluccas by the westward route, and Joris Spilbergen was appointed to
  the command as admiral, with a commission from the States-General. He
  was furnished with four ships of Amsterdam, two of Rotterdam and one
  from Zeeland. On the 6th of May 1615 Spilbergen entered the Pacific
  Ocean, and touched at several places on the coast of Chile and Peru,
  defeating the Spanish fleet in a naval engagement off Chilca. After
  plundering Payta and making requisitions at Acapulco, the Dutch fleet
  crossed the Pacific and reached the Moluccas in March 1616.

  The Dutch now resolved to discover a passage into the Pacific to the
  south of Tierra del Fuego, the insular nature of which had been
  ascertained by Sir Francis Drake. The vessels fitted out for this
  purpose were the "Eendracht," of 360 tons, commanded by Jacob Lemaire,
  and the "Hoorn," of 110 tons, under Willem Schouten. They sailed from
  the Texel on the 14th of June 1615, and by the 20th of January 1616
  they were south of the entrance of Magellan's Strait. Passing through
  the strait of Lemaire they came to the southern extremity of Tierra
  del Fuego, which was named Cape Horn, in honour of the town of Hoorn
  in West Friesland, of which Schouten was a native. They passed the
  cape on the 31st of January, encountering the usual westerly winds.
  The great merit of this discovery of a second passage into the South
  sea lies in the fact that it was not accidental or unforeseen, but was
  due to the sagacity of those who designed the voyage. On the 1st of
  March the Dutch fleet sighted the island of Juan Fernandez; and,
  having crossed the Pacific, the explorers sailed along the north coast
  of New Guinea and arrived at the Moluccas on the 17th of September
  1616.

  There were several early indications of the existence of the great
  Australian continent, and the Dutch endeavoured to obtain further
  knowledge concerning the country and its extent; but only its northern
  and western coasts had been visited before the time of Governor van
  Diemen. Dirk Hartog had been on the west coast in latitude 26° 30' S.
  in 1616. Pelsert struck on a reef called "Houtman's Abrolhos" on the
  4th of June 1629. In 1697 the Dutch captain Vlamingh landed on the
  west coast of Australia, then called New Holland, in 31° 43' S., and
  named the Swan river from the black swans he discovered there. In 1642
  the governor and council of Batavia fitted out two ships to prosecute
  the discovery of the south land, then believed to be part of a vast
  Antarctic continent, and entrusted the command to Captain Abel Jansen
  Tasman. This voyage proved to be the most important to geography that
  had been undertaken since the first circumnavigation of the globe.
  Tasman sailed from Batavia in 1642, and on the 24th of November
  sighted high land in 42° 30' S., which was named van Diemen's Land,
  and after landing there proceeded to the discovery of the western
  coast of New Zealand; at first called Staten Land, and supposed to be
  connected with the Antarctic continent from which this voyage proved
  New Holland to be separated. He then reached Tongatabu, one of the
  Friendly Islands of Cook; and returned by the north coast of New
  Guinea to Batavia. In 1644 Tasman made a second voyage to effect a
  fuller discovery of New Guinea.


    French in North America.

  The French directed their enterprise more in the direction of North
  America than of the Indies. One of their most distinguished explorers
  was Samuel Champlain, a captain in the navy, who, after a remarkable
  journey through Mexico and the West Indies from 1599 to 1602,
  established his historic connexion with Canada, to the geographical
  knowledge of which he made a very large addition.


    Missionaries in the East.

  The principles and methods of surveying and position finding had by
  this time become well advanced, and the most remarkable example of the
  early application of these improvements is to be found in the survey
  of China by Jesuit missionaries. They first prepared a map of the
  country round Peking, which was submitted to the emperor Kang-hi, and,
  being satisfied with the accuracy of the European method of surveying,
  he resolved to have a survey made of the whole empire on the same
  principles. This great work was begun in July 1708, and the completed
  maps were presented to the emperor in 1718. The records preserved in
  each city were examined, topographical information was diligently
  collected, and the Jesuit fathers checked their triangulation by
  meridian altitudes of the sun and pole star and by a system of
  remeasurements. The result was a more accurate map of China than
  existed, at that time, of any country in Europe. Kang-hi next ordered
  a similar map to be made of Tibet, the survey being executed by two
  lamas who were carefully trained as surveyors by the Jesuits at
  Peking. From these surveys were constructed the well-known maps which
  were forwarded to Duhalde, and which D'Anville utilized for his atlas.


    The 18th century.

    Asia.

  Several European missionaries had previously found their way from
  India to Tibet. Antonio Andrada, in 1624, was the first European to
  enter Tibet since the visit of Friar Odoric in 1325. The next journey
  was that of Fathers Grueber and Dorville about 1660, who succeeded in
  passing from China, through Tibet, into India. In 1715 Fathers
  Desideri and Freyre made their way from Agra, across the Himalayas, to
  Lhasa, and the Capuchin Friar Orazio della Penna resided in that city
  from 1735 until 1747. But the most remarkable journey in this
  direction was performed by a Dutch traveller named Samuel van de
  Putte. He left Holland in 1718, went by land through Persia to India,
  and eventually made his way to Lhasa, where he resided for a long
  time. He went thence to China, returned to Lhasa, and was in India in
  time to be an eye-witness of the sack of Delhi by Nadir Shah in 1737.
  In 1743 he left India and died at Batavia on the 27th of September
  1745. The premature death of this illustrious traveller is the more to
  be lamented because his vast knowledge died with him. Two English
  missions sent by Warren Hastings to Tibet, one led by George Bogle in
  1774, and the other by Captain Turner in 1783, complete Tibetan
  exploration in the 18th century.

  From Persia much new information was supplied by Jean Chardin, Jean
  Tavernier, Charles Hamilton, Jean de Thévenot and Father Jude
  Krusinski, and by English traders on the Caspian. In 1738 John Elton
  traded between Astrakhan and the Persian port of Enzelî on the
  Caspian, and undertook to build a fleet for Nadir Shah. Another
  English merchant, named Jonas Hanway, arrived at Astrabad from Russia,
  and travelled to the camp of Nadir at Kazvin. One lasting and valuable
  result of Hanway's wanderings was a charming book of travels. In 1700
  Guillaume Delisle published his map of the continents of the Old
  World; and his successor D'Anville produced his map of India in 1752.
  D'Anville's map contained all that was then known, but ten years
  afterwards Major Rennell began his surveying labours, which extended
  over the period from 1763 to 1782. His survey covered an area 900 m.
  long by 300 wide, from the eastern confines of Bengal to Agra, and
  from the Himalayas to Calpi. Rennell was indefatigable in collecting
  geographical information; his Bengal atlas appeared in 1781, his
  famous map of India in 1788 and the memoir in 1792. Surveys were also
  made along the Indian coasts.

  Arabia received very careful attention, in the 18th century, from the
  Danish scientific mission, which included Carsten Niebuhr among its
  members. Niebuhr landed at Loheia, on the coast of Yemen, in December
  1762, and went by land to Sana. All the other members of the mission
  died, but he proceeded from Mokha to Bombay. He then made a journey
  through Persia and Syria to Constantinople, returning to Copenhagen in
  1767. His valuable work, the _Description of Arabia_, was published in
  1772, and was followed in 1774-1778 by two volumes of travels in Asia.
  The great traveller survived until 1815, when he died at the age of
  eighty-two.


    Africa.

  James Bruce of Kinnaird, the contemporary of Niebuhr, was equally
  devoted to Eastern travel; and his principal geographical work was the
  tracing of the Blue Nile from its source to its junction with the
  White Nile. Before the death of Bruce an African Association was
  formed, in 1788, for collecting information respecting the interior of
  that continent, with Major Rennell and Sir Joseph Banks as leading
  members. The association first employed John Ledyard (who had
  previously made an extraordinary journey into Siberia) to cross Africa
  from east to west on the parallel of the Niger, and William Lucas to
  cross the Sahara to Fezzan. Lucas went from Tripoli to Mesurata,
  obtained some information respecting Fezzan and returned in 1789. One
  of the chief problems the association wished to solve was that of the
  existence and course of the river Niger, which was believed by some
  authorities to be identical with the Congo. Mungo Park, then an
  assistant surgeon of an Indiaman, volunteered his services, which were
  accepted by the association, and in 1795 he succeeded in reaching the
  town of Segu on the Niger, but was prevented from continuing his
  journey to Timbuktu. Five years later he accepted an offer from the
  government to command an expedition into the interior of Africa, the
  plan being to cross from the Gambia to the Niger and descend the
  latter river to the sea. After losing most of his companions he
  himself and the rest perished in a rapid on the Niger at Busa, having
  been attacked from the shore by order of a chief who thought he had
  not received suitable presents. His work, however, had established the
  fact that the Niger was not identical with the Congo.

  While the British were at work in the direction of the Niger, the
  Portuguese were not unmindful of their old exploring fame. In 1798 Dr
  F.J.M. de Lacerda, an accomplished astronomer, was appointed to
  command a scientific expedition of discovery to the north of the
  Zambesi. He started in July, crossed the Muchenja Mountains, and
  reached the capital of the Cazembe, where he died of fever. Lacerda
  left a valuable record of his adventurous journey; but with Mungo Park
  and Lacerda the history of African exploration in the 18th century
  closes.


    South America.

  In South America scientific exploration was active during this period.
  The great geographical event of the century, as regards that
  continent, was the measurement of an arc of the meridian. The
  undertaking was proposed by the French Academy as part of an
  investigation with the object of ascertaining the length of the degree
  near the equator and near the pole respectively so as to determine the
  figure of the earth. A commission left Paris in 1735, consisting of
  Charles Marie de la Condamine, Pierre Bouguer, Louis Godin and Joseph
  de Jussieu the naturalist. Spain appointed two accomplished naval
  officers, the brothers Ulloa, as coadjutors. The operations were
  carried on during eight years on a plain to the south of Quito; and,
  in addition to his memoir on this memorable measurement, La Condamine
  collected much valuable geographical information during a voyage down
  the Amazon. The arc measured was 3° 7' 3" in length; and the work
  consisted of two measured bases connected by a series of triangles,
  one north and the other south of the equator, on the meridian of
  Quito. Contemporaneously, in 1738, Pierre Louis Moreau de Maupertuis,
  Alexis Claude Clairaut, Charles Etienne Louis Camus, Pierre Charles
  Lemonnier and the Swedish physicist Celsius measured an arc of the
  meridian in Lapland.


    The Pacific Ocean.

  The British and French governments despatched several expeditions of
  discovery into the Pacific and round the world during the 18th
  century. They were preceded by the wonderful and romantic voyages of
  the buccaneers. The narratives of such men as Woodes Rogers, Edward
  Davis, George Shelvocke, Clipperton and William Dampier, can never
  fail to interest, while they are not without geographical value. The
  works of Dampier are especially valuable, and the narratives of
  William Funnell and Lionel Wafer furnished the best accounts then
  extant of the Isthmus of Darien. Dampier's literary ability eventually
  secured for him a commission in the king's service; and he was sent on
  a voyage of discovery, during which he explored part of the coasts of
  Australia and New Guinea, and discovered the strait which bears his
  name between New Guinea and New Britain, returning in 1701. In 1721
  Jacob Roggewein was despatched on a voyage of some importance across
  the Pacific by the Dutch West India Company, during which he
  discovered Easter Island on the 6th of April 1722.

  The voyage of Lord Anson to the Pacific in 1740-1744 was of a
  predatory character, and he lost more than half his men from scurvy;
  while it is not pleasant to reflect that at the very time when the
  French and Spaniards were measuring an arc of the meridian at Quito,
  the British under Anson were pillaging along the coast of the Pacific
  and burning the town of Payta. But a romantic interest attaches to the
  wreck of the "Wager," one of Anson's fleet, on a desert island near
  Chiloe, for it bore fruit in the charming narrative of Captain John
  Byron, which will endure for all time. In 1764 Byron himself was sent
  on a voyage of discovery round the world, which led immediately after
  his return to the despatch of another to complete his work, under the
  command of Captain Samuel Wallis.

  The expedition, consisting of the "Dolphin" commanded by Wallis, and
  the "Swallow" under Captain Philip Carteret, sailed in September 1766,
  but the ships were separated on entering the Pacific from the Strait
  of Magellan. Wallis discovered Tahiti on the 19th of June 1767, and he
  gave a detailed account of that island. He returned to England in May
  1768. Carteret discovered the Charlotte and Gloucester Islands, and
  Pitcairn Island on the 2nd of July 1767; revisited the Santa Cruz
  group, which was discovered by Mendaña and Quiros; and discovered the
  strait separating New Britain from New Ireland. He reached Spithead
  again in February 1769. Wallis and Carteret were followed very closely
  by the French expedition of Bougainville, which sailed from Nantes in
  November 1766. Bougainville had first to perform the unpleasant task
  of delivering up the Falkland Islands, where he had encouraged the
  formation of a French settlement, to the Spaniards. He then entered
  the Pacific, and reached Tahiti in April 1768. Passing through the New
  Hebrides group he touched at Batavia, and arrived at St Malo after an
  absence of two years and four months.


    Captain Cook.

  The three voyages of Captain James Cook form an era in the history of
  geographical discovery. In 1767 he sailed for Tahiti, with the object
  of observing the transit of Venus, accompanied by two naturalists, Sir
  Joseph Banks and Dr Solander, a pupil of Linnaeus, as well as by two
  astronomers. The transit was observed on the 3rd of June 1769. After
  exploring Tahiti and the Society group, Cook spent six months
  surveying New Zealand, which he discovered to be an island, and the
  coast of New South Wales from latitude 38° S. to the northern
  extremity. The belief in a vast Antarctic continent stretching far
  into the temperate zone had never been abandoned, and was vehemently
  asserted by Charles Dalrymple, a disappointed candidate nominated by
  the Royal Society for the command of the Transit expedition of 1769.
  In 1772 the French explorer Yves Kerguelen de Tremarec had discovered
  the land that bears his name in the South Indian Ocean without
  recognizing it to be an island, and naturally believed it to be part
  of the southern continent.

  Cook's second voyage was mainly intended to settle the question of the
  existence of such a continent once for all, and to define the limits
  of any land that might exist in navigable seas towards the Antarctic
  circle. James Cook at his first attempt reached a south latitude of
  57° 15'. On a second cruise from the Society Islands, in 1773, he,
  first of all men, crossed the Antarctic circle, and was stopped by ice
  in 71° 10' S. During the second voyage Cook visited Easter Island,
  discovered several islands of the New Hebrides and New Caledonia; and
  on his way home by Cape Horn, in March 1774, he discovered the
  Sandwich Island group and described South Georgia. He proved
  conclusively that any southern continent that might exist lay under
  the polar ice. The third voyage was intended to attempt the passage
  from the Pacific to the Atlantic by the north-east. The "Resolution"
  and "Discovery" sailed in 1776, and Cook again took the route by the
  Cape of Good Hope. On reaching the North American coast, he proceeded
  northward, fixed the position of the western extremity of America and
  surveyed Bering Strait. He was stopped by the ice in 70° 41' N., and
  named the farthest visible point on the American shore Icy Cape. He
  then visited the Asiatic shore and discovered Cape North. Returning to
  Hawaii, Cook was murdered by the natives. On the 14th of February
  1779, his second, Captain Edward Clerke, took command, and proceeding
  to Petropavlovsk in the following summer, he again examined the edge
  of the ice, but only got as far as 70° 33' N. The ships returned to
  England in October 1780.

  In 1785 the French government carefully fitted out an expedition of
  discovery at Brest, which was placed under the command of François La
  Pérouse, an accomplished and experienced officer. After touching at
  Concepcion in Chile and at Easter Island, La Pérouse proceeded to
  Hawaii and thence to the coast of California, of which he has given a
  very interesting account. He then crossed the Pacific to Macao, and in
  July 1787 he proceeded to explore the Gulf of Tartary and the shores
  of Sakhalin, remaining some time at Castries Bay, so named after the
  French minister of marine. Thence he went to the Kurile Islands and
  Kamchatka, and sailed from the far north down the meridian to the
  Navigator and Friendly Islands. He was in Botany Bay in January 1788;
  and sailing thence, the explorer, his ship and crew were never seen
  again. Their fate was long uncertain. In September 1791 Captain
  Antoine d'Entrecasteaux sailed from Brest with two vessels to seek for
  tidings. He visited the New Hebrides, Santa Cruz, New Caledonia and
  Solomon Islands, and made careful though rough surveys of the
  Louisiade Archipelago, islands north of New Britain and part of New
  Guinea. D'Entrecasteaux died on board his ship on the 20th of July
  1793, without ascertaining the fate of La Pérouse. Captain Peter
  Dillon at length ascertained, in 1828, that the ships of La Pérouse
  had been wrecked on the island of Vanikoro during a hurricane.

  The work of Captain Cook bore fruit in many ways. His master, Captain
  William Bligh, was sent in the "Bounty" to convey breadfruit plants
  from Tahiti to the West Indies. He reached Tahiti in October 1788, and
  in April 1789 a mutiny broke out, and he, with several officers and
  men, was thrust into an open boat in mid-ocean. During the remarkable
  voyage he then made to Timor, Bligh passed amongst the northern
  islands of the New Hebrides, which he named the Banks Group, and made
  several running surveys. He reached England in March 1790. The
  "Pandora," under Captain Edwards, was sent out in search of the
  "Bounty," and discovered the islands of Cherry and Mitre, east of the
  Santa Cruz group, but she was eventually lost on a reef in Torres
  Strait. In 1796-1797 Captain Wilson, in the missionary ship "Duff,"
  discovered the Gambier and other islands, and rediscovered the islands
  known to and seen by Quiros, but since called the Duff Group. Another
  result of Captain Cook's work was the colonization of Australia. On
  the 18th of January 1788 Admiral Phillip and Captain Hunter arrived in
  Botany Bay in the "Supply" and "Sirius," followed by six transports,
  and established a colony at Port Jackson. Surveys were then undertaken
  in several directions. In 1795 and 1796 Matthew Flinders and George
  Bass were engaged on exploring work in a small boat called the "Tom
  Thumb." In 1797 Bass, who had been a surgeon, made an expedition
  southwards, continued the work of Cook from Ram Head, and explored the
  strait which bears his name, and in 1798 he and Flinders were
  surveying on the east coast of Van Diemen's land.

  Yet another outcome of Captain Cook's work was the voyage of George
  Vancouver, who had served as a midshipman in Cook's second and third
  voyages. The Spaniards under Quadra had begun a survey of
  north-western America and occupied Nootka Sound, which their
  government eventually agreed to surrender. Captain Vancouver was sent
  out to receive the cession, and to survey the coast from Cape
  Mendocino northwards. He commanded the old "Discovery," and was at
  work during the seasons of 1792, 1793 and 1794, wintering at Hawaii.
  Returning home in 1795, he completed his narrative and a valuable
  series of charts.


    Arctic regions.

  The 18th century saw the Arctic coast of North America reached at two
  points, as well as the first scientific attempt to reach the North
  Pole. The Hudson Bay Company had been incorporated in 1670, and its
  servants soon extended their operations over a wide area to the north
  and west of Canada. In 1741 Captain Christopher Middleton was ordered
  to solve the question of a passage from Hudson Bay to the westward.
  Leaving Fort Churchill in July 1742, he discovered the Wager river and
  Repulse Bay. He was followed by Captain W. Moor in 1746, and Captain
  Coats in 1751, who examined the Wager Inlet up to the end. In November
  1769 Samuel Hearne was sent by the Hudson Bay Company to discover the
  sea on the north side of America, but was obliged to return. In
  February 1770 he set out again from Fort Prince of Wales; but, after
  great hardships, he was again forced to return to the fort. He started
  once more in December 1771, and at length reached the Coppermine
  river, which he surveyed to its mouth, but his observations are
  unreliable. With the same object Alexander Mackenzie, with a party of
  Canadians, set out from Fort Chippewyan on the 3rd of June 1789, and
  descending the great river which now bears the explorer's name reached
  the Arctic sea.

  In February 1773 the Royal Society submitted a proposal to the king
  for an expedition towards the North Pole. The expedition was fitted
  out under Captains Constantine Phipps and Skeffington Lutwidge, and
  the highest latitude reached was 80° 48' N., but no opening was
  discovered in the heavy Polar pack. The most important Arctic work in
  the 18th century was performed by the Russians, for they succeeded in
  delineating the whole of the northern coast of Siberia. Some of this
  work was possibly done at a still earlier date. The Cossack Simon
  Dezhneff is thought to have made a voyage, in the summer of 1648, from
  the river Kolyma, through Bering Strait (which was rediscovered by
  Vitus Bering in 1728) to Anadyr. Between 1738 and 1750 Manin and
  Sterlegoff made their way in small sloops from the mouth of the
  Yenesei as far north as 75° 15' N. The land from Taimyr to Cape
  Chelyuskin, the most northern extremity of Siberia, was mapped in many
  years of patient exploration by Chelyuskin, who reached the extreme
  point (77° 34' N.) in May 1742. To the east of Cape Chelyuskin the
  Russians encountered greater difficulties. They built small vessels at
  Yakutsk on the Lena, 900 m. from its mouth, whence the first
  expedition was despatched under Lieut. Prontschichev in 1735. He
  sailed from the mouth of the Lena to the mouth of the Olonek, where he
  wintered, and on the 1st of September 1736 he got as far as 77° 29'
  N., within 5 m. of Cape Chelyuskin. Both he and his young wife died of
  scurvy, and the vessel returned. A second expedition, under Lieut.
  Laptyev, started from the Lena in 1739, but encountered masses of
  drift ice in Chatanga bay, and with this ended the voyages to the
  westward of the Lena. Several attempts were also made to navigate the
  sea from the Lena to the Kolyma. In 1736 Lieut. Laptyev sailed, but
  was stopped by the drift ice in August, and in 1739, during another
  trial, he reached the mouth of the Indigirka, where he wintered. In
  the season of 1740 he continued his voyage to beyond the Kolyma,
  wintering at Nizhni Kolymsk. In September 1740 Vitus Bering sailed
  from Okhotsk on a second Arctic voyage with George William Steller on
  board as naturalist. In June 1741 he named the magnificent peak on the
  coast of North America Mount St Elias and explored the Aleutian
  Islands. In November the ship was wrecked on Bering Island; and the
  gallant Dane, worn out with scurvy, died there on the 8th of December
  1741. In March 1770 a merchant named Liakhov saw a large herd of
  reindeer coming from the north to the Siberian coast, which induced
  him to start in a sledge in the direction whence they came. Thus he
  reached the New Siberian or Liakhov Islands, and for years afterwards
  the seekers for fossil ivory resorted to them. The Russian Captain
  Vassili Chitschakov in 1765 and 1766 made two persevering attempts to
  penetrate the ice north of Spitsbergen, and reached 80° 30' N., while
  Russian parties twice wintered at Bell Sound.


    Geographical societies.

  In reviewing the progress of geographical discovery thus far, it has
  been possible to keep fairly closely to a chronological order. But in
  the 19th century and after exploring work was so generally and
  steadily maintained in all directions, and was in so many cases
  narrowed down from long journeys to detailed surveys within relatively
  small areas, that it becomes desirable to cover the whole period at
  one view for certain great divisions of the world. (See AFRICA; ASIA;
  AUSTRALIA; POLAR REGIONS; &c.) Here, however, may be noticed the
  development of geographical societies devoted to the encouragement of
  exploration and research. The first of the existing geographical
  societies was that of Paris, founded in 1825 under the title of La
  Société de Géographie. The Berlin Geographical Society (Gesellschaft
  für Erdkunde) is second in order of seniority, having been founded in
  1827. The Royal Geographical Society, which was founded in London in
  1830, comes third on the list; but it may be viewed as a direct result
  of the earlier African Association founded in 1788. Sir John Barrow,
  Sir John Cam Hobhouse (Lord Broughton), Sir Roderick Murchison, Mr
  Robert Brown and Mr Bartle Frere formed the foundation committee of
  the Royal Geographical Society, and the first president was Lord
  Goderich. The action of the society in supplying practical instruction
  to intending travellers, in astronomy, surveying and the various
  branches of science useful to collectors, has had much to do with
  advancement of discovery. Since the war of 1870 many geographical
  societies have been established on the continent of Europe. At the
  close of the 19th century there were upwards of 100 such societies in
  the world, with more than 50,000 members, and over 150 journals were
  devoted entirely to geographical subjects.[11] The great development
  of photography has been a notable aid to explorers, not only by
  placing at their disposal a faithful and ready means of recording the
  features of a country and the types of inhabitants, but by supplying a
  method of quick and accurate topographical surveying.


  THE PRINCIPLES OF GEOGRAPHY

  As regards the scope of geography, the order of the various
  departments and their inter-relation, there is little difference of
  opinion, and the principles of geography[12] are now generally
  accepted by modern geographers. The order in which the various
  subjects are treated in the following sketch is the natural succession
  from fundamental to dependent facts, which corresponds also to the
  evolution of the diversities of the earth's crust and of its
  inhabitants.


    Mathematical geography.

  The fundamental geographical conceptions are mathematical, the
  relations of space and form. The figure and dimensions of the earth
  are the first of these. They are ascertained by a combination of
  actual measurement of the highest precision on the surface and angular
  observations of the positions of the heavenly bodies. The science of
  geodesy is part of mathematical geography, of which the arts of
  surveying and cartography are applications. The motions of the earth
  as a planet must be taken into account, as they render possible the
  determination of position and direction by observations of the
  heavenly bodies. The diurnal rotation of the earth furnishes two fixed
  points or poles, the axis joining which is fixed or nearly so in its
  direction in space. The rotation of the earth thus fixes the
  directions of north and south and defines those of east and west. The
  angle which the earth's axis makes with the plane in which the planet
  revolves round the sun determines the varying seasonal distribution of
  solar radiation over the surface and the mathematical zones of
  climate. Another important consequence of rotation is the deviation
  produced in moving bodies relatively to the surface. In the form known
  as Ferrell's Law this runs: "If a body moves in any direction on the
  earth's surface, there is a deflecting force which arises from the
  earth's rotation which tends to deflect it to the right in the
  northern hemisphere but to the left in the southern hemisphere." The
  deviation is of importance in the movement of air, of ocean currents,
  and to some extent of rivers.[13]


    Physical geography.

  In popular usage the words "physical geography" have come to mean
  geography viewed from a particular standpoint rather than any special
  department of the subject. The popular meaning is better conveyed by
  the word physiography, a term which appears to have been introduced by
  Linnaeus, and was reinvented as a substitute for the cosmography of
  the middle ages by Professor Huxley. Although the term has since been
  limited by some writers to one particular part of the subject, it
  seems best to maintain the original and literal meaning. In the
  stricter sense, physical geography is that part of geography which
  involves the processes of contemporary change in the crust and the
  circulation of the fluid envelopes. It thus draws upon physics for the
  explanation of the phenomena with the space-relations of which it is
  specially concerned. Physical geography naturally falls into three
  divisions, dealing respectively with the surface of the
  lithosphere--geomorphology; the hydrosphere--oceanography; and the
  atmosphere--climatology. All these rest upon the facts of mathematical
  geography, and the three are so closely inter-related that they cannot
  be rigidly separated in any discussion.


    Geomorphology.

  Geomorphology is the part of geography which deals with terrestrial
  relief, including the submarine as well as the subaërial portions of
  the crust. The history of the origin of the various forms belongs to
  geology, and can be completely studied only by geological methods. But
  the relief of the crust is not a finished piece of sculpture; the
  forms are for the most part transitional, owing their characteristic
  outlines to the process by which they are produced; therefore the
  geographer must, for strictly geographical purposes, take some account
  of the processes which are now in action modifying the forms of the
  crust. Opinion still differs as to the extent to which the
  geographer's work should overlap that of the geologist.

  The primary distinction of the forms of the crust is that between
  elevations and depressions. Granting that the geoid or mean surface of
  the ocean is a uniform spheroid, the distribution of land and water
  approximately indicates a division of the surface of the globe into
  two areas, one of elevation and one of depression. The increasing
  number of measurements of the height of land in all continents and
  islands, and the very detailed levellings in those countries which
  have been thoroughly surveyed, enable the average elevation of the
  land above sea-level to be fairly estimated, although many vast gaps
  in accurate knowledge remain, and the estimate is not an exact one.
  The only part of the sea-bed the configuration of which is at all well
  known is the zone bordering the coasts where the depth is less than
  about 100 fathoms or 200 metres, i.e. those parts which sailors speak
  of as "in soundings." Actual or projected routes for telegraph cables
  across the deep sea have also been sounded with extreme accuracy in
  many cases; but beyond these lines of sounding the vast spaces of the
  ocean remain unplumbed save for the rare researches of scientific
  expeditions, such as those of the "Challenger," the "Valdivia," the
  "Albatross" and the "Scotia." Thus the best approximation to the
  average depth of the ocean is little more than an expert guess; yet a
  fair approximation is probable for the features of sub-oceanic relief
  are so much more uniform than those of the land that a smaller number
  of fixed points is required to determine them.


    Crustal relief.

  The chief element of uncertainty as to the largest features of the
  relief of the earth's crust is due to the unexplored area in the
  Arctic region and the larger regions of the Antarctic, of which we
  know nothing. We know that the earth's surface if unveiled of water
  would exhibit a great region of elevation arranged with a certain
  rough radiate symmetry round the north pole, and extending southwards
  in three unequal arms which taper to points in the south. A depression
  surrounds the little-known south polar region in a continuous ring and
  extends northwards in three vast hollows lying between the arms of the
  elevated area. So far only is it possible to speak with certainty, but
  it is permissible to take a few steps into the twilight of dawning
  knowledge and indicate the chief subdivisions which are likely to be
  established in the great crust-hollow and the great crust-heap. The
  boundary between these should obviously be the mean surface of the
  sphere.

  Sir John Murray deduced the mean height of the land of the globe as
  about 2250 ft. above sea-level, and the mean depth of the oceans as
  2080 fathoms or 12,480 ft. below sea-level.[14] Calculating the area
  of the land at 55,000,000 sq. m. (or 28.6% of the surface), and that
  of the oceans as 137,200,000 sq. m. (or 71.4% of the surface), he
  found that the volume of the land above sea-level was 23,450,000 cub.
  m., the volume of water below sea-level 323,800,000, and the total
  volume of the water equal to about 1/666th of the volume of the whole
  globe. From these data, as revised by A. Supan,[15] H.R. Mill
  calculated the position of mean sphere-level at about 10,000 ft. or
  1700 fathoms below sea-level. He showed that an imaginary spheroidal
  shell, concentric with the earth and cutting the slope between the
  elevated and depressed areas at the contour-line of 1700 fathoms,
  would not only leave above it a volume of the crust equal to the
  volume of the hollow left below it, but would also divide the surface
  of the earth so that the area of the elevated region was equal to that
  of the depressed region.[16]


    Areas of the crust according to Murray.

  A similar observation was made almost simultaneously by Romieux,[17]
  who further speculated on the equilibrium between the weight of the
  elevated land mass and that of the total waters of the ocean, and
  deduced some interesting relations between them. Murray, as the result
  of his study, divided the earth's surface into three zones--the
  _continental area_ containing all dry land, the _transitional area_
  including the submarine slopes down to 1000 fathoms, and the _abysmal
  area_ consisting of the floor of the ocean beyond that depth; and Mill
  proposed to take the line of mean-sphere level, instead of the
  empirical depth of 1000 fathoms, as the boundary between the
  transitional and abysmal areas.

  An elaborate criticism of all the existing data regarding the volume
  relations of the vertical relief of the globe was made in 1894 by
  Professor Hermann Wagner, whose recalculations of volumes and mean
  heights--the best results which have yet been obtained--led to the
  following conclusions.[18]


    Areas of the crust according to Wagner.

  The area of the dry land was taken as 28.3% of the surface of the
  globe, and that of the oceans as 71.7%. The mean height deduced for
  the land was 2300 ft. above sea-level, the mean depth of the sea
  11,500 ft. below, while the position of mean-sphere level comes out as
  7500 ft. (1250 fathoms) below sea-level. From this it would appear
  that 43% of the earth's surface was above and 57% below the mean
  level. It must be noted, however, that since 1895 the soundings of
  Nansen in the north polar area, of the "Valdivia," "Belgica," "Gauss"
  and "Scotia" in the Southern Ocean, and of various surveying ships in
  the North and South Pacific, have proved that the mean depth of the
  ocean is considerably greater than had been supposed, and mean-sphere
  level must therefore lie deeper than the calculations of 1895 show;
  possibly not far from the position deduced from the freer estimate of
  1888. The whole of the available data were utilized by the prince of
  Monaco in 1905 in the preparation of a complete bathymetrical map of
  the oceans on a uniform scale, which must long remain the standard
  work for reference on ocean depths.

  By the device of a hypsographic curve co-ordinating the vertical
  relief and the areas of the earth's surface occupied by each zone of
  elevation, according to the system introduced by Supan,[19] Wagner
  showed his results graphically.

  This curve with the values reduced from metres to feet is reproduced
  below.

  Wagner subdivides the earth's surface, according to elevation, into
  the following five regions:


    _Wagner's Divisions of the Earth's Crust:_

    +---------------------+-----------+-------------+-------------+
    |        Name.        |Per cent of|    From     |     To      |
    |                     |  Surface. |             |             |
    +---------------------+-----------+-------------+-------------+
    | Depressed area      |     3     |   Deepest.  |-16,400 feet.|
    | Oceanic plateau     |    54     |-16,400 feet.|- 7,400   "  |
    | Continental slope   |     9     |- 7,400   "  |-   660   "  |
    | Continental plateau |    28     |-   660   "  |+ 3,000   "  |
    | Culminating area    |     6     |+ 3,300   "  |   Highest.  |
    +---------------------+-----------+-------------+-------------+

  [Illustration]

  The continental plateau might for purposes of detailed study be
  divided into the _continental shelf_ from -660 ft. to sea-level, and
  _lowlands_ from sea-level to +660 ft. (corresponding to the mean level
  of the whole globe).[20] _Uplands_ reaching from 660 ft. to 2300 (the
  approximate mean level of the land), and _highlands_, from 2300
  upwards, might also be distinguished.


    Arrangement of world-ridges and hollows.

  A striking fact in the configuration of the crust is that each
  continent, or elevated mass of the crust, is diametrically opposite to
  an ocean basin or great depression; the only partial exception being
  in the case of southern South America, which is antipodal to eastern
  Asia. Professor C. Lapworth has generalized the grand features of
  crustal relief in a scheme of attractive simplicity. He sees
  throughout all the chaos of irregular crust-forms the recurrence of a
  certain harmony, a succession of folds or waves which build up all the
  minor features.[21] One great series of crust waves from east to west
  is crossed by a second great series of crust waves from north to
  south, giving rise by their interference to six great elevated masses
  (the continents), arranged in three groups, each consisting of a
  northern and a southern member separated by a minor depression. These
  elevated masses are divided from one another by similar great
  depressions.


    Lapworth's fold-theory.

  He says: "The surface of each of our great continental masses of land
  resembles that of a long and broad arch-like form, of which we see the
  simplest type in the New World. The surface of the North American arch
  is sagged downwards in the middle into a central depression which lies
  between two long marginal plateaus, and these plateaus are finally
  crowned by the wrinkled crests which form its two modern mountain
  systems. The surface of each of our ocean floors exactly resembles
  that of a continent turned upside down. Taking the Atlantic as our
  simplest type, we may say that the surface of an ocean basin resembles
  that of a mighty trough or syncline, buckled up more or less centrally
  in a medial ridge, which is bounded by two long and deep marginal
  hollows, in the cores of which still deeper grooves sink to the
  profoundest depths. This complementary relationship descends even to
  the minor features of the two. Where the great continental sag sinks
  below the ocean level, we have our gulfs and our Mediterraneans, seen
  in our type continent, as the Mexican Gulf and Hudson Bay. Where the
  central oceanic buckle attains the water-line we have our oceanic
  islands, seen in our type ocean, as St Helena and the Azores. Although
  the apparent crust-waves are neither equal in size nor symmetrical in
  form, this complementary relationship between them is always
  discernible. The broad Pacific depression seems to answer to the broad
  elevation of the Old World--the narrow trough of the Atlantic to the
  narrow continent of America."


    Suess's theory.

  The most thorough discussion of the great features of terrestrial
  relief in the light of their origin is that by Professor E. Suess,[22]
  who points out that the plan of the earth is the result of two
  movements of the crust--one, subsidence over wide areas, giving rise
  to oceanic depressions and leaving the continents protuberant; the
  other, folding along comparatively narrow belts, giving rise to
  mountain ranges. This theory of crust blocks dropped by subsidence is
  opposed to Lapworth's theory of vast crust-folds, but geology is the
  science which has to decide between them.

  Geomorphology is concerned, however, in the suggestions which have
  been made as to the cause of the distribution of heap and hollow in
  the larger features of the crust. Élie de Beaumont, in his
  speculations on the relation between the direction of mountain ranges
  and their geological age and character, was feeling towards a
  comprehensive theory of the forms of crustal relief; but his ideas
  were too geometrical, and his theory that the earth is a spheroid
  built up on a rhombic dodecahedron, the pentagonal faces of which
  determined the direction of mountain ranges, could not be proved.[23]
  The "tetrahedral theory" brought forward by Lowthian Green,[24] that
  the form of the earth is a spheroid based on a regular tetrahedron, is
  more serviceable, because it accounts for three very interesting facts
  of the terrestrial plan--(1) the antipodal position of continents and
  ocean basins; (2) the triangular outline of the continents; and (3)
  the excess of sea in the southern hemisphere. Recent investigations
  have recalled attention to the work of Lowthian Green, but the
  question is still in the controversial stage.[25] The study of tidal
  strain in the earth's crust by Sir George Darwin has led that
  physicist to indicate the possibility of the triangular form and
  southerly direction of the continents being a result of the
  differential or tidal attraction of the sun and moon. More recently
  Professor A.E.H. Love has shown that the great features of the relief
  of the lithosphere may be expressed by spherical harmonics of the
  first, second and third degrees, and their formation related to
  gravitational action in a sphere of unequal density.[26]

  In any case it is fully recognized that the plan of the earth is so
  clear as to leave no doubt as to its being due to some general cause
  which should be capable of detection.


    The continents.

  If the level of the sea were to become coincident with the mean level
  of the lithosphere, there would result one tri-radiate land-mass of
  nearly uniform outline and one continuous sheet of water broken by
  few islands. The actual position of sea-level lies so near the summit
  of the crust-heap that the varied relief of the upper portion leads to
  the formation of a complicated coast-line and a great number of
  detached portions of land. The hydrosphere is, in fact, continuous,
  and the land is all in insular masses: the largest is the Old World of
  Europe, Asia and Africa; the next in size, America; the third,
  possibly, Antarctica; the fourth, Australia; the fifth, Greenland.
  After this there is a considerable gap before New Guinea, Borneo,
  Madagascar, Sumatra and the vast multitude of smaller islands
  descending in size by regular gradations to mere rocks. The contrast
  between island and mainland was natural enough in the days before the
  discovery of Australia, and the mainland of the Old World was
  traditionally divided into three continents. These "continents,"
  "parts of the earth," or "quarters of the globe," proved to be
  convenient divisions; America was added as a fourth, and subsequently
  divided into two, while Australia on its discovery was classed
  sometimes as a new continent, sometimes merely as an island, sometimes
  compromisingly as an island-continent, according to individual
  opinion. The discovery of the insularity of Greenland might again give
  rise to the argument as to the distinction between island and
  continent. Although the name of continent was not applied to large
  portions of land for any physical reasons, it so happens that there is
  a certain physical similarity or homology between them which is not
  shared by the smaller islands or peninsulas.


    Homology of continents.

  The typical continental form is triangular as regards its sea-level
  outline. The relief of the surface typically includes a central plain,
  sometimes dipping below sea-level, bounded by lateral highlands or
  mountain ranges, loftier on one side than on the other, the higher
  enclosing a plateau shut in by mountains. South America and North
  America follow this type most closely; Eurasia (the land mass of
  Europe and Asia) comes next, while Africa and Australia are farther
  removed from the type, and the structure of Antarctica and Greenland
  is unknown.

  If the continuous, unbroken, horizontal extent of land in a continent
  is termed its _trunk_,[27] and the portions cut up by inlets or
  channels of the sea into islands and peninsulas the _limbs_, it is
  possible to compare the continents in an instructive manner.

  The following table is from the statistics of Professor H. Wagner,[28]
  his metric measurements being transposed into British units:


    _Comparison of the Continents._

    +---------------+-------+-------+-------+------+--------+------+------+
    |               |       |       |       | Area |        |      |      |
    |               | Area  | Mean  |  Area |penin-|  Area  | Area | Area |
    |               | total |height,| trunk,|sulas,|islands,|limbs,|limbs,|
    |               |  mil. | feet. |  mil. | mil. |  mil.  | mil. | per  |
    |               | sq. m.|       | sq. m.|sq. m.| sq. m. |sq. m.| cent.|
    +---------------+-------+-------+-------+------+--------+------+------+
    | Old World     | 35.8  | 2360  |       |      |        |      |      |
    | New World     | 16.2  | 2230  |       |      |        |      |      |
    | Eurasia       | 20.85 | 2620  | 15.42 | 4.09 |  1.34  | 5.43 | 26   |
    | Africa        | 11.46 | 2130  | 11.22 |  ..  |  0.24  | 0.24 |  2.1 |
    | North America |  9.26 | 2300  |  6.92 | 0.78 |  1.56  | 2.34 | 25   |
    | South America |  6.84 | 1970  |  6.76 | 0.02 |  0.06  | 0.08 |  1.1 |
    | Australia     |  3.43 | 1310  |  2.77 | 0.16 |  0.50  | 0.66 | 19   |
    | Asia          | 17.02 | 3120  | 12.93 | 3.05 |  1.04  | 4.09 | 24   |
    | Europe        |  3.83 |  980  |  2.49 | 1.04 |  0.30  | 1.34 | 35   |
    +---------------+-------+-------+-------+------+--------+------+------+


    Islands.

  The usual classification of islands is into continental and oceanic.
  The former class includes all those which rise from the continental
  shelf, or show evidence in the character of their rocks of having at
  one time been continuous with a neighbouring continent. The latter
  rise abruptly from the oceanic abysses. Oceanic islands are divided
  according to their geological character into volcanic islands and
  those of organic origin, including coral islands. More elaborate
  subdivisions according to structure, origin and position have been
  proposed.[29] In some cases a piece of land is only an island at high
  water, and by imperceptible gradation the form passes into a
  peninsula. The typical peninsula is connected with the mainland by a
  relatively narrow isthmus; the name is, however, extended to any limb
  projecting from the trunk of the mainland, even when, as in the Indian
  peninsula, it is connected by its widest part.


    Coasts.

  Small peninsulas are known as promontories or headlands, and the
  extremity as a cape. The opposite form, an inlet of the sea, is known
  when wide as a gulf, bay or bight, according to size and degree of
  inflection, or as a fjord or ria when long and narrow. It is
  convenient to employ a specific name for a projection of a coast-line
  less pronounced than a peninsula, and for an inlet less pronounced
  than a bay or bight; outcurve and incurve may serve the turn. The
  varieties of coast-lines were reduced to an exact classification by
  Richthofen, who grouped them according to the height and slope of the
  land into cliff-coasts (_Steilküsten_)--narrow beach coasts with
  cliffs, wide beach coasts with cliffs, and low coasts, subdividing
  each group according as the coast-line runs parallel to or crosses the
  line of strike of the mountains, or is not related to mountain
  structure. A further subdivision depends on the character of the
  inter-relation of land and sea along the shore producing such types as
  a fjord-coast, ria-coast or lagoon-coast. This extremely elaborate
  subdivision may be reduced, as Wagner points out, to three types--the
  continental coast where the sea comes up to the solid rock-material of
  the land; the marine coast, which is formed entirely of soft material
  sorted out by the sea; and the composite coast, in which both forms
  are combined.


    Coast-lines.

  On large-scale maps it is necessary to show two coast-lines, one for
  the highest, the other for the lowest tide; but in small-scale maps a
  single line is usually wider than is required to represent the whole
  breadth of the inter-tidal zone. The measurement of a coast-line is
  difficult, because the length will necessarily be greater when
  measured on a large-scale map where minute irregularities can be taken
  into account. It is usual to distinguish between the general
  coast-line measured from point to point of the headlands disregarding
  the smaller bays, and the detailed coast-line which takes account of
  every inflection shown by the map employed, and follows up river
  entrances to the point where tidal action ceases. The ratio between
  these two coast-lines represents the "coastal development" of any
  region.


    Submarine forms.

  While the forms of the sea-bed are not yet sufficiently well known to
  admit of exact classification, they are recognized to be as a rule
  distinct from the forms of the land, and the importance of using a
  distinctive terminology is felt. Efforts have been made to arrive at a
  definite international agreement on this subject, and certain terms
  suggested by a committee were adopted by the Eighth International
  Geographical Congress at New York in 1904.[30] The forms of the ocean
  floor include the "shelf," or shallow sea margin, the "depression," a
  general term applied to all submarine hollows, and the "elevation." A
  depression when of great extent is termed a "basin," when it is of a
  more or less round form with approximately equal diameters, a "trough"
  when it is wide and elongated with gently sloping borders, and a
  "trench" when narrow and elongated with steeply sloping borders, one
  of which rises higher than the other. The extension of a trough or
  basin penetrating the land or an elevation is termed an "embayment"
  when wide, and a "gully" when long and narrow; and the deepest part of
  a depression is termed a "deep." A depression of small extent when
  steep-sided is termed a "caldron," and a long narrow depression
  crossing a part of the continental border is termed a "furrow." An
  elevation of great extent which rises at a very gentle angle from a
  surrounding depression is termed a "rise," one which is relatively
  narrow and steep-sided a "ridge," and one which is approximately equal
  in length and breadth but steep-sided a "plateau," whether it springs
  direct from a depression or from a rise. An elevation of small extent
  is distinguished as a "dome" when it is more than 100 fathoms from the
  surface, a "bank" when it is nearer the surface than 100 fathoms but
  deeper than 6 fathoms, and a "shoal" when it comes within 6 fathoms of
  the surface and so becomes a serious danger to shipping. The highest
  point of an elevation is termed a "height," if it does not form an
  island or one of the minor forms.


    Land forms.

  The forms of the dry land are of infinite variety, and have been
  studied in great detail.[31] From the descriptive or topographical
  point of view, geometrical form alone should be considered; but the
  origin and geological structure of land forms must in many cases be
  taken into account when dealing with the function they exercise in the
  control of mobile distributions. The geographers who have hitherto
  given most attention to the forms of the land have been trained as
  geologists, and consequently there is a general tendency to make
  origin or structure the basis of classification rather than form
  alone.


    The six elementary land forms.

  The fundamental form-elements may be reduced to the six proposed by
  Professor Penck as the basis of his double system of classification by
  form and origin.[32] These may be looked upon as being all derived by
  various modifications or arrangements of the single form-unit, the
  _slope_ or inclined plane surface. No one form occurs alone, but
  always grouped together with others in various ways to make up
  districts, regions and lands of distinctive characters. The
  form-elements are:

  1. The _plain_ or gently inclined uniform surface.

  2. The _scarp_ or steeply inclined slope; this is necessarily of small
  extent except in the direction of its length.

  3. The _valley_, composed of two lateral parallel slopes inclined
  towards a narrow strip of plain at a lower level which itself slopes
  downwards in the direction of its length. Many varieties of this
  fundamental form may be distinguished.

  4. The _mount_, composed of a surface falling away on every side from
  a particular place. This place may either be a point, as in a volcanic
  cone, or a line, as in a mountain range or ridge of hills.

  5. The _hollow_ or form produced by a land surface sloping inwards
  from all sides to a particular lowest place, the converse of a mount.

  6. The _cavern_ or space entirely surrounded by a land surface.


    Geology and land forms.

  These forms never occur scattered haphazard over a region, but always
  in an orderly subordination depending on their mode of origin. The
  dominant forms result from crustal movements, the subsidiary from
  secondary reactions during the action of the primitive forms on mobile
  distributions. The geological structure and the mineral composition of
  the rocks are often the chief causes determining the character of the
  land forms of a region. Thus the scenery of a limestone country
  depends on the solubility and permeability of the rocks, leading to
  the typical Karst-formations of caverns, swallow-holes and underground
  stream courses, with the contingent phenomena of dry valleys and
  natural bridges. A sandy beach or desert owes its character to the
  mobility of its constituent sand-grains, which are readily drifted and
  piled up in the form of dunes. A region where volcanic activity has
  led to the embedding of dykes or bosses of hard rock amongst softer
  strata produces a plain broken by abrupt and isolated eminences.[33]


    Classification of mountains.

  It would be impracticable to go fully into the varieties of each
  specific form; but, partly as an example of modern geographical
  classification, partly because of the exceptional importance of
  mountains amongst the features of the land, one exception may be made.
  The classification of mountains into types has usually had regard
  rather to geological structure than to external form, so that some
  geologists would even apply the name of a mountain range to a region
  not distinguished by relief from the rest of the country if it bear
  geological evidence of having once been a true range. A mountain may
  be described (it cannot be defined) as an elevated region of irregular
  surface rising comparatively abruptly from lower ground. The actual
  elevation of a summit above sea-level does not necessarily affect its
  mountainous character; a gentle eminence, for instance, rising a few
  hundred feet above a tableland, even if at an elevation of say 15,000
  ft., could only be called a hill.[34] But it may be said that any
  abrupt slope of 2000 ft. or more in vertical height may justly be
  called a mountain, while abrupt slopes of lesser height may be called
  hills. Existing classifications, however, do not take account of any
  difference in kind between mountain and hills, although it is common
  in the German language to speak of _Hügelland_, _Mittelgebirge_ and
  _Hochgebirge_ with a definite significance.

  The simple classification employed by Professor James Geikie[35] into
  mountains of accumulation, mountains of elevation and mountains of
  circumdenudation, is not considered sufficiently thorough by German
  geographers, who, following Richthofen, generally adopt a
  classification dependent on six primary divisions, each of which is
  subdivided. The terms employed, especially for the subdivisions,
  cannot be easily translated into other languages, and the English
  equivalents in the following table are only put forward tentatively:--

    RICHTHOFEN'S CLASSIFICATION OF MOUNTAINS[36]

      I. _Tektonische Gebirge_--Tectonic mountains.
        (a) _Bruchgebirge oder Schollengebirge_--Block mountains.
            1. _Einseitige Schollengebirge oder Schollenrandgebirge_--
                  Scarp or tilted block mountains.
                    (i.) _Tafelscholle_--Table blocks.
                   (ii.) _Abrasionsscholle_--Abraded blocks.
                  (iii.) _Transgressionsscholle_--Blocks of unconformable
                              strata.
            2. _Flexurgebirge_--Flexure mountains.
            3. _Horstgebirge_--Symmetrical block mountains.
        (b) _Faltungsgebirge_--Fold mountains.
            1. _Homöomorphe Faltungsgebirge_--Homomorphic fold mountains.
            2. _Heteromorphe Faltungsgebirge_--Heteromorphic fold
                    mountains.

     II. _Rumpfgebirge oder Abrasionsgebirge_--Trunk or abraded mountains.

    III. _Ausbruchsgebirge_--Eruptive mountains.

     IV. _Aufschüttungsgebirge_--Mountains of accumulation.

      V. _Flachböden_--Plateaux.
        (a) _Abrasionsplatten_--Abraded plateaux.
        (b) _Marines Flachland_--Plain of marine erosion.
        (c) _Schichtungstafelland_--Horizontally stratified tableland.
        (d) _Übergusstafelland_--Lava plain.
        (e) _Stromflachland_--River plain.
        (f) _Flachböden der atmosphärischen Aufschüttung_--Plains of
                   aeolian formation.

     VI. _Erosionsgebirge_--Mountains of erosion.


    Mountain forms.

  From the morphological point of view it is more important to
  distinguish the associations of forms, such as the _mountain mass_ or
  group of mountains radiating from a centre, with the valleys furrowing
  their flanks spreading towards every direction; the _mountain chain_
  or line of heights, forming a long narrow ridge or series of ridges
  separated by parallel valleys; the _dissected plateau_ or highland,
  divided into mountains of circumdenudation by a system of deeply-cut
  valleys; and the _isolated peak_, usually a volcanic cone or a hard
  rock mass left projecting after the softer strata which embedded it
  have been worn away (Monadnock of Professor Davis).


    Distribution of mountains.

  The geographical distribution of mountains is intimately associated
  with the great structural lines of the continents of which they form
  the culminating region. Lofty lines of fold mountains form the
  "backbones" of North America in the Rocky Mountains and the west coast
  systems, of South America in the Cordillera of the Andes, of Europe in
  the Pyrenees, Alps, Carpathians and Caucasus, and of Asia in the
  mountains of Asia Minor, converging on the Pamirs and diverging thence
  in the Himalaya and the vast mountain systems of central and eastern
  Asia. The remarkable line of volcanoes around the whole coast of the
  Pacific and along the margin of the Caribbean and Mediterranean seas
  is one of the most conspicuous features of the globe.


    Functions of land forms.

    Land waste.

    Glaciers.

  If land forms may be compared to organs, the part they serve in the
  economy of the earth may, without straining the term, be characterized
  as functions. The first and simplest function of the land surface is
  that of guiding loose material to a lower level. The downward pull of
  gravity suffices to bring about the fall of such material, but the
  path it will follow and the distance it will travel before coming to
  rest depend upon the land form. The loose material may, and in an arid
  region does, consist only of portions of the higher parts of the
  surface detached by the expansion and contraction produced by heating
  and cooling due to radiation. Such broken material rolling down a
  uniform scarp would tend to reduce its steepness by the loss of
  material in the upper part and by the accumulation of a mound or scree
  against the lower part of the slope. But where the side is not a
  uniform scarp, but made up of a series of ridges and valleys, the
  tendency will be to distribute the detritus in an irregular manner,
  directing it away from one place and collecting it in great masses in
  another, so that in time the land form assumes a new appearance. Snow
  accumulating on the higher portions of the land, when compacted into
  ice and caused to flow downwards by gravity, gives rise, on account of
  its more coherent character, to continuous glaciers, which mould
  themselves to the slopes down which they are guided, different
  ice-streams converging to send forward a greater volume. Gradually
  coming to occupy definite beds, which are deepened and polished by the
  friction, they impress a characteristic appearance on the land, which
  guides them as they traverse it, and, although the ice melts at lower
  levels, vast quantities of clay and broken stones are brought down and
  deposited in terminal moraines where the glacier ends.


    Rain.

    River systems.

    Adjustment of rivers to land.

  Rain is by far the most important of the inorganic mobile
  distributions upon which land forms exercise their function of
  guidance and control. The precipitation of rain from the aqueous
  vapour of the atmosphere is caused in part by vertical movements of
  the atmosphere involving heat changes and apparently independent of
  the surface upon which precipitation occurs; but in greater part it is
  dictated by the form and altitude of the land surface and the
  direction of the prevailing winds, which itself is largely influenced
  by the land. It is on the windward faces of the highest ground, or
  just beyond the summit of less dominant heights upon the leeward side,
  that most rain falls, and all that does not evaporate or percolate
  into the ground is conducted back to the sea by a route which depends
  only on the form of the land. More mobile and more searching than ice
  or rock rubbish, the trickling drops are guided by the deepest lines
  of the hillside in their incipient flow, and as these lines converge,
  the stream, gaining strength, proceeds in its torrential course to
  carve its channel deeper and entrench itself in permanent occupation.
  Thus the stream-bed, from which at first the water might be blown away
  into a new channel by a gale of wind, ultimately grows to be the
  strongest line of the landscape. As the main valley deepens, the
  tributary stream-beds are deepened also, and gradually cut their way
  headwards, enlarging the area whence they draw their supplies. Thus
  new land forms are created--valleys of curious complexity, for
  example--by the "capture" and diversion of the water of one river by
  another, leading to a change of watershed.[37] The minor tributaries
  become more numerous and more constant, until the system of torrents
  has impressed its own individuality on the mountain side. As the river
  leaves the mountain, ever growing by the accession of tributaries, it
  ceases, save in flood time, to be a formidable instrument of
  destruction; the gentler slope of the land surface gives to it only
  power sufficient to transport small stones, gravel, sand and
  ultimately mud. Its valley banks are cut back by the erosion of minor
  tributaries, or by rain-wash if the climate be moist, or left steep
  and sharp while the river deepens its bed if the climate be arid. The
  outline of the curve of a valley's sides ultimately depends on the
  angle of repose of the detritus which covers them, if there has been
  no subsequent change, such as the passage of a glacier along the
  valley, which tends to destroy the regularity of the cross-section.
  The slope of the river bed diminishes until the plain compels the
  river to move slowly, swinging in _meanders_ proportioned to its size,
  and gradually, controlled by the flattening land, ceasing to transport
  material, but raising its banks and silting up its bed by the dropped
  sediment, until, split up and shoaled, its distributaries struggle
  across its delta to the sea. This is the typical river of which there
  are infinite varieties, yet every variety would, if time were given,
  and the land remained unchanged in level relatively to the sea,
  ultimately approach to the type. Movements of the land either of
  subsidence or elevation, changes in the land by the action of erosion
  in cutting back an escarpment or cutting through a col, changes in
  climate by affecting the rainfall and the volume of water, all tend to
  throw the river valley out of harmony with the actual condition of its
  stream. There is nothing more striking in geography than the
  perfection of the adjustment of a great river system to its valleys
  when the land has remained stable for a very lengthened period. Before
  full adjustment has been attained the river bed may be broken in
  places by waterfalls or interrupted by lakes; after adjustment the bed
  assumes a permanent outline, the slope diminishing more and more
  gradually, without a break in its symmetrical descent. Excellent
  examples of the indecisive drainage of a new land surface, on which
  the river system has not had time to impress itself, are to be seen in
  northern Canada and in Finland, where rivers are separated by scarcely
  perceptible divides, and the numerous lakes frequently belong to more
  than one river system.


    The geographical cycle.

  The action of rivers on the land is so important that it has been made
  the basis of a system of physical geography by Professor W.M. Davis,
  who classifies land surfaces in terms of the three factors--structure,
  process and time.[38] Of these time, during which the process is
  acting on the structure, is the most important. A land may thus be
  characterized by its position in the "geographical cycle", or cycle of
  erosion, as young, mature or old, the last term being reached when the
  base-level of erosion is attained, and the land, however varied its
  relief may have been in youth or maturity, is reduced to a nearly
  uniform surface or peneplain. By a re-elevation of a peneplain the
  rivers of an old land surface may be restored to youthful activity,
  and resume their shaping action, deepening the old valleys and
  initiating new ones, starting afresh the whole course of the
  geographical cycle. It is, however, not the action of the running
  water on the land, but the function exercised by the land on the
  running water, that is considered here to be the special province of
  geography. At every stage of the geographical cycle the land forms, as
  they exist at that stage, are concerned in guiding the condensation
  and flow of water in certain definite ways. Thus, for example, in a
  mountain range at right angles to a prevailing sea-wind, it is the
  land forms which determine that one side of the range shall be richly
  watered and deeply dissected by a complete system of valleys, while
  the other side is dry, indefinite in its valley systems, and sends
  none of its scanty drainage to the sea. The action of rain, ice and
  rivers conspires with the movement of land waste to strip the layer of
  soil from steep slopes as rapidly as it forms, and to cause it to
  accumulate on the flat valley bottoms, on the graceful flattened cones
  of alluvial fans at the outlet of the gorges of tributaries, or in the
  smoothly-spread surface of alluvial plains.

  The whole question of the régime of rivers and lakes is sometimes
  treated under the name hydrography, a name used by some writers in the
  sense of marine surveying, and by others as synonymous with
  oceanography. For the study of rivers alone the name potamology[39]
  has been suggested by Penck, and the subject being of much practical
  importance has received a good deal of attention.[40]


    Lakes and internal drainage.

  The study of lakes has also been specialized under the name of
  limnology (see LAKE).[41] The existence of lakes in hollows of the
  land depends upon the balance between precipitation and evaporation. A
  stream flowing into a hollow will tend to fill it up, and the water
  will begin to escape as soon as its level rises high enough to reach
  the lowest part of the rim. In the case of a large hollow in a very
  dry climate the rate of evaporation may be sufficient to prevent the
  water from ever rising to the lip, so that there is no outflow to the
  sea, and a basin of internal drainage is the result. This is the case,
  for instance, in the Caspian sea, the Aral and Balkhash lakes, the
  Tarim basin, the Sahara, inner Australia, the great basin of the
  United States and the Titicaca basin. These basins of internal
  drainage are calculated to amount to 22% of the land surface. The
  percentages of the land surface draining to the different oceans are
  approximately--Atlantic, 34.3%; Arctic sea, 16.5%; Pacific, 14.4%;
  Indian Ocean, 12.8%.[42]


    Terminology of river systems.

  The parts of a river system have not been so clearly defined as is
  desirable, hence the exaggerated importance popularly attached to "the
  source" of a river. A well-developed river system has in fact many
  equally important and widely-separated sources, the most distant from
  the mouth, the highest, or even that of largest initial volume not
  being necessarily of greater geographical interest than the rest. The
  whole of the land which directs drainage towards one river is known as
  its basin, catchment area or drainage area--sometimes, by an incorrect
  expression, as its valley or even its watershed. The boundary line
  between one drainage area and others is rightly termed the watershed,
  but on account of the ambiguity which has been tolerated it is better
  to call it water-parting or, as in America, divide. The only other
  important term which requires to be noted here is _talweg_, a word
  introduced from the German into French and English, and meaning the
  deepest line along the valley, which is necessarily occupied by a
  stream unless the valley is dry.

  The functions of land forms extend beyond the control of the
  circulation of the atmosphere, the hydrosphere and the water which is
  continually being interchanged between them; they are exercised with
  increased effect in the higher departments of biogeography and
  anthropogeography.


    Biogeography.

  The sum of the organic life on the globe is termed by some geographers
  the biosphere, and it has been estimated that the whole mass of living
  substance in existence at one time would cover the surface of the
  earth to a depth of one-fifth of an inch.[43] The distribution of
  living organisms is a complex problem, a function of many factors,
  several of which are yet but little known. They include the biological
  nature of the organism and its physical environment, the latter
  involving conditions in which geographical elements, direct or
  indirect, preponderate. The direct geographical elements are the
  arrangement of land and sea (continents and islands standing in sharp
  contrast) and the vertical relief of the globe, which interposes
  barriers of a less absolute kind between portions of the same land
  area or oceanic depression. The indirect geographical elements, which,
  as a rule, act with and intensify the direct, are mainly climatic; the
  prevailing winds, rainfall, mean and extreme temperatures of every
  locality depending on the arrangement of land and sea and of land
  forms. Climate thus guided affects the weathering of rocks, and so
  determines the kind and arrangement of soil. Different species of
  organisms come to perfection in different climates; and it may be
  stated as a general rule that a species, whether of plant or animal,
  once established at one point, would spread over the whole zone of the
  climate congenial to it unless some barrier were interposed to its
  progress. In the case of land and fresh-water organisms the sea is the
  chief barrier; in the case of marine organisms, the land. Differences
  in land forms do not exert great influence on the distribution of
  living creatures directly, but indirectly such land forms as mountain
  ranges and internal drainage basins are very potent through their
  action on soil and climate. A snow-capped mountain ridge or an arid
  desert forms a barrier between different forms of life which is often
  more effective than an equal breadth of sea. In this way the surface
  of the land is divided into numerous natural regions, the flora and
  fauna of each of which include some distinctive species not shared by
  the others. The distribution of life is discussed in the various
  articles in this _Encyclopaedia_ dealing with biological, botanical
  and zoological subjects.[44]


    Floral zones.

  The classification of the land surface into areas inhabited by
  distinctive groups of plants has been attempted by many
  phyto-geographers, but without resulting in any scheme of general
  acceptance. The simplest classification is perhaps that of Drude
  according to climatic zones, subdivided according to continents. This
  takes account of--(1) the _Arctic-Alpine_ zone, including all the
  vegetation of the region bordering on perpetual snow; (2) the _Boreal_
  zone, including the temperate lands of North America, Europe and Asia,
  all of which are substantially alike in botanical character; (3) the
  _Tropical_ zone, divided sharply into (a) the tropical zone of the New
  World, and (b) the tropical zone of the Old World, the forms of which
  differ in a significant degree; (4) the _Austral_ zone, comprising all
  continental land south of the equator, and sharply divided into three
  regions the floras of which are strikingly distinct--(a) South
  American, (b) South African and (c) Australian; (5) the _Oceanic_,
  comprising all oceanic islands, the flora of which consists
  exclusively of forms whose seeds could be drifted undestroyed by ocean
  currents or carried by birds. To these might be added the antarctic,
  which is still very imperfectly known. Many subdivisions and
  transitional zones have been suggested by different authors.


    Vegetation areas.

  From the point of view of the economy of the globe this classification
  by species is perhaps less important than that by mode of life and
  physiological character in accordance with environment. The following
  are the chief areas of vegetational activity usually recognized: (1)
  The ice-deserts of the arctic and antarctic and the highest mountain
  regions, where there is no vegetation except the lowest forms, like
  that which causes "red snow." (2) The tundra or region of intensely
  cold winters, forbidding tree-growth, where mosses and lichens cover
  most of the ground when unfrozen, and shrubs occur of species which in
  other conditions are trees, here stunted to the height of a few
  inches. A similar zone surrounds the permanent snow on lofty mountains
  in all latitudes. The tundra passes by imperceptible gradations into
  the moor, bog and heath of warmer climates. (3) The temperate forests
  of evergreen or deciduous trees, according to circumstances, which
  occupy those parts of both temperate zones where rainfall and sunlight
  are both abundant. (4) The grassy steppes or prairies where the
  rainfall is diminished and temperatures are extreme, and grass is the
  prevailing form of vegetation. These pass imperceptibly into--(5) the
  arid desert, where rainfall is at a minimum, and the only plants are
  those modified to subsist with the smallest supply of water. (6) The
  tropical forest, which represents the maximum of plant luxuriance,
  stimulated by the heaviest rainfall, greatest heat and strongest
  light. These divisions merge one into the other, and admit of almost
  indefinite subdivision, while they are subject to great modifications
  by human interference in clearing and cultivating. Plants exhibit the
  controlling power of environment to a high degree, and thus vegetation
  is usually in close adjustment to the bolder geographical features of
  a region.


    Faunal realms.

  The divisions of the earth into faunal regions by Dr P.L. Sclater have
  been found to hold good for a large number of groups of animals as
  different in their mode of life as birds and mammals, and they may
  thus be accepted as based on nature. They are six in number: (1)
  _Palaearctic_, including Europe, Asia north of the Himalaya, and
  Africa north of the Sahara; (2) _Ethiopian_, consisting of Africa
  south of the Atlas range, and Madagascar; (3) _Oriental_, including
  India, Indo-China and the Malay Archipelago north of Wallace's line,
  which runs between Bali and Lombok; (4) _Australian_, including
  Australia, New Zealand, New Guinea and Polynesia; (5) _Nearctic_ or
  North America, north of Mexico; and (6) _Neotropical_ or South
  America. Each of these divisions is the home of a special fauna, many
  species of which are confined to it alone; in the Australian region,
  indeed, practically the whole fauna is peculiar and distinctive,
  suggesting a prolonged period of complete biological isolation. In
  some cases, such as the Ethiopian and Neotropical and the Palaearctic
  and Nearctic regions, the faunas, although distinct, are related,
  several forms on opposite sides of the Atlantic being analogous, e.g.
  the lion and puma, ostrich and rhea. Where two of the faunal realms
  meet there is usually, though not always, a mixing of faunas. These
  facts have led some naturalists to include the Palaearctic and
  Nearctic regions in one, termed _Holarctic_, and to suggest
  transitional regions, such as the _Sonoran_, between North and South
  America, and the _Mediterranean_, between Europe and Africa, or to
  create sub-regions, such as Madagascar and New Zealand. Oceanic
  islands have, as a rule, distinctive faunas and floras which resemble,
  but are not identical with, those of other islands in similar
  positions.


    Biological distribution as a means of geographical research.

  The study of the evolution of faunas and the comparison of the faunas
  of distant regions have furnished a trustworthy instrument of
  pre-historic geographical research, which enables earlier geographical
  relations of land and sea to be traced out, and the approximate
  period, or at least the chronological order of the larger changes, to
  be estimated. In this way, for example, it has been suggested that a
  land, "Lemuria," once connected Madagascar with the Malay Archipelago,
  and that a northern extension of the antarctic land once united the
  three southern continents.

  The distribution of fossils frequently makes it possible to map out
  approximately the general features of land and sea in long-past
  geological periods, and so to enable the history of crustal relief to
  be traced.[45]


    Reaction of organisms on environment.

  While the tendency is for the living forms to come into harmony with
  their environment and to approach the state of equilibrium by
  successive adjustments if the environment should happen to change, it
  is to be observed that the action of organisms themselves often tends
  to change their environment. Corals and other quick-growing calcareous
  marine organisms are the most powerful in this respect by creating new
  land in the ocean. Vegetation of all sorts acts in a similar way,
  either in forming soil and assisting in breaking up rocks, in filling
  up shallow lakes, and even, like the mangrove, in reclaiming wide
  stretches of land from the sea. Plant life, utilizing solar light to
  combine the inorganic elements of water, soil and air into living
  substance, is the basis of all animal life. This is not by the supply
  of food alone, but also by the withdrawal of carbonic acid from the
  atmosphere, by which vegetation maintains the composition of the air
  in a state fit for the support of animal life. Man in the primitive
  stages of culture is scarcely to be distinguished from other animals
  as regards his subjection to environment, but in the higher grades of
  culture the conditions of control and reaction become much more
  complicated, and the department of anthropogeography is devoted to
  their consideration.


    Anthropogeography.

  The first requisites of all human beings are food and protection, in
  their search for which men are brought into intimate relations with
  the forms and productions of the earth's surface. The degree of
  dependence of any people upon environment varies inversely as the
  degree of culture or civilization, which for this purpose may perhaps
  be defined as the power of an individual to exercise control over the
  individual and over the environment for the benefit of the community.
  The development of culture is to a certain extent a question of race,
  and although forming one species, the varieties of man differ in
  almost imperceptible gradations with a complexity defying
  classification (see ANTHROPOLOGY). Professor Keane groups man round
  four leading types, which may be named the black, yellow, red and
  white, or the Ethiopic, Mongolic, American and Caucasic. Each may be
  subdivided, though not with great exactness, into smaller groups,
  either according to physical characteristics, of which the form of the
  head is most important, or according to language.


    Types of man.

  The black type is found only in tropical or sub-tropical countries,
  and is usually in a primitive condition of culture, unless educated by
  contact with people of the white type. They follow the most primitive
  forms of religion (mainly fetishism), live on products of the woods or
  of the chase, with the minimum of work, and have only a loose
  political organization. The red type is peculiar to America,
  inhabiting every climate from polar to equatorial, and containing
  representatives of many stages of culture which had apparently
  developed without the aid or interference of people of any other race
  until the close of the 15th century. The yellow type is capable of a
  higher culture, cherishes higher religious beliefs, and inhabits as a
  rule the temperate zone, although extending to the tropics on one side
  and to the arctic regions on the other. The white type, originating in
  the north temperate zone, has spread over the whole world. They have
  attained the highest culture, profess the purest forms of monotheistic
  religion, and have brought all the people of the black type and many
  of those of the yellow under their domination.

  The contrast between the yellow and white types has been softened by
  the remarkable development of the Japanese following the assimilation
  of western methods.

  The actual number of human inhabitants in the world has been
  calculated as follows:

                            By Continents.[46]

    Asia                       875,000,000
    Europe                     392,000,000
    Africa                     170,000,000
    America                    143,000,000
    Australia and Polynesia      7,000,000
                             -------------
      Total                  1,587,000,000

                               By Race.[47]

    White (Caucasic)           770,000,000
    Yellow (Mong.)             540,000,000
    Black (Ethiopic)           175,000,000
    Red (American)              22,000,000
                             -------------
      Total                  1,507,000,000

  In round numbers the population of the world is about 1,600,000,000,
  and, according to an estimate by Ravenstein,[48] the maximum
  population which it will be possible for the earth to maintain is 6000
  millions, a number which, if the average rate of increase in 1891
  continued, would be reached within 200 years.

  While highly civilized communities are able to evade many of the
  restrictions of environment, to overcome the barriers to
  intercommunication interposed by land or sea, to counteract the
  adverse influence of climate, and by the development of trade even to
  inhabit countries which cannot yield a food-supply, the mass of
  mankind is still completely under the control of those conditions
  which in the past determined the distribution and the mode of life of
  the whole human race.


    Influence of environment on man.

  In tropical forests primitive tribes depend on the collection of wild
  fruits, and in a minor degree on the chase of wild animals, for their
  food. Clothing is unnecessary; hence there is little occasion for
  exercising the mental faculties beyond the sense of perception to
  avoid enemies, or the inventive arts beyond what is required for the
  simplest weapons and the most primitive fortifications. When the
  pursuit of game becomes the chief occupation of a people there is of
  necessity a higher development of courage, skill, powers of
  observation and invention; and these qualities are still further
  enhanced in predatory tribes who take by force the food, clothing and
  other property prepared or collected by a feebler people. The
  fruit-eating savage cannot stray beyond his woods which bound his life
  as the water bounds that of a fish; the hunter is free to live on the
  margin of forests or in open country, while the robber or warrior from
  some natural stronghold of the mountains sweeps over the adjacent
  plains and carries his raids into distant lands. Wide grassy steppes
  lead to the organization of the people as nomads whose wealth consists
  in flocks and herds, and their dwellings are tents. The nomad not only
  domesticates and turns to his own use the gentler and more powerful
  animals, such as sheep, cattle, horses, camels, but even turns some
  predatory creatures, like the dog, into a means of defending their
  natural prey. They hunt the beasts of prey destructive to their
  flocks, and form armed bands for protection against marauders or for
  purposes of aggression on weaker sedentary neighbours. On the fertile
  low grounds along the margins of rivers or in clearings of forests,
  agricultural communities naturally take their rise, dwelling in
  villages and cultivating the wild grains, which by careful nurture and
  selection have been turned into rich cereals. The agriculturist as a
  rule is rooted to the soil. The land he tills he holds, and acquires a
  closer connexion with a particular patch of ground than either the
  hunter or the herdsman. In the temperate zone, where the seasons are
  sharply contrasted, but follow each other with regularity, foresight
  and self-denial were fostered, because if men did not exercise these
  qualities seed-time or harvest might pass into lost opportunities and
  the tribes would suffer. The more extreme climates of arid regions on
  the margins of the tropics, by the unpredictable succession of
  droughts and floods, confound the prevision of uninstructed people,
  and make prudence and industry qualities too uncertain in their
  results to be worth cultivating. Thus the civilization of agricultural
  peoples of the temperate zone grew rapidly, yet in each community a
  special type arose adapted to the soil, the crop and the climate. On
  the seashore fishing naturally became a means of livelihood, and
  dwellers by the sea, in virtue of the dangers to which they are
  exposed from storm and unseaworthy craft, are stimulated to a higher
  degree of foresight, quicker observation, prompter decision and more
  energetic action in emergencies than those who live inland. The
  building and handling of vessels also, and the utilization of such
  uncontrollable powers of nature as wind and tide, helped forward
  mechanical invention. To every type of coast there may be related a
  special type of occupation and even of character; the deep and gloomy
  fjord, backed by almost impassable mountains, bred bold mariners whose
  only outlet for enterprise was seawards towards other lands--the
  _viks_ created the vikings. On the gently sloping margin of the
  estuary of a great river a view of tranquil inland life was equally
  presented to the shore-dweller, and the ocean did not present the only
  prospect of a career. Finally the mountain valley, with its patches of
  cultivable soil on the alluvial fans of tributary torrents, its narrow
  pastures on the uplands only left clear of snow in summer, its
  intensified extremes of climates and its isolation, almost equal to
  that of an island, has in all countries produced a special type of
  brave and hardy people, whose utmost effort may bring them comfort,
  but not wealth, by honest toil, who know little of the outer world,
  and to whom the natural outlet for ambition is marauding on the
  fertile plains. The highlander and viking, products of the valleys
  raised high amid the mountains or half-drowned in the sea, are
  everywhere of kindred spirit.

  It is in some such manner as these that the natural conditions of
  regions, which must be conformed to by prudence and utilized by labour
  to yield shelter and food, have led to the growth of peoples differing
  in their ways of life, thought and speech. The initial differences so
  produced are confirmed and perpetuated by the same barriers which
  divide the faunal or floral regions, the sea, mountains, deserts and
  the like, and much of the course of past history and present politics
  becomes clear when the combined results of differing race and
  differing environment are taken into account.[49]


    Density of population.

  The specialization which accompanies the division of labour has
  important geographical consequences, for it necessitates communication
  between communities and the interchange of their products. Trade
  makes it possible to work mineral resources in localities where food
  can only be grown with great difficulty and expense, or which are even
  totally barren and waterless, entirely dependent on supplies from
  distant sources.

  The population which can be permanently supported by a given area of
  land differs greatly according to the nature of the resources and the
  requirements of the people. Pastoral communities are always scattered
  very thinly over large areas; agricultural populations may be almost
  equally sparse where advanced methods of agriculture and labour-saving
  machinery are employed; but where a frugal people are situated on a
  fertile and inexhaustible soil, such as the deltas and river plains of
  Egypt, India and China, an enormous population may be supported on a
  small area. In most cases, however, a very dense population can only
  be maintained in regions where mineral resources have fixed the site
  of great manufacturing industries. The maximum density of population
  which a given region can support is very difficult to determine; it
  depends partly on the race and standard of culture of the people,
  partly on the nature and origin of the resources on which they depend,
  partly on the artificial burdens imposed and very largely on the
  climate. Density of population is measured by the average number of
  people residing on a unit of area; but in order to compare one part of
  the world with another the average should, strictly speaking, be taken
  for regions of equal size or of equal population; and the portions of
  the country which are permanently uninhabitable ought to be excluded
  from the calculation.[50] Considering the average density of
  population within the political limits of countries, the following
  list is of some value; the figures for a few smaller divisions of
  large countries are added (in brackets) for comparison:

    _Average Population on 1 sq. m._ (_For 1900 or 1901._)

    +--------------------+---------+-------------------+---------+
    |      Country.      | Density |      Country.     | Density |
    |                    | of pop. |                   | of pop. |
    +--------------------+---------+-------------------+---------+
    | (Saxony)           |   743*  | Ceylon            |  141**  |
    | Belgium            |   589*  | Greece            |   97    |
    | Java               |   568** | European Turkey   |   90    |
    | (England and Wales)|   558   | Spain             |   97    |
    | (Bengal)           |   495** | European Russia   |   55**  |
    | Holland            |   436   | Sweden            |   30    |
    | United Kingdom     |   344   | United States     |   25    |
    | Japan              |   317   | Mexico            |   18    |
    | Italy              |   293   | Norway            |   18    |
    | China proper       |   270** | Persia            |   15    |
    | German Empire      |   270   | New Zealand       |    7    |
    | Austria            |   226   | Argentina         |    5    |
    | Switzerland        |   207   | Brazil            |    4.5  |
    | France             |   188   | Eastern States of |         |
    | Indian Empire      |   167** |   Australia       |    3    |
    | Denmark            |   160** | Dominion of Canada|    1.5  |
    | Hungary            |   154** | Siberia           |    1    |
    | Portugal           |   146   | West Australia    |    0.2  |
    +--------------------+---------+-------------------+---------+
      * Almost exclusively industrial.
     ** Almost exclusively agricultural.


    Migration.

  The movement of people from one place to another without the immediate
  intention of returning is known as migration, and according to its
  origin it may be classed as centrifugal (directed _from_ a particular
  area) and centripetal (directed _towards_ a particular area).
  Centrifugal migration is usually a matter of compulsion; it may be
  necessitated by natural causes, such as a change of climate leading to
  the withering of pastures or destruction of agricultural land, to
  inundation, earthquake, pestilence or to an excess of population over
  means of support; or to artificial causes, such as the wholesale
  deportation of a conquered people; or to political or religious
  persecution. In any case the people are driven out by some adverse
  change; and when the urgency is great they may require to drive out in
  turn weaker people who occupy a desirable territory, thus propagating
  the wave of migration, the direction of which is guided by the forms
  of the land into inevitable channels. Many of the great historic
  movements of peoples were doubtless due to the gradual change of
  geographical or climatic conditions; and the slow desiccation of
  Central Asia has been plausibly suggested as the real cause of the
  peopling of modern Europe and of the medieval wars of the Old World,
  the theatres of which were critical points on the great natural lines
  of communication between east and west.

  In the case of centripetal migrations people flock to some particular
  place where exceptionally favourable conditions have been found to
  exist. The rushes to gold-fields and diamond-fields are typical
  instances; the growth of towns on coal-fields and near other sources
  of power, and the rapid settlement of such rich agricultural districts
  as the wheat-lands of the American prairies and great plains are other
  examples.

  There is, however, a tendency for people to remain rooted to the land
  of their birth, when not compelled or induced by powerful external
  causes to seek a new home.


    Political geography.

  Thus arises the spirit of patriotism, a product of purely geographical
  conditions, thereby differing from the sentiment of loyalty, which is
  of racial origin. Where race and soil conspire to evoke both loyalty
  and patriotism in a people, the moral qualities of a great and
  permanent nation are secured. It is noticeable that the patriotic
  spirit is strongest in those places where people are brought most
  intimately into relation with the land; dwellers in the mountain or by
  the sea, and, above all, the people of rugged coasts and mountainous
  archipelagoes, have always been renowned for love of country, while
  the inhabitants of fertile plains and trading communities are
  frequently less strongly attached to their own land.

  Amongst nomads the tribe is the unit of government, the political bond
  is personal, and there is no definite territorial association of the
  people, who may be loyal but cannot be patriotic. The idea of a
  country arises only when a nation, either homogeneous or composed of
  several races, establishes itself in a region the boundaries of which
  may be defined and defended against aggression from without. Political
  geography takes account of the partition of the earth amongst
  organized communities, dealing with the relation of races to regions,
  and of nations to countries, and considering the conditions of
  territorial equilibrium and instability.


    Boundaries.

  The definition of boundaries and their delimitation is one of the most
  important parts of political geography. Natural boundaries are always
  the most definite and the strongest, lending themselves most readily
  to defence against aggression. The sea is the most effective of all,
  and an island state is recognized as the most stable. Next in
  importance comes a mountain range, but here there is often difficulty
  as to the definition of the actual crest-line, and mountain ranges
  being broad regions, it may happen that a small independent state,
  like Switzerland or Andorra, occupies the mountain valleys between two
  or more great countries. Rivers do not form effective international
  boundaries, although between dependent self-governing communities they
  are convenient lines of demarcation. A desert, or a belt of country
  left purposely without inhabitants, like the mark, marches or
  debatable lands of the middle ages, was once a common means of
  separating nations which nourished hereditary grievances. The
  "buffer-state" of modern diplomacy is of the same ineffectual type. A
  less definite though very practical boundary is that formed by the
  meeting-line of two languages, or the districts inhabited by two
  races. The line of fortresses protecting Austria from Italy lies in
  some places well back from the political boundary, but just inside the
  linguistic frontier, so as to separate the German and Italian races
  occupying Austrian territory. Arbitrary lines, either traced from
  point to point and marked by posts on the ground, or defined as
  portions of meridians and parallels, are now the most common type of
  boundaries fixed by treaty. In Europe and Asia frontiers are usually
  strongly fortified and strictly watched in times of peace as well as
  during war. In South America strictly defined boundaries are still the
  exception, and the claims of neighbouring nations have very frequently
  given rise to war, though now more commonly to arbitration.[51]


    Forms of government.

  The modes of government amongst civilized peoples have little
  influence on political geography; some republics are as arbitrary and
  exacting in their frontier regulations as some absolute monarchies. It
  is, however, to be noticed that absolute monarchies are confined to
  the east of Europe and to Asia, Japan being the only established
  constitutional monarchy east of the Carpathians. Limited monarchies
  are (with the exception of Japan) peculiar to Europe, and in these the
  degree of democratic control may be said to diminish as one passes
  eastwards from the United Kingdom. Republics, although represented in
  Europe, are the peculiar form of government of America and are unknown
  in Asia.

  The forms of government of colonies present a series of transitional
  types from the autocratic administration of a governor appointed by
  the home government to complete democratic self-government. The latter
  occurs only in the temperate possessions of the British empire, in
  which there is no great preponderance of a coloured native population.
  New colonial forms have been developed during the partition of Africa
  amongst European powers, the sphere of influence being especially
  worthy of notice. This is a vaguer form of control than a
  protectorate, and frequently amounts merely to an agreement amongst
  civilized powers to respect the right of one of their number to
  exercise government within a certain area, if it should decide to do
  so at any future time.

  The central governments of all civilized countries concerned with
  external relations are closely similar in their modes of action, but
  the internal administration may be very varied. In this respect a
  country is either centralized, like the United Kingdom or France, or
  federated of distinct self-governing units like Germany (where the
  units include kingdoms, at least three minor types of monarchies,
  municipalities and a crown land under a nominated governor), or the
  United States, where the units are democratic republics. The ultimate
  cause of the predominant form of federal government may be the
  geographical diversity of the country, as in the cantons occupying the
  once isolated mountain valleys of Switzerland, the racial diversity of
  the people, as in Austria-Hungary, or merely political expediency, as
  in republics of the American type.

  The minor subdivisions into provinces, counties and parishes, or
  analogous areas, may also be related in many cases to natural features
  or racial differences perpetuated by historical causes. The
  territorial divisions and subdivisions often survive the conditions
  which led to their origin; hence the study of political geography is
  allied to history as closely as the study of physical geography is
  allied to geology, and for the same reason.


    Towns.

  The aggregation of population in towns was at one time mainly brought
  about by the necessity for defence, a fact indicated by the defensive
  sites of many old towns. In later times, towns have been more often
  founded in proximity to valuable mineral resources, and at critical
  points or nodes on lines of communication. These are places where the
  mode of travelling or of transport is changed, such as seaports, river
  ports and railway termini, or natural resting-places, such as a ford,
  the foot of a steep ascent on a road, the entrance of a valley leading
  up from a plain into the mountains, or a crossing-place of roads or
  railways.[52] The existence of a good natural harbour is often
  sufficient to give origin to a town and to fix one end of a line of
  land communication.


    Lines of communication.

  In countries of uniform surface or faint relief, roads and railways
  may be constructed in any direction without regard to the
  configuration. In places where the low ground is marshy, roads and
  railways often follow the ridge-lines of hills, or, as in Finland, the
  old glacial eskers, which run parallel to the shore. Wherever the
  relief of the land is pronounced, roads and railways are obliged to
  occupy the lowest ground winding along the valleys of rivers and
  through passes in the mountains. In exceptional cases obstructions
  which it would be impossible or too costly to turn are overcome by a
  bridge or tunnel, the magnitude of such works increasing with the
  growth of engineering skill and financial enterprise. Similarly the
  obstructions offered to water communication by interruption through
  land or shallows are overcome by cutting canals or dredging out
  channels. The economy and success of most lines of communication
  depend on following as far as possible existing natural lines and
  utilizing existing natural sources of power.[53]


    Commercial geography.

  Commercial geography may be defined as the description of the earth's
  surface with special reference to the discovery, production, transport
  and exchange of commodities. The transport concerns land routes and
  sea routes, the latter being the more important. While steam has been
  said to make a ship independent of wind and tide, it is still true
  that a long voyage even by steam must be planned so as to encounter
  the least resistance possible from prevailing winds and permanent
  currents, and this involves the application of oceanographical and
  meteorological knowledge. The older navigation by utilizing the power
  of the wind demands a very intimate knowledge of these conditions, and
  it is probable that a revival of sailing ships may in the present
  century vastly increase the importance of the study of maritime
  meteorology.

  The discovery and production of commodities require a knowledge of the
  distribution of geological formations for mineral products, of the
  natural distribution, life-conditions and cultivation or breeding of
  plants and animals and of the labour market. Attention must also be
  paid to the artificial restrictions of political geography, to the
  legislative restrictions bearing on labour and trade as imposed in
  different countries, and, above all, to the incessant fluctuations of
  the economic conditions of supply and demand and the combinations of
  capitalists or workers which affect the market.[54] The term "applied
  geography" has been employed to designate commercial geography, the
  fact being that every aspect of scientific geography may be applied to
  practical purposes, including the purposes of trade. But apart from
  the applied science, there is an aspect of pure geography which
  concerns the theory of the relation of economics to the surface of the
  earth.


    Conclusion.

  It will be seen that as each successive aspect of geographical science
  is considered in its natural sequence the conditions become more
  numerous, complex, variable and practically important. From the
  underlying abstract mathematical considerations all through the
  superimposed physical, biological, anthropological, political and
  commercial development of the subject runs the determining control
  exercised by crust-forms acting directly or indirectly on mobile
  distributions; and this is the essential principle of geography.
       (H. R. M.)


FOOTNOTES:

  [1] A concise sketch of the whole history of geographical method or
    theory as distinguished from the history of geographical discovery
    (see later section of this article) is only to be found in the
    introduction to H. Wagner's _Lehrbuch der Geographie_, vol. i.
    (Leipzig, 1900), which is in every way the most complete treatise on
    the principles of geography.

  [2] _History of Ancient Geography_ (Cambridge, 1897), p. 70.

  [3] See J.L. Myres, "An Attempt to reconstruct the Maps used by
    Herodotus," _Geographical Journal_, viii. (1896), p. 605.

  [4] _Geschichte der wissenschaftlichen Erdkunde der Griechen_
    (Leipzig, 1891), Abt. 3, p. 60.

  [5] Bunbury's _History of Ancient Geography_ (2 vols., London, 1879),
    Müller's _Geographi Graeci minores_ (2 vols., Paris, 1855, 1861) and
    Berger's _Geschichte der wissenschaftlichen Erdkunde der Griechen_ (4
    vols., Leipzig, 1887-1893) are standard authorities on the Greek
    geographers.

  [6] The period of the early middle ages is dealt with in Beazley's
    _Dawn of Modern Geography_ (London; part i., 1897; part ii., 1901;
    part iii., 1906); see also Winstedt, _Cosmos Indicopleustes_ (1910).

  [7] From translator's preface to the English version by Mr Dugdale
    (1733), entitled _A Complete System of General Geography_, revised by
    Dr Peter Shaw (London, 1756).

  [8] Printed in _Schriften zur physischen Geographie_, vol. vi. of
    Schubert's edition of the collected works of Kant (Leipzig, 1839).
    First published with notes by Rink in 1802.

  [9] _History of Civilization_, vol. i. (1857).

  [10] See H.J. Mackinder in _British Association Report_ (Ipswich),
    1895, p. 738, for a summary of German opinion, which has been
    expressed by many writers in a somewhat voluminous literature.

  [11] H. Wagner's year-book, _Geographische Jahrbuch_, published at
    Gotha, is the best systematic record of the progress of geography in
    all departments; and Haack's _Geographen Kalender_, also published
    annually at Gotha, gives complete lists of the geographical societies
    and geographers of the world.

  [12] This phrase is old, appearing in one of the earliest English
    works on geography, William Cuningham's _Cosmographical Glasse
    conteinyng the pleasant Principles of Cosmographie, Geographie,
    Hydrographie or Navigation_ (London, 1559).

  [13] See also S. Günther, _Handbuch der mathematischen Geographie_
    (Stuttgart, 1890).

  [14] "On the Height of the Land and the Depth of the Ocean," _Scot.
    Geog. Mag._ iv. (1888), p. 1. Estimates had been made previously by
    Humboldt, De Lapparent, H. Wagner, and subsequently by Penck and
    Heiderich, and for the oceans by Karstens.

  [15] _Petermanns Mitteilungen_, xxv. (1889), p. 17.

  [16] _Proc. Roy. Soc. Edin._ xvii. (1890) p. 185.

  [17] _Comptes rendus Acad. Sci._ (Paris, 1890), vol. iii. p. 994.

  [18] "Areal und mittlere Erhebung der Landflächen sowie der
    Erdkruste" in Gerland's _Beiträge zur Geophysik_, ii. (1895) p. 667.
    See also _Nature_, 54 (1896), p. 112.

  [19] _Petermanns Mitteilungen_, xxxv. (1889) p. 19.

  [20] The areas of the continental shelf and lowlands are
    approximately equal, and it is an interesting circumstance that,
    taken as a whole, the actual coast-line comes just midway on the most
    nearly level belt of the earth's surface, excepting the ocean floor.
    The configuration of the continental slope has been treated in detail
    by Nansen in _Scientific Results of Norwegian North Polar
    Expedition_, vol. iv. (1904), where full references to the literature
    of the subject will be found.

  [21] _British Association Report_ (Edinburgh, 1892), p. 699.

  [22] _Das Antlitz der Erde_ (4 vols., Leipzig, 1885, 1888, 1901).
    Translated under the editorship of E. de Margerie, with much
    additional matter, as _La Face de la terre_, vols. i. and ii. (Paris,
    1897, 1900), and into English by Dr Hertha Sollas as _The Face of the
    Earth_, vols. i. and ii. (Oxford, 1904, 1906).

  [23] Élie de Beaumont, _Notice sur les systèmes de montagnes_ (3
    vols., Paris, 1852).

  [24] _Vestiges of the Molten Globe_ (London, 1875).

  [25] See J.W. Gregory, "The Plan of the Earth and its Causes," _Geog.
    Journal_, xiii. (1899) p. 225; Lord Avebury, _ibid._ xv. (1900) p.
    46; Marcel Bertrand, "Déformation tétraédrique de la terre et
    déplacement du pôle," _Comptes rendus Acad. Sci._ (Paris, 1900), vol.
    cxxx. p. 449; and A. de Lapparent, _ibid._ p. 614.

  [26] See A.E.H. Love, "Gravitational Stability of the Earth," _Phil.
    Trans._ ser. A. vol. ccvii. (1907) p. 171.

  [27] _Rumpf_, in German, the language in which this distinction was
    first made.

  [28] _Lehrbuch der Geographie_ (Hanover and Leipzig, 1900), Bd. i. S.
    245, 249.

  [29] See, for example, F.G. Hahn's _Insel-Studien_ (Leipzig, 1883).

  [30] See _Geographical Journal_, xxii. (1903) pp. 191-194.

  [31] The most important works on the classification of land forms are
    F. von Richthofen, _Führer für Forschungsreisende_ (Berlin, 1886); G.
    de la Noë and E. de Margerie, _Les Formes du terrain_ (Paris, 1888);
    and above all A. Penck, _Morphologie der Erdoberfläche_ (2 vols.,
    Stuttgart, 1894). Compare also A. de Lapparent, _Leçons de géographie
    physique_ (2nd ed., Paris, 1898), and W.M. Davis, _Physical
    Geography_ (Boston, 1899).

  [32] "Geomorphologie als genetische Wissenschaft," in _Report of
    Sixth International Geog. Congress_ (London, 1895), p. 735 (English
    Abstract, p. 748).

  [33] On this subject see J. Geikie, _Earth Sculpture_ (London, 1898);
    J.E. Marr, _The Scientific Study of Scenery_ (London, 1900); Sir A.
    Geikie, _The Scenery and Geology of Scotland_ (London, 2nd ed.,
    1887); Lord Avebury (Sir J. Lubbock), _The Scenery of Switzerland_
    (London, 1896) and _The Scenery of England_ (London, 1902).

  [34] Some geographers distinguish a mountain from a hill by origin;
    thus Professor Seeley says "a mountain implies elevation and a hill
    implies denudation, but the external forms of both are often
    identical." _Report VI. Int. Geog. Congress_ (London, 1895), p. 751.

  [35] "Mountains," in _Scot. Geog. Mag._ ii. (1896) p. 145.

  [36] _Führer für Forschungsreisende_, pp. 652-685.

  [37] See, for a summary of river-action, A. Phillipson, _Studien über
    Wasserscheiden_ (Leipzig, 1886); also I.C. Russell, _River
    Development_, (London, 1898) (published as _The Rivers of North
    America_, New York, 1898).

  [38] W.M. Davis, "The Geographical Cycle," _Geog. Journ._ xiv. (1899)
    p. 484.

  [39] A. Penck, "Potamology as a Branch of Physical Geography," _Geog.
    Journ._ x. (1897) p. 619.

  [40] See, for instance, E. Wisotzki, _Hauptfluss und Nebenfluss_
    (Stettin, 1889). For practical studies see official reports on the
    Mississippi, Rhine, Seine, Elbe and other great rivers.

  [41] F.A. Forel, _Handbuch der Seenkunde: allgemeine Limnologie_
    (Stuttgart, 1901); F.A. Forel, "La Limnologie, branche de la
    géographie," _Report VI. Int. Geog. Congress_ (London, 1895), p. 593;
    also _Le Léman_ (2 vols., Lausanne, 1892, 1894); H. Lullies, "Studien
    über Seen," _Jubiläumsschrift der Albertus-Universität_ (Königsberg,
    1894); and G.R. Credner, "Die Reliktenseen," _Petermanns
    Mitteilungen_, Ergänzungshefte 86 and 89 (Gotha., 1887, 1888).

  [42] J. Murray, "Drainage Areas of the Continents," _Scot. Geog.
    Mag._ ii. (1886) p. 548.

  [43] Wagner, _Lehrbuch der Geographie_ (1900), i. 586.

  [44] For details, see A.R. Wallace, _Geographical Distribution of
    Animals and Island Life_; A. Heilprin, _Geographical and Geological
    Distribution of Animals_ (1887); O. Drude, _Handbuch der
    Pflanzengeographie_; A. Engler, _Entwickelungsgeschichte der
    Pflanzenwelt_; also Beddard, _Zoogeography_ (Cambridge, 1895); and
    Sclater, _The Geography of Mammals_ (London, 1899).

  [45] See particularly A. de Lapparent, _Traité de géologie_ (4th ed.,
    Paris, 1900).

  [46] Estimate for 1900. H. Wagner, _Lehrbuch der Geographie_, i. P.
    658.

  [47] Estimate for year not stated. A.H. Keane in _International
    Geography_, p. 108.

  [48] In _Proc. R. G. S._ xiii. (1891) p. 27.

  [49] On the influence of land on people see Shaler, _Nature and Man
    in America_ (New York and London, 1892); and Ellen C. Semple's
    _American History and its Geographic Conditions_ (Boston, 1903).

  [50] See maps of density of population in Bartholomew's great
    large-scale atlases, _Atlas of Scotland_ and _Atlas of England_.

  [51] For the history of territorial changes in Europe, see Freeman,
    _Historical Geography of Europe_, edited by Bury (Oxford), 1903; and
    for the official definition of existing boundaries, see Hertslet,
    _The Map of Europe by Treaty_ (4 vols., London, 1875, 1891); _The Map
    of Africa by Treaty_ (3 vols., London, 1896). Also Lord Curzon's
    Oxford address on _Frontiers_ (1907).

  [52] For numerous special instances of the determining causes of town
    sites, see G.G. Chisholm, "On the Distribution of Towns and Villages
    in England," _Geographical Journal_ (1897), ix. 76, x. 511.

  [53] The whole subject of anthropogeography is treated in a masterly
    way by F. Ratzel in his _Anthropogeographie_ (Stuttgart, vol. i. 2nd
    ed., 1899, vol. ii. 1891), and in his _Politische Geographie_
    (Leipzig, 1897). The special question of the reaction of man on his
    environment is handled by G.P. Marsh in _Man and Nature, or Physical
    Geography as modified by Human Action_ (London, 1864).

  [54] For commercial geography see G.G. Chisholm, _Manual of
    Commercial Geography_ (1890).




GEOID (from Gr. [Greek: gê], the earth), an imaginary surface employed
by geodesists which has the property that every element of it is
perpendicular to the plumb-line where that line cuts it. Compared with
the "spheroid of reference" the surface of the geoid is in general
depressed over the oceans and raised over the great land masses. (See
EARTH, FIGURE OF THE.)




GEOK-TEPE, a former fortress of the Turkomans, in Russian Transcaspia,
in the oasis of Akhal-tekke, on the Transcaspian railway, 28 m. N.W. of
Askabad. It consisted of a walled enclosure 1¾ m. in circuit, the wall
being 18 ft. high and 20 to 30 ft. thick. In December 1880 the place was
attacked by 6000 Russians under General Skobelev, and after a siege of
twenty-three days was carried by storm, although the defenders numbered
25,000. A monument and a small museum commemorate the event.




GEOLOGY (from Gr. [Greek: gê], the earth, and [Greek: logos], science),
the science which investigates the physical history of the earth. Its
object is to trace the structural progress of our planet from the
earliest beginnings of its separate existence, through its various
stages of growth, down to the present condition of things. It seeks to
determine the manner in which the evolution of the earth's great surface
features has been effected. It unravels the complicated processes by
which each continent has been built up. It follows, even into detail,
the varied sculpture of mountain and valley, crag and ravine. Nor does
it confine itself merely to changes in the inorganic world. Geology
shows that the present races of plants and animals are the descendants
of other and very different races which once peopled the earth. It
teaches that there has been a progressive development of the
inhabitants, as well as one of the globe on which they have dwelt; that
each successive period in the earth's history, since the introduction of
living things, has been marked by characteristic types of the animal and
vegetable kingdoms; and that, however imperfectly the remains of these
organisms have been preserved or may be deciphered, materials exist for
a history of life upon the planet. The geographical distribution of
existing faunas and floras is often made clear and intelligible by
geological evidence; and in the same way light is thrown upon some of
the remoter phases in the history of man himself. A subject so
comprehensive as this must require a wide and varied basis of evidence.
It is one of the characteristics of geology to gather evidence from
sources which at first sight seem far removed from its scope, and to
seek aid from almost every other leading branch of science. Thus, in
dealing with the earliest conditions of the planet, the geologist must
fully avail himself of the labours of the astronomer. Whatever is
ascertainable by telescope, spectroscope or chemical analysis, regarding
the constitution of other heavenly bodies, has a geological bearing. The
experiments of the physicist, undertaken to determine conditions of
matter and of energy, may sometimes be taken as the starting-points of
geological investigation. The work of the chemical laboratory forms the
foundation of a vast and increasing mass of geological inquiry. To the
botanist, the zoologist, even to the unscientific, if observant,
traveller by land or sea, the geologist turns for information and
assistance.

But while thus culling freely from the dominions of other sciences,
geology claims as its peculiar territory the rocky framework of the
globe. In the materials composing that framework, their composition and
arrangement, the processes of their formation, the changes which they
have undergone, and the terrestrial revolutions to which they bear
witness, lie the main data of geological history. It is the task of the
geologist to group these elements in such a way that they may be made to
yield up their evidence as to the march of events in the evolution of
the planet. He finds that they have in large measure arranged themselves
in chronological sequence,--the oldest lying at the bottom and the
newest at the top. Relics of an ancient sea-floor are overlain by traces
of a vanished land-surface; these are in turn covered by the deposits of
a former lake, above which once more appear proofs of the return of the
sea. Among these rocky records lie the lavas and ashes of long-extinct
volcanoes. The ripple left upon the shore, the cracks formed by the
sun's heat upon the muddy bottom of a dried-up pool, the very imprint of
the drops of a passing rainshower, have all been accurately preserved,
and yield their evidence as to geographical conditions often widely
different from those which exist where such markings are now found.

But it is mainly by the remains of plants and animals imbedded in the
rocks that the geologist is guided in unravelling the chronological
succession of geological changes. He has found that a certain order of
appearance characterizes these organic remains, that each great group of
rocks is marked by its own special types of life, and that these types
can be recognized, and the rocks in which they occur can be correlated
even in distant countries, and where no other means of comparison would
be possible. At one moment he has to deal with the bones of some large
mammal scattered through a deposit of superficial gravel, at another
time with the minute foraminifers and ostracods of an upraised
sea-bottom. Corals and crinoids crowded and crushed into a massive
limestone where they lived and died, ferns and terrestrial plants matted
together into a bed of coal where they originally grew, the scattered
shells of a submarine sand-bank, the snails and lizards which lived and
died within a hollow-tree, the insects which have been imprisoned within
the exuding resin of old forests, the footprints of birds and
quadrupeds, the trails of worms left upon former shores--these, and
innumerable other pieces of evidence, enable the geologist to realize in
some measure what the faunas and floras of successive periods have been,
and what geographical changes the site of every land has undergone.

It is evident that to deal successfully with these varied materials, a
considerable acquaintance with different branches of science is needful.
Especially necessary is a tolerably wide knowledge of the processes now
at work in changing the surface of the earth, and of at least those
forms of plant and animal life whose remains are apt to be preserved in
geological deposits, or which in their structure and habitat enable us
to realize what their forerunners were. It has often been insisted that
the present is the key to the past; and in a wide sense this assertion
is eminently true. Only in proportion as we understand the present,
where everything is open on all sides to the fullest investigation, can
we expect to decipher the past, where so much is obscure, imperfectly
preserved or not preserved at all. A study of the existing economy of
nature ought thus to be the foundation of the geologist's training.

While, however, the present condition of things is thus employed, we
must obviously be on our guard against the danger of unconsciously
assuming that the phase of nature's operations which we now witness has
been the same in all past time, that geological changes have always or
generally taken place in former ages in the manner and on the scale
which we behold to-day, and that at the present time all the great
geological processes, which have produced changes in the past eras of
the earth's history, are still existent and active. As a working
hypothesis we may suppose that the nature of geological processes has
remained constant from the beginning; but we cannot postulate that the
action of these processes has never varied in energy. The few centuries
wherein man has been observing nature obviously form much too brief an
interval by which to measure the intensity of geological action in all
past time. For aught we can tell the present is an era of quietude and
slow change, compared with some of the eras which have preceded it. Nor
perhaps can we be quite sure that, when we have explored every
geological process now in progress, we have exhausted all the causes of
change which, even in comparatively recent times, have been at work.

In dealing with the geological record, as the accessible solid part of
the globe is called, we cannot too vividly realize that at the best it
forms but an imperfect chronicle. Geological history cannot be compiled
from a full and continuous series of documents. From the very nature of
its origin the record is necessarily fragmentary, and it has been
further mutilated and obscured by the revolutions of successive ages.
And even where the chronicle of events is continuous, it is of very
unequal value in different places. In one case, for example, it may
present us with an unbroken succession of deposits many thousands of
feet in thickness, from which, however, only a few meagre facts as to
geological history can be gleaned. In another instance it brings before
us, within the compass of a few yards, the evidence of a most varied and
complicated series of changes in physical geography, as well as an
abundant and interesting suite of organic remains. These and other
characteristics of the geological record become more apparent and
intelligible as we proceed in the study of the science.

_Classification._--For systematic treatment the subject may be
conveniently arranged in the following parts:--

1. _The Historical Development of Geological Science._--Here a brief
outline will be given of the gradual growth of geological conceptions
from the days of the Greeks and Romans down to modern times, tracing the
separate progress of the more important branches of inquiry and noting
some of the stages which in each case have led up to the present
condition of the science.

2. _The Cosmical Aspects of Geology._--This section embraces the
evidence supplied by astronomy and physics regarding the form and
motions of the earth, the composition of the planets and sun, and the
probable history of the solar system. The subjects dealt with under this
head are chiefly treated in separate articles.

3. _Geognosy._--An inquiry into the materials of the earth's substance.
This division, which deals with the parts of the earth, its envelopes of
air and water, its solid crust and the probable condition of its
interior, especially treats of the more important minerals of the crust,
and the chief rocks of which that crust is built up. Geognosy thus lays
a foundation of knowledge regarding the nature of the materials
constituting the mass of the globe, and prepares the way for an
investigation of the processes by which these materials are produced and
altered.

4. _Dynamical Geology_ studies the nature and working of the various
geological processes whereby the rocks of the earth's crust are formed
and metamorphosed, and by which changes are effected upon the
distribution of sea and land, and upon the forms of terrestrial
surfaces. Such an inquiry necessitates a careful examination of the
existing geological economy of nature, and forms a fitting introduction
to an inquiry into the geological changes of former periods.

5. _Geotectonic or Structural Geology_ has for its object the
architecture of the earth's crust. It embraces an inquiry into the
manner in which the various materials composing this crust have been
arranged. It shows that some have been formed in beds or strata of
sediment on the floor of the sea, that others have been built up by the
slow aggregation of organic forms, that others have been poured out in a
molten condition or in showers of loose dust from subterranean sources.
It further reveals that, though originally laid down in almost
horizontal beds, the rocks have subsequently been crumpled, contorted
and dislocated, that they have been incessantly worn down, and have
often been depressed and buried beneath later accumulations.

6. _Palaeontological Geology._--This branch of the subject, starting
from the evidence supplied by the organic forms which are found
preserved in the crust of the earth, includes such questions as the
relations between extinct and living types, the laws which appear to
have governed the distribution of life in time and in space, the
relative importance of different genera of animals in geological
inquiry, the nature and use of the evidence from organic remains
regarding former conditions of physical geography. Some of these
problems belong also to zoology and botany, and are more fully discussed
in the articles PALAEONTOLOGY and PALAEOBOTANY.

7. _Stratigraphical Geology._--This section might be called geological
history. It works out the chronological succession of the great
formations of the earth's crust, and endeavours to trace the sequence of
events of which they contain the record. More particularly, it
determines the order of succession of the various plants and animals
which in past time have peopled the earth, and thus ascertains what has
been the grand march of life upon this planet.

8. _Physiographical Geology_, proceeding from the basis of fact laid
down by stratigraphical geology regarding former geographical changes,
embraces an inquiry into the origin and history of the features of the
earth's surface--continental ridges and ocean basins, plains, valleys
and mountains. It explains the causes on which local differences of
scenery depend, and shows under what very different circumstances, and
at what widely separated intervals, the hills and mountains, even of a
single country, have been produced.

Most of the detail embraced in these several sections is relegated to
separate articles, to which references are here inserted. The following
pages thus deal mainly with the general principles and historical
development of the science:--


  PART I.--HISTORICAL DEVELOPMENT

  _Geological Ideas among the Greeks and Romans._--Many geological
  phenomena present themselves in so striking a form that they could
  hardly fail to impress the imagination of the earliest and rudest
  races of mankind. Such incidents as earthquakes and volcanic
  eruptions, destructive storms on land and sea, disastrous floods and
  landslips suddenly strewing valleys with ruin, must have awakened the
  terror of those who witnessed them. Prominent features of landscape,
  such as mountain-chains with their snows, clouds and thunderstorms,
  dark river-chasms that seem purposely cleft open in order to give
  passage to the torrents that rush through them, crags with their
  impressive array of pinnacles and recesses must have appealed of old,
  as they still do, to the awe and wonder of those who for the first
  time behold them. Again, banks of sea-shells in far inland districts
  would, in course of time, arrest the attention of the more intelligent
  and reflective observers, and raise in their minds some kind of
  surmise as to how such shells could ever have come there. These and
  other conspicuous geological problems found their earliest solution in
  legends and myths, wherein the more striking terrestrial features and
  the elemental forces of nature were represented to be the
  manifestation of the power of unseen supernatural beings.

  The basin of the Mediterranean Sea was especially well adapted, from
  its physical conditions, to be the birth-place of such fables. It is a
  region frequently shaken by earthquakes, and contains two distinct
  centres of volcanic activity, one in the Aegean Sea and one in Italy.
  It is bounded on the north by a long succession of lofty snow-capped
  mountain-ranges, whence copious rivers, often swollen by heavy rains
  or melted snows, carry the drainage into the sea. On the south it
  boasts the Nile, once so full of mystery; likewise wide tracts of arid
  desert with their dreaded dust storms. The Mediterranean itself,
  though an inland sea, is subject to gales, which, on exposed coasts,
  raise breakers quite large enough to give a vivid impression of the
  power of ocean waves. The countries that surround this great sheet of
  water display in many places widely-spread deposits full of sea
  shells, like those that still live in the neighbouring bays and gulfs.
  Such a region was not only well fitted to supply subjects for
  mythology, but also to furnish, on every side, materials which, in
  their interest and suggestiveness, would appeal to the reason of
  observant men.

  It was natural, therefore, that the early philosophers of Greece
  should have noted some of these geological features, and should have
  sought for other explanations of them than those to be found in the
  popular myths. The opinions entertained in antiquity on these subjects
  may be conveniently grouped under two heads: (1) Geological processes
  now in operation, and (2) geological changes in the past.


    Earthquakes and volcanoes.

  1. _Contemporary Processes._--The geological processes of the present
  time are partly at work underground and partly on the surface of the
  earth. The former, from their frequently disastrous character,
  received much attention from Greek and Roman authors. Aristotle, in
  his _Meteorics_, cites the speculations of several of his predecessors
  which he rejects in favour of his own opinion to the effect that
  earthquakes are due to the generation of wind within the earth, under
  the influence of the warmth of the sun and the internal heat. Wind,
  being the lightest and most rapidly moving body, is the cause of
  motion in other bodies, and fire, united with wind, becomes flame,
  which is endowed with great rapidity of motion. Aristotle looked upon
  earthquakes and volcanic eruptions as closely connected with each
  other, the discharge of hot materials to the surface being the result
  of a severe earthquake, when finally the wind rushes out with
  violence, and sometimes buries the surrounding country under sparks
  and cinders, as had happened at Lipari. These crude conceptions of
  the nature of volcanic action, and the cause of earthquakes, continued
  to prevail for many centuries. They are repeated by Lucretius, who,
  however, following Anaximenes, includes as one of the causes of
  earthquakes the fall of mountainous masses of rock undermined by time,
  and the consequent propagation of gigantic tremors far and wide
  through the earth. Strabo, having travelled through the volcanic
  districts of Italy, was able to recognize that Vesuvius had once been
  an active volcano, although no eruption had taken place from it within
  human memory. He continued to hold the belief that volcanic energy
  arose from the movement of subterranean wind. He believed that the
  district around the Strait of Messina, which had formerly suffered
  from destructive earthquakes, was seldom visited by them after the
  volcanic vents of that region had been opened, so as to provide an
  escape for the subterranean fire, wind, water and burning masses. He
  cites in his _Geography_ a number of examples of widespread as well as
  local sinkings of land, and alludes also to the uprise of the
  sea-bottom. He likewise regards some islands as having been thrown up
  by volcanic agency, and others as torn from the mainland by such
  convulsions as earthquakes.

  The most detailed account of earthquake phenomena which has come down
  to us from antiquity is that of Seneca in his _Quaestiones Naturales_.
  This philosopher had been much interested in the accounts given him by
  survivors and witnesses of the earthquake which convulsed the district
  of Naples in February A.D. 63. He distinguished several distinct
  movements of the ground: 1st, the up and down motion (_succussio_);
  2nd, the oscillatory motion (_inclinatio_); and probably a third, that
  of trembling or vibration. While admitting that some earthquakes may
  arise from the collapse of the walls of subterranean cavities, he
  adhered to the old idea, held by the most numerous and important
  previous writers, that these commotions are caused mainly by the
  movements of wind imprisoned within the earth. As to the origin of
  volcanic outbursts he supposed that the subterranean wind in
  struggling for an outlet, and whirling through the chasms and
  passages, meets with great store of sulphur and other combustible
  substances, which by mere friction are set on fire. The elder Pliny
  reiterates the commonly accepted opinion as to the efficacy of wind
  underground. In discussing the phenomena of earthquakes he remarks
  that towns with many culverts and houses with cellars suffer less than
  others, and that at Naples those houses are most shaken which stand on
  hard ground. It thus appears that with regard to subterranean
  geological operations, no advance was made during the time of the
  Greeks and Romans as to the theoretical explanation of these
  phenomena; but a considerable body of facts was collected, especially
  as to the effects of earthquakes and the occurrence of volcanic
  eruptions.


    Action of rivers.

  The superficial processes of geology, being much less striking than
  those of subterranean energy, naturally attracted less attention in
  antiquity. The operations of rivers, however, which so intimately
  affect a human population, were watched with more or less care.
  Herodotus, struck by the amount of alluvial silt brought down annually
  by the Nile and spread over the flat inundated land, inferred that
  "Egypt is the gift of the river." Aristotle, in discussing some of the
  features of rivers, displays considerable acquaintance with the
  various drainage-systems on the north side of the Mediterranean basin.
  He refers to the mountains as condensers of the atmospheric moisture,
  and shows that the largest rivers rise among the loftiest high
  grounds. He shows how sensibly the alluvial deposits carried down to
  the sea increase the breadth of the land, and cites some parts of the
  shores of the Black Sea, where, in sixty years, the rivers had brought
  down such a quantity of material that the vessels then in use required
  to be of much smaller draught than previously, the water shallowing so
  much that the marshy ground would, in course of time, become dry land.
  Strabo supplies further interesting information as to the work of
  rivers in making their alluvial plains and in pushing their deltas
  seaward. He remarks that these deltas are prevented from advancing
  farther outward by the ebb and flow of the tides.


    Occurrences of fossils.

  2. _Past Processes._--The abundant well-preserved marine shells
  exposed among the upraised Tertiary and post-Tertiary deposits in the
  countries bordering the Mediterranean are not infrequently alluded to
  in Greek and Latin literature. Xenophanes of Colophon (614 B.C.)
  noticed the occurrence of shells and other marine productions inland
  among the mountains, and inferred from them that the land had risen
  out of the sea. A similar conclusion was drawn by Xanthus the Lydian
  (464 B.C.) from shells like scallops and cockles, which were found far
  from the sea in Armenia and Lower Phrygia. Herodotus, Eratosthenes,
  Strato and Strabo noted the vast quantities of fossil shells in
  different parts of Egypt, together with beds of salt, as evidence that
  the sea had once spread over the country. But by far the most
  philosophical opinions on the past mutations of the earth's surface
  are those expressed by Aristotle in the treatise already cited.
  Reviewing the evidence of these changes, he recognized that the sea
  now covers tracts that were once dry land, and that land will one day
  reappear where there is now sea. These alternations are to be regarded
  as following each other in a certain order and periodicity. But they
  are apt to escape our notice because they require successive periods
  of time, which, compared with our brief existence, are of enormous
  duration, and because they are brought about so imperceptibly that we
  fail to detect them in progress. In a celebrated passage in his
  _Metamorphoses_, Ovid puts into the mouth of the philosopher
  Pythagoras an account of what was probably regarded as the Pythagorean
  view of the subject in the Augustan age. It affirms the interchange of
  land and sea, the erosion of valleys by descending rivers, the washing
  down of mountains into the sea, the disappearance of the rivers and
  the submergence of land by earthquake movements, the separation of
  some islands from, and the union of others with, the mainland, the
  uprise of hills by volcanic action, the rise and extinction of burning
  mountains. There was a time before Etna began to glow, and the time is
  coming when the mountain will cease to burn.

  From this brief sketch it will be seen that while the ancients had
  accumulated a good deal of information regarding the occurrence of
  geological changes, their interpretations of the phenomena were to a
  considerable extent mere fanciful speculation. They had acquired only
  a most imperfect conception of the nature and operation of the
  geological processes; and though many writers realized that the
  surface of the earth has not always been, and will not always remain,
  as it is now, they had no glimpse of the vast succession of changes of
  that surface which have been revealed by geology. They built
  hypotheses on the slenderest basis of fact, and did not realize the
  necessity of testing or verifying them.

  _Progress of Geological Conceptions in the Middle Ages._--During the
  centuries that succeeded the fall of the Western empire little
  progress was made in natural science. The schoolmen in the monasteries
  and other seminaries were content to take their science from the
  literature of Greece and Rome. The Arabs, however, not only collected
  and translated that literature, but in some departments made original
  observations themselves. To one of the most illustrious of their
  number, Avicenna, the translator of Aristotle, a treatise has been
  ascribed, in which singularly modern ideas are expressed regarding
  mountains, some of which are there stated to have been produced by an
  uplifting of the ground, while others have been left prominent, owing
  to the wearing away of the softer rocks around them. In either case,
  it is confessed that the process would demand long tracts of time for
  its completion.

  After the revival of learning the ancient problem presented by fossil
  shells imbedded in the rocks of the interior of many countries
  received renewed attention. But the conditions for its solution were
  no longer what they had been in the days of the philosophers of
  antiquity. Men were not now free to adopt and teach any doctrine they
  pleased on the subject. The Christian church had meanwhile arisen to
  power all over Europe, and adjudged as heretics all who ventured to
  impugn any of her dogmas. She taught that the land and the sea had
  been separated on the third day of creation, before the appearance of
  any animal life, which was not created until the fifth day. To assert
  that the dry land is made up in great part of rocks that were formed
  in the sea, and are crowded with the remains of animals, was plainly
  to impugn the veracity of the Bible. Again, it had come to be the
  orthodox belief that only somewhere about 6000 years had elapsed since
  the time of Adam and Eve. If any thoughtful observer, impressed with
  the overwhelming force of the evidence that the fossiliferous
  formations of the earth's crust must have taken long periods of time
  for their accumulation, ventured to give public expression to his
  conviction, he ran considerable risk of being proceeded against as a
  heretic. It was needful, therefore, to find some explanation of the
  facts of nature, which would not run counter to the ecclesiastical
  system of the day. Various such interpretations were proposed,
  doubtless in an honest endeavour at reconciliation. Three of these
  deserve special notice: (1) Many able observers and diligent
  collectors of fossils persuaded themselves that these objects never
  belonged to organisms of any kind, but should be regarded as mere
  "freaks of nature," having no more connexion with any once living
  creature than the frost patterns on a window. They were styled
  "formed" or "figured" stones, "lapides sui generis," and were asserted
  to be due to some inorganic imitative process within the earth or to
  the influence of the stars. (2) Observers who could not resist the
  evidence of their senses that the fossil shells once belonged to
  living animals, and who, at the same time, felt the necessity of
  accounting for the presence of marine organisms in the rocks of which
  the dry land is largely built up, sought a way out of the difficulty
  by invoking the Deluge of Noah. Here was a catastrophe which, they
  said, extended over the whole globe, and by which the entire dry land
  was submerged even up to the tops of the high hills. True, it only
  lasted one hundred and fifty days, but so little were the facts then
  appreciated that no difficulty seems to have been generally felt in
  crowding the accumulation of the thousands of feet of fossiliferous
  formations into that brief space of time. (3) Some more intelligent
  men in Italy, recognizing that these interpretations could not be
  upheld, fell back upon the idea that the rocks in which fossil shells
  are imbedded might have been heaped up by repeated and vigorous
  eruptions from volcanic centres. Certain modern eruptions in the
  Aegean Sea and in the Bay of Naples had drawn attention to the
  rapidity with which hills of considerable size could be piled around
  an active crater. It was argued that if Monte Nuovo near Naples could
  have been accumulated to a height of nearly 500 ft. in two days, there
  seemed to be no reason against believing that, during the time of the
  Flood, and in the course of the centuries that have elapsed since
  that event, the whole of the fossiliferous rocks might have been
  deposited. Unfortunately for this hypothesis it ignored the fact that
  these rocks do not consist of volcanic materials.


    Leonardo da Vinci; Fracastorio; Falloppio.

  So long as the fundamental question remained in dispute as to the true
  character and history of the stratified portion of the earth's crust
  containing organic remains, geology as a science could not begin its
  existence. The diluvialists (those who relied on the hypothesis of the
  Flood) held the field during the 16th, 17th and a great part of the
  18th century. They were looked on as the champions of orthodoxy; and,
  on that account, they doubtless wielded much more influence than would
  have been gained by them from the force of their arguments. Yet during
  those ages there were not wanting occasional observers who did good
  service in combating the prevalent misconceptions, and in preparing
  the way for the ultimate triumph of truth. It was more especially in
  Italy, where many of the more striking phenomena of geology are
  conspicuously displayed, that the early pioneers of the science arose,
  and that for several generations the most marked progress was made
  towards placing the investigations of the past history of the earth
  upon a basis of careful observation and scientific deduction. One of
  the first of these leaders was Leonardo da Vinci (1452-1519), who,
  besides his achievements in painting, sculpture, architecture and
  engineering, contributed some notable observations regarding the great
  problem of the origin of fossil shells. He ridiculed the notion that
  these objects could have been formed by the influence of the stars,
  and maintained that they had once belonged to living organisms, and
  therefore that what is now land was formerly covered by the sea.
  Girolamo Fracastorio (1483-1553) claimed that the shells could never
  have been left by the Flood, which was a mere temporary inundation,
  but that they proved the mountains, in which they occur, to have been
  successively uplifted out of the sea. On the other hand, even an
  accomplished anatomist like Gabriello Falloppio (1523-1562) found it
  easier to believe that the bones of elephants, teeth of sharks, shells
  and other fossils were mere earthy inorganic concretions, than that
  the waters of Noah's Flood could ever nave reached as far as Italy.


    Nicolas Steno.

  By much the most important member of this early band of Italian
  writers was undoubtedly Nicolas Steno (1631-1687), who, though born in
  Copenhagen, ultimately settled in Florence. Having made a European
  reputation as an anatomist, his attention was drawn to geological
  problems by finding that the rocks of the north of Italy contained
  what appeared to be sharks' teeth closely resembling those of a
  dog-fish, of which he had published the anatomy. Cautiously at first,
  for fear of offending orthodox opinions, but afterwards more boldly,
  he proclaimed his conviction that those objects had once been part of
  living animals, and that they threw light on some of the past history
  of the earth. He published in 1669 a small tract, _De solido intra
  solidum naturaliter contento_, in which he developed the ideas he had
  formed of this history from an attentive study of the rocks. He showed
  that the stratified formations of the hills and valleys consist of
  such materials as would be laid down in the form of sediment in turbid
  water; that where they contain marine productions this water is proved
  to have been the sea; that diversities in their composition point to
  commingling of currents, carrying different kinds of sediment of which
  the heaviest would first sink to the bottom. He made original and
  important observations on stratification, and laid down some of the
  fundamental axioms in stratigraphy. He reasoned that as the original
  position of strata was approximately horizontal, when they are found
  to be steeply inclined or vertical, or bent into arches, they have
  been disrupted by subterranean exhalations, or by the falling in of
  the roofs of underground cavernous spaces. It is to this alteration of
  the original position of the strata that the inequalities of the
  earth's surface, such as mountains, are to be ascribed, though some
  have been formed by the outburst of fire, ashes and stones from inside
  the earth. Another effect of the dislocation has been to provide
  fissures, which serve as outlets for springs. Steno's anatomical
  training peculiarly fitted him for dealing authoritatively with the
  question of the nature and origin of the fossils contained in the
  rocks. He had no hesitation in affirming that, even if no shells had
  ever been found living in the sea, the internal structure of these
  fossils would demonstrate that they once formed parts of living
  animals. And not only shells, but teeth, bones and skeletons of many
  kinds of fishes had been quarried out of the rocks, while some of the
  strata had skulls, horns and teeth of land-animals. Illustrating his
  general principles by a sketch of what he supposed to have been the
  past history of Tuscany, he added a series of diagrams which show how
  clearly he had conceived the essential elements of stratigraphy. He
  thought he could perceive the records of six successive phases in the
  evolution of the framework of that country, and was inclined to
  believe that a similar chronological sequence would be found all over
  the world. He anticipated the objections that would be brought against
  his views on account of the insuperable difficulty in granting the
  length of time that would be required for all the geographical
  vicissitudes which his interpretation required. He thought that many
  of the fossils must be as old as the time of the general deluge, but
  he was careful not to indulge in any speculation as to the antiquity
  of the earth.


    Lazzaro Moro.

  To the Italian school, as especially typified in Steno, must be
  assigned the honour of having thus begun to lay firmly and truly the
  first foundation stones of the modern science of geology. The same
  school included Antonio Vallisneri (1661-1730), who surpassed his
  predecessors in his wider and more exact knowledge of the
  fossiliferous rocks that form the backbone of the Italian peninsula,
  which he contended were formed during a wide and prolonged submergence
  of the region, altogether different from the brief deluge of Noah.
  There was likewise Lazzaro Moro (1687-1740), who did good service
  against the diluvialists, but the fundamental feature of his system of
  nature lay in the preponderant part which, unaware of the great
  difference between volcanic materials and ordinary sediment, he
  assigned to volcanic action in the production of the sedimentary rocks
  of the earth's crust. He supposed that in the beginning the globe was
  completely surrounded with water, beneath which the solid earth lay as
  a smooth ball. On the third day of creation, however, vast fires were
  kindled inside the globe, whereby the smooth surface of stone was
  broken up, and portions of it, appearing above the water, formed the
  earliest land. From that time onward, volcanic eruptions succeeded
  each other, not only on the emerged land, but on the sea-floor, over
  which the ejected material spread in an ever augmenting thickness of
  sedimentary strata. In this way Moro carried the history of the
  stratified rocks beyond the time of the Flood back to the Creation,
  which was supposed to have been some 1600 years earlier; and he
  brought it down to the present day, when fresh sedimentary deposits
  are continually accumulating. He thus incurred no censure from the
  ecclesiastical guardians of the faith, and he succeeded in attracting
  increased public attention to the problems of geology. The influence
  of his teaching, however, was subsequently in great part due to the
  Carmelite friar Generelli, who published an eloquent exposition of
  Moro's views.

  _The Cosmogonists and Theories of the Earth._--While in Italy
  substantial progress was made in collecting information regarding the
  fossiliferous formations of that country, and in forming conclusions
  concerning them based upon more or less accurate observations, the
  tendency to mere fanciful speculation, which could not be wholly
  repressed in any country, reached a remarkable extravagance in
  England. In proportion as materials were yet lacking from which to
  construct a history of the evolution of our planet in accordance with
  the teaching of the church, imagination supplied the place of
  ascertained fact, and there appeared during the last twenty years of
  the 18th century a group of English cosmogonists, who, by the
  sensational character of their speculations, aroused general attention
  both in Britain and on the continent. It may be doubted, however,
  whether the effect of their writings was not to hinder the advance of
  true science by diverting men from the observation of nature into
  barren controversy over unrealities. It is not needful here to do more
  than mention the names of Thomas Burnet, whose _Sacred Theory of the
  Earth_ appeared in 1681, and William Whiston, whose New Theory of the
  Earth was published in 1696. Hardly less fanciful than these writers,
  though his practical acquaintance with rocks and fossils was
  infinitely greater, was John Woodward, whose _Essay towards a Natural
  History of the Earth_ dates from 1695. More important as a
  contribution to science was the catalogue of the large collection of
  fossils, which he had made from the rocks of England and which he
  bequeathed to the university of Cambridge. This catalogue appeared in
  1728-1729 with the title of _An attempt towards a Natural History of
  the Fossils of England_.


    Descartes.

  A striking contrast to these cosmogonists is furnished by another
  group, which arose in France and Germany, and gave to the world the
  first rational ideas concerning the probable primeval evolution of our
  globe. The earliest of these pioneers was the illustrious philosopher
  René Descartes (1596-1650). He propounded a scheme of cosmical
  development in which he represented the earth, like the other planets,
  to have been originally a mass of glowing material like the sun, and
  to have gradually cooled on the outside, while still retaining an
  incandescent, self-luminous nucleus. Yet with this noble conception,
  which modern science has accepted, Descartes could not shake himself
  free from the time-honoured error in regard to the origin of volcanic
  action. He thought that certain exhalations within the earth condense
  into oil, which, when in violent motion, enters into the subterranean
  cavities, where it passes into a kind of smoke. This smoke is from
  time to time ignited by a spark of fire and, pressing violently
  against its containing walls, gives rise to earthquakes. If the flame
  breaks through to the surface at the top of a mountain, it may escape
  with enormous energy, hurling forth much earth mingled with sulphur or
  bitumen, and thus producing a volcano. The mountain might burn for a
  long time until at last its store of fuel in the shape of sulphur or
  bitumen would be exhausted. Not only did the philosopher refrain from
  availing himself of the high internal temperature of the globe as the
  source of volcanic energy, he even did not make use of it as the cause
  of the ignition of his supposed internal fuel, but speculated on the
  kindling of the subterranean fires by the spirits or gases setting
  fire to the exhalations, or by the fall of masses of rock and the
  sparks produced by their friction or percussion.


    Leibnitz.

  The ideas of Descartes regarding planetary evolution were enlarged and
  made more definite by Wilhelm Gottfried Leibnitz (1646-1716), whose
  teaching has largely influenced all subsequent speculation on the
  subject. In his great tract, the _Protogaea_ (published in 1749,
  thirty-three years after his death), he traced the probable passage of
  our earth from an original condition of incandescent vapour into that
  of a smooth molten globe, which, by continuous cooling, acquired an
  external solid crust and rugose surface. He thought that the more
  ancient rocks, such as granite and gneiss, might be portions of the
  earliest outer crust; and that as the external solidification
  advanced, immense subterranean cavities were left which were filled
  with air and water. By the collapse of the roofs of these caverns,
  valleys might be originated at the surface, while the solid
  intervening walls would remain in place and form mountains. By the
  disruption of the crust, enormous bodies of water were launched over
  the surface of the earth, which swept vast quantities of sediment
  together, and thus gave rise to sedimentary deposits. After many
  vicissitudes of this kind, the terrestrial forces calmed down, and a
  more stable condition of things was established.

  An important feature in the cosmogony of Leibnitz is the prominent
  place which he assigned to organic remains in the stratified rocks of
  the crust. Ridiculing the foolish attempts to account for the presence
  of these objects by calling them "sports of nature," he showed that
  they are to be regarded as historical monuments; and he adduced a
  number of instances wherein successive platforms of strata, containing
  organic remains, bear witness to a series of advances and retreats of
  the sea. He recognized that some of the fossils appeared to have
  nothing like them in the living world of to-day, but some analogous
  forms might yet be found, he thought, in still unexplored parts of the
  earth; and even if no living representatives should ever be
  discovered, many types of animals might have undergone transformation
  during the great changes which had affected the surface of the earth.
  In spite of his clear realization of the vast store of potential
  energy residing within the highly heated interior of the earth,
  Leibnitz continued to regard volcanic action as due to the combustion
  of inflammable substances enclosed within the terrestrial crust, such
  as stone-coal, naphtha and sulphur.


    Buffon.

  Appealing to a much wider public than Descartes or Leibnitz, and
  basing his speculations on a wider acquaintance with the organic and
  inorganic realms of nature, G.L.L. de Buffon (1707-1788) was
  undoubtedly one of the most influential forces that in Europe guided
  the growth of geological ideas during the 18th century. He published
  in 1749 a _Theory of the Earth_, in which he adopted views similar to
  those of Descartes and Leibnitz as to planetary evolution; but though
  he realized the importance of fossils as records of former conditions
  of the earth's surface, he accounted for them by supposing that they
  had been deposited from a universal ocean, a large part of which had
  subsequently been engulfed into caverns in the interior of the globe.
  Thirty years later, after having laboured with skill and enthusiasm in
  all branches of natural history, he published another work, his famous
  _Époques de la nature_ (1778), which is specially remarkable as the
  first attempt to deal with the history of the earth in a chronological
  manner, and to compute, on a basis of experiment, the antiquity of the
  several stages of this history. His experiments were made with globes
  of cast iron, and could not have yielded results of any value for his
  purpose; but in so far as his calculations were not mere random
  guesses but had some kind of foundation on experiment, they deserve
  respectful recognition. He divided the history of our earth into six
  periods of unequal duration, the whole comprising a period of some
  70,000 or 75,000 years. He supposed that the stage of incandescence,
  before the globe had consolidated to the centre, lasted 2936 years,
  and that about 35,000 years elapsed before the surface had cooled
  sufficiently to be touched, and therefore to be capable of supporting
  living things. Terrestrial animal life, however, was not introduced
  until 55,000 or 60,000 years after the beginning of the world or about
  15,000 years before our time. Looking into the future, he foresaw
  that, by continued refrigeration, our globe will eventually become
  colder than ice, and this fair face of nature, with its manifold
  varieties of plant and animal life, will perish after having existed
  for 132,000 years.

  Buffon's conception of the operation of the geological agents did not
  become broader or more accurate in the interval between the appearance
  of his two treatises. He still continued to believe in the lowering of
  the ocean by subsidence into vast subterranean cavities, with a
  consequent emergence of land. He still looked on volcanoes as due to
  the burning of "pyritous and combustible stones," though he now called
  in the co-operation of electricity. He calculated that the first
  volcanoes could not arise until some 50,000 years after the beginning
  of the world, by which time a sufficient extent of dense vegetation
  had been buried in the earth to supply them with fuel. He appears to
  have had but an imperfect acquaintance with the literature of his own
  time. At least there can be little doubt that had he availed himself
  of the labours of his own countryman, Jean Etienne Guettard
  (1715-1786), of Giovanni Arduíno (1714-1795) in Italy, and of Johann
  Gottlob Lehmann (d. 1767) and George Christian Füchsel (1722-1773) in
  Germany, he would have been able to give to his "epochs" a more
  definite succession of events and a greater correspondence with the
  facts of nature.


    James Hutton.

  Among the writers of the 18th century, who formed philosophical
  conceptions of the system of processes by which the life of our earth
  as a habitable globe is carried on, a foremost place must be assigned
  to James Hutton (1726-1797). Educated for the medical profession, he
  studied at Edinburgh and at Paris, and took his doctor's degree at
  Leiden. But having inherited a small landed property in Berwickshire,
  he took to agriculture, and after putting his land into excellent
  order, let his farm and betook himself to Edinburgh, there to gratify
  the scientific tastes which he had developed early in life. He had
  been more especially led to study minerals and rocks, and to meditate
  on the problems which they suggest as to the constitution and history
  of the earth. His journeys in Britain and on the continent of Europe
  had furnished him with material for reflection; and he had gradually
  evolved a system or theory in which all the scattered facts could be
  arranged so as to show their mutual dependence and their place in the
  orderly mechanism of the world. He used to discuss his views with one
  or two of his friends, but refrained from publishing them to the world
  until, on the foundation of the Royal Society of Edinburgh, he
  communicated an outline of his doctrine to that learned body in 1785.
  Some years later he expanded this first essay into a larger work in
  two volumes, which were published in 1795 with the title of _Theory of
  the Earth, with Proofs and Illustrations_.


    John Playfair.

  Hutton's teaching has exercised a profound influence on modern
  geology. This influence, however, has arisen less from his own
  writings than from the account of his doctrines given by his friend
  John Playfair in the classic work entitled _Illustrations of the
  Huttonian Theory_, published in 1802. Hutton wrote in so prolix and
  obscure a style as rather to repel than attract readers. Playfair, on
  the other hand, expressed himself in such clear and graceful language
  as to command general attention, and to gain wide acceptance for his
  master's views. Unlike the older cosmogonists, Hutton refrained from
  trying to explain the origin of things, and from speculations as to
  what might possibly have been the early history of our globe. He
  determined from the outset to interpret the past by what can be seen
  to be the present order of nature; and he refused to admit the
  operation of causes which cannot be shown to be part of the actual
  terrestrial system. Like other observers who had preceded him, he
  recognized in the various rocks composing the dry land evidence of
  former geographical conditions very different from those which now
  prevail. He saw that the vast majority of rocks consist of hardened
  sediments and must have been deposited in the sea. He could
  distinguish among them an older or Primary series, and a younger or
  Secondary series; and did not dispute the existence of a Tertiary
  series claimed by Peter Simon Pallas (1741-1811). He believed that
  these various aqueous accumulations had been consolidated by
  subterranean heat, that the oldest and lowest rocks had suffered most
  from this action, that into these more deep-seated masses subsequent
  veins and larger bodies of molten matter were injected from below, and
  thus that what was originally loose detritus eventually became changed
  in such crystalline schists as are now found in mountain-chains. In
  the course of these terrestrial revolutions sedimentary strata,
  originally more or less nearly horizontal, have been pushed upward,
  dislocated, crumpled, placed on end, and even elevated to form ranges
  of lofty mountains. Hutton looked upon these disturbances as due to
  the expansive power of subterranean heat; but he did not attempt to
  sketch the mechanism of the process, and he expressly declined to
  offer any conjecture as to how the land so elevated remains in that
  position. He thought that the interior of our planet may "be a fluid
  mass, melted, but unchanged by the action of heat"; and, far from
  connecting volcanoes with the combustion of inflammable substances, as
  had been the prevalent belief for so many centuries, he looked upon
  them as a beneficent provision of "spiracles to the subterranean
  furnace, in order to prevent the unnecessary elevation of land and
  fatal effects of earthquakes."

  A distinguishing feature of the Huttonian philosophy is to be seen in
  the breadth of its conceptions regarding the geological operations
  continually in progress on the surface of the globe. Hutton saw that
  the land is undergoing a ceaseless process of degradation, through the
  influence of the air, frost, rain, rivers and the sea, and that in
  course of time, if no countervailing agency should intervene, the
  whole of the dry land will be washed away into the sea. But he also
  perceived that this universal erosion is not everywhere carried on at
  the same rate; that it is specially active along the channels of
  torrents and rivers, and that, owing to this difference these channels
  are gradually deepened and widened, until the complicated
  valley-system of a country is carved out. He recognized that the
  detritus worn away from the land must be spread out over the floor of
  the sea, so as to form there strata similar to those that compose most
  of the dry land. As he could detect in the structure of land
  convincing evidence that former sea floors had been elevated to form
  the continents and islands of to-day, he could look forward to future
  ages, when the same subterranean agency which had raised up the
  present land would again be employed to uplift the bed of the existing
  ocean, thus to renew the surface of our earth as a habitable globe,
  and to start a fresh cycle of erosion and deposition.


    Lamarck.

  Though Hutton was not unaware that organic remains abound in many of
  the stratified rocks, he left them out of consideration in the
  elaboration of his theory. It was otherwise with one of his French
  contemporaries, the illustrious J.B. Lamarck (1744-1829), who, after
  having attained great eminence as a botanist, turned to zoology when
  he was nearly fifty years of age, and before long rose to even greater
  distinction in that department of science. His share in the
  classification and description of the mollusca and in founding
  invertebrate palaeontology, his theory of organic evolution and his
  philosophical treatment of many biological questions have been tardily
  recognized, but his contributions to geology have been less generally
  acknowledged. When he accepted the "professorship of zoology; of
  insects, of worms and of microscopic animals" at the Museum of Natural
  History, Paris, in 1793, he at once entered with characteristic ardour
  and capacity into the new field of research then opened to him. In
  dealing with the mollusca he considered not merely the living but also
  the extinct forms, especially the abundant, varied and well-preserved
  genera and species furnished by the Tertiary deposits of the Paris
  basin, of which he published descriptions and plates that proved of
  essential service in the stratigraphical work of Cuvier and Alexandre
  Brongniart (1770-1847). His labours among these relics of ancient seas
  and lakes led him to ponder over the past history of the globe, and as
  he was seldom dilatory in making known the opinions he had formed, he
  communicated some of his conclusions to the National Institute in
  1799. These, including a further elaboration of his views, he
  published in 1802 in a small volume entitled Hydrogéologie.

  This treatise, though it did not reach a second edition and has never
  been reprinted, deserves an honourable place in geological literature.
  Its object, the author states, was to present some important and novel
  considerations, which he thought should form the basis of a true
  theory of the earth. He entirely agreed with the doctrine of the
  subaerial degradation of the land and the erosion of valleys by
  running water. Not even Playfair could have stated this doctrine more
  emphatically, and it is worthy of notice that Playfair's
  _Illustrations of the Huttonian Theory_ appeared in the same year with
  Lamarck's book. The French naturalist, however, carried his
  conclusions so far as to take no account of any great movements of the
  terrestrial crust, which might have produced or modified the main
  physical features of the surface of the globe. He thought that all
  mountains, except such as were thrown up by volcanic agency or local
  accidents, have been cut out of plains, the original surfaces of which
  are indicated by the crests and summits of these elevations.

  Lamarck, in reflecting upon the wide diffusion of fossil shells and
  the great height above the sea at which they are found, conceived the
  extraordinary idea that the ocean basin has been scoured out by the
  sea, and that, by an impulse communicated to the waters through the
  influence chiefly of the moon, the sea is slowly eating away the
  eastern margins of the continents, and throwing up detritus on their
  western coasts, and is thus gradually shifting its basin round the
  globe. He would not admit the operation of cataclysms; but insisted as
  strongly as Hutton on the continuity of natural processes, and on the
  necessity of explaining former changes of the earth's surface by
  causes which can still be seen to be in operation. As might be
  anticipated from his previous studies, he brought living things and
  their remains into the forefront of his theory of the earth. He looked
  upon fossils as one of the chief means of comprehending the
  revolutions which the surface of the earth has undergone; and in his
  little volume he again and again dwells on the vast antiquity to which
  these revolutions bear witness. He acutely argues, from the condition
  of fossil shells, that they must have lived and died where their
  remains are now found.

  In the last part of his treatise Lamarck advances some peculiar
  opinions in physics and chemistry, which he had broached eighteen
  years before, but which had met with no acceptance among the
  scientific men of his time. He believed that the tendency of all
  compound substances is to decay, and thereby to be resolved into their
  component constituents. Yet he saw that the visible crust of the earth
  consists almost wholly of compound bodies. He therefore set himself to
  solve the problem thus presented. Perceiving that the biological
  action of living organisms is constantly forming combinations of
  matter, which would never have otherwise come into existence, he
  proceeded to draw the extraordinary conclusion that the action of
  plant and animal life (the _Pouvoir de la vie_) upon the inorganic
  world is so universal and so potent, that the rocks and minerals which
  form the outer part of the earth's crust are all, without exception,
  the result of the operations of once living bodies. Though this
  sweeping deduction must be allowed to detract from the value of
  Lamarck's work, there can be no doubt that he realized, more fully
  than any one had done before him, the efficacy of plants and animals
  as agents of geological change.


    Cuvier.

  The last notable contributor to the cosmological literature of geology
  was another illustrious Frenchman, the comparative anatomist Cuvier
  (1769-1832). He was contemporary with Lamarck, but of a very different
  type of mind. The brilliance of his speculations, and the charm with
  which he expounded them, early gained for him a prominent place in the
  society of Paris. He too was drawn by his zoological studies to
  investigate fossil organic remains, and to consider the former
  conditions of the earth's surface, of which they are memorials. It was
  among the vertebrate organisms of the Paris basin that he found his
  chief material, and from them that he prepared the memoirs which led
  to him being regarded as the founder of vertebrate palaeontology. But
  beyond their biological interest, they awakened in him a keen desire
  to ascertain the character and sequence of the geographical
  revolutions to which they bear witness. He approached the subject from
  an opposite and less philosophical point of view than that of Lamarck,
  coming to it with certain preconceived notions, which affected all
  his subsequent writings. While Lamarck was by instinct an
  evolutionist, who sought to trace in the history of the past the
  operation of the same natural processes as are still at work, Cuvier,
  on the other hand, was a catastrophist, who invoked a succession of
  vast cataclysms to account for the interruptions in the continuity of
  the geological record.

  In a preliminary _Discourse_ prefixed to his _Recherches sur les
  ossemens fossiles_ (1821) Cuvier gave an outline of what he conceived
  to have been the past history of our globe, so far as he had been able
  to comprehend it from his investigations of the Tertiary formations of
  France. He believed that in that history evidence can be recognized of
  the occurrence of many sudden and disastrous revolutions, which, to
  judge from their effects on the animal life of the time, must have
  exceeded in violence anything we can conceive at the present day, and
  must have been brought about by other agencies than those which are
  now in operation. Yet, in spite of these catastrophes, he saw that
  there has been an upward progress in the animal forms inhabiting the
  globe, until the series ended in the advent of man. He could not,
  however, find any evidence that one species has been developed from
  another, for in that case there should have been traces of
  intermediate forms among the stratified formations, where he affirmed
  that they had never been found. A prominent position in the
  _Discourse_ is given to a strenuous argument to disprove the alleged
  antiquity of some nations, and to show that the last great catastrophe
  occurred not more than some 5000 or 6000 years ago. Cuvier thus linked
  himself with those who in previous generations had contended for the
  efficacy of the Deluge. But his researches among fossil animals had
  given him a far wider outlook into the geological past, and had opened
  up to him a succession of deeply interesting problems in the history
  of life upon the earth, which, though he had not himself material for
  their solution, he could foresee would be cleared up in the future.

  _Gradual Shaping of Geology into a Distinct Branch of Science._--It
  will be seen from the foregoing historical sketch that it was only
  after the lapse of long centuries, and from the labours of many
  successive generations of observers and writers, that what we now know
  as the science of geology came to be recognized as a distinct
  department of natural knowledge, founded upon careful and extended
  study of the structure of the earth, and upon observation of the
  natural processes, which are now at work in changing the earth's
  surface. The term "geology,"[1] descriptive of this branch of the
  investigation of nature, was not proposed until the last quarter of
  the 18th century by Jean André De Luc (1727-1817) and Horace Benedict
  De Saussure (1740-1749). But the science was then in a markedly
  half-formed condition, theoretical speculation still in large part
  supplying the place of deductions from a detailed examination of
  actual fact. In 1807 a few enterprising spirits founded the Geological
  Society of London for the special purpose of counteracting the
  prevalent tendency and confining their intention "to investigate the
  mineral structure of the earth." The cosmogonists and framers of
  Theories of the Earth were succeeded by other schools of thought. The
  Catastrophists saw in the composition of the crust of the earth
  distinct evidence that the forces of nature were once much more
  stupendous in their operation than they now are, and that they had
  from time to time devastated the earth's surface; extirpating the
  races of plants and animals, and preparing the ground for new
  creations of organized life. Then came the Uniformitarians, who,
  pushing the doctrines of Hutton to an extreme which he did not
  propose, saw no evidence that the activity of the various geological
  causes has ever seriously differed from what it is at present. They
  were inclined to disbelieve that the stratified formations of the
  earth's crust furnish conclusive evidence of a gradual progression,
  from simple types of life in the oldest strata to the most highly
  developed forms in the youngest; and saw no reason why remains of the
  higher vertebrates should not be met with among the Palaeozoic
  formations. Sir Charles Lyell (1797-1875) was the great leader of this
  school. His admirably clear and philosophical presentations of
  geological facts which, with unwearied industry, he collected from the
  writings of observers in all parts of the world, impressed his views
  upon the whole English-speaking world, and gave to geological science
  a coherence and interest which largely accelerated its progress. In
  his later years, however, he frankly accepted the views of Darwin in
  regard to the progressive character of the geological record.

  The youngest of the schools of geological thought is that of the
  Evolutionists. Pointing to the whole body of evidence from inorganic
  and organic nature, they maintain that the history of our planet has
  been one of continual and unbroken development from the earliest
  cosmical beginnings down to the present time, and that the crust of
  the earth contains an abundant, though incomplete, record of the
  successive stages through which the plant and animal kingdoms have
  reached their existing organization. The publication of Darwin's
  _Origin of Species_ in 1859, in which evolution was made the key to
  the history of the animal and vegetable kingdoms, produced an
  extraordinary revolution in geological opinion. The older schools of
  thought rapidly died out, and evolution became the recognized creed of
  geologists all over the world.


    Werner.

  _Development of Opinion regarding Igneous Rocks._--So long as the idea
  prevailed that volcanoes are caused by the combustion of inflammable
  substances underground, there could be no rational conception of
  volcanic action and its products. Even so late as the middle of the
  18th century, as above remarked, such a good observer as Lazzaro Moro
  drew so little distinction between volcanic and other rocks that he
  could believe the fossiliferous formations to have been mainly formed
  of materials ejected from eruptive vents. After his time the notion
  continued to prevail that all the rocks which form the dry land were
  laid down under water. Even streams of lava, which were seen to flow
  from an active crater, were regarded only as portions of sedimentary
  or other rocks, which had been melted by the fervent heat of the
  burning inflammable materials that had been kindled underground. In
  spite of the speculations of Descartes and Leibnitz, it was not yet
  generally comprehended that there exists beneath the terrestrial crust
  a molten magma, which, from time to time, has been injected into that
  crust, and has pierced through it, so as to escape at the surface with
  all the energy of an active volcano. What we now recognize to be
  memorials of these former injections and propulsions were all
  confounded with the rocks of unquestionably aqueous origin. The last
  great teacher by whom these antiquated doctrines were formulated into
  a system and promulgated to the world was Abraham Gottlob Werner
  (1749-1815), the most illustrious German mineralogist and geognost of
  the second half of the 18th century. While still under twenty-six
  years of age, he was appointed teacher of mining and mineralogy at the
  Mining Academy of Freiberg in Saxony--a post which he continued to
  fill up to the end of his life. Possessed of great enthusiasm for his
  subject, clear, methodical and eloquent in his exposition of it, he
  soon drew around him men from all parts of the world, who repaired to
  study under the great oracle of what he called geognosy (Gr. [Greek:
  gê], the earth, [Greek: gnôsis], knowledge) or earth-knowledge.
  Reviving doctrines that had been current long before his time, he
  taught that the globe was once completely surrounded with an ocean,
  from which the rocks of the earth's crust were deposited as chemical
  precipitates, in a certain definite order over the whole planet. Among
  these "universal formations" of aqueous origin were included many
  rocks, which have long been recognized to have been once molten, and
  to have risen from below into the upper parts of the terrestrial
  crust. Werner, following the old tradition, looked upon volcanoes as
  modern features in the history of the planet, which could not have
  come into existence until a sufficient amount of vegetation had been
  buried to furnish fuel for their maintenance. Hence he attached but
  little importance to them, and did not include in his system of rocks
  any division of volcanic or igneous materials. From the predominant
  part assigned by him to the sea in the accumulation of the materials
  of the visible part of the earth, Werner and his school were known as
  "Neptunists."


    Origin of basalt.

  But many years before the Saxon professor began to teach, clear
  evidence had been produced from central France that basalt, one of the
  rocks claimed by him as a chemical precipitate and a universal
  formation, is a lava which has been poured out in a molten state at
  various widely separated periods of time and at many different places.
  So far back as 1752 J.E. Guettard (1715-1786) had shown that the
  basaltic rocks of Auvergne are true lavas, which have flowed out in
  streams from groups of once active cones. Eleven years later the
  observation was confirmed and greatly extended by Nicholas Desmarest
  (1725-1815), who, during a long course of years, worked out and mapped
  the complicated volcanic records of that interesting region, and
  demonstrated to all who were willing impartially to examine the
  evidence the true volcanic nature of basalt. These views found
  acceptance from some observers, but they were vehemently opposed by
  the followers of Werner, who, by the force of his genius, made his
  theoretical conceptions predominate all over Europe. The controversy
  as to the origin of basalt was waged with great vigour during the
  later decades of the 18th century. Desmarest took no part in it. He
  had accumulated such conclusive proof of the correctness of his
  deductions, and had so fully expounded the clearness of the evidence
  in their favour furnished by the region of Auvergne, that, when any
  one came to consult him on the subject, he contented himself with
  giving the advice to "go and see." While the debate was in progress on
  the continent, the subject was approached from a new and independent
  point of view by Hutton in Scotland. This illustrious philosopher, as
  already stated, realized the importance of the internal heat of the
  globe in consolidating the sedimentary rocks, and believed that molten
  material from the earth's interior has been protruded from below into
  the overlying crust. Some of the material thus injected could be
  recognized, he thought, in granite and in the various dark massive
  rocks which, known in Scotland under the name of "whinstone," were
  afterwards called "Trap," and are now grouped under various names,
  such as basalt, dolerite and diorite. So important a share did Hutton
  thus assign to the internal heat in the geological evolution of the
  planet, that he and those who adopted the same opinions were styled
  "Plutonists," or, especially where they concerned themselves with the
  volcanic origin of basalt, "Vulcanists." The geological world was thus
  divided into two hostile camps, that of the Neptunists or Wernerians,
  and that of the Plutonists, Vulcanists or Huttonians.

  After many years of futile controversy the first serious weakening of
  the position of the dominant Neptunist school arose from the defection
  of some of the most prominent of Werner's pupils. In particular Jean
  François D'Aubuisson de Voisins (1769-1819), who had written a
  treatise on the aqueous origin of the basalts of Saxony, went
  afterwards to Auvergne, where he was speedily a convert to the views
  expounded by Desmarest as to the volcanic nature of basalt. Having
  thus to relinquish one of the fundamental articles of the Freiberg
  faith, he was subsequently led to modify his adherence to others
  until, as he himself confessed, his views came almost wholly to agree
  with those of Hutton. Not less complete, and even more important, was
  the conversion of the great Leopold von Buch (1774-1853). He, too, was
  trained by Werner himself, and proved to be the most illustrious pupil
  of the Saxon professor. Full of admiration for the Neptunism in which
  he had been reared, he, in his earliest separate work, maintained the
  aqueous origin of basalt, and contrasted the wide field opened up to
  the spirit of observation by his master's teaching with the narrower
  outlook offered by "the volcanic theory." But a little further
  acquaintance with the facts of nature led Von Buch also to abandon his
  earlier prepossessions. It was a personal visit to the volcanic region
  of Auvergne that first opened his eyes, and led him to recant what he
  had believed and written about basalt. But the abandonment of so
  essential a portion of the Wernerian creed prepared the way for
  further relinquishments. When a few years later he went to Norway and
  found to his astonishment that granite, which he had been taught to
  regard as the oldest chemical precipitate from the universal ocean,
  could there be seen to have broken through and metamorphosed
  fossiliferous limestones, and to have sent veins into them, his faith
  in Werner's order of the succession of the rocks in the earth's crust
  received a further momentous shock. While one after another of the
  Freiberg doctrines crumbled away before him, he was now able to
  interrogate nature on a wider field than the narrow limits of Saxony,
  and he was thus gradually led to embrace the tenets of the opposite
  school. His commanding position, as the most accomplished geologist on
  the continent, gave great importance to his recantation of the
  Neptunist creed. His defection indeed was the severest blow that this
  creed had yet sustained. It may be said to have rung the knell of
  Wernerianism, which thereafter rapidly declined in influence, while
  Plutonism came steadily to the front, where it has ever since
  remained.

  Although Desmarest had traced in Auvergne a long succession of
  volcanic eruptions, of which the oldest went back to a remote period
  of time, and although he had shown that this succession, coupled with
  the records of contemporaneous denudation, might be used in defining
  epochs of geological history, it was not until many years after his
  day that volcanic action came to be recognized as a normal part of the
  mechanism of our globe, which had been in operation from the remotest
  past, and which had left numerous records among the rocks of the
  terrestrial crust. During the progress of the controversy between the
  two great opposing factions in the later portion of the 18th and the
  first three decades of the 19th century, those who espoused the
  Vulcanist cause were intent on proving that certain rocks, which are
  intercalated among the stratified formations and which were claimed by
  the Neptunists as obviously formed by water, are nevertheless of truly
  igneous origin. These observers fixed their eyes on the evidence that
  the material of such rocks, instead of having been deposited from
  aqueous solution, had once been actually molten, and had in that
  condition been thrust between the strata, had enveloped portions of
  them, and had indurated or otherwise altered them. They spoke of these
  masses as "unerupted lavas"; and undoubtedly in innumerable instances
  they were right. But their zeal to establish an intrusive origin led
  them to overlook the proofs that some intercalated sheets of igneous
  material had not been injected into the strata, but had been poured
  out at the surface as truly volcanic discharges, and therefore
  belonged to the ancient periods represented by the strata between
  which they are interposed. It may readily be supposed that any proofs
  of the contemporaneous intercalation of such sheets would be eagerly
  seized upon by the Neptunists in favour of their aqueous theory. The
  influence of the ancient belief that "burning mountains" could only
  rise from the combustion of subterranean inflammable materials
  extended even into the ranks of the Vulcanists, so far at least as to
  lead to a general acquiescence in the assumption that volcanoes
  appeared to belong to a late phase in the history of the planet. It
  was not until after considerable progress had been made in determining
  the palaeontological distinctions and order of succession of the
  stratified formations of the earth's crust that it became possible to
  trace among these formations a succession of volcanic episodes which
  were contemporaneous with them. In no part of the world has an ampler
  record of such episodes been preserved than in the British Isles. It
  was natural, therefore, that the subject should there receive most
  attention. As far back as 1820 Ami Boué (1794-1881) showed that the
  Old Red Sandstone of Scotland includes a great series of volcanic
  rocks, and that other rocks of volcanic origin are associated with the
  Carboniferous formations. H.T. de la Beche (1796-1855) afterwards
  traced proofs of contemporaneous eruptions among the Devonian rocks of
  the south-west of England. Adam Sedgwick (1785-1873) showed, first in
  the Lake District, and afterwards in North Wales, the presence of
  abundant volcanic sheets among the oldest divisions of the Palaeozoic
  series; while Roderick Impey Murchison (1792-1871) made similar
  discoveries among the Lower Silurian rocks. From the time of these
  pioneers the volcanic history of the country has been worked out by
  many observers until it is now known with a fulness as yet unattained
  in any other region.

  _Growth of Opinion regarding Earthquakes._--We have seen how crude
  were the conceptions of the ancients regarding the causes of volcanic
  action, and that they connected volcanoes and earthquakes as results
  of the commotion of wind imprisoned within subterranean caverns and
  passages. One of the earliest treatises, in which the phenomena of
  terrestrial movements were discussed in the spirit of modern science,
  was the posthumous collection of papers by Robert Hooke (1635-1703),
  entitled _Lectures and Discourses of Earthquakes and Subterranean
  Eruptions_, where the probable agency of earthquakes in upheaving and
  depressing land is fully considered, but without any definite
  pronouncement as to the author's conception of its origin. Hooke still
  associated earthquakes with volcanic action, and connected both with
  what he called "the general congregation of sulphurous subterraneous
  vapours." He conceived that some kind of "fermentation" takes place
  within the earth, and that the materials which catch fire and give
  rise to eruptions or earthquakes are analogous to those that
  constitute gunpowder. The first essay wherein earthquakes are treated
  from the modern point of view as the results of a shock that sends
  waves through the crust of the earth was written by the Rev. John
  Michell, and communicated to the Royal Society in the year 1760. Still
  under the old misconception that volcanoes are due to the combustion
  of inflammable materials, which he thought might be set on fire by the
  spontaneous combustion of pyritous strata, he supposed that, by the
  sudden access of large bodies of water to these subterranean fires,
  vapour is produced in such quantity and with such force as to give
  rise to the shock. From the centre of origin of this shock waves, he
  thought, are propagated through the earth, which are largest at the
  start and gradually diminish as they travel outwards. By drawing lines
  at different places in the direction of the track of these waves, he
  believed that the place of common intersection of these lines would be
  nearly the centre of the disturbance. In this way he showed that the
  great Lisbon earthquake of 1755 had its focus under the Atlantic,
  somewhere between the latitudes of Lisbon and Oporto, and he estimated
  that the depth at which it originated could not be much less than 1
  m., and probably did not exceed 3 m. Michell, however, misconceived
  the character of the waves which he described, seeing that he believed
  them to be due to the actual propagation of the vapour itself
  underneath the surface of the earth. A century had almost passed after
  the date of his essay before modern scientific methods of observation
  and the use of recording instruments began to be applied to the study
  of earthquake phenomena. In 1846 Robert Mallet (1810-1881) published
  an important paper "On the Dynamics of Earthquakes" in the
  _Transactions of the Royal Irish Academy_. From that time onward he
  continued to devote his energies to the investigation, studying the
  effects of the Calabrian earthquake of 1857, experimenting on the
  transmission of waves of shock through various materials, caused by
  exploding charges of gunpowder, and collecting all the information to
  be obtained on the subject. His writings, and especially his work in
  two volumes on _The First Principles of Observational Seismology_,
  must be regarded as having laid the foundations of this branch of
  modern geology (see EARTHQUAKE; SEISMOMETER).

  _History of the Evolution of Stratigraphical Geology._--Men had long
  been familiar with the evidence that the present dry land once lay
  under the sea, before they began to realize that the rocks, of which
  the land consists, contain a record of many alternations of land and
  sea, and relics of a long succession of plants and animals from early
  and simple types up to the manifold and complex forms of to-day. In
  countries where coal-mining had been prosecuted for generations, it
  had been recognized that the rocks consist of strata superposed on
  each other in a definite order, which was found to extend over the
  whole of a district. As far back as 1719 John Strachey drew attention
  to this fact in a communication published in the _Philosophical
  Transactions_. John Michell (1760), in the paper on earthquakes
  already cited, showed that he had acquired a clear understanding of
  the order of succession among stratified formations, and perceived
  that to disturbances of the terrestrial crust must be ascribed the
  fact that the lower or older and more inclined strata form the
  mountains, while the younger and more horizontal strata are spread
  over the plains.

  In Italy G. Arduíno (1713-1795) classified the rocks in the north of
  the peninsula as Primitive, Secondary, Tertiary and Volcanic. A
  similar threefold order was announced for the Harz and Erzgebirge by
  J.G. Lehmann in 1756. He recognized in that region an ancient series
  of rocks in inclined or vertical strata, which rise to the tops of the
  hills and descend to an unknown depth into the interior. These masses,
  he thought, were contemporaneous with the making of the world. Next
  came the Flötzgebirge, consisting of younger sediments, disposed in
  flat or gently inclined sheets which overlie the first and more
  disturbed series, and are full of petrified remains of plants and
  animals. Lastly he included the mountains which have from time to time
  been formed by local accidents. Still more advanced were the
  conceptions of G.C. Füchsel, who in the year 1762 published in Latin
  _A History of the Earth and the Sea, based on a History of the
  Mountains of Thuringia_; and in 1773, in German, a _Sketch of the most
  Ancient History of the Earth and Man_. In these works he described the
  stratigraphical relations and general characters of the various
  geological formations in his little principality; and taking them as
  indicative of a general order of succession, he traced what he
  believed to have been a series of revolutions through which the earth
  has passed. In interpreting this geological history, he laid great
  stress on the evidence of the fossils contained in the rocks. He
  recognized that the various formations differ from each other in their
  enclosed organic remains, and that from these differences the
  existence of former sea-bottoms and land surfaces can be determined.

  The labours of these pioneers paved the way for the advent of Werner.
  Though the system evolved by this teacher claimed to discard theory
  and to be established on a basis of observed facts, it rested on a
  succession of hypotheses, for which no better foundation could be
  shown than the belief of their author in their validity. Starting from
  the extremely limited stratigraphical range displayed in the
  geological structure of Saxony, he took it as a type for the rest of
  the globe, persuading himself and impressing upon his followers that
  the rocks of that small kingdom were to be taken as examples of his
  "universal formations." The oldest portion of the series, classed by
  him as "Primitive," consisted of rocks which he maintained had been
  deposited from chemical solution. Yet they included granite, gneiss,
  basalt, porphyry and serpentine, which, even in his own day, were by
  many observers correctly regarded as of igneous origin. A later group
  of rocks, to which he gave the name of "Transition," comprised, in his
  belief, partly chemical, partly mechanical sediments, and contained
  the earliest fossil organic remains. A third group, for which he
  reserved Lehmann's name "Flötz," was made up chiefly of mechanical
  detritus, while youngest of all came the "Alluvial" series of loams,
  clays, sands, gravels and peat. It was by the gradual subsidence of
  the ocean that, as he believed, the general mass of the dry land
  emerged, the first-formed rocks being left standing up, sometimes on
  end, to form the mountains, while those of later date, less steeply
  inclined, occupied successively lower levels down to the flat alluvial
  accumulations of the plains. Neither Werner, nor any of his followers,
  ventured to account for what became of the water as the sea-level
  subsided, though, in despite of their antipathy to anything like
  speculation, they could not help suggesting, as an answer to the
  cogent arguments of their opponents, that "one of the celestial bodies
  which sometimes approach near to the earth may have been able to
  withdraw a portion of our atmosphere and of our ocean." Nor was any
  attempt made to explain the extraordinary nature of the supposed
  chemical precipitates of the universal ocean. The progress of inquiry
  even in Werner's lifetime disproved some of the fundamental portions
  of his system. Many of the chemical precipitates were shown to be
  masses that had been erupted in a molten state from below. His order
  of succession was found not to hold good; and though he tried to
  readjust his sequence and to introduce into it modifications to suit
  new facts, its inherent artificiality led to its speedy decline after
  his death. It must be conceded, however, that the stress which he laid
  upon the fact that the rocks of the earth's crust were deposited in a
  definite order had an important influence in directing attention to
  this subject, and in preparing the way for a more natural system,
  based not on mere mineralogical characters, but having regard to the
  organic remains, which were now being gathered in ever-increasing
  numbers and variety from stratified formations of many different ages
  and from all parts of the globe.

  It was in France and in England that the foundations of stratigraphy,
  based upon a knowledge of organic remains, were first successfully
  laid. Abbé J.L. Giraud-Soulavie (1752-1813), in his _Histoire
  naturelle de la France méridionale_, which appeared in seven volumes,
  subdivided the limestones of Vivarais into five ages, each marked by a
  distinct assemblage of shells. In the lowest strata, representing the
  first age, none of the fossils were believed by him to have any living
  representatives, and he called these rocks "Primordial." In the next
  group a mingling of living with extinct forms was observable. The
  third age was marked by the presence of shells of still existing
  species. The strata of the fourth series were characterized by
  carbonaceous shales or slates, containing remains of primordial
  vegetation, and perhaps equivalents of the first three calcareous
  series. The fifth age was marked by recent deposits containing remains
  of terrestrial vegetation and of land animals. It is remarkable that
  these sagacious conclusions should have been formed and published at a
  time when the geologists of the Continent were engaged in the
  controversy about the origin of basalt, or in disputes about the
  character and stratigraphical position of the supposed universal
  formations, and when the interest and importance of fossil organic
  remains still remained unrecognized by the vast majority of the
  combatants.

  The rocks of the Paris basin display so clearly an orderly
  arrangement, and are so distinguished for the variety and perfect
  preservation of their enclosed organic remains, that they could not
  fail to attract the early notice of observers. J. É. Guettard, G.F.
  Rouelle (1703-1770), N. Desmarest, A.L. Lavoisier (1743-1794) and
  others made observations in this interesting district. But it was
  reserved for Cuvier (1769-1832) and A. Brongniart (1770-1847) to work
  out the detailed succession of the Tertiary formations, and to show
  how each of these is characterized by its own peculiar assemblage of
  organic remains. The later progress of investigation has slightly
  corrected and greatly amplified the tabular arrangement established by
  these authors in 1808, but the broad outlines of the Tertiary
  stratigraphy of the Paris basin remain still as Cuvier and Brongniart
  left them. The most important subsequent change in the classification
  of the Tertiary formations was made by Sir Charles Lyell, who,
  conceiving in 1828 the idea of a classification of these rocks by
  reference to their relative proportions of living and extinct species
  of shells, established, in collaboration with G.P. Deshayes, the now
  universally accepted divisions Eocene, Miocene and Pliocene.

  Long before Cuvier and Brongniart published an account of their
  researches, another observer had been at work among the Secondary
  formations of the west of England, and had independently discovered
  that the component members of these formations were each distinguished
  by a peculiar group of organic remains; and that this distinction
  could be used to discriminate them over all the region through which
  he had traced them. The remarkable man who arrived at this
  far-reaching generalization was William Smith (1769-1839), a land
  surveyor who, in the prosecution of his professional business, found
  opportunities of traversing a great part of England, and of putting
  his deductions to the test. As the result of these journeys he
  accumulated materials enough to enable him to produce a geological map
  of the country, on which the distribution and succession of the rocks
  were for the first time delineated. Smith's labours laid the
  foundation of stratigraphical geology in England and he was styled
  even in his lifetime the "Father of English geology." From his day
  onward the significance of fossil organic remains gained rapidly
  increasing recognition. Thus in England the outlines traced by him
  among the Secondary and Tertiary formations were admirably filled in
  by Thomas Webster (1773-1844); while the Cretaceous series was worked
  out in still greater detail in the classic memoirs of William Henry
  Fitton (1780-1861).

  There was one stratigraphical domain, however, into which William
  Smith did not enter. He traced his sequence of rocks down into the
  Coal Measures, but contented himself with only a vague reference to
  what lay underneath that formation. Though some of these underlying
  rocks had in various countries yielded abundant fossils, they had
  generally suffered so much from terrestrial disturbances, and their
  order of succession was consequently often so much obscured throughout
  western Europe, that they remained but little known for many years
  after the stratigraphy of the Secondary and Tertiary series had been
  established. At last in 1831 Murchison began to attack this _terra
  incognita_ on the borders of South Wales, working into it from the Old
  Red Sandstone, the stratigraphical position of which was well known.
  In a few years he succeeded in demonstrating the existence of a
  succession of formations, each distinguished by its own peculiar
  assemblage of organic remains which were distinct from those in any of
  the overlying strata. To these formations he gave the name of Silurian
  (q.v.). From the key which his researches supplied, it was possible to
  recognize in other countries the same order of formations and the same
  sequence of fossils, so that, in the course of a few years,
  representatives of the Silurian system were found far and wide over
  the globe. While Murchison was thus engaged, Sedgwick devoted himself
  to the more difficult task of unravelling the complicated structure of
  North Wales. He eventually made out the order of the several
  formations there, with their vast intercalations of volcanic material.
  He named them the Cambrian system (q.v.), and found them to contain
  fossils, which, however, lay for some time unexamined by him. He at
  first believed, as Murchison also did, that his rocks were all older
  than any part of the Silurian series. It was eventually discovered
  that a portion of them was equivalent to the lower part of that
  series. The oldest of Sedgwick's groups, containing distinctive
  fossils, retain the name Cambrian, and are of high interest, as they
  enclose the remains of the earliest faunas which are yet well known.
  Sedgwick and Murchison rendered yet another signal service to
  stratigraphical geology by establishing, in 1839, on a basis of
  palaeontological evidence supplied by W. Lonsdale, the independence of
  the Devonian system (q.v.).

  For many years the rocks below the oldest fossiliferous deposits
  received comparatively little attention. They were vaguely described
  as the "crystalline schists" and were often referred to as parts of
  the primeval crust in which no chronology was to be looked for. W.E.
  Logan (1798-1875) led the way, in Canada, by establishing there
  several vast series of rocks, partly of crystalline schists and
  gneisses (Laurentian) and partly of slates and conglomerates
  (Huronian). Later observers, both in Canada and the United States,
  have greatly increased our knowledge of these rocks, and have shown
  their structure to be much more complex than was at first supposed
  (see ARCHEAN SYSTEM).

  During the latter half of the 19th century the most important
  development of stratigraphical geology was the detailed working out
  and application of the principle of zonal classification to the
  fossiliferous formations--that is, the determination of the sequence
  and distribution of organic remains in these formations, and the
  arrangement of the strata into zones, each of which is distinguished
  by a peculiar assemblage of fossil species (see under Part VI.). The
  zones are usually named after one especially characteristic species.
  This system of classification was begun in Germany with reference to
  the members of the Jurassic system (q.v.) by A. Oppel (1856-1858) and
  F.A. von Quenstedt (1858), and it has since been extended through the
  other Mesozoic formations. It has even been found to be applicable to
  the Palaeozoic rocks, which are now subdivided into palaeontological
  zones. In the Silurian system, for example, the graptolites have been
  shown by C. Lapworth to furnish a useful basis for zonal subdivisions.
  The lowest fossiliferous horizon in the Cambrian rocks of Europe and
  North America is known as the _Olenellus_ zone, from the prominence in
  it of that genus of trilobite.

  Another conspicuous feature in the progress of stratigraphy during the
  second half of the 19th century was displayed by the rise and rapid
  development of what is known as Glacial geology. The various deposits
  of "drift" spread over northern Europe, and the boulders scattered
  across the surface of the plains had long attracted notice, and had
  even found a place in popular legend and superstition. When men began
  to examine them with a view to ascertain their origin, they were
  naturally regarded as evidences of the Noachian deluge. The first
  observer who drew attention to the smoothed and striated surfaces of
  rock that underlie the Drifts was Hutton's friend, Sir James Hall, who
  studied them in the lowlands of Scotland and referred them to the
  action of great debacles of water, which, in the course of some
  ancient terrestrial convulsion, had been launched across the face of
  the country. Playfair, however, pointed out that the most potent
  geological agents for the transportation of large blocks of stone are
  the glaciers. But no one was then bold enough to connect the travelled
  boulders with glaciers on the plains of Germany and of Britain. Yet
  the transporting agency of ice was invoked in explanation of their
  diffusion. It came to be the prevalent belief among the geologists of
  the first half of the 19th century, that the fall of temperature,
  indicated by the gradual increase in the number of northern species of
  shells in the English Crag deposits, reached its climax during the
  time of the Drift, and that much of the north and centre of Europe was
  then submerged beneath a sea, across which floating icebergs and floes
  transported the materials of the Drift and dropped the scattered
  boulders. As the phenomena are well developed around the Alps, it was
  necessary to suppose that the submergence involved the lowlands of the
  Continent up to the foot of that mountain chain--a geographical change
  so stupendous as to demand much more evidence than was adduced in its
  support. At last Louis Agassiz (1807-1873), who had varied his
  palaeontological studies at Neuchâtel by excursions into the Alps, was
  so much struck by the proofs of the former far greater extension of
  the Swiss glaciers, that he pursued the investigation and satisfied
  himself that the ice had formerly extended from the Alpine valleys
  right across the great plain of Switzerland, and had transported huge
  boulders from the central mountains to the flanks of the Jura. In the
  year 1840 he visited Britain and soon found evidence of similar
  conditions there. He showed that it was not by submergence in a sea
  cumbered with floating ice, but by the former presence of vast
  glaciers or sheets of ice that the Drift and erratic blocks had been
  distributed. The idea thus propounded by him did not at once command
  complete approval, though traces of ancient glaciers in Scotland and
  Wales were soon detected by native geologists, particularly by W.
  Buckland, Lyell, J.D. Forbes and Charles Maclaren. Robert Chambers
  (1802-1871) did good service in gathering additional evidence from
  Scotland and Norway in favour of Agassiz's views, which steadily
  gained adherents until, after some quarter of a century, they were
  adopted by the great majority of geologists in Britain, and
  subsequently in other countries. Since that time the literature of
  geology has been swollen by a vast number of contributions in which
  the history of the Glacial period, and its records both in the Old and
  New World, have been fully discussed.

  _Rise and Progress of Palaeontological Geology._--As this branch of
  the science deals with the evidence furnished by fossil organic
  remains as to former geographical conditions, it early attracted
  observers who, in the superficial beds of marine shells found at some
  distance from the coast, saw proofs of the former submergence of the
  land under the sea. But the occurrence of fossils embedded in the
  heart of the solid rocks of the mountains offered much greater
  difficulties of explanation, and further progress was consequently
  slow. Especially baneful was the belief that these objects were mere
  sports of nature, and had no connexion with any once living organisms.
  So long as the true organic origin of the fossil plants and animals
  contained in the rocks was in dispute, it was hardly possible that
  much advance could be made in their systematic study, or in the
  geological deductions to be drawn from them. One good result of the
  controversy, however, was to be seen in the large collections of these
  "formed stones" that were gathered together in the cabinets and
  museums of the 17th and 18th centuries. The accumulation and
  comparison of these objects naturally led to the production of
  treatises in which they were described and not unfrequently
  illustrated by good engravings. Switzerland was more particularly
  noted for the number and merit of its works of this kind, such as
  that of K.N. Lang (_Historia lapidum figuratorum Helvetiae_, 1708) and
  those of Johann Jacob Scheuchzer (1672-1733). In England, also,
  illustrated treatises were published both by men who looked on fossils
  as mere freaks of nature, and by those who regarded them as proofs of
  Noah's flood. Of the former type were the works of Martin Lister
  (1638-1712) and Robert Plot (_Natural History of Oxfordshire_, 1677).
  The Celtic scholar Edward Llwyd (1660-1709) wrote a Latin treatise
  containing good plates of a thousand fossils in the Ashmolean Museum,
  Oxford, and J. Woodward, in 1728-1729, published his _Natural History
  of the Fossils of England_, already mentioned, wherein he described
  his own extensive collection, which he bequeathed to the University of
  Cambridge, where it is still carefully preserved. The most voluminous
  and important of all these works, however, appeared at a later date at
  Nuremberg. It was begun by G.W. Knorr (1705-1761), who himself
  engraved for it a series of plates, which for beauty and accuracy have
  seldom been surpassed. After his death the work was continued by
  J.E.I. Walch (1725-1778), and ultimately consisted of four massive
  folio volumes and nearly 300 plates under the title of _Lapides
  diluvii universalis testes_. Although the authors supposed their
  fossils to be relics of Noah's flood, their work must be acknowledged
  to mark a distinct onward stage in the palaeontological department of
  geology.

  It was in France that palaeontological geology began to be cultivated
  in a scientific spirit. The potter Bernard Palissy, as far back as
  1580, had dwelt on the importance of fossil shells as monuments of
  revolutions of the earth's surface; but the observer who first
  undertook the detailed study of the subject was Jean Etienne Guettard,
  who began in 1751 to publish his descriptions of fossils in the form
  of memoirs presented to the Academy of Sciences of Paris. To him they
  were not only of deep interest as monuments of former types of
  existence, but they had an especial value as records of the changes
  which the country had undergone from sea to land and from land to sea.
  More especially noteworthy was a monograph by him which appeared in
  1765 bearing the title "On the accidents that have befallen Fossil
  Shells compared with those which are found to happen to shells now
  living in the Sea." In this treatise he showed that the fossils have
  been encrusted with barnacles and serpulae, have been bored into by
  other organisms, and have often been rounded or broken before final
  entombment; and he inferred that these fossils must have lived and
  died on the sea-floor under similar conditions to those which obtain
  on the sea-floor to-day. His argument was the most triumphant that had
  ever been brought against the doctrine of _lusus naturae_, and that of
  the efficacy of Noah's flood--doctrines which still held their ground
  in Guettard's day. When Soulavie, Cuvier and Brongniart in France, and
  William Smith in England, showed that the rock formations of the
  earth's crust could be arranged in chronological order, and could be
  recognized far and wide by means of their enclosed organic remains,
  the vast significance of these remains in geological research was
  speedily realized, and palaeontological geology at once entered on a
  new and enlarged phase of development. But apart from their value as
  chronological monuments, and as witnesses of former conditions of
  geography, fossils presented in themselves a wide field of
  investigation as types of life that had formerly existed, but had now
  passed away. It was in France that this subject first took definite
  shape as an important branch of science. The mollusca of the Tertiary
  deposits of the Paris basin became, in the hands of Lamarck, the basis
  on which invertebrate palaeontology was founded. The same series of
  strata furnished to Cuvier the remains of extinct land animals, of
  which, by critical study of their fragmentary bones and skeletons, he
  worked out restorations that may be looked on as the starting-point of
  vertebrate palaeontology. These brilliant researches, rousing
  widespread interest in such studies, showed how great a flood of light
  could be thrown on the past history of the earth and its inhabitants.
  But the full significance of these extinct types of life could not be
  understood so long as the doctrine of the immutability of species, so
  strenuously upheld by Cuvier, maintained its sway among naturalists.
  Lamarck, as far back as the year 1800, had begun to propound his
  theory of evolution and the transformation of species; but his views,
  strongly opposed by Cuvier and the great body of naturalists of the
  day, fell into neglect. Not until after the publication in 1859 of the
  _Origin of Species_ by Charles Darwin were the barriers of old
  prejudice in this matter finally broken down. The possibility of
  tracing the ancestry of living forms back into the remotest ages was
  then perceived; the time-honoured fiction that the stratified
  formations record a series of catastrophes and re-creations was
  finally dissipated; and the earth's crust was seen to contain a noble,
  though imperfect, record of the grand evolution of organic types of
  which our planet has been the theatre.

  _Development of Petrographical Geology._--Theophrastus, the favourite
  pupil of Aristotle, wrote a treatise _On Stones_, which has come down
  to our own day, and may be regarded as the earliest work on
  petrography. At a subsequent period Pliny, in his _Natural History_,
  collected all that was known in his day regarding the occurrence and
  uses of minerals and rocks. But neither of these works is of great
  scientific importance, though containing much interesting information.
  Minerals from their beauty and value attracted notice before much
  attention was paid to rocks, and their study gave rise to the science
  of mineralogy long before geology came into existence. When rocks
  began to be more particularly scrutinized, it was chiefly from the
  side of their usefulness for building and other economic purposes. The
  occurrence of marine shells in many of them had early attracted
  attention to them. But their varieties of composition and origin did
  not become the subject of serious study until after Linnaeus and J.G.
  Wallerius in the 18th century had made a beginning. The first
  important contribution to this department of the science was that of
  Werner, who in 1786 published a classification and description of
  rocks in which he arranged them in two divisions, simple and compound,
  and further distinguished them by various external characters and by
  their relative age. The publication of this scheme may be said to mark
  the beginning of scientific petrography. Werner's system, however, had
  the serious defect that the chronological order in which he grouped
  the rocks, and the hypothesis by which he accounted for them as
  chemical precipitates from the original ocean, were both alike
  contrary to nature. It was hardly possible indeed that much progress
  could be made in this branch of geology until chemistry and mineralogy
  had made greater advances; and especially until it was possible to
  ascertain the intimate chemical and mineralogical composition, and the
  minute structure of rocks. The study, however, continued to be pursued
  in Germany, where the influence of Werner's enthusiasm still led men
  to enter the petrographical rather than the palaeontological domain.
  The resources of modern chemistry were pressed into the service, and
  analyses were made and multiplied to such a degree that it seemed as
  if the ultimate chemical constitution of every type of rock had now
  been thoroughly revealed. The condition of the science in the middle
  of the 19th century was well shown by J.L.A. Roth, who in 1861
  collected about 1000 trustworthy analyses which up to that time had
  been made. But though the chemical elements of the rocks had been
  fairly well determined, the manner in which they were combined in the
  compound rocks could for the most part be only more or less plausibly
  conjectured. As far back as 1831 an account was published of a process
  devised by William Nicol of Edinburgh, whereby sections of fossil wood
  could be cut, mounted on glass, and reduced to such a degree of
  transparency as to be easily examined under a microscope. Henry Sorby,
  of Sheffield, having seen Nicol's preparations, perceived how
  admirably adapted the process was for the study of the minute
  structure and composition of rocks. In 1858 he published in the
  _Quarterly Journal of the Geological Society_ a paper "On the
  Microscopical Structure of Crystals." This essay led to a complete
  revolution of petrographical methods and gave a vast impetus to the
  study of rocks. Petrology entered upon a new and wider field of
  investigation. Not only were the mineralogical constituents of the
  rocks detected, but minute structures were revealed which shed new
  light on the origin and history of these mineral masses, and opened up
  new paths in theoretical geology. In the hands of H. Vogelsang, F.
  Zirkel, H. Rosenbusch, and a host of other workers in all civilized
  countries, the literature of this department of the science has grown
  to a remarkable extent. Armed with the powerful aid of modern optical
  instruments, geologists are now able with far more prospect of success
  to resume the experiments begun a century before by de Saussure and
  Hall. G.A. Daubrée, C. Friedel, E. Sarasin, F. Fouqué and A. Michel
  Lévy in France, C. Doelter y Cisterich and E. Hussak of Gratz, J.
  Morozewicz of Warsaw and others, have greatly advanced our knowledge
  by their synthetical analyses, and there is every reason to hope that
  further advances will be made in this field of research.

  _Rise of Physiographical Geology._--Until stratigraphical geology had
  advanced so far as to show of what a vast succession of rocks the
  crust of the earth is built up, by what a long and complicated series
  of revolutions these rocks have come to assume their present
  positions, and how enormous has been the lapse of time which all these
  changes represent, it was not possible to make a scientific study of
  the surface features of our globe. From ancient times it had been
  known that many parts of the land had once been under the sea; but
  down even to the beginning of the 19th century the vaguest conceptions
  continued to prevail as to the operations concerned in the submergence
  and elevation of land, and as to the processes whereby the present
  outlines of terrestrial topography were determined. We have seen, for
  instance, that according to the teaching of Werner the oldest rocks
  were first precipitated from solution in the universal ocean to form
  the mountains, that the vertical position of their strata was
  original, that as the waters subsided successive formations were
  deposited and laid bare, and that finally the superfluous portion of
  the ocean was whisked away into space by some unexplained co-operation
  of another planetary body. Desmarest, in his investigation of the
  volcanic history of Auvergne, was the first observer to perceive by
  what a long process of sculpture the present configuration of the land
  has been brought about. He showed conclusively that the valleys have
  been carved out by the streams that flow in them, and that while they
  have sunk deeper and deeper into the framework of the land, the spaces
  of ground between them have been left as intervening ridges and hills.
  De Saussure learnt a similar lesson from his studies of the Alps, and
  Hutton and Playfair made it a cardinal feature in their theory of the
  earth. Nevertheless the idea encountered so much opposition that it
  made but little way until after the middle of the 19th century.
  Geologists preferred to believe in convulsions of nature, whereby
  valleys were opened and mountains were upheaved. That the main
  features of the land, such as the great mountain-chains, had been
  produced by gigantic plication of the terrestrial crust was now
  generally admitted, and also that minor fractures and folds had
  probably initiated many of the valleys. But those who realized most
  vividly the momentous results achieved by ages of subaerial denudation
  perceived that, as Hutton showed, even without the aid of underground
  agency, the mere flow of water in streams across a mass of land must
  in course of time carve out just such a system of valleys as may
  anywhere be seen. It was J.B. Jukes who, in 1862, first revived the
  Huttonian doctrine, and showed how completely it explained the
  drainage-lines in the south of Ireland. Other writers followed in
  quick succession until, in a few years, the doctrine came to be widely
  recognized as one of the established principles of modern geology.
  Much help was derived from the admirable illustrations of
  land-sculpture and river-erosion supplied from the Western Territories
  and States of the American Union.

  Another branch of physiographical geology which could only come into
  existence after most of the other departments of the science had made
  large progress, deals with the evolution of the framework of each
  country and of the several continents and oceans of the globe. It is
  now possible, with more or less confidence, to trace backward the
  history of every terrestrial area, to see how sea and land have there
  succeeded each other, how rivers and lakes have come and gone, how the
  crust of the earth has been ridged up at widely separated intervals,
  each movement determining some line of mountains or plains, how the
  boundaries of the oceans have shifted again and again in the past, and
  thus how, after so prolonged a series of revolutions, the present
  topography of each country, and of the globe as a whole, has been
  produced. In the prosecution of this subject maps have been
  constructed to show what is conjectured to have been the distribution
  of sea and land during the various geological periods in different
  parts of the world, and thus to indicate the successive stages through
  which the architecture of the land has been gradually evolved. The
  most noteworthy contribution to this department of the science is the
  _Antlitz der Erde_ of Professor Suess of Vienna. This important and
  suggestive work has been translated into French and English.


PART II.--COSMICAL ASPECTS

Before geology had attained to the position of an inductive science, it
was customary to begin investigations into the history of the earth by
propounding or adopting some more or less fanciful hypothesis in
explanation of the origin of our planet, or even of the universe. Such
preliminary notions were looked upon as essential to a right
understanding of the manner in which the materials of the globe had been
put together. One of the distinguishing features of Hutton's Theory of
the Earth consisted in his protest that it is no part of the province of
geology to discuss the origin of things. He taught that in the materials
from which geological evidence is to be compiled there can be found "no
traces of a beginning, no prospect of an end." In England, mainly to the
influence of the school which he founded, and to the subsequent rise of
the Geological Society of London, which resolved to collect facts
instead of fighting over hypotheses, is due the disappearance of the
crude and unscientific cosmologies by which the writings of the earlier
geologists were distinguished.

But there can now be little doubt that in the reaction against those
visionary and often grotesque speculations, geologists were carried too
far in an opposite direction. In allowing themselves to believe that
geology had nothing to do with questions of cosmogony, they gradually
grew up in the conviction that such questions could never be other than
mere speculation, interesting or amusing as a theme for the employment
of the fancy, but hardly coming within the domain of sober and inductive
science. Nor would they soon have been awakened out of this belief by
anything in their own science. It is still true that in the data with
which they are accustomed to deal, as comprising the sum of geological
evidence, there can be found no trace of a beginning, though the
evidence furnished by the terrestrial crust shows a general evolution of
organic forms from some starting-point which cannot be seen. The oldest
rocks which have been discovered on any part of the globe have probably
been derived from other rocks older than themselves. Geology by itself
has not yet revealed, and is little likely ever to reveal, a trace of
the first solid crust of our globe. If, then, geological history is to
be compiled from direct evidence furnished by the rocks of the earth, it
cannot begin at the beginning of things, but must be content to date
its first chapter from the earliest period of which any record has been
preserved among the rocks.

Nevertheless, though geology in its usual restricted sense has been, and
must ever be, unable to reveal the earliest history of our planet, it no
longer ignores, as mere speculation, what is attempted in this subject
by its sister sciences. Astronomy, physics and chemistry have in late
years all contributed to cast light on the earlier stages of the earth's
existence, previous to the beginning of what is commonly regarded as
geological history. But whatever extends our knowledge of the former
conditions of our globe may be legitimately claimed as part of the
domain of geology. If this branch of inquiry, therefore, is to continue
worthy of its name as the science of the earth, it must take cognizance
of these recent contributions from other sciences. It must no longer be
content to begin its annals with the records of the oldest rocks, but
must endeavour to grope its way through the ages which preceded the
formation of any rocks. Thanks to the results achieved with the
telescope, the spectroscope and the chemical laboratory, the story of
these earliest ages of our earth is every year becoming more definite
and intelligible.

Up to the present time no definite light has been thrown by physics on
the origin and earliest condition of our globe. The famous nebular
theory (q.v.) of Kant and Laplace sketched the supposed evolution of the
solar system from a gaseous nebula, slowly rotating round a more
condensed central portion of its mass, which eventually became the sun.
As a consequence of increased rapidity of rotation resulting from
cooling and contraction, the nebula acquired a more and more lenticular
form, until at last it threw off from its equatorial protuberance a ring
of matter. Subsequently the same process was repeated, and other similar
rings successively separated from the parent mass. Each ring went
through a corresponding series of changes until it ultimately became a
planet, with or without one or more attendant satellites. The intimate
relationship of our earth to the sun and the other planets was, in this
way, shown. But there are some serious physical difficulties in the way
of the acceptance of the nebular hypothesis. Another explanation is
given by the meteoritic hypothesis, according to which, out of the
swarms of meteorites with which the regions of space are crowded, the
sun and planets have been formed by gradual accretion.

According to these theoretical views we should expect to find a general
uniformity of composition in the constituent matter of the solar system.
For many years the only available evidence on this point was derived
from the meteorites (q.v.) which so constantly fall from outer space
upon the surface of the earth. These bodies were found to consist of
elements, all of which had been recognized as entering into the
constitution of the earth. But the discoveries of spectroscopic research
have made known a far more widely serviceable method of investigation,
which can be applied even to the luminous stars and nebulae that lie far
beyond the bounds of the solar system. By this method information has
been obtained regarding the constitution of the sun, and many of our
terrestrial metals, such as iron, nickel and magnesium, have been
ascertained to exist in the form of incandescent vapour in the solar
atmosphere. The present condition of the sun probably represents one of
the phases through which stars and planets pass in their progress
towards becoming cool and dark bodies in space. If our globe was at
first, like its parent sun, an incandescent mass of probably gaseous
matter, occupying much more space than it now fills, we can conceive
that it has ever since been cooling and contracting until it has reached
its present form and dimensions, and that it still retains a high
internal temperature. Its oblately spheroidal form is such as would be
assumed by a rotating mass of matter in the transition from a vaporous
and self-luminous or liquid condition to one of cool and dark solidity.
But it has been claimed that even a solid spherical globe might develop,
under the influence of protracted rotation, such a shape as the earth at
present possesses.

The observed increase of temperature downwards in our planet has
hitherto been generally accepted as a relic and proof of an original
high temperature and mobility of substance. Recently, however, the
validity of this proof has been challenged on the ground that the
ascertained amount of radium in the rocks of the outer crust is more
than sufficient to account for the observed downward increase of
temperature. Too little, however, is known of the history and properties
of what is called radium to afford a satisfactory ground on which to
discard what has been, and still remains, the prevalent belief on this
subject.

An important epoch in the geological history of the earth was marked by
the separation of the moon from its mass (see TIDE). Whether the
severance arose from the rupture of a surrounding ring or the gradual
condensation of matter in such a ring, or from the ejection of a single
mass of matter from the rapidly rotating planet, it has been shown that
our satellite was only a few thousand miles from the earth's surface,
since when it has retreated to its present distance of 240,000 m. Hence
the influence of the moon's attraction, and all the geological effects
to which it gives rise, attained their maximum far back in the
development of the globe, and have been slowly diminishing throughout
geological history.

The sun by virtue of its vast size has not yet passed out of the
condition of glowing gas, and still continues to radiate heat beyond the
farthest planet of the solar system. The earth, however, being so small
a body in comparison, would cool down much more quickly. Underneath its
hot atmosphere a crust would conceivably begin to form over its molten
surface, though the interior might still possess a high temperature and,
owing to the feeble conducting power of rocks, would remain intensely
hot for a protracted series of ages.

Full information regarding the form and size of the earth, and its
relations to the other planetary members of the solar system, will be
found in the articles PLANET and SOLAR SYSTEM. For the purposes of
geological inquiry the reader will bear in mind that the equatorial
diameter of our globe is estimated to be about 7925 m., and the polar
diameter about 7899 m.; the difference between these two sums
representing the amount of flattening at the poles (about 26½ m.). The
planet has been compared in shape to an orange, but it resembles an
orange which has been somewhat squeezed, for its equatorial
circumference is not a regular circle but an ellipse, of which the major
axis lies in long. 8° 15' W.--on a meridian which cuts the north-west
corner of America, passing through Portugal and Ireland, and the
north-east corner of Asia in the opposite hemisphere.

The rotation of the earth on its axis exerts an important influence on
the movements of the atmosphere, and thereby affects the geological
operations connected with these movements. The influence of rotation is
most marked in the great aerial circulation between the poles and the
equator. Currents of air, which set out in a meridional direction from
high latitudes towards the equator, come from regions where the velocity
due to rotation is small to where it is greater, and they consequently
fall behind. Thus, in the northern hemisphere a north wind, as it moves
away from its northern source of origin, is gradually deflected more and
more towards the west and becomes a north-east current; while in the
opposite hemisphere a wind making from high southern latitudes towards
the equator becomes, from the same cause, a south-east current. Where,
on the other hand, the air moves from the equatorial to the polar
regions its higher velocity of rotation carries it eastward, so that on
the south side of the equator it becomes a north-west current and on the
north side a south-west current. It is to this cause that the easting
and westing of the great atmospheric currents are to be attributed, as
is familiarly exemplified in the trade winds.

The atmospheric circulation thus deflected influences the circulation of
the ocean. The winds which persistently blow from the north-east on the
north side of the equator, and from the south-east on the south side,
drive the superficial waters onwards, and give rise to converging
oceanic currents which unite to form the great westerly equatorial
current.

A more direct effect of terrestrial rotation has been claimed in the
case of rivers which flow in a meridional direction. It has been
asserted that those, which in the northern hemisphere flow from north to
south, like the Volga, by continually passing into regions where the
velocity of rotation is increasingly greater, are thrown more against
their western than their eastern banks, while those whose general course
is in an opposite direction, like the Irtisch and Yenesei, press more
upon their eastern sides. There cannot be any doubt that the tendency of
the streams must be in the directions indicated. But when the
comparatively slow current and constantly meandering course of most
rivers are taken into consideration, it may be doubted whether the
influence of rotation is of much practical account so far as
river-erosion is concerned.

One of the cosmical relations of our planet which has been more
especially prominent in geological speculations relates to the position
of the earth's axis of rotation. Abundant evidence has now been obtained
to prove that at a comparatively late geological period a rich flora,
resembling that of warm climates at the present day, existed in high
latitudes even within less than 9° of the north pole, where, with an
extremely low temperature and darkness lasting for half of the year, no
such vegetation could possibly now exist. It has accordingly been
maintained by many geologists that the axis of rotation must have
shifted, and that when the remarkable Arctic assemblage of fossil plants
lived the region of their growth must have lain in latitudes much nearer
to the equator of the time.

The possibility of any serious displacement of the rotational axis since
a very early period in the earth's history has been strenuously denied
by astronomers, and their arguments have been generally, but somewhat
reluctantly, accepted by geologists, who find themselves confronted with
a problem which has hitherto seemed insoluble. That the axis is not
rigidly stable, however, has been postulated by some physicists, and has
now been demonstrated by actual observation and measurement. It is
admitted that by the movement of large bodies of water the air over the
surface of the globe, and more particularly by the accumulation of vast
masses of snow and ice in different regions, the position of the axis
might be to some extent shifted; more serious effects might follow from
widespread upheavals or depressions of the surface of the lithosphere.
On the assumption of the extreme rigidity of the earth's interior,
however, the general result of mathematical calculation is to negative
the supposition that in any of these ways within the period represented
by what is known as the "geological record," that is, since the time of
the oldest known sedimentary formations, the rotational axis has ever
been so seriously displaced as to account for such stupendous geological
events as the spread of a luxuriant vegetation far up into polar
latitudes. If, however, the inside of the globe possesses a great
plasticity than has been allowed, the shifting of the axis might not be
impossible, even to such an extent as would satisfy the geological
requirements. This question is one on which the last word has not been
said, and regarding which judgment must remain in suspense.

In recent years fresh information bearing on the minor devagations of
the pole has been obtained from a series of several thousand careful
observations made in Europe and North America. It has thus been
ascertained that the pole wanders with a curiously irregular but
somewhat spiral movement, within an amplitude of between 40 and 50 ft.,
and completes its erratic circuit in about 428 days. It was not supposed
that its movement had any geological interest, but Dr John Milne has
recently pointed out that the times of sharpest curvature in the path of
the pole coincide with the occurrence of large earthquakes, and has
suggested that, although it can hardly be assumed that this coincidence
shows any direct connexion between earthquake frequency and changes in
the position of the earth's axis, both effects may not improbably arise
from the same redistribution of surface material by ocean currents and
meteorological causes.

If for any reason the earth's centre of gravity were sensibly displaced,
momentous geological changes would necessarily ensue. That the centre of
gravity does not coincide with the centre of figure of the globe, but
lies to the south of it, has long been known. This greater aggregation
of dense material in the southern hemisphere probably dates from the
early ages of the earth's consolidation, and it is difficult to believe
that any readjustment of the distribution of this material in the
earth's interior is now possible. But certain rearrangements of the
hydrosphere on the surface of the globe may, from time to time, cause a
shifting of the centre of gravity, which will affect the level of the
ocean. The accumulation of enormous masses of ice around the pole will
give rise to such a displacement, and will thus increase the body of
oceanic water in the glaciated hemisphere. Various calculations have
been made of the effect of the transference of the ice-cap from one pole
to the other, a revolution which may possibly have occurred more than
once in the past history of the globe. James Croll estimated that if the
mass of ice in the southern hemisphere be assumed to be 1000 ft. thick
down to lat. 60°, its removal to the opposite hemisphere would raise the
level of the sea 80 ft. at the north pole, while the Rev. Osmond Fisher
made the rise as much as 409 ft. The melting of the ice would still
further raise the sea-level by the addition of so large a volume of
water to the ocean. To what extent superficial changes of this kind have
operated in geological history remains an unsolved problem, but their
probable occurrence in the past has to be recognized as one of the
factors that must be considered in tracing the revolutions of the
earth's surface.

_The Age of the Earth._--Intimately connected with the relations of our
globe to the sun and the other members of the solar system is the
question of the planet's antiquity--a subject of great geological
importance, regarding which much discussion has taken place since the
middle of the 19th century. Though an account of this discussion
necessarily involves allusion to departments of geology which are more
appropriately referred to in later parts of this article, it may perhaps
be most conveniently included here.

Geologists were for many years in the habit of believing that no limit
could be assigned to the antiquity of the planet, and that they were at
liberty to make unlimited drafts on the ages of the past. In 1862 and
subsequent years, however, Lord Kelvin (then Sir William Thomson)
pointed out that these demands were opposed to known physical facts, and
that the amount of time required for geological history was not only
limited, but must have been comprised within a comparatively narrow
compass. His argument rested on three kinds of evidence: (1) the
internal heat and rate of cooling of the earth; (2) the tidal
retardation of the earth's rotation; and (3) the origin and age of the
sun's heat.

1. Applying Fourier's theory of thermal conductivity, Lord Kelvin
contended that in the known rate of increase of temperature downward and
beneath the surface, and the rate of loss of heat from the earth, we
have a limit to the antiquity of the planet. He showed, from the data
available at the time, that the superficial consolidation of the globe
could not have occurred less than 20 million years ago, or the
underground heat would have been greater than it is; nor more than 400
million years ago, otherwise the underground temperature would have
shown no sensible increase downwards. He admitted that very wide limits
were necessary. In subsequently discussing the subject, he inclined
rather towards the lower than the higher antiquity, but concluded that
the limit, from a consideration of all the evidence, must be placed
within some such period of past time as 100 millions of years.

2. The argument from tidal retardation proceeds on the admitted fact
that, owing to the friction of the tide-wave, the rotation of the earth
is retarded, and is, therefore, much slower now than it must have been
at one time. Lord Kelvin affirmed that had the globe become solid some
10,000 million years ago, or indeed any high antiquity beyond 100
million years, the centrifugal force due to the more rapid rotation must
have given the planet a very much greater polar flattening than it
actually possesses. He admitted, however, that, though 100 million years
ago that force must have been about 3% greater than now, yet "nothing
we know regarding the figure of the earth, and the disposition of land
and water, would justify us in saying that a body consolidated when
there was more centrifugal force by 3% than now, might not now be in all
respects like the earth, so far as we know it at present."

3. The third argument, based upon the age of the sun's heat, is
confessedly less to be relied on than the two previous ones. It proceeds
upon calculations as to the amount of heat which would be available by
the falling together of masses from space, which gave rise by their
impact to our sun. The vagueness of the data on which this argument
rests may be inferred from the fact that in one passage P.G. Tait placed
the limit of time during which the sun has been illuminating the earth
as, "on the very highest computation, not more than about 15 or 20
millions of years"; while, in another sentence of the same volume, he
admitted that, "by calculations in which there is no possibility of
large error, this hypothesis [of the origin of the sun's heat by the
falling together of masses of matter] is thoroughly competent to explain
100 millions of years' solar radiation at the present rate, perhaps
more." In more recently reviewing his argument, Lord Kelvin expressed
himself in favour of more strictly limiting geological time than he had
at first been disposed to do. He insists that the time "was more than 20
and less than 40 millions of years and probably much nearer 20 than 40."
Geologists appear to have reluctantly brought themselves to believe that
perhaps, after all, 100 millions of years might suffice for the
evolution of geological history. But when the time was cut down to 15 or
20 millions they protested that such a restricted period was
insufficient for that evolution, and though they did not offer any
effective criticism of the arguments of the physicists they felt
convinced that there must be some flaw in the premises on which these
arguments were based.

By degrees, however, there have arisen among the physicists themselves
grave doubts as to the validity of the physical evidence on which the
limitation of the earth's age has been founded, and at the same time
greater appreciation has been shown of the signification and strength of
the geological proofs of the high antiquity of our planet. In an address
from the chair of the Mathematical Section of the British Association in
1886, Professor (afterwards Sir) George Darwin reviewed the controversy,
and pronounced the following deliberate judgment in regard to it: "In
considering these three arguments I have adduced some reasons against
the validity of the first [tidal friction], and have endeavoured to show
that there are elements of uncertainty surrounding the second [secular
cooling of the earth]; nevertheless, they undoubtedly constitute a
contribution of the first importance to physical geology. Whilst, then,
we may protest against the precision with which Professor Tait seeks to
deduce results from them, we are fully justified in following Sir
William Thomson, who says that 'the existing state of things on the
earth, life on the earth--all geological history showing continuity of
life--must be limited within some such period of past time as 100
million years'." Lord Kelvin has never dealt with the geological and
palaeontological objections against the limitation of geological time to
a few millions of years. But Professor Darwin, in the address just
cited, uttered the memorable warning: "At present our knowledge of a
definite limit to geological time has so little precision that we should
do wrong summarily to reject theories which appear to demand longer
periods of time than those which now appear allowable." In his
presidential address to the British Association at Cape Town in 1905 he
returned to the subject, remarking that the argument derived from the
increase of underground temperature "seems to be entirely destroyed" by
the discovery of the properties of radium. He thinks that "it does not
seem extravagant to suppose that 500 to 1000 million years may have
elapsed since the birth of the moon." He has "always believed that the
geologists were more nearly correct than the physicists, notwithstanding
the fact that appearances were so strongly against them," and he
concludes thus: "It appears, then, that the physical argument is not
susceptible of a greater degree of certainty than that of the
geologists, and the scale of geological time remains in great measure
unknown" (see also Tide, chap. viii.).

In an address to the mathematical section of the American Association
for the Advancement of Science in 1889, the vice-president of the
section, R.S. Woodward, thus expressed himself with regard to the
physical arguments brought forward by Lord Kelvin and Professor Tait in
limitation of geological time: "Having been at some pains to look into
this matter, I feel bound to state that, although the hypothesis appears
to be the best which can be formulated at present, the odds are against
its correctness. Its weak links are the unverified assumptions of an
initial uniform temperature and a constant diffusivity. Very likely
these are approximations, but of what order we cannot decide.
Furthermore, if we accept the hypothesis, the odds appear to be against
the present attainment of trustworthy numerical results, since the data
for calculation, obtained mostly from observations on continental areas,
are far too meagre to give satisfactory average values for the entire
mass of the earth."

Still more emphatic is the protest made from the physical side by
Professor John Perry. He has attacked each of the three lines of
argument of Lord Kelvin, and has impugned the validity of the
conclusions drawn from them. The argument from tidal retardation he
dismisses as fallacious, following in this contention the previous
criticism of the Rev. Maxwell Close and Sir George Darwin. In dealing
with the argument based on the secular cooling of the earth, he holds it
to be perfectly allowable to assume a much higher conductivity for the
interior of the globe, and that such a reasonable assumption would
enable us greatly to increase our estimate of the earth's antiquity. As
for the third argument, from the age of the sun's heat, he points out
that the sun may have been repeatedly fed by a supply of meteorites from
outside, while the earth may have been protected from radiation, and
been able to retain much of its heat by being enveloped in a dense
atmosphere. Remarking that "almost anything is possible as to the
present internal state of the earth," he concludes thus: "To sum up, we
can find no published record of any lower maximum age of life on the
earth, as calculated by physicists, than 400 millions of years. From the
three physical arguments Lord Kelvin's higher limits are 1000, 400 and
500 million years. I have shown that we have reasons for believing that
the age, from all these, may be very considerably underestimated. It is
to be observed that if we exclude everything but the arguments from mere
physics, the _probable_ age of life on the earth is much less than any
of the above estimates; but if the palaeontologists have good reasons
for demanding much greater times, I see nothing from the physicists'
point of view which denies them four times the greatest of these
estimates."

A fresh line of argument against Lord Kelvin's limitation of the
antiquity of our globe has recently been started by the remarkable
discoveries in radio-activity. From the ascertained properties of radium
it appears to be possible that our estimates of solar heat, as derived
from the theory of gravitation, may have to be augmented ten or twenty
times; that stores of radium and similar bodies within the earth may
have indefinitely deferred the establishment of the present temperature
gradient from the surface inward; that consequently the earth may have
remained for long ages at a temperature not greatly different from that
which it now possesses, and hence that the times during which our globe
has supported animal and vegetable life may be very much longer than
that allowed in the estimates previously made by physicists from other
data (see RADIOACTIVITY).

The arguments from the geological side against the physical contention
that would limit the age of our globe to some 10 or 20 millions of years
are mainly based on the observed rates of geological and biological
changes at the present time upon land and sea, and on the nature,
physical history and organic contents of the stratified crust of the
earth. Unfortunately, actual numerical data are not obtainable in many
departments of geological activity, and even where they can be procured
they do not yet rest on a sufficiently wide collection of accurate and
co-ordinated observations. But in some branches of dynamical geology,
material exists for, at least, a preliminary computation of the rate of
change. This is more especially the case in respect of the wide domain
of denudation. The observational records of the action of the sea, of
springs, rivers and glaciers are becoming gradually fuller and more
trustworthy. A method of making use of these records for estimating the
rate of denudation of the land has been devised. Taking the Mississippi
as a general type of river action, it has been shown that the amount of
material conveyed by this stream into the sea in one year is equivalent
to the lowering of the general surface of the drainage basin of the
river by 1/6000 of a foot. This would amount to one foot in 6000 years
and 1000 ft. in 6 million years. So that at the present rate of waste in
the Mississippi basin a whole continent might be worn away in a few
millions of years.

It is evident that as deposition and denudation are simultaneous
processes, the ascertainment of the rate at which solid material is
removed from the surface of the land supplies some necessary information
for estimating the rate at which new sedimentary formations are being
accumulated on the floor of the sea, and for a computation of the length
of time that would be required at the present rate of change for the
deposition of all the stratified rocks that enter into the composition
of the crust of our globe. If the thickness of these rocks be assumed to
be 100,000 ft., and if we could suppose them to have been laid down over
as wide an area as that of the drainage basins from the waste of which
they were derived, then at the present rate of denudation their
accumulation would require some 600 millions of years. But, as Dr A.R.
Wallace has justly pointed out, the tract of sea-floor over which the
material derived from the waste of the terrestrial surface is laid down
is at present much less than that from which this material is worn away.
We have no means, however, of determining what may have been the ratio
between the two areas in past time. Certainly ancient marine sedimentary
rocks cover at the present day a much more extensive area than that in
which they are now being elaborated. If we take the ratio postulated by
Dr Wallace--1 to 19--the 100,000 ft. of sedimentary strata would require
31 millions of years for their accumulation. It is quite possible,
however, that this ratio may be much too high. There are reasons for
believing that the proportion of coast-line to land area has been
diminishing during geological time; in other words, that in early times
the land was more insular and is now more continental. So that the 31
millions of years may be much less than the period that would be
required, even on the supposition of continuous uninterrupted denudation
and sedimentation, during the whole of the time represented by the
stratified formations.

But no one who has made himself familiar with the actual composition of
these formations and the detailed structure of the terrestrial crust can
fail to recognize how vague, imperfect and misleading are the data on
which such computations are founded. It requires no prolonged
acquaintance with the earth's crust to impress upon the mind that one
all-important element is omitted, and indeed can hardly be allowed for
from want of sufficiently precise data, but the neglect of which must
needs seriously impair the value of all numerical calculations made
without it. The assumption that the stratified formations can be treated
as if they consisted of a continuous unbroken sequence of sediments,
indicating a vast and uninterrupted process of waste and deposition, is
one that is belied on every hand by the actual structure of these
formations. It can only give us a minimum of the time required; for,
instead of an unbroken series, the sedimentary formations are full of
"unconformabilities"--gaps in the sequence of the chronological
records--as if whole chapters and groups of chapters had been torn out
of a historical work. It can often be shown that these breaks of
continuity must have been of vast duration, and actually exceeded in
chronological importance thick groups of strata lying below and above
them (see Part VI.). Moreover, even among the uninterrupted strata,
where no such unconformabilities exist, but where the sediments follow
each other in apparently uninterrupted sequence, and might be thought to
have been deposited continuously at the same general rate, and without
the intervention of any pause, it can be demonstrated that sometimes an
inch or two of sediment might, on certain horizons, represent the
deposit of an enormously longer period than a hundred or a thousand
times the same amount of sediment on other horizons. A prolonged study
of these questions leads to a profound conviction that in many parts of
the geological record the time represented by sedimentary deposits may
be vastly less than the time which is not so represented.

It has often been objected that the present rate of geological change
ought not to be taken as a measure of the rate in past time, because the
total sum of terrestrial energy has been steadily diminishing, and
geological processes must consequently have been more vigorous in former
ages than they are now. Geologists do not pretend to assert that there
has been no variation or diminution in the activities of the various
processes which they have to study. What they do insist on is that the
present rate of change is the only one which we can watch and measure,
and which will thus supply a statistical basis for any computations on
the subject. But it has been dogmatically affirmed that because
terrestrial energy has been diminishing therefore all kinds of
geological work must have been more vigorously and more rapidly carried
on in former times than now; that there were far more abundant and more
stupendous volcanoes, more frequent and more destructive earthquakes,
more gigantic upheavals and subsidences, more powerful oceanic waves and
tides, more violent atmospheric disturbances with heavier rainfall and
more active denudation.

It is easy to make these assertions, and they look plausible; but, after
all, they rest on nothing stronger than assumption. They can be tested
by an appeal to the crust of the earth, in which the geological history
of our planet has been so fully recorded. Had such portentous
manifestations of geological activity ever been the normal condition of
things since the beginning of that history, there ought to be a record
of them in the rocks. But no evidence for them has been found there,
though it has been diligently sought for in all quarters of the globe.
We may confidently assert that while geological changes may quite
possibly have taken place on a gigantic scale in the earliest ages of
the earth's existence, of which no geological record remains, there is
no proof that they have ever done so since the time when the very oldest
of the stratified formations were deposited. There is no need to
maintain that they have always been conducted precisely on the same
scale as now, or to deny that they may have gradually become less
vigorous as the general sum of terrestrial energy has diminished. But we
may unhesitatingly affirm that no actual evidence of any such
progressive diminution of activity has been adduced from the geological
record in the crust of the earth: that, on the contrary, no appearances
have been detected there which necessarily demand the assumption of
those more powerful operations postulated by physicists, or which are
not satisfactorily explicable by reference to the existing scale of
nature's processes.

That this conclusion is warranted even with regard to the innate energy
of the globe itself will be seen if we institute a comparison between
the more ancient and the more recent manifestations of that energy.
Take, for example, the proofs of gigantic plication, fracture and
displacement within the terrestrial crust. These, as they have affected
the most ancient rocks of Europe, have been worked out in great detail
in the north-west of Scotland. But they are not essentially different
from or on a greater scale than those which have been proved to have
affected the Alps, and to have involved strata of so recent a date as
the older Tertiary formations. On the contrary, it may be doubted
whether any denuded core of an ancient mountain-chain reveals traces of
such stupendous disturbances of the crust as those which have given rise
to the younger mountain-chains of the globe. It may, indeed, quite well
have been the rule that instead of diminishing in intensity of effect,
the consequences of terrestrial contraction have increased in magnitude,
the augmenting thickness of the crust offering greater resistance to
the stresses, and giving rise to vaster plications, faults,
thrust-planes and metamorphism, as this growing resistance had to be
overcome.

The assertion that volcanic action must have been more violent and more
persistent in ancient times than it is now has assuredly no geological
evidence in its support. It is quite true that there are vastly more
remains of former volcanoes scattered over the surface of the globe than
there are active craters now, and that traces of copious eruptions of
volcanic material can be followed back into some of the oldest parts of
the geological record. But we have no proof that ever at any one time in
geological history there have been more or larger or more vigorous
volcanoes than those of recent periods. It may be said that the absence
of such proof ought not to invalidate the assertion until a far wider
area of the earth's surface has been geologically studied. But most
assuredly, as far as geological investigation has yet gone, there is an
overwhelming body of evidence to show that from the earliest epochs in
geological history, as registered in the stratified rocks, volcanic
action has manifested itself very much as it does now, but on a less
rather than on a greater scale. Nowhere can this subject be more
exhaustively studied than in the British Isles, where a remarkably
complete series of volcanic eruptions has been chronicled ranging from
the earliest Palaeozoic down to older Tertiary time. The result of a
prolonged study of British volcanic geology has demonstrated that, even
to minute points of detail, there has been a singular uniformity in the
phenomena from beginning to end. The oldest lavas and ashes differ in no
essential respect from the youngest. Nor have they been erupted more
copiously or more frequently. Many successive volcanic periods have
followed each other after prolonged intervals of repose, each displaying
the same general sequence of phenomena and similar evidence of gradual
diminution and extinction. The youngest, instead of being the feeblest,
were the most extensive outbursts in the whole of this prolonged series.

If now we turn for evidence of the alleged greater activity of all the
epigene or superficial forces, and especially for proofs of more rapid
denudation and deposition on the earth's surface, we search for it in
vain among the stratified formations of the terrestrial crust. Had the
oldest of these rocks been accumulated in a time of great atmospheric
perturbation, of torrential rains, colossal tides and violent storms, we
might surely expect to find among the sediments some proof of such
disturbed meteorological and geographical conditions. We should look, on
the one hand, for tumultuous accumulations of coarse unworn detritus,
rapidly swept by rains, floods and waves from land to sea, and on the
other hand, for an absence of any evidence of the tranquil and
continuous deposit of such fine laminated silt as could only settle in
quiet water. But an appeal to the geological record is made in vain for
any such proofs. The oldest sediments, like the youngest, reveal the
operation only of such agents and such rates of activity as are still to
be witnessed in the accumulation of the same kind of deposits. If, for
instance, we search the most ancient thick sedimentary formation in
Britain--the Torridon Sandstone of north-west Scotland, which is older
than the oldest fossiliferous deposits--we meet with nothing which might
not be found in any Palaeozoic, Mesozoic or Cainozoic group of similar
sediments. We see an accumulation, at least 8000 or 10,000 ft. thick, of
consolidated sand, gravel and mud, such as may be gathering now on the
floor of any large mountain-girdled lake. The conglomerates of this
ancient series are not pell-mell heaps of angular detritus, violently
swept away from the land and huddled promiscuously on the sea-floor.
They are, in general, built up of pebbles that have been worn smooth,
rounded and polished by prolonged attrition in running water, and they
follow each other on successive platforms with intervening layers of
finer sediment. The sandstones are composed of well water-worn sand,
some of which has been laid down so tranquilly that its component grains
have been separated out in layers according to their specific gravity,
in such manner that they now present dark laminae in which particles of
magnetic iron, zircon and other heavy minerals have been sifted out
together, just as iron-sand may be seen gathered into thin sheets on
sandy beaches at the present day. Again, the same series of primeval
sediments includes intercalations of fine silt, which has been deposited
as regularly and intermittently there as it has been among the most
recent formations. These bands of shale have been diligently searched
for fossils, as yet without success; but they may eventually disclose
organic remains older than any hitherto found in Europe.

We now come to the consideration of the palaeontological evidence as to
the value of geological time. Here the conclusions derived from a study
of the structure of the sedimentary formations are vastly strengthened
and extended. In the first place, the organization of the most ancient
plants and animals furnishes no indication that they had to contend with
any greater violence of storm, flood, wave or ocean-current than is
familiar to their modern descendants. The oldest trees, shrubs, ferns
and club-mosses display no special structures that suggest a difference
in the general conditions of their environment. The most ancient
crinoids, sponges, crustaceans, arachnids and molluscs were as
delicately constructed as those of to-day, and their remains are often
found in such perfect preservation as to show that neither during their
lifetime nor after their death were they subject to any greater violence
of the elements than their living representatives now experience. Of
much more cogency, however, is the evidence supplied by the grand upward
succession of organic forms, from the most ancient stratified rocks up
to the present day. No biologist now doubts for a moment that this
marvellous succession is the result of a gradual process of evolution
from lower to higher types of organization. There may be differences of
opinion as to the causes which have governed this process and the order
of the steps through which it has advanced, but no one who is conversant
with the facts will now venture to deny that it has taken place, and
that, on any possible explanation of its progress, it must have demanded
an enormous lapse of time. In the Cambrian or oldest fossiliferous
formations there is already a large and varied fauna, in which the
leading groups of invertebrate life are represented. On no tenable
hypothesis can these be regarded as the first organisms that came into
being on our planet. They must have had a long ancestry, and as Darwin
first maintained, the time required for their evolution may have been
"as long as, or probably far longer than, the whole interval from the
Silurian [Cambrian] age to the present day." The records of these
earliest eras of organic development have unfortunately not survived the
geological revolutions of the past; at least, they have not yet been
recovered. But it cannot be doubted that they once existed and
registered their testimony to the prodigious lapse of time prior to the
deposition of the most ancient fossiliferous formations which have
escaped destruction.

The impressive character of the evidence furnished by the sequence of
organic forms throughout the great series of fossiliferous strata can
hardly be fully realized without a detailed and careful study of the
subject. Professor E.B. Poulton, in an address to the zoological section
of the British Association at the Liverpool Meeting in 1896, showed how
overwhelming are the demands which this evidence makes for long periods
of time, and how impossible it is of comprehension unless these demands
be conceded. The history of life upon the earth, though it will probably
always be surrounded with great and even insuperable difficulties,
becomes broadly comprehensible in its general progress when sufficient
time is granted for the evolution which it records; but it remains
unintelligible on any other conditions.

Taken then as a whole, the body of evidence, geological and
palaeontological, in favour of the high antiquity of our globe is so
great, so manifold, and based on such an ever-increasing breadth of
observation and reflection, that it may be confidently appealed to in
answer to the physical arguments which would seek to limit that
antiquity to ten or twenty millions of years. In the present state of
science it is out of our power to state positively what must be the
lowest limit of the age of the earth. But we cannot assume it to be much
less, and it may possibly have been much more, than the 100 millions of
years which Lord Kelvin was at one time willing to concede.[2]


PART III.--GEOGNOSY. THE INVESTIGATION OF THE NATURE AND COMPOSITION OF
THE MATERIALS OF WHICH THE EARTH CONSISTS

This division of the science is devoted to a description of the parts of
the earth--of the atmosphere and ocean that surround the planet, and
more especially of the solid materials that underlie these envelopes and
extend downwards to an unknown distance into the interior. These various
constituents of the globe are here considered as forms of matter capable
of being analysed, and arranged according to their composition and the
place they take in the general composition of the globe.

Viewed in the simplest way the earth may be regarded as made up of three
distinct parts, each of which ever since an early period of planetary
history has been the theatre of important geological operations. (1) An
envelope of air, termed the _atmosphere_, which surrounds the whole
globe; (2) A lower and less extensive envelope of water, known as the
_hydrosphere_ (Gr. [Greek: hydôr], water) which, constituting the oceans
and seas, covers nearly three-fourths of the underlying solid surface of
the planet; (3) A globe, called the _lithosphere_ (Gr. [Greek: lithos],
stone), the external part of which, consisting of solid stone, forms the
_crust_, while underneath, and forming the vast mass of the interior,
lies the _nucleus_, regarding the true constitution of which we are
still ignorant.

1. _The Atmosphere._--The general characters of the atmosphere are
described in separate articles (see especially ATMOSPHERE; METEOROLOGY).
Only its relations to geology have here to be considered. As this
gaseous envelope encircles the whole globe it is the most universally
present and active of all the agents of geological change. Its efficacy
in this respect arises partly from its composition, and the chemical
reactions which it effects upon the surface of the land, partly from its
great variations in temperature and moisture, and partly from its
movements.

  Many speculations have been made regarding the chemical composition of
  the atmosphere during former geological periods. There can indeed be
  little doubt that it must originally have differed greatly from its
  present condition. If the whole mass of the planet originally existed
  in a gaseous state, there would be practically no atmosphere. The
  present outer envelope of air may be considered to be the surviving
  relic of this condition, after all the other constituents have been
  incorporated into the hydrosphere and lithosphere. The oxygen, which
  now forms fully a half of the outer crust of the earth, was doubtless
  originally, whether free or in combination, part of the atmosphere.
  So, too, the vast beds of coal found all over the world, in geological
  formations of many different ages, represent so much carbonic acid
  once present in the air. The chlorides and other salts in the sea may
  likewise partly represent materials carried down out of the atmosphere
  in the primitive condensation of the aqueous vapour, though they have
  been continually increased ever since by contributions from the
  drainage of the land. It has often been suggested that, during the
  Carboniferous period, the atmosphere must have been warmer and more
  charged with aqueous vapour and carbon dioxide than at the present
  day, to admit of so luxuriant a flora as that from which the
  coal-seams were formed. There seems, however, to be at present no
  method of arriving at any certainty on this subject. Lastly, the
  amount of carbonic acid absorbed in the weathering of rocks at the
  surface, and the consequent production of carbonates, represents an
  enormous abstraction of this gas.

  As at present constituted, the atmosphere is regarded as a mechanical
  mixture of nearly four volumes of nitrogen and one of oxygen, together
  with an average of 3.5 parts of carbon dioxide in every 10,000 parts
  of air, and minute quantities of various other gases and solid
  particles. Of the vapours contained in it by far the most important is
  that of water which, although always present, varies greatly in amount
  according to variations in temperature. By condensation the water
  vapour appears in visible form as dew, mist, cloud, rain, hail, snow
  and ice, and in these forms includes and carries down some of the
  other vapours, gases and solid particles present in the air. The
  circulation of water from the atmosphere to the land, from the land to
  the sea, and again from the sea to the land, forms the great
  geological process whereby the habitable condition of the planet is
  maintained and the surface of the land is sculptured (Part IV.).

2. _The Hydrosphere._--The water envelope covers nearly three-fourths of
the surface of the earth, and forms the various oceans and seas which,
though for convenience of reference distinguished by separate names, are
all linked together in one great body. The physical characters of this
vast envelope are discussed in separate articles (see OCEAN and
OCEANOGRAPHY). Viewed from the geological standpoint, the features of
the sea that specially deserve attention are first the composition of
its waters, and secondly its movements.

  Sea-water is distinguished from that of ordinary lakes and rivers by
  its greater specific gravity and its saline taste. Its average density
  is about 1.026, but it varies even within the same ocean, being least
  where large quantities of fresh water are added from rain or melting
  snow and ice, and greatest where evaporation is most active. That
  sea-water is heavier than fresh arises from the greater proportion of
  salts which it contains in solution. These salts constitute about
  three and a half parts in every hundred of water. They consist mainly
  of chlorides of sodium and magnesium, the sulphates of magnesium,
  calcium and potassium, with minuter quantities of magnesium bromide
  and calcium carbonate. Still smaller proportions of other substances
  have been detected, gold for example having been found in the
  proportion of 1 part in 15,180,000.

  That many of the salts have existed in the sea from the time of its
  first condensation out of the primeval atmosphere appears to be
  probable. It is manifest, however, that, whatever may have been the
  original composition of the oceans, they have for a vast section of
  geological time been constantly receiving mineral matter in solution
  from the land. Every spring, brook and river removes various salts
  from the rocks over which it moves, and these substances, thus
  dissolved, eventually find their way into the sea. Consequently
  sea-water ought to contain more or less traceable proportions of every
  substance which the terrestrial waters can remove from the land, in
  short, of probably every element present in the outer shell of the
  globe, for there seems to be no constituent of this earth which may
  not, under certain circumstances, be held in solution in water.
  Moreover, unless there be some counteracting process to remove these
  mineral ingredients, the ocean water ought to be growing, insensibly
  perhaps, but still assuredly, saltier, for the supply of saline matter
  from the land is incessant.

  To the geologist the presence of mineral solutions in sea-water is a
  fact of much importance, for it explains the origin of a considerable
  part of the stratified rocks of the earth's crust. By evaporation the
  water has given rise to deposits of rock-salt, gypsum and other
  materials. The lime contained in solution, whether as sulphate or
  carbonate, has been extracted by many tribes of marine animals, which
  have thus built up out of their remains vast masses of solid
  limestone, of which many mountain-chains largely consist.

  Another important geological feature of the sea is to be seen in the
  fact that its basins form the great receptacles for the detritus worn
  away from the land. Besides the limestones, the visible parts of the
  terrestrial crust are, in large measure, composed of sedimentary rocks
  which were originally laid down on the sea-bottom. Moreover, by its
  various movements, the sea occupies a prominent place among the
  epigene or superficial agents which produce geological changes on the
  surface of the globe.

3. _The Lithosphere._--Beneath the gaseous and liquid envelopes lies the
solid part of the planet, which is conveniently regarded as consisting
of two parts,--(a) the crust, and (b) the interior or nucleus.


  The crust.

It was for a long time a prevalent belief that the interior of the globe
is a molten mass round which an outer shell has gradually formed through
cooling. Hence the term "crust" was applied to this external solid
envelope, which was variously computed to be 10, 20, or more miles in
thickness. The portion of this crust accessible to human observation was
seen to afford abundant evidence of vast plications and corrugations of
its substance, which were regarded as only explicable on the supposition
of a thin solid collapsible shell floating on a denser liquid interior.
When, however, physical arguments were adduced to show the great
rigidity of the earth as a whole, the idea of a thin crust enclosing a
molten nucleus was reluctantly abandoned by geologists, who found the
problem of the earth's interior to be incapable of solution by any
evidence which their science could produce. They continued, however, to
use the term "crust" as a convenient word to denote the cool outer layer
of the earth's mass, the structure and history of which form the main
subjects of geological investigation. More recently, however, various
lines of research have concurred in suggesting that, whatever may be the
condition of the interior, its substance must differ greatly from that
of the outer shell, and that there may be more reason than appeared for
the retention of the name of crust. Observations on earthquake motion by
Dr John Milne and others, show that the rate and character of the waves
transmitted through the interior of the earth differ in a marked degree
from those propagated along the crust. This difference indicates that
rocky material, such as we know at the surface, may extend inwards for
some 30 m., below which the earth's interior rapidly becomes fairly
homogeneous and possesses a high rigidity. From measurements of the
force of gravity in India by Colonel S.G. Burrard, it has been inferred
that the variations in density of the outer parts of the earth do not
descend farther than 30 or 40 m., which might be assumed to be the limit
of the thickness of the crust. Recent researches in regard to the
radio-active substances present in rocks suggest that the crust is not
more than 50 m. thick, and that the interior differs from it in
possessing little or no radio-active material.


  The interior.

Though we cannot hope ever to have direct acquaintance with more than
the mere outside skin of our planet, we may be led to infer the
irregular distribution of materials within the crust from the present
distribution of land and water, and the observed differences in the
amount of deflection of the plumb-line near the sea and near
mountain-chains. The fact that the southern hemisphere is almost wholly
covered with water appears explicable only on the assumption of an
excess of density in the mass of that portion of the planet. The
existence of such a vast sheet of water as that of the Pacific Ocean is
to be accounted for, as Archdeacon J.H. Pratt pointed out, by the
presence of "some excess of matter in the solid parts of the earth
between the Pacific Ocean and the earth's centre, which retains the
water in its place, otherwise the ocean would flow away to the other
parts of the earth." A deflection of the plumb-line towards the sea,
which has in a number of cases been observed, indicates that "the
density of the crust beneath the mountains must be less than that below
the plains, and still less than that below the ocean-bed." Apart
therefore from the depression of the earth's surface in which the oceans
lie, we must regard the internal density, whether of crust or nucleus,
to be somewhat irregularly arranged, there being an excess of heavy
materials in the water hemisphere, and beneath the ocean-beds, as
compared with the continental masses.

In our ignorance regarding the chemical constitution of the nucleus of
our planet, an argument has sometimes been based upon the known fact
that the specific gravity of the globe as a whole is about double that
of the crust. This has been held by some writers to prove that the
interior must consist of much heavier material and is therefore probably
metallic. But the effect of pressure ought to make the density of the
nucleus much higher, even if the interior consisted of matter no heavier
than the crust. That the total density of the planet does not greatly
exceed its observed amount seems only explicable on the supposition that
some antagonistic force counteracts the effects of pressure. The only
force we can suppose capable of so acting is heat. But comparatively
little is yet known regarding the compression of gases, liquids and
solids under such vast pressures as must exist within the nucleus.

That the interior of the earth possesses a high temperature is inferred
from the evidence of various sources. (1) Volcanoes, which are openings
that constantly, or intermittently, give out hot vapours and molten lava
from reservoirs beneath the crust. Besides active volcanoes, it is known
that former eruptive vents have been abundantly and widely distributed
over the globe from the earliest geological periods down to our own day.
(2) Hot springs are found in many parts of the globe, with temperatures
varying up to the boiling point of water. (3) From mines, tunnels and
deep borings into the earth it has been ascertained that in all quarters
of the globe below the superficial zone of invariable temperature, there
is a progressive increase of heat towards the interior. The rate of this
increase varies, being influenced, among other causes, by the varying
conductivity of the rocks. But the average appears to be about 1° Fahr.
for every 50 or 60 ft. of descent, as far down as observations have
extended. Though the increase may not advance in the same proportion at
great depths, the inference has been confidently drawn that the
temperature of the nucleus must be exceedingly high.

The probable condition of the earth's interior has been a fruitful
source of speculation ever since geology came into existence; but no
general agreement has been arrived at on the subject. Three chief
hypotheses have been propounded: (1) that the nucleus is a molten mass
enclosed within a solid shell; (2) that, save in local vesicular spaces
which may be filled with molten or gaseous material, the globe is solid
and rigid to the centre; (3) that the great body of the nucleus consists
of incandescent vapours and gases, especially vaporous iron, which under
the gigantic pressure within the earth are so compressed as to confer
practical rigidity on the globe as a whole, and that outside this main
part of the nucleus the gases pass into a shell of molten magma, which,
in turn, shades off outwards into the comparatively thin, cool
solidified crust. Recent seismological observations have led to the
inference that the outer crust, some 30 to 45 m. thick, must rapidly
merge into a fairly homogeneous nucleus which, whatever be its
constitution, transmits undulatory movements through its substance with
uniform velocity and is believed to possess a high rigidity.

The origin of the earth's high internal temperature has been variously
accounted for. Most usually it has been assumed to be the residue of the
original "tracts of fluent heat" out of which the planet shaped itself
into a globe. According to another supposition the effects of the
gradual gravitational compression of the earth's mass have been the main
source of the high temperature. Recent researches in radio-activity, to
which reference has already been made, have indicated another possible
source of the internal heat in the presence of radium in the rocks of
the crust. This substance has been detected in all igneous rocks,
especially among the granites, in quantity sufficient, according to the
Hon. R.J. Strutt, to account for the observed temperature-gradient in
the crust, and to indicate that this crust cannot be more than 45 m.
thick, otherwise the outflow of heat would be greater than the amount
actually ascertained. Inside this external crust containing radio-active
substances, it is supposed, as already stated, that the nucleus consists
of some totally different matter containing little or no radium.

  _Constitution of the Earth's Crust._--As the crust of the earth
  contains the "geological record," or stony chronicle from which
  geology interprets the history of our globe, it forms the main subject
  of study to the geologist. The materials of which this crust consists
  are known as minerals and rocks. From many chemical analyses, which
  have been made of these materials, the general chemical constitution
  of, at least, the accessible portion of the crust has been
  satisfactorily ascertained. This information becomes of much
  importance in speculations regarding the early history of the globe.
  Of the elements known to the chemist the great majority form but a
  small proportion of the composition of the crust, which is mainly
  built up of about twenty of them. Of these by far the most important
  are the non-metallic elements oxygen and silicon. The former forms
  about 47% and the latter rather more than 28% of the original crust,
  so that these two elements make up about three-fourths of the whole.
  Next after them come the metals aluminium (8.16%), iron (4.64),
  calcium (3.50), magnesium (2.62), sodium (2.63), and potassium (2.35).
  The other twelve elements included in the twenty vary in amount from a
  proportion of 0.41% in the case of titanium, to not more than 0.01% of
  chlorine, fluorine, chromium, nickel and lithium. The other fifty or
  more elements exist in such minute proportions in the crust that,
  probably, not one of them amounts to as much as 0.01%, though they
  include the useful metals, except iron. Taking the crust, and the
  external envelopes of the ocean and the air, we thus perceive that
  these outer parts of our planet consist of more than three-fourths of
  non-metals and less than one-fourth of metals.

  The combinations of the elements which are of most importance in the
  constitution of the terrestrial crust consist of oxides. From the mean
  of a large number of analyses of the rocks of the lower or primitive
  portion of the crust, it has been ascertained that silica (SiO2) forms
  almost 60% and alumina (Al2O3) upwards of 15% of the whole. The other
  combinations in order of importance are lime (CaO) 4.90%, magnesia
  (MgO) 4.36, soda (Na2O) 3.55, ferrous oxide (FeO) 3.52, potash (K2O)
  2.80, ferric oxide (Fe2O3) 2.63, water (H2O) 1.52, titanium oxide
  (TiO2) 0.60, phosphoric acid (P2O5) 0.22; the other combinations of
  elements thus form less than 1% of the crust.

  These different combinations of the elements enter into further
  combinations with each other so as to produce the wide assortment of
  simple minerals (see MINERALOGY). Thus, silica and alumina are
  combined to form the aluminous silicates, which enter so largely into
  the composition of the crust of the earth. The silicates of magnesia,
  potash and soda constitute other important families of minerals. A
  mass of material composed of one, but more usually of more than one
  mineral, is known as a _rock_. Under this term geologists are
  accustomed to class not only solid stone, such as granite and
  limestone, but also less coherent materials such as clay, peat and
  even loose sand. The accessible portion of the earth's crust consists
  of various kinds of rocks, which differ from each other in structure,
  composition and origin, and are therefore susceptible of diverse
  classifications according to the point of view from which they are
  considered. The details of this subject will be found in the article
  PETROLOGY.

  _Classification of Rocks._--Various systems of classification of rocks
  have been proposed, but none of them is wholly satisfactory. The most
  useful arrangement for most purposes of the geologist is one based on
  the broad differences between them in regard to their mode of origin.
  From this point of view they may be ranged in three divisions:

  1. In the first place, a large number of rocks may be described as
  original or underived, for it is not possible to trace them back to
  any earlier source. They belong to the primitive constitution of the
  planet, and, as they have all come up from below through the crust,
  they serve to show the nature of the material which lies immediately
  below the outer parts of that crust. They include the numerous
  varieties of lava, which have been poured out in a molten state from
  volcanic vents, also a great series of other rocks which, though they
  may never have been erupted to the surface, have been forced upward in
  a melted condition into the other rocks of the crust and have
  solidified there. From their mode of origin this great class of rocks
  has been called "igneous" or "eruptive." As they generally show no
  definite internal structure save such as may result from joints, they
  have been termed "massive" or "unstratified," to distinguish them from
  those of the second division which are strongly marked out by the
  presence of a stratified structure. The igneous rocks present a
  considerable range of composition. For the most part they consist
  mainly of aluminous silicates, some of them being highly acid
  compounds with 75% or more of silica. But they also include highly
  basic varieties wherein the proportion of silica sinks to 40%, and
  where magnesia greatly predominates over alumina. The textures of
  igneous rocks likewise comprise a wide series of varieties. On the one
  hand, some are completely vitreous, like obsidian, which is a natural
  glass. From this extreme every gradation may be traced through gradual
  increase of the products of devitrification, until the mass may become
  completely crystalline. Again, some crystalline igneous rocks are so
  fine in grain as not to show their component crystals save under the
  microscope, while in others the texture is so coarse as to present the
  component minerals in separate crystals an inch or more in length.
  These differences indicate that, at first, the materials of the rock
  may have been as completely molten as artificial glass, and that the
  crystalline condition has been subsequently developed by cooling, and
  the separation of the chemical constituents into definite crystalline
  minerals. Many of the characters of igneous rocks have been reproduced
  experimentally by fusing together their minerals, or the constituents
  of their minerals, in the proper proportion. But it has not yet been
  found possible to imitate the structure of such rocks as granite.
  Doubtless these rocks consolidated with extreme slowness at great
  depths below the surface, under vast pressures and probably in the
  presence of water or water-vapour--conditions which cannot be
  adequately imitated in a laboratory.

  Though the igneous rocks occupy extensive areas in some countries,
  they nevertheless cover a much smaller part of the whole surface of
  the land than is taken up by the second division or stratified rocks.
  But they increase in quantity downwards and probably extend
  continuously round the globe below the other rocks. This important
  series brings before us the relations of the molten magma within the
  earth to the overlying crust and to the outer surface. On the one
  hand, it includes the oldest and most deep-seated extravasations of
  that magma, which have been brought to light by ruptures and upheavals
  of the crust and prolonged denudation. On the other, it presents to
  our study the varied outpourings of molten and fragmentary materials
  in the discharges of modern and ancient volcanoes. Between these two
  extremes of position and age, we find that the crust has been, as it
  were, riddled with injections of the magma from below. These features
  will be further noticed in Part V. of this article.

  2. The "sedimentary" or "stratified rocks" form by much the larger
  part of the dry land of the globe, and they are prolonged to an
  unknown distance from the shores under the bed of the sea. They
  include those masses of mineral matter which, unlike the igneous
  rocks, can be traced back to a definite origin on the surface of the
  earth. Three distinct types may be recognized among them: (a) By far
  the largest proportion of them consists of different kinds of sediment
  derived from the disintegration of pre-existing rocks. In this
  "fragmental" group are placed all the varieties of shingle, gravel,
  sand, clay and mud, whether these materials remain in a loose
  incoherent condition, or have been compacted into solid stone. (b)
  Another group consists of materials that have been deposited by
  chemical precipitation from solution in water. The white sinter laid
  down by calcareous springs is a familiar example on a small scale.
  Beds of rock-salt, gypsum and dolomite have, in some regions, been
  accumulated to a thickness of many thousand feet, by successive
  precipitations of the salt contained in the water of inland seas. (c)
  An abundant and highly important series of sedimentary formations has
  been formed from the remains of plants and animals. Such accumulations
  may arise either from the transport and deposit of these remains, as
  in the case of sheets of drift-wood, and banks of drifted sea-shells,
  or from the growth and decay of the organisms on the spot, as happens
  in peat bogs and in coral-reefs.

  As the sedimentary rocks have for the most part been laid down under
  water, and more especially on the sea-floor, they are often spoken of
  as "aqueous," in contradistinction to the igneous rocks. Some of them,
  however, are accumulated by the drifting action of wind upon loose
  materials, and are known as "aeolian" formations. Familiar instances
  of such wind-formed deposits are the sand-dunes along many parts of
  the sea coast. Much more extensive in area are the sands of the great
  deserts in the arid regions of the globe.

  It is from the sedimentary rocks that the main portion of geological
  history is derived. They have been deposited one over another in
  successive strata from a remote period in the development of the globe
  down to the present time. From this arrangement they have been termed
  "stratified," in contrast to the unstratified or igneous series. They
  have preserved memorials of the geographical revolutions which the
  surface of the earth has undergone; and above all, in the abundant
  fossils which they have enclosed, they furnish a momentous record of
  the various tribes of plants and animals which have successively
  flourished on land and sea. Their investigation is thus the most
  important task which devolves upon the geologist.

  3. In the third place comes a series of rocks which are not now in
  their original condition, but have undergone such alteration as to
  have acquired new characters that more or less conceal their first
  structures. Some of them can be readily recognized as altered igneous
  masses; others are as manifestly of sedimentary origin; while of many
  it is difficult to decide what may have been their pristine character.
  To this series the term "metamorphic" has been applied. Its members
  are specially distinguished by a prevailing fissile, or schistose,
  structure which they did not at first possess, and which differs from
  anything found in unaltered igneous or sedimentary rocks. This
  fissility is combined with a more or less pronounced crystalline
  structure. These changes are believed to be the result of movements
  within the crust of the earth, whereby the most solid rocks were
  crushed and sheared, while, at the same time, under the influence of a
  high temperature and the presence of water, they underwent internal
  chemical reactions, which led to a rearrangement and recomposition of
  their mineral constituents and the production of a crystalline
  structure (see METAMORPHISM).

  Among the less altered metamorphic rocks of sedimentary origin, the
  successive laminae of deposit of the original sediment can be easily
  observed; but they are also traversed by a new set of divisional
  planes, along which they split across the original bedding. Together
  with this superinduced cleavage there have been developed in them
  minute hairs, scales and rudimentary crystals. Further stages of
  alteration are marked by the increase of micaceous scales, garnets and
  other minerals, especially along the planes of cleavage, until the
  whole rock becomes crystalline, and displays its chief component
  minerals in successive discontinuous folia which merge into each
  other, and are often crumpled and puckered. Massive igneous rocks can
  be observed to have undergone intense crushing and cleavage, and to
  have ultimately assumed a crystalline foliated character. Rocks which
  present this aspect are known as schists (q.v.). They range from the
  finest silky slates, or phyllites, up to the coarsest gneisses, which
  in hand-specimens can hardly be distinguished from granites. There is
  indeed every reason to believe that such gneisses were probably
  originally true granites, and that their foliation and
  recrystallization have been the result of metamorphism.

  The schists are more especially to be found in the heart of
  mountain-chains, and in regions where the lowest and oldest parts of
  the earth's crust have, in the course of geological revolutions, been
  exposed to the light of day. They have been claimed by some writers
  to be part of the original or primitive surface of our globe that
  first consolidated on the molten nucleus. But the progress of
  investigation all over the world has shown that this supposition
  cannot be sustained. The oldest known rocks present none of the
  characters of molten material that has cooled and hardened in the air,
  like the various forms of recent lava. On the contrary, they possess
  many of the features characteristic of bodies of eruptive material
  that have been injected into the crust at some depth underground, and
  are now visible at the surface, owing to the removal by denudation of
  the rocks under which they consolidated. In their less foliated
  portions they can be recognized as true eruptive rocks. In many places
  gneisses that possess a thoroughly typical foliation have been found
  to pierce ancient sedimentary formations as intrusive bosses and
  veins.


PART IV.--DYNAMICAL GEOLOGY

This section of the science includes the investigation of those
processes of change which are at present in progress upon the earth,
whereby modifications are made on the structure and composition of the
crust, on the relations between the interior and the surface, as shown
by volcanoes, earthquakes and other terrestrial disturbances, on the
distribution of oceans and continents, on the outlines of the land, on
the form and depth of the sea-bottom, on climate, and on the races of
plants and animals by which the earth is tenanted. It brings before us,
in short, the whole range of activities which it is the province of
geology to study, and leads us to precise notions regarding their
relations to each other and the results which they achieve. A knowledge
of this branch of the subject is thus the essential groundwork of a true
and fruitful acquaintance with the principles of geology, seeing that it
necessitates a study of the present order of nature, and thus provides a
key for the interpretation of the past.

The whole range of operations included within the scope of inquiry in
this branch of the science may be regarded as a vast cycle of change,
into which we may break at any point, and round which we may travel,
only to find ourselves brought back to our starting-point. It is a
matter of comparatively small moment at what part of the cycle we begin
our inquiries. We shall always find that the changes we see in action
have resulted from some that preceded, and give place to others which
follow them.

At an early time in the earth's history, anterior to any of the periods
of which a record remains in the visible rocks, the chief sources of
geological action probably lay within the earth itself. If, as is
generally supposed, the planet still retained a great store of its
initial heat, it was doubtless the theatre of great chemical changes,
giving rise, perhaps, to manifestations of volcanic energy somewhat like
those which have so marvellously roughened the surface of the moon. As
the outer layers of the globe cooled, and the disturbances due to
internal heat and chemical action became less marked, the conditions
would arise in which the materials for geological history were
accumulated. The influence of the sun, which must always have operated,
would then stand out more clearly, giving rise to that wide circle of
superficial changes wherein variations of temperature and the
circulation of air and water over the surface of the earth come into
play.

In the pursuit of his inquiries into the past history and into the
present _régime_ of the earth, the geologist must needs keep his mind
ever open to the reception of evidence for kinds and especially for
degrees of action which he had not before imagined. Human experience has
been too short to allow him to assume that all the causes and modes of
geological change have been definitively ascertained. On the earth
itself there may remain for future discovery evidence of former
operations by heat, magnetism, chemical change or otherwise, which may
explain many of the phenomena with which geology has to deal. Of the
influences, so many and profound, which the sun exerts upon our planet,
we can as yet only perceive a little. Nor can we tell what other
cosmical influences may have lent their aid in the evolution of
geological changes.

Much useful information regarding many geological processes has been
obtained from experimental research in laboratories and elsewhere, and
much more may be confidently looked for from future extensions of this
method of inquiry. The early experiments of Sir James Hall, already
noticed, formed the starting-point for numerous subsequent researches,
which have elucidated many points in the origin and history of rocks. It
is true that we cannot hope to imitate those operations of nature which
demand enormous pressures and excessively high temperatures combined
with a long lapse of time. But experience has shown that in regard to a
large number of processes, it is possible to imitate nature's working
with sufficient accuracy to enable us to understand them, and so to
modify and control the results as to obtain a satisfactory solution of
some geological problems.

In the present state of our knowledge, all the geological energy upon
and within the earth must ultimately be traced back to the primeval
energy of the parent nebula or sun. There is, however, a certain
propriety and convenience in distinguishing between that part of it
which is due to the survival of some of the original energy of the
planet and that part which arises from the present supply of energy
received day by day from the sun. In the former case we have to deal
with the interior of the earth, and its reaction upon the surface; in
the latter, we deal with the surface of the earth and to some extent
with its reaction on the interior. This distinction allows of a broad
treatment of the subject under two divisions:

I. Hypogene or Plutonic Action: The changes within the earth caused by
internal heat, mechanical movement and chemical rearrangements.

II. Epigene or Surface Action: The changes produced on the superficial
parts of the earth, chiefly by the circulation of air and water set in
motion by the sun's heat.


_DIVISION I.--HYPOGENE OR PLUTONIC ACTION_

In the discussion of this branch of the subject we must carry in our
minds the conception of a globe still possessing a high internal
temperature, radiating heat into space and consequently contracting in
bulk. Portions of molten rocks from inside are from time to time poured
out at the surface. Sudden shocks are generated by which destructive
earthquakes are propagated through the diameter of the globe as well as
to and along its surface. Wide geographical areas are pushed up or sink
down. In the midst of these movements remarkable changes are produced
upon the rocks of the crust; they are plicated, fractured, crushed,
rendered crystalline and even fused.


  (A) _Volcanoes and Volcanic Action._

  This subject is discussed in the article VOLCANO, and only a general
  view of its main features will be given here. Under the term volcanic
  action (vulcanism, vulcanicity) are embraced all the phenomena
  connected with the expulsion of heated materials from the interior of
  the earth to the surface. A volcano may be defined as a conical hill
  or mountain, built up wholly or mainly of materials which have been
  ejected from below, and which have accumulated around the central vent
  of eruption. As a rule its truncated summit presents a cup-shaped
  cavity, termed the crater, at the bottom of which is the opening of
  the main funnel or pipe whereby communication is maintained with the
  heated interior. From time to time, however, in large volcanoes rents
  are formed on the sides of the cone, whence steam and other hot
  vapours and also streams of molten lava are poured forth. On such
  rents smaller or parasitic cones are often formed, which imitate the
  operations of the parent cone and, after repeated eruptions, may rise
  to hills hundreds of feet in height. In course of centuries the result
  of the constant outpouring of volcanic materials may be to build up a
  large mountain like Etna, which towers above the sea to a height of
  10,840 feet, and has some 200 minor cones along its flanks.

  But all volcanic eruptions do not proceed from central orifices. In
  Iceland it has been observed that, from fissures opened in the ground
  and extending for long distances, molten material has issued in such
  abundance as to be spread over the surrounding country for many miles,
  while along the lines of fissure small cones or hillocks of
  fragmentary material have accumulated round more active parts of the
  rent. There is reason to believe that in the geological past this
  fissure-type of eruption has repeatedly been developed, as well as the
  more common form of central cones like Vesuvius or Etna.

  In the operations of existing volcanoes only the superficial
  manifestations of volcanic action are observable. But when the rocks
  of the earth's crust are studied, they are found to enclose the relics
  of former volcanic eruptions. The roots of ancient volcanoes have thus
  been laid bare by geological revolutions; and some of the
  subterranean phases of volcanic action are thereby revealed which are
  wholly concealed in an active volcano. Hence to obtain as complete a
  conception as possible of the nature and history of volcanic action,
  regard must be had, not merely to modern volcanoes, but to the records
  of ancient eruptions which have been preserved within the crust.

  The substances discharged from volcanic vents consist of--(1) Gases
  and vapours: which, dissolved in the molten magma of the interior,
  take the chief share in volcanic activity. They include in greatest
  abundance water-gas, which condenses into the clouds of steam so
  conspicuous in volcanic eruptions. Hydrochloric acid and sulphuretted
  hydrogen are likewise plentiful, together with many other substances
  which, sublimed by the high internal temperature, take a solid form on
  cooling at the surface. (2) Molten rock or lava: which ranges from the
  extremely acid type of the obsidians and rhyolites with 70% or more of
  silica, to the more basic and heavy varieties such as basalts and
  leucite-lavas with much iron, and sometimes no more than 45% of
  silica. The specific gravity of lavas varies between 2.37 and 3.22,
  and the texture ranges from nearly pure glass, like obsidian, to a
  coarse granitoid compound, as in some rhyolites. (3) Fragmentary
  materials, which are sometimes discharged in enormous quantity and
  dispersed over a wide extent of country, the finer particles being
  transported by upper air-currents for hundreds of miles. These
  materials arise either from the explosion of lava by the sudden
  expansion of the dissolved vapours and gases, as the molten rock rises
  to the surface, or from the breaking up and expulsion of portions of
  the walls of the vent, or of the lava, which happens to have
  solidified within these walls. They vary from the finest impalpable
  dust and ashes, through increasing stages of coarseness up to huge
  "bombs" torn from the upper surface of the molten rock in the vent,
  and large blocks of already solidified lava, or of non-volcanic rock
  detached from the sides of the pipe up which the eruptions take place.

  Nothing is yet known as to the determining cause of any particular
  volcanic eruption. Some vents, like that of Stromboli, in the
  Mediterranean, are continually active, and have been so ever since man
  has observed them. Others again have been only intermittently in
  eruption, with intervals of centuries between their periods of
  activity. We are equally in the dark as to what has determined the
  sites on which volcanic action has manifested itself. There is reason,
  indeed, to believe that extensive fractures of the terrestrial crust
  have often provided passages up which the vapours, imprisoned in the
  internal magma, have been able to make their way, accompanied by other
  products. Where chains of volcanoes rise along definite lines, like
  those of Sumatra, Java, and many other tracts both in the Old and the
  New World, there appears to be little doubt that their linear
  distribution should be attributed to this cause. But where a volcano
  has appeared by itself, in a region previously exempt from volcanic
  action, the existence of a contributing fissure cannot be so
  confidently presumed. The study of certain ancient volcanoes, the
  roots of which have been exposed by long denudation, has shown an
  absence of any visible trace of their having availed themselves of
  fractures in the crust. The inference has been drawn that volcanic
  energy is capable of itself drilling an orifice through the crust,
  probably at some weaker part, and ejecting its products at the
  surface. The source of this energy is to be sought in the enormous
  expansive force of the vapours and gases dissolved in the magma. They
  are kept in solution by the enormous pressure within the earth; but as
  the lava approaches the surface and this pressure is relieved these
  dissolved vapours and gases rush out with explosive violence, blowing
  the upper part of the lava column into dust, and allowing portions of
  the liquid mass below to rise and escape, either from the crater or
  from some fissure which the vigour of explosion has opened on the side
  of the cone. So gigantic is the energy of these pent-up vapours, that,
  after a long period of volcanic quiescence, they sometimes burst forth
  with such violence as to blow off the whole of the upper part or even
  one side of a large cone. The history of Vesuvius, and the great
  eruptions of Krakatoa in 1883 and of Bandaizan in 1888 furnish
  memorable examples of great volcanic convulsions. It has been observed
  that such stupendous discharges of aeriform and fragmentary matter may
  be attended with the emission of little or no lava. On the other hand,
  some of the largest outflows of lava have been accompanied by
  comparatively little fragmentary material. Thus, the great lava-floods
  of Iceland in 1783 spread for 40 m. away from their parent fissure,
  which was marked only by a line of little cones of slag.

  The temperature of lava as it issues from underground has been
  measured more or less satisfactorily, and affords an indication of
  that existing within the earth. At Vesuvius it has been ascertained to
  be more than 2000° Fahr. At first the molten rock glows with a white
  light, which rapidly reddens, and disappears under the rugged brown
  and black crust that forms on the surface. Underneath this badly
  conducting crust, the lava cools so slowly that columns of steam have
  been noticed rising from its surface more than 80 years after its
  eruption.

  Considerable alteration in the topography of volcanic regions may be
  produced by successive eruptions. The fragmentary materials are
  sometimes discharged in such abundance as to cover the ground for many
  miles around with a deposit of loose ashes, cinders and slag. Such a
  deposit accumulating to a depth of many feet may completely bury
  valleys and water-courses, and thus greatly affect the drainage. The
  coarsest materials accumulate nearest to the vent that emits them. The
  finer dust is not infrequently hurled forth with such an impetus as to
  be carried for thousands of feet into the tracks of upper
  air-currents, whereby it may be borne for hundreds of miles away from
  the vent so as ultimately to fall to the ground in countries far
  removed from any active volcano. Outflows of lava, from their greater
  solidity and durability, produce still more serious and lasting
  changes in the external features of the ground over which they flow.
  As they naturally seek the lowest levels, they find their way into the
  channels of streams. If they keep along the channels, they seal them
  up under a mass of compact stone which the running water, if not
  wholly diverted elsewhere, will take many long centuries to cut
  through. If, on the other hand, the lava crosses a stream, it forms a
  massive dam, above which the water is ponded back so as to form a
  lake.

  As the result of prolonged activity a volcanic cone is gradually built
  up by successive outflows of lava and showers of dust and stones.
  These materials are arranged in beds, or sheets, inclined outwards
  from the central vent. On surrounding level ground the alternating
  beds are flat. In course of time, deep gullies are cut on the outer
  slopes of the cone by rain, and by the heavy showers that arise from
  the condensation of the copious discharges of steam during eruptions.
  Along the sides of these ravines instructive sections may be studied
  of the volcanic strata. The larger rivers of some volcanic regions
  have likewise eroded vast gorges in the more horizontal lavas and
  ashes of the flatter country, and have thus laid bare stupendous
  cliffs, along which the successive volcanic sheets can be seen piled
  above each other for many hundred feet. On a small scale, some of
  these features are well displayed among the rivers that drain the
  volcanic tracts of central France; on a great scale, they are
  presented in the course of the Snake river, and other streams that
  traverse the great volcanic country of western North America. Similar
  volcanic scenery has been produced in western Europe by the action of
  denudation in dissecting the flat Tertiary lavas of Scotland, the
  Faeroe Isles and Iceland.

  Of special interest to the geologist are those volcanoes which have
  taken their rise on the sea-bottom; for the volcanic intercalations
  among the stratified formations of the earth's crust are almost
  entirely of submarine origin. Many active volcanoes situated on
  islands have begun their eruptions below sea-level. Both Vesuvius and
  Etna sprang up on the floor of the Mediterranean sea, and have
  gradually built up their cones into conspicuous parts of the dry land.
  Examples of a similar history are to be found among the volcanic
  islands of the Pacific Ocean. In some of these cases a movement of
  elevation has carried the submarine lavas, tuffs and agglomerates
  above sea-level, and has furnished opportunities of comparing these
  materials with those of recent subaerial origin, and also with the
  ancient records of submarine eruptions which have been preserved among
  the stratified formations. From the evidence thus supplied, it can be
  shown that the materials ejected from modern submarine volcanic vents
  closely resemble those accumulated by subaerial volcanoes; that the
  dust, ashes and stones become intermingled or interstratified with
  coral-mud, or other non-volcanic deposit of the sea-bottom, that
  vesicular lavas may be intercalated among them as on land, and that
  between the successive sheets of volcanic origin, layers of limestone
  may be laid down which are composed chiefly, or wholly, of the remains
  of calcareous marine organisms.

  Though active volcanoes are widely distributed over the globe, and are
  especially abundant around the vast basin of the Pacific Ocean, they
  afford an incomplete picture of the extent to which volcanic action
  has displayed itself on the surface of our planet. When the rocks of
  the land are attentively studied they disclose proofs of that action
  in many districts where there is now no outward sign of it. Not only
  so, but they reveal that volcanoes have been in eruption in some of
  these districts during many different periods of the past, back to the
  beginnings of geological history. The British Islands furnish a
  remarkable example of such a series of ancient eruptions. From the
  Cambrian period all through Palaeozoic times there rose at intervals
  in that country a succession of volcanic centres from some of which
  thousands of feet of lavas and tuffs were discharged. Again in older
  Tertiary times the same region witnessed a stupendous outpouring of
  basalt, the surviving relics of which are more than 3000 ft. thick,
  and cover many hundreds of square miles. Similar evidence is supplied
  in other countries both in the Old and the New world. Hence it is
  proved that, in the geological past, volcanic action has been vigorous
  at long intervals on the same sites during a vast series of ages,
  though no active vents are to be seen there now. The volcanoes now
  active form but a small proportion of the total number which has
  appeared on the surface of the earth.

  With regard to the cause of volcanic action much has been speculated,
  but little can be confidently affirmed. That water in the form of
  occluded gas plays the chief part in forcing the lava column up a
  volcanic chimney, and in the violent explosions that accompany the
  rise of the molten material, is generally admitted. But opinions
  differ as to the source of this water. According to some
  investigators, it should be regarded as in large measure of meteoric
  origin, derived from the descent of rain into the earth, and its
  absorption by the molten magma in the interior. Others, contending
  that the supply so furnished, even if it could reach and be dissolved
  in the magma, would yet be insufficient to furnish the prodigious
  quantity of aqueous vapour discharged during an eruption, maintain
  that the water belongs to the magma itself. They point to the admitted
  fact that many substances, particularly metals in a state of fusion,
  can absorb large quantities of vapours and gases without chemical
  combination, and on cooling discharge them with eruptive phenomena
  somewhat like those of volcanoes. This question must be regarded as
  one of the still unsolved problems of geology.


  (B) _Movements of the Earth's Crust._

  Among the hypogene forces in geological dynamics an important place
  must be assigned to movements of the terrestrial crust. Though the
  expression "the solid earth" has become proverbial, it appears
  singularly inappropriate in the light of the results obtained in
  recent years by the use of delicate instruments of observation. With
  the facilities supplied by these instruments (see SEISMOMETER), it has
  been ascertained that the ground beneath our feet is subject to
  continual slight tremors, and feeble pulsations of longer duration,
  some of which may be due to daily or seasonal variations of
  temperature, atmospheric pressure or other meteorological causes. The
  establishment of self-recording seismometers all over the world has
  led to the detection of many otherwise imperceptible shocks, over and
  above the appreciable earth-waves propagated from earthquake centres
  of disturbance. Moreover, it has been ascertained that some parts of
  the surface of the land are slowly rising, while others are falling
  with reference to the sea-level. From time to time the surface suffers
  calamitous devastation from earthquakes, when portions of the crust
  under great strain suddenly give way. Lastly, at intervals, probably
  separated from each other by vast periods of time, the terrestrial
  crust undergoes intense plication and fracture, and is consequently
  ridged up into mountain-chains. No event of this kind has been
  witnessed since man began to record his experiences. But from the
  structure of mountains, as laid open by prolonged denudation, it is
  possible to form a vivid conception of the nature and effects of these
  most stupendous of all geological revolutions.

  In considering this department of geological inquiry it will be
  convenient to treat it under the following heads: (1) Slow depression
  and upheaval; (2) Earthquakes; (3) Mountain-making; (4) Metamorphism
  of rocks.

  1. _Slow Depression and Upheaval._--On the west side of Japan the land
  is believed to be sinking below the sea, for fields are replaced by
  beaches of sand or shingle, while the depth of the sea off shore has
  perceptibly increased. A subsidence of the south of Sweden has taken
  place in comparatively recent times, for streets and foundations of
  houses at successive levels are found below high-water mark. The west
  coast of Greenland over an extent of more than 600 m. is sinking, and
  old settlements are now submerged. Proofs of submergence of land are
  furnished by "submerged forests," and beds of terrestrial peat now
  lying at various depths below the level of the sea, of which many
  examples have been collected along the shores of the British Isles,
  Holland and France. Interesting evidence that the west of Europe now
  stands at a lower level than it did at a late geological period is
  supplied in the charts of the North Sea and Atlantic, which show that
  the valleys of the land are prolonged under the sea. These valleys
  have been eroded out of the rocks by the streams which flow in them,
  and the depth of their submerged portions below the sea level affords
  an indication of the extent of the subsidence.

  The uprise of land has been detected in various parts of the world.
  One of the most celebrated instances is that of the shores of the Gulf
  of Bothnia, where, at Stockholm, the elevation, between the years 1774
  and 1875, appears to have been 48 centimetres (18½ in.) in a century.
  But on the west side of Sweden, fronting the Skager Rak, the coast,
  between the years 1820 and 1870, rose 30 centimetres, which is at the
  rate of 60 centimetres, or nearly 2 ft. in a century. In the region of
  the Great Lakes in the interior of Canada and the United States it has
  been ascertained that the land is undergoing a slow tilt towards the
  south-west, of which the mean rate appears to be rather less than 6
  in. in a century. If this rate of change should continue the waters of
  Lake Michigan, owing to the progress of the tilt, will, in some 500 or
  600 years, submerge the city of Chicago, and eventually the drainage
  of the lakes will be diverted into the basin of the Mississippi. Proof
  of recent emergence of land is supplied by what are called "raised
  beaches" or "strand-lines," that is, lines of former shores marked by
  sheets of littoral deposits, or platforms cut by shore-waves in rock
  and flanked by old sea-cliffs and lines of sea-worn caves. Admirable
  examples of these features are to be seen along the west coast of
  Europe from the south of England to the north of Norway. These lines
  of old shores become fainter in proportion to their antiquity. In
  Britain they occur at various heights, the platforms at 25, 50 and 100
  ft. being well marked.

  The cause of these slow upward and downward movements of the crust of
  the earth is still imperfectly understood. Upheaval might conceivably
  be produced by an ascent of the internal magma, and the consequent
  expansion of the overlying crust by heat; while depression might
  follow any subsidence of the magma, or its displacement to another
  district. If, as is generally believed, the globe is still
  contracting, the shrinkage of the surface may cause both these
  movements. Subsidence will be in excess, but between subsiding tracts
  lateral thrust may suffice to push upward intervening more solid and
  stable ground; but no solution of the problem yet proposed is wholly
  satisfactory.

  2. _Earthquakes._--As this subject is discussed in a separate article
  it will be sufficient here to take note of its more important
  geological bearings. It was for many centuries taken for granted that
  earthquakes and volcanoes are due to a common cause. We have seen that
  in classical antiquity they were looked on as the results of the
  movements of wind imprisoned within the earth. Long after this notion
  was discarded, and a more scientific appreciation of volcanic action
  was reached, it was still thought that earthquakes should be regarded
  as manifestations of the same source of energy as that which displays
  itself in volcanic eruptions. It is true that earthquakes are frequent
  in districts of active volcanoes, and they may undoubtedly be often
  due there to the explosions of the magma, or to the rupture of rocks
  caused by its ascent towards the surface. But such shocks are
  comparatively local in their range and feeble in their effects. There
  is now a general agreement that between the great world-shaking
  earthquakes and volcanic phenomena, no immediate and intimate
  relationship can be traced, though they may be connected in ways which
  are not yet perceived. Some of the more recent great earthquakes on
  land have proved that the waves of shock are produced by the sudden
  rupture or collapse of rocks under great strain, either along lines of
  previous fracture or of new rents in the terrestrial crust; and that
  such ruptures may occur at a remote distance from any volcano. Thus
  the recent disastrous San Francisco earthquake has been recognized to
  have resulted from a slipping of ground along the line of an old
  fault, which has been traced for a long distance in California
  generally parallel to the coast. The position of this fault at the
  surface has long been clearly followed by its characteristic
  topography. After the earthquake these superficial features were found
  to have been removed by the same cause that had originated them. For
  some 300 m. on the track of this old fault-line a renewed slipping was
  seen to have taken place along one or both sides, and the ground at
  the surface was ruptured as well as displaced horizontally. Obviously,
  the jar occasioned by the sudden and simultaneous subsidence of a
  portion of the earth's crust several hundred miles long, must be far
  more serious than could be produced by an earthquake radiating from a
  single local volcanic focus.

  From their disastrous effects on buildings and human lives, an
  exaggerated importance has been imputed to earthquakes as agents of
  geological change. Experience shows that even after a severe shock
  which may have destroyed numerous towns and villages, together with
  thousands of their inhabitants, the face of the country has suffered
  scarcely any perceptible change, and that, in the course of a year or
  two, when the ruined houses and prostrate trees have been cleared
  away, little or no obvious trace of the catastrophe may remain. Among
  the more enduring records of a great earthquake may be enumerated (a)
  landslips, which lay bare hillsides, and sometimes pond back the
  drainage of valleys so as to give rise to lakes; (b) alterations of
  the topography, as in fissuring of the ground, or in the production of
  inequalities whereby the drainage is affected; new valleys and new
  lakes may thus be formed, while previously existing lakes may be
  emptied; (c) permanent changes of level, either in an upward or
  downward direction.

  3. _Mountain-making._--This subject may be referred to here for the
  striking evidence which it supplies of the importance of movements of
  the earth's crust among geological processes. The structure of a great
  mountain-chain such as the Alps proves that the crust of the earth has
  been intensely plicated, crumpled and fractured. Vast piles of
  sedimentary strata have been folded to such an extent as to occupy now
  only half of their original horizontal extent. This compression in the
  case of the Alps has been computed to amount to as much as 120,000
  metres or 74 English miles, so that two points on the opposite sides
  of that chain have been brought by so much nearer to each other than
  they were originally before the movements. Besides such intense
  plication, extensive rupturing of the crust has taken place in the
  same range of mountains. Not only have the most ancient rocks been
  squeezed up into the central axis of the chain, but huge slices of
  them have been torn away from the main body, and thrust forward for
  many miles, so as now actually to form the summits of mountains, which
  are almost entirely composed of much younger formations. If these
  colossal disturbances occurred rapidly, they would give rise to
  cataclysms of inconceivable magnitude over the surface of the globe.
  No record has been discovered of such accompanying devastation. But
  whether sudden and violent, or prolonged and gradual, such stupendous
  upturnings of the crust did undoubtedly take place, as is clearly
  revealed in innumerable natural sections, which have been laid open by
  the denudation of the crests and sides of the mountains.

  4. _Metamorphism of Rocks_ (see METAMORPHISM).--During the movements
  to which the crust of the earth has been subject, not only have the
  rocks been folded and fractured, but they have likewise, in many
  regions, acquired new internal structures, and have thus undergone a
  process of "regional metamorphism." This rearrangement of their
  substance has been governed by conditions which are probably not yet
  all recognized, but among them we should doubtless include a high
  temperature, intense pressure, mechanical movement resulting in
  crushing, shearing and foliation, and the presence of water in their
  pores. It is among igneous rocks that the progressive stages of
  metamorphism can be most easily traced. Their definite original
  structure and mineral composition afford a starting-point from which
  the investigation may be begun and pursued. Where an igneous rock has
  been invaded by metamorphic changes, it may be observed to have been
  first broken down into separate lenticles, the cores of which may
  still retain, with little or no alteration, the original
  characteristic minerals and crystalline structure of the rock. Between
  these lenticles, the intervening portions have been crushed down into
  a powder or paste, which seems to have been squeezed round and past
  them, and shows a laminated arrangement that resembles the
  flow-structure in lavas. As the degree of metamorphism increases, the
  lenticles diminish in size, and the intervening crushed and foliated
  matrix increases in amount, until at last it may form the entire mass
  of the rock. While the original minerals are thus broken down, new
  varieties make their appearance. Of these, among the earliest to
  present themselves are usually the micas, that impart their
  characteristic silvery sheen to the surfaces of the folia along which
  they spread. Younger felspars, as well as mica, are developed, and
  there arise also sillimanite, garnet, andalusite and many others. The
  texture becomes more coarsely crystalline, and the segregation of the
  constituent minerals more definite along the lines of foliation. From
  the finest silky phyllites a graduation may be traced through
  successively coarser mica-schists, until we reach the almost granitic
  texture of the coarsest gneisses.

  Regional metamorphism has arisen in the heart of mountain-chains, and
  in any other district where the deformation of the crust has been
  sufficiently intense. There is another type of alteration termed
  "contact-metamorphism," which is developed around masses of igneous
  rock, especially where these have been intruded in large bosses among
  stratified formations. It is particularly displayed around masses of
  granite, where sandstones are found altered into quartzite, shales and
  grits into schistose compounds, and where sometimes fossils are still
  recognizable among the metamorphic minerals.


_DIVISION II.--EPIGENE OR SUPERFICIAL ACTION_

It is on the surface of the globe, and by the operation of agents
working there, that at present the chief amount of visible geological
change is effected. In considering this branch of inquiry, we are not
involved in a preliminary difficulty regarding the very nature of the
agencies as is the case in the investigation of plutonic action. On the
contrary, the surface agents are carrying on their work under our very
eyes. We can watch it in all its stages, measure its progress, and mark
in many ways how accurately it represents similar changes which, for
long ages previously, must have been effected by the same means. But in
the systematic treatment of this subject we encounter a difficulty of
another kind. We discover that while the operations to be discussed are
numerous and readily observable, they are so interwoven into one great
network that any separation of them under different subdivisions is sure
to be more or less artificial and to convey an erroneous impression.
While, therefore, under the unavoidable necessity of making use of such
a classification of subjects, we must always bear in mind that it is
employed merely for convenience, and that in nature superficial
geological action must be continually viewed as a whole, since the work
of each agent has constant reference to that of the others, and is not
properly intelligible unless that connexion be kept in view.

The movements of the air; the evaporation from land and sea; the fall of
rain, hail and snow; the flow of rivers and glaciers; the tides,
currents and waves of the ocean; the growth and decay of organized
existence, alike on land and in the depths of the sea;--in short, the
whole circle of movement, which is continually in progress upon the
surface of our planet, are the subjects now to be examined. It is
desirable to adopt some general term to embrace the whole of this range
of inquiry. For this end the word epigene (Gr. [Greek: epi], upon) has
been suggested as a convenient term, and antithetical to hypogene (Gr.
[Greek: hypo], under), or subterranean action.

A simple arrangement of this part of Geological Dynamics is in three
sections:

A. _Air._--The influence of the atmosphere in destroying and forming
rocks.

B. _Water._--The geological functions of the circulation of water
through the air and between sea and land, and the action of the sea.

C. _Life._--The part taken by plants and animals in preserving,
destroying or reproducing geological formations.

The words destructive, reproductive and conservative, employed in
describing the operations of the epigene agents, do not necessarily
imply that anything useful to man is destroyed, reproduced or preserved.
On the contrary, the destructive action of the atmosphere may turn
barren rock into rich soil, while its reproductive effects sometimes
turn rich land into barren desert. Again, the conservative influence of
vegetation has sometimes for centuries retained as barren morass what
might otherwise have become rich meadow or luxuriant woodland. The
terms, therefore, are used in a strictly geological sense, to denote the
removal and re-deposition of material, and its agency in preserving what
lies beneath it.


  (A) _The Air._

  As a geological agent, the air brings about changes partly by its
  component gases and partly by its movements. Its destructive action is
  both chemical and mechanical. The chemical changes are probably
  mainly, if not entirely, due to the moisture of the air, and
  particularly to the gases, vapours and organic matter which the
  moisture contains. Dry air seems to have little or no appreciable
  influence in promoting these reactions. As the changes in question are
  similar to those much more abundantly brought about by rain they are
  described in the following section under the division on rain.

  Among the more recognizable mechanical changes effected in the
  atmosphere, one of considerable importance is to be seen in the result
  of great and rapid changes of temperature. Heat expands rocks, while
  cold contracts them. In countries with a great annual range of
  temperature, considerable difficulty is sometimes experienced in
  selecting building materials liable to be little affected by the
  alternate expansion and contraction, which prevents the joints of
  masonry from remaining close and tight. In dry tropical climates,
  where the days are intensely hot and the nights extremely cold, the
  rapid nocturnal contraction produces a strain so great as to rival
  frost in its influence upon the surface of exposed rocks,
  disintegrating them into sand, or causing them to crack or peel off in
  skins or irregular pieces. Dr Livingstone found in Africa (12° S.
  lat., 34° E. long.) that surfaces of rock which during the day were
  heated up to 137° Fahr., cooled so rapidly by radiation at night that,
  unable to sustain the strain of contraction, they split and threw off
  sharp angular fragments from a few ounces to 100 or 200 [lb] in
  weight. In temperate regions this action, though much less pronounced,
  still makes itself felt. In these climates, however, and still more in
  high latitudes, somewhat similar results are brought about by frost.

  By its motion in wind the air drives loose sand over rocks, and in
  course of time abrades and smoothes them. "Desert polish" is the name
  given to the characteristic lustrous surface thus imparted. Holes are
  said to be drilled in window glass at Cape Cod by the same agency.
  Cavities are now and then hollowed out of rocks by the gyration in
  them of little fragments of stone or grains of sand kept in motion by
  the wind. Hurricanes form important geological agents upon land in
  uprooting trees, and thus sometimes impeding the drainage of a country
  and giving rise to the formation of peat mosses.

  The reproductive action of the air arises partly from the effect of
  the chemical and mechanical disintegration involved in the process of
  "weathering," and partly from the transporting power of wind and of
  aerial currents. The layer of soil, which covers so much of the
  surface of the land, is the result of the decay of the underlying
  rocks, mingled with mineral matter blown over the ground by wind, or
  washed thither by rain, and with the mouldering remains of plants and
  animals. The extent to which fine dust may be transported over the
  surface of the land can hardly be realized in countries clothed with a
  covering of vegetation, though even there, in dry weather during
  spring, clouds of dust may often be seen blown away by wind from bare
  ploughed fields. Intercepted by the leaves of plants and washed down
  to their roots by rain, this dust goes to increase the soil below. In
  arid climates, where dust clouds are dense and frequent, enormous
  quantities of fine mineral particles are thus borne along and
  accumulated. The remarkable deposit of "Loess," which is sometimes
  more than 1500 ft. thick and covers extensive areas in China and other
  countries, is regarded as due to the drifting of dust by wind. Again
  the dunes of sand so abundant along the inner side of sandy
  sea-beaches in many different parts of the world are attributable to
  the same action.


  (B) _Water._

  In treating of the epigene action of water in geological processes it
  will be convenient to deal first with its operations in traversing the
  land, and then with those which it performs in the sea. The
  circulation of water from land to sea and again from sea to land
  constitutes the fundamental cause of most of the daily changes by
  which the surface of the land is affected.

  1. _Rain._--Rain effects two kinds of changes upon the surface of the
  land. It acts _chemically_ upon soils and stones, and sinking under
  ground continues a great series of similar reactions there. It acts
  _mechanically_, by washing away loose materials, and thus powerfully
  affecting the contours of the land. Its chemical action depends mainly
  upon the nature and proportion of the substances which, in descending
  to the earth, it abstracts from the atmosphere. Rain always absorbs a
  little air, which, in addition to its nitrogen and oxygen, contains
  carbonic acid, and in minute proportions, sodium chloride, sulphuric
  acid and other ingredients, especially inorganic dust, organic
  particles and living germs. Probably the most generally efficient of
  these constituents are oxygen, carbonic acid and organic matter. Armed
  with these reagents, rain effects a chemical decomposition of the
  rocks on which it falls, and through which it sinks underground. The
  principal changes thus produced are as follows: (a) Oxidation.--Owing
  to the prominence of oxygen in rain-water, and its readiness to unite
  with any substance which can contain more of it, a thin oxidized
  pellicle is formed on the surface of many rocks on which rain falls,
  and this oxidized layer if not at once washed off, sinks deeper until
  a crust is formed over the stone. A familiar illustration of this
  action is afforded by the rust, or oxide, which forms on iron when
  exposed to moisture, though this iron may be kept long bright if
  allowed to remain screened from moist air and rain. (b)
  Deoxidation.--Organic matter having an affinity for more oxygen
  decomposes peroxides by depriving them of some part of their share of
  that element and reducing them to protoxides. These changes are
  especially noticeable among the iron oxides so abundantly diffused
  among rocks. Hence rain-water, in sinking through soil and obtaining
  such organic matter, becomes thereby a reducing agent. (c)
  Solution.--This may take place either by the simple action of the
  water, as in the solution of rock-salt, or by the influence of the
  carbonic acid present in the rain. (d) Formation of Carbonates.--A
  familiar example of the action of carbonic acid in rain is to be seen
  in the corrosion of exposed marble slabs. The carbonic acid dissolves
  some of the lime, which, as a bicarbonate, is held in solution in the
  carbonated water, but is deposited again when the water loses its
  carbonic acid or evaporates. It is not merely carbonates, however,
  which are liable to this kind of destruction. Even silicates of lime,
  potash and soda, combinations existing abundantly as constituents of
  rocks, are attacked; their silica is liberated, and their alkalis or
  alkaline earths, becoming carbonates, are removed in solution. (e)
  Hydration.--Some minerals, containing little or no water, and
  therefore called anhydrous, when exposed to the action of the
  atmosphere, absorb water, or become hydrous, and are then usually more
  prone to further change. Hence the rocks of which they form part
  become disintegrated.

  Besides the reactions here enumerated, a considerable amount of decay
  may be observed as the result of the presence of sulphuric and nitric
  acid in the air, especially in that of large towns and manufacturing
  districts, where much coal is consumed. Metallic surfaces, as well as
  various kinds of stone, are there corroded, while the mortar of walls
  may often be observed to be slowly swelling out and dropping off,
  owing to the conversion of the lime into sulphate. Great injury is
  likewise done from a similar cause to marble monuments in exposed
  graveyards.

  The general result of the disintegrating action of the air and of
  rain, including also that of plants and animals, to be noticed in the
  sequel, is denoted by the term "weathering." The amount of decay
  depends partly on conditions of climate, especially the range of
  temperature, the abundance of moisture, height above the sea and
  exposure to prevalent winds. Many rocks liable to be saturated with
  rain and rapidly dried under a warm sun are apt to disintegrate at the
  surface with comparative rapidity. The nature and progress of the
  weathering are mainly governed by the composition and texture of the
  rocks exposed to it. Rocks composed of particles liable to little
  chemical change from the influence of moisture are best fitted to
  resist weathering, provided they possess sufficient cohesion to
  withstand the mechanical processes of disintegration. Siliceous
  sandstones are excellent examples of this permanence. Consisting
  wholly or mainly of the durable mineral quartz, they are sometimes
  able so to withstand decay that buildings made of them still retain,
  after the lapse of centuries, the chisel-marks of the builders. Some
  rocks, which yield with comparative rapidity to the chemical attacks
  of moisture, may show little or no mark of disintegration on their
  surface. This is particularly the case with certain calcareous rocks.
  Limestone when pure is wholly soluble in acidulated water. Rain
  falling on such a rock removes some of it in solution, and will
  continue to do so until the whole is dissolved away. But where a
  limestone is full of impurities, a weathered crust of more or less
  insoluble particles remains after the solution of the calcareous part
  of the stone. Hence the relative purity of limestones may be roughly
  determined by examining their weathered surfaces, where, if they
  contain much sand, the grains will be seen projecting from the
  calcareous matrix, and where, should the rock be very ferruginous, the
  yellow hydrous peroxide, or ochre, will be found as a powdery crust.
  In limestones containing abundant encrinites, shells, or other organic
  remains, the weathered surface commonly presents the fossils standing
  out in relief. The crystalline arrangement of the lime in the organic
  structures enables them to resist disintegration better than the
  general mechanically aggregated matrix of the rock. An experienced
  fossil collector will always search well such weathered surfaces, for
  he often finds there, delicately picked out by the weather, minute
  and frail fossils which are wholly invisible on a freshly broken
  surface of the stone. Many rocks weather with a thick crust, or even
  decay inwards for many feet or yards. Basalt, for example, often shows
  a yellowish-brown ferruginous layer on its surface, formed by the
  conversion of its felspar into kaolin, and the removal of its calcium
  silicate as carbonate, by the hydration of its olivine and augite and
  their conversion into serpentine, or some other hydrous magnesian
  silicate, and by the conversion of its magnetite into limonite.
  Granite sometimes shows in a most remarkable way the distance to which
  weathering can reach. It may occasionally be dug into for a depth of
  20 or 30 ft., the quartz crystals and veins retaining their original
  positions, while the felspar is completely kaolinized. It is to the
  endlessly varied effects of weathering that the abundant fantastic
  shapes assumed by crags and other rocky masses are due. Most varieties
  of rock have their own characteristic modes of weathering, whereby
  they may be recognized even from a distance. To some of these features
  reference will be made in Part VIII.

  The mechanical action of rain, which is intimately bound up with its
  chemical action, consists in washing off the fine superficial
  particles of rocks which have been corroded and loosened by the
  process of weathering, and in thus laying open fresh portions to the
  same influences of decay. The detritus so removed is partly carried
  down into the soil which is thereby enriched, partly held in
  suspension in the little runnels into which the rain-drops gather as
  they begin to flow over the land, partly pushed downwards along the
  surface of sloping ground. A good deal of it finds its way into the
  nearest brooks and rivers, which are consequently made muddy by heavy
  rain.

  It is natural that a casual consideration of the subject should lead
  to an impression that, though the general result of the fall of rain
  upon a land-surface must lead to some amount of disintegration and
  lowering of that surface, the process must be so slow and slight as
  hardly to be considered of much importance among geological
  operations. But further attention will show such an impression to be
  singularly erroneous. It loses sight of the fact that a change which
  may be hardly appreciable within a human lifetime, or even within the
  comparatively brief span of geological time embraced in the compass of
  human history, may nevertheless become gigantic in its results in the
  course of immensely protracted periods. An instructive lesson in the
  erosive action of rain may be found in the pitted and channelled
  surface of ground lying under the drip of the eaves of a cottage. The
  fragments of stone and pebbles of gravel that form part of the soil
  can there be seen sticking out of the ground, because being hard they
  resist the impetus of the falling drops, protecting for a time the
  earth beneath them, while that which surrounded and covered them is
  washed away. From this familiar illustration the observer may advance
  through every stage in the disappearance of material which once
  covered the surface, until he comes to examples where once continuous
  and thick sheets of solid rock have been reduced to a few fragments or
  have been entirely removed. Since the whole land surface over which
  rain falls is exposed to this waste, the superficial covering of
  decayed rock or soil, as Hutton insisted, is constantly, though
  imperceptibly, travelling outward and downward to the sea. In this
  process of transport rain is an important carrying agent, while at the
  same time it serves to connect the work of the other disintegrating
  forces, and to make it conducive to the general degradation of the
  land. Though this decay is general and constant, it is obviously not
  uniform. In some places where, from the nature of the rock, from the
  flatness of the ground, or from other causes, rain works under great
  difficulties, the rate of waste may be extremely slow. In other places
  it may be rapid enough to be appreciable from year to year. A survey
  of this department of geological activity shows how unequal wasting by
  rain, combined with the operations of brooks and rivers, has produced
  the details of the present relief of the land, those tracts where the
  destruction has been greatest forming hollows and valleys, others,
  where it has been less, rising into ridges and hills (Part VIII.).

  Rain-action is not merely destructive, but is accompanied with
  reproductive effects, chief of which is the formation of soil. In
  favourable situations it has gathered together accumulations of loam
  and earth from neighbouring higher ground, such as the "brick-earth,"
  "head," and "rain-wash" of the south of England--earthy deposits,
  sometimes full of angular stones, derived from the subaerial waste of
  the rocks of the neighbourhood.

  2. _Underground Water._--Of the rain which falls upon the land one
  portion flows off into brooks and rivers by which the water is
  conducted back to the ocean; the larger part, however, sinks into the
  ground and disappears. It is this latter part which has now to be
  considered. Over and above the proportion of the rainfall which is
  absorbed by living vegetation and by the soil, there is a continual
  filtering down of the water from the surface into the rocks that lie
  below, where it partly lodges in pores and interstices, and partly
  finds its way into subterranean joints and fissures, in which it
  performs an underground circulation, and ultimately issues once more
  at the surface in the form of springs (q.v.). In the course of this
  circulation the water performs an important geological task. Not only
  carrying down with it the substances which the rain has abstracted
  from the air, but obtaining more acids and organic matter from the
  soil, it is enabled to effect chemical changes in the rocks
  underneath, and especially to dissolve limestone and other calcareous
  formations. So considerable is the extent of this solution in some
  places that the springs which come to the surface, and begin there to
  evaporate and lose some of their carbonic acid, contain more dissolved
  lime than they can hold. They consequently deposit it in the form of
  calcareous tuff or sinter (q.v.). Other subterranean waters issue with
  a large proportion of iron-salts in solution which form deposits of
  ochre. The various mineral springs so largely made use of for the
  mitigation or cure of diseases owe their properties to the various
  salts which they have dissolved out of rocks underground. As the
  result of prolonged subterranean solution in limestone districts,
  passages and caves (q.v.), sometimes of great width and length, are
  formed. When these lie near the surface their roofs sometimes fall in
  and engulf brooks and rivers, which then flow for some way underground
  until the tunnels conduct them back again to daylight on some lower
  ground.

  Besides its chemical activity water exerts among subterranean rocks a
  mechanical influence which leads to important changes in the
  topography of the surface. In removing the mineral matter, either in
  solution or as fine sediment, it sometimes loosens the support of
  overlying masses of rock which may ultimately give way on sloping
  ground, and rush down the declivities in the form of landslips. These
  destructive effects are specially frequent on the sides of valleys in
  mountainous countries and on lines of sea-cliff.

  3. _Brooks and Rivers._--As geological agents the running waters on
  the face of the land play an important part in epigene changes. Like
  rain and springs they have both a chemical and a mechanical action.
  The latter receives most attention, as it undoubtedly is the more
  important; but the former ought not to be omitted in any survey of the
  general waste of the earth's surface. The water of rivers must possess
  the powers of a chemical solvent like rain and springs, though its
  actual work in this respect can be less easily measured, seeing that
  river water is directly derived from rain and springs, and necessarily
  contains in solution mineral substances supplied to it by them and not
  by its own operation. Nevertheless, it is sometimes easy to prove that
  streams dissolve chemically the rocks of their channels. Thus, in
  limestone districts the base of the cliffs of river ravines may be
  found eaten away into tunnels, arches, and overhanging projections,
  presenting in their smooth surfaces a great contrast to the angular
  jointed faces of the same rock, where now exposed to the influence
  only of the weather on the higher parts of the cliff.

  The mechanical action of rivers consists (a) in transporting mud,
  sand, gravel and blocks of stone from higher to lower levels; (b) in
  using these loose materials to widen and deepen their channels by
  erosion; (c) in depositing their load of detritus wherever possible
  and thus to make new geological formations.

  (a) _Transporting Power._--River-water is distinguished from that of
  springs by being less transparent, because it contains more or less
  mineral matter in suspension, derived mainly from what is washed down
  by rain, or carried in by brooks, but partly also from the abrasion of
  the water-channels by the erosive action of the rivers themselves. The
  progress of this burden of detritus may be instructively followed from
  the mountain-tributaries of a river down to the mouth of the main
  stream. In the high grounds the water-courses may be observed to be
  choked with large fragments of rock disengaged from the cliffs and
  crags on either side. Traced downwards the blocks are seen to become
  gradually smaller and more rounded. They are ground against each
  other, and upon the rocky sides and bottom of the channel, getting
  more and more reduced as they descend, and at the same time abrading
  the rocks over or against which they are driven. Hence a great deal of
  débris is produced, and is swept along by the onward and downward
  movement of the water. The finer portions, such as mud and fine sand,
  are carried in suspension, and impart the characteristic turbidity to
  river-water; the coarser sand and gravel are driven along the
  river-bottom. The proportion of suspended mineral matter has been
  ascertained with more or less precision for a number of rivers. As an
  illustrative example of a river draining a vast area with different
  climates, forms of surface and geological structure the Mississippi
  may be cited. The average proportion of sediment in its water was
  ascertained by Humphreys and Abbot to be 1/1500 by weight or 1/2900 by
  volume. These engineers found that, in addition to this suspended
  material, coarse detritus is constantly being pushed forward along the
  bed of the river into the Gulf of Mexico, to an amount which they
  estimated at about 750,000,000 cubic ft. of sand, earth and gravel;
  they concluded that the Mississippi carries into the gulf every year
  an amount of mechanically transported sediment sufficient to make a
  prism one square mile in area and 268 ft. in height.

  (b) _Excavating Power._--It is by means of the sand, gravel and stones
  which they drive against the sides and bottoms of their channels that
  streams have hollowed out the beds in which they flow. Not only is the
  coarse detritus reduced in size by the friction of the stones against
  each other, but, at the same time, these materials abrade the rocks
  against which they are driven by the current. Where, owing to the
  shape of the bottom of the channel, the stones are caught in eddies,
  and are kept whirling round there, they become more and more worn down
  themselves, and at the same time scour out basin-shaped cavities, or
  "pot-holes," in the solid rock below. The uneven bed of a swiftly
  flowing stream may in this way be honeycombed with such eroded basins
  which coalesce and thus appreciably lower the surface of the bed. The
  steeper the channel, other conditions being equal, the more rapid will
  be the erosion. Geological structure also affects the character and
  rate of the excavation. Where the rocks are so arranged as to favour
  the formation and persistence of a waterfall, a long chasm may be
  hollowed out like that of the Niagara below the falls, where a hard
  thick bed of nearly flat limestone lies on softer and more easily
  eroded shales. The latter are scooped out from underneath the
  limestone, which from time to time breaks off in large masses and the
  waterfall gradually retreats up stream, while the ravine is
  proportionately lengthened. To the excavating power of rivers the
  origin of the valley systems of the dry land must be mainly assigned
  (see Part VIII.).

  (c) _Reproductive Power._--So long as a stream flows over a steep
  declivity its velocity suffices to keep the sediment in suspension,
  but when from any cause, such as a diminution of slope, the velocity
  is checked, the transporting power is lessened and the sediment begins
  to fall to the bottom and to remain there. Hence various river-formed
  or "alluvial" deposits are laid down. These sometimes cover
  considerable spaces at the foot of mountains. The floors of valleys
  are strewn with detritus, and their level may thereby be sensibly
  raised. In floods the ground inundated on either side of a stream
  intercepts some part of the detritus, which is then spread over the
  flood-plain and gradually heightens it. At the same time the stream
  continues to erode the channel, and ultimately is unable to reach the
  old flood-plain. It consequently forms a new plain at a lower level,
  and thus, by degrees, it comes to be flanked on either side by a
  series of successive terraces or platforms, each of which marks one of
  its former levels. Where a river enters a large body of water its
  current is checked. Some of its sediment is consequently dropped, and
  by slow accumulation forms a delta (q.v.). On land, every lake in
  mountain districts furnishes instances of this kind of alluvium. But
  the most important deltas are those formed in the sea at the mouths of
  the larger rivers of the globe. Off many coast-lines the detritus
  washed from the land gathers into bars, which enclose long strips of
  water more or less completely separated from the sea outside and known
  as lagoons. A chain of such lagoon-barriers stretches for hundreds of
  miles round the Gulf of Mexico and the eastern shores of the United
  States.

  4. _Lakes._--These sheets of water, considered as a whole, do not
  belong to the normal system of drainage on the land whereby valleys
  are excavated. On the contrary they are exceptional to it; for the
  constant tendency of running water is to fill them up, or to drain
  them by wearing down the barriers that contain them at their outflow.
  Some of them are referable to movements of the terrestrial crust
  whereby depressions arise on the surface of the land, as has been
  noted after earthquakes. Others have arisen from solution such as that
  of rock-salt or of limestone, the removal of which by underground
  water causes a subsidence of the ground above. A third type of
  lake-basin occurs in regions that are now or have once been subject to
  the erosive action of glaciers (see under next subdivision,
  _Terrestrial Ice_). Many small lakes or tarns have been caused by the
  deposit of débris across a valley as by landslips or moraines.
  Considered from a geological point of view, lakes perform an important
  function in regulating the drainage of the ground below their outfall
  and diminishing the destructive effects of floods, in filtering the
  water received from their affluent streams, and in providing
  undisturbed areas of deposit in which thick and extensive lacustrine
  formations may be accumulated. In the inland basins of some dry
  climates the lakes are salt, owing to excess of evaporation, and their
  bottoms become the sites of chemical deposits, particularly of
  chlorides of sodium and magnesium, and calcium sulphate and carbonate.

  5. _Terrestrial Ice._--Each of the forms assumed by frozen water has
  its own characteristic action in geological processes. Frost has a
  powerful influence in breaking up damp soils and surfaces of stone in
  the pores or cracks of which moisture has lodged. The water in
  freezing expands, and in so doing pushes asunder the component
  particles of soil or stone, or widens the space between the walls of
  joints or crevices. When the ice melts the loosened grains remain
  apart ready to be washed away by rain or blown off by wind, while by
  the widening of joints large blocks of rock are detached from the
  faces of cliffs. Where rivers or lakes are frozen over the ice exerts
  a marked pressure on their banks; and when it breaks up large sheets
  of it are driven ashore, pushing up quantities of gravel and stones
  above the level of the water. The piling up of the disrupted ice
  against obstructions in rivers ponds back the water, and often leads
  to destructive floods when the ice barriers break. Where the ice has
  formed round boulders in shallow water, or at the bottom
  ("anchor-ice"), it may lift these up when the frost gives way, and may
  transport them for some distance. Ice formed in the atmosphere, and
  descending to the ground in the form of hail, often causes great
  destruction to vegetation and not infrequently to animal life. Where
  the frozen moisture reaches the earth as snow, it serves to protect
  rock, soil and vegetation from the effects of frost; but on sloping
  ground it is apt to give rise to destructive avalanches or landslips,
  while indirectly, by its rapid melting, it may cause serious floods in
  rivers.

  But the most striking geological work performed by terrestrial ice is
  that achieved by glaciers (q.v.) and ice-sheets. These vast masses of
  moving ice, when they descend from mountains where the steeper rocks
  are clear of snow, receive on their surface the débris detached by
  frost from the declivities above, and bear these materials to lower
  levels or to the sea. Enormous quantities of rock-rubbish are thus
  transported in the Alps and other high mountain ranges. When the ice
  retreats the boulders carried by it are dropped where it melts, and
  left there as memorials of the former extension of the glaciers.
  Evidence of this nature proves the much wider extent of the Alpine ice
  at a comparatively recent geological date. It can also be shown that
  detritus from Scandinavia has been ice-borne to the south-east of
  England and far into the heart of Europe.

  The ice, by means of grains of sand and pieces of stone which it drags
  along, scores, scratches and polishes the surfaces of rock underneath
  it, and, in this way, produces the abundant fine sediment that gives
  the characteristic milky appearance to the rivers that issue from the
  lower ends of glaciers. By such long-continued attrition the rocks are
  worn down, portions of them of softer nature, or where the ice acts
  with especial vigour, are hollowed out into cavities which, on the
  disappearance of the ice, may be filled with water and become tarns or
  lakes. Rocks over which land-ice has passed are marked by a peculiar
  smooth, flowing outline, which forms a contrast to the more rugged
  surface produced by ordinary weathering. They are covered with
  groovings, which range from the finest striae left by sharp grains of
  sand to deep ruts ground out by blocks of stone. The trend of these
  markings shows the direction in which the ice flowed. By their
  evidence the position and movement of former glaciers in countries
  from which the ice has entirely vanished may be clearly determined
  (see GLACIAL PERIOD).

  6. _The Sea._--The physical features of the sea are discussed in
  separate articles (see OCEAN AND OCEANOGRAPHY). The sea must be
  regarded as the great regulator of temperature and climate over the
  globe, and as thus exerting a profound influence on the distribution
  of plant and animal life. Its distinctly geological work is partly
  erosive and partly reproductive. As an eroding agent it must to some
  extent effect chemical decompositions in the rocks and sediments over
  which it spreads; but these changes have not yet been satisfactorily
  studied. Undoubtedly, its chief destructive power is of a mechanical
  kind, and arises from the action of its waves in beating upon
  shore-cliffs. By the alternate compression and expansion of the air in
  crevices of the rocks on which heavy breakers fall, and by the
  hydraulic pressure which these masses of sea-water exert on the walls
  of the fissures into which they rush, large masses of rock are
  loosened and detached, and caves and tunnels are drilled along the
  base of sea-cliffs. Probably still more efficacious are the blows of
  the loose shingle, which, caught up and hurled forward by the waves,
  falls with great force upon the shore rocks, battering them as with a
  kind of artillery until they are worn away. The smooth surfaces of the
  rocks within reach of the waves contrasted with their angular forms
  above that limit bear witness to the amount of waste, while the
  rounded forms of the boulders and shingle show that they too are being
  continually reduced in size. Thus the sea, by its action on the
  coasts, produces much sediment, which is swept away by its waves and
  currents and strewn over its floor. Besides this material, it is
  constantly receiving the fine silt and sand carried down by rivers. As
  the floor of the ocean is thus the final receptacle for the waste of
  the land, it becomes the chief era on the surface of the globe for the
  accumulation of new stratified formations. And such has been one of
  its great functions since the beginning of geological time, as is
  proved by the rocks that form the visible part of the earth's crust,
  and consist in great part of marine deposits. Chemical precipitates
  take place more especially in enclosed parts of the sea, where
  concentration of the water by evaporation can take place, and where
  layers of sodium chloride, calcium sulphate and carbonate, and other
  salts are laid down. But the chief marine accumulations are of
  detrital origin. Near the land and for a variable distance extending
  sometimes to 200 or 300 m. from shore the deposits consist chiefly of
  sediments derived from the waste of the land, the finer silts being
  transported farthest from their source. At greater depths and
  distances the ocean floor receives a slow deposit of exceedingly fine
  clay, which is believed to be derived from the decomposition of pumice
  and volcanic dust from insular or submarine volcanoes. Wide tracts of
  the bottom are covered with various forms of ooze derived from the
  accumulation of the remains of minute organisms.


  (C) _Life._

  Among the agents by which geological changes are carried on upon the
  surface of the globe living organisms must be enumerated. Both plants
  and animals co-operate with the inorganic agents in promoting the
  degradation of the land. In some cases, on the other hand, they
  protect rocks from decay, while, by the accumulation of their remains,
  they give rise to extensive formations both upon the land and in the
  sea. Their operations may hence be described as alike destructive,
  conservative and reproductive. Under this heading also the influence
  of Man as a geological agent deserves notice.

  (a) _Plants._--Vegetation promotes the disintegration of rocks and
  soil in the following ways: (1) By keeping the surfaces of stone
  moist, and thus promoting both mechanical and chemical dissolution, as
  is especially shown by liverworts, mosses and other moisture-loving
  plants. (2) By producing through their decay carbonic and other
  acids, which, together with decaying organic matter taken up by
  passing moisture, become potent in effecting the chemical
  decomposition of rocks and in promoting the disintegration of soils.
  (3) By inserting their roots or branches between joints of rock, which
  are thereby loosened, so that large slices may be eventually wedged
  off. (4) By attracting rain, as thick woods, forests and peat-mosses
  do, and thus accelerating the general waste of a country by running
  water. (5) By promoting the decay of diseased and dead plants and
  animals, as when fungi overspread a damp rotting tree or the carcase
  of a dead animal.

  That plants also exert a conservative influence on the surface of the
  land is shown in various ways. (1) The formation of a stratum of turf
  protects the soil and rocks underneath from being rapidly
  disintegrated and washed away by atmospheric action. (2) Many plants,
  even without forming a layer of turf, serve by their roots or branches
  to protect the loose sand or soil on which they grow from being
  removed by wind. The common sand-carex and other arenaceous plants
  bind the loose sand-dunes of our coasts, and give them a permanence,
  which would at once be destroyed were the sand laid bare again to
  storms. The growth of shrubs and brushwood along the course of a
  stream not only keeps the alluvial banks from being so easily
  undermined and removed as would otherwise be the case, but serves to
  arrest the sediment in floods, filtering the water and thereby adding
  to the height of the flood plain. (3) Some marine plants, like the
  calcareous nullipores, afford protection to shore rocks by covering
  them with a hard incrustation. The tangles and smaller Fuci which grow
  abundantly on the littoral zone break the force of the waves or
  diminish the effects of ground swell. (4) Forests and brushwood
  protect the soil, especially on slopes, from being washed away by rain
  or ploughed up by avalanches.

  Plants contribute by the aggregation of their remains to the formation
  of stratified deposits. Some marine algae which secrete carbonate of
  lime not only encrust rocks but give rise to sheets of submarine
  limestone. An analogous part is played in fresh-water lakes by various
  lime-secreting plants, such as _Chara_. Long-continued growth of
  vegetation has, in some regions, produced thick accumulations of a
  dark loam, as in the black cotton soil (_regur_) of India, and the
  black earth (_tchernozom_) of Russia. Peat-mosses are formed in
  temperate and arctic climates by the growth of marsh-loving plants,
  sometimes to a thickness of 40 or 50 ft. In tropical regions the
  mangrove swamps on low moist shores form a dense jungle, sometimes 20
  m. broad, which protects these shores from the sea until, by the
  arrest of sediment and the constant contribution of decayed
  vegetation, the spongy ground is at last turned into firm soil. Some
  plants (diatoms) can abstract silica and build it into their
  framework, so that their remains form a siliceous deposit or ooze
  which covers spaces of the deep sea-floor estimated at more than ten
  millions of square miles in extent.

  (b) _Animals._--These exert a destructive influence in the following
  ways: (1) By seriously affecting the composition and arrangement of
  the vegetable soil. Worms bring up the lower portions of the soil to
  the surface, and while thus promoting its fertility increase its
  liability to be washed away by rain. Burrowing animals, by throwing up
  the soil and subsoil, expose these to be dried and blown away by the
  wind. At the same time their subterranean passages serve to drain off
  the superficial water and to injure the stability of the surface of
  the ground above them. In Britain the mole and rabbit are familiar
  examples. (2) By interfering with or even diverting the flow of
  streams. Thus beaver-dams check the current of water-courses,
  intercept floating materials, and sometimes turn streams into new
  channels. The embankments of the Mississippi are sometimes weakened to
  such an extent by the burrowings of the cray-fish as to give way and
  allow the river to inundate the surrounding country. Similar results
  have happened in Europe from subterranean operations of rats. (3) Some
  mollusca bore into stone or wood and by the number of contiguous
  perforations greatly weaken the material. (4) Many animals exercise a
  ruinously destructive influence upon vegetation. Of the numerous
  plagues of this kind the locust, phylloxera and Colorado beetle may be
  cited.

  The most important geological function performed by animals is the
  formation of new deposits out of their remains. It is chiefly by the
  lower grades of the animal kingdom that this work is accomplished,
  especially by molluscs, corals and foraminifera. Shell-banks are
  formed abundantly in such comparatively shallow and enclosed basins as
  that of the North Sea, and on a much more extensive scale on the floor
  of the West Indian seas. By the coral polyps thick masses of
  limestones have been built up in the warmer seas of the globe (see
  CORAL REEFS). The floor of the Atlantic and other oceans is covered
  with a fine calcareous ooze derived mainly from the remains of
  foraminifera, while in other regions the bottom shows a siliceous ooze
  formed almost entirely of radiolaria. Vertebrate animals give rise to
  phosphatic deposits formed sometimes of their excrement, as in guano
  and coprolites, sometimes of an accumulation of their bones.

  (c) _Man._--No survey of the geological workings of plant and animal
  life upon the surface of the globe can be complete which does not take
  account of the influence of man--an influence of enormous and
  increasing consequence in physical geography, for man has introduced,
  as it were, an element of antagonism to nature. His interference shows
  itself in his relations to climate, where he has affected the
  meteorological conditions of different countries: (1) By removing
  forests, and laying bare to the sun and winds areas which were
  previously kept cool and damp under trees, or which, lying on the lee
  side, were protected from tempests. It is supposed that the wholesale
  destruction of the woodlands formerly existing in countries bordering
  the Mediterranean has been in part the cause of the present
  desiccation of these districts. (2) By drainage, whereby the
  discharged rainfall is rapidly removed, and the evaporation is
  lessened, with a consequent diminution of rainfall and some increase
  in the general temperature of a country. (3) By the other processes of
  agriculture, such as the transformation of moor and bog into
  cultivated land, and the clothing of bare hillsides with green crops
  or plantations of coniferous and hardwood trees.

  Still more obvious are the results of human interference with the flow
  of water: (1) By increasing or diminishing the rainfall man directly
  affects the volume of rivers. (2) By his drainage operations he makes
  the rain to run off more rapidly than before, and thereby increases
  the magnitude of floods and of the destruction caused by them. (3) By
  wells, bores, mines, or other subterranean works he interferes with
  the underground waters, and consequently with the discharge of
  springs. (4) By embanking rivers he confines them to narrow channels,
  sometimes increasing their scour, and enabling them to carry their
  sediment further seaward, sometimes causing them to deposit it over
  the plains and raise their level. (5) By his engineering operations
  for water-supply he abstracts water from its natural basins and
  depletes the streams.

  In many ways man alters the aspect of a country: (1) By changing
  forest into bare mountain, or clothing bare mountains with forest. (2)
  By promoting the growth or causing the removal of peat-mosses. (3) By
  heedlessly uncovering sand-dunes, and thereby setting in motion a
  process of destruction which may convert hundreds of acres of fertile
  land into waste sand, or by prudently planting the dunes with
  sand-loving vegetation and thus arresting their landward progress. (4)
  By so guiding the course of rivers as to make them aid him in
  reclaiming waste land, and bringing it under cultivation. (5) By piers
  and bulwarks, whereby the ravages of the sea are stayed, or by the
  thoughtless removal from the beach of stones which the waves had
  themselves thrown up, and which would have served for a time to
  protect the land. (6) By forming new deposits either designedly or
  incidentally. The roads, bridges, canals, railways, tunnels, villages
  and towns with which man has covered the surface of the land will in
  many cases form a permanent record of his presence. Under his hand the
  whole surface of civilized countries is very slowly covered with a
  stratum, either formed wholly by him or due in great measure to his
  operations and containing many relics of his presence. The soil of
  ancient towns has been increased to a depth of many feet by their
  successive destructions and renovations.

  Perhaps the most subtle of human influences are to be seen in the
  distribution of plant and animal life upon the globe. Some of man's
  doings in this domain are indeed plain enough, such as the extirpation
  of wild animals, the diminution or destruction of some forms of
  vegetation, the introduction of plants and animals useful to himself,
  and especially the enormous predominance given by him to the cereals
  and to the spread of sheep and cattle. But no such extensive
  disturbance of the normal conditions of the distribution of life can
  take place without carrying with it many secondary effects, and
  setting in motion a wide cycle of change and of reaction in the animal
  and vegetable kingdoms. For example, the incessant warfare waged by
  man against birds and beasts of prey in districts given up to the
  chase leads sometimes to unforeseen results. The weak game is allowed
  to live, which would otherwise be killed off and give more room for
  the healthy remainder. Other animals which feed perhaps on the same
  materials as the game are by the same cause permitted to live
  unchecked, and thereby to act as a further hindrance to the spread of
  the protected species. But the indirect results of man's interference
  with the régime of plants and animals still require much prolonged
  observation.


PART V.--GEOTECTONIC OR STRUCTURAL GEOLOGY

From a study of the nature and composition of minerals and rocks, and an
investigation of the different agencies by which they are formed and
modified, the geologist proceeds to inquire how these materials have
been put together so as to build up the visible part of the earth's
crust. He soon ascertains that they have not been thrown together wholly
at random, but that they show a recognizable order of arrangement. Some
of them, especially those of most recent growth, remain in their
original condition and position, but, in proportion to their antiquity,
they generally present increasing alteration, until it may no longer be
possible to tell what was their pristine state. As by far the largest
accessible portion of the terrestrial crust consists of stratified
rocks, and as these furnish clear evidence of most of the modifications
to which they have been subjected in the long course of geological
history, it is convenient to take them into consideration first. They
possess a number of structures which belong to the original conditions
in which they were accumulated. They present in addition other
structures which have been superinduced upon them, and which they share
with the unstratified or igneous rocks.


1. ORIGINAL STRUCTURES

(a) _Stratified Rocks._--This extensive and important series is above
all distinguished by possessing a prevailing stratified arrangement.
Their materials have been laid down in laminae, layers and strata, or
beds, pointing generally to the intermittent deposition of the sediments
of which they consist. As this stratification was, as a rule, originally
nearly or quite horizontal, it serves as a base from which to measure
any subsequent disturbance which the rocks have undergone. The
occurrence of false-bedding, i.e. bands of inclined layers between the
normal planes of stratification, does not form any real exception; but
indicates the action of shifting currents whereby the sediment was
transported and thrown down. Other important records of the original
conditions of deposit are supplied by ripple-marks, sun-cracks,
rain-prints and concretions.

  From the nature of the material further light is cast on the
  geographical conditions in which the strata were accumulated. Thus,
  conglomerates indicate the proximity of old shore-lines, sandstones
  mark deposits in comparatively shallow water, clays and shales point
  to the tranquil accumulation of fine silt at a greater depth and
  further from land, while fossiliferous limestones bear witness to
  clearer water in which organisms flourished at some distance from
  deposits of sand and mud. Again, the alternation of different kinds of
  sediment suggests a variability in the conditions of deposition, such
  as a shifting of the sediment-bearing currents and of the areas of
  muddy and clear water. A thick group of conformable strata, that is, a
  series of deposits which show no discordance in their stratification,
  may usually be regarded as having been laid down on a sea-floor that
  was gently sinking. Here and there evidence is obtainable of the
  limits or of the progress of the subsidence by what is called
  "overlap." Of the absolute length of time represented by any strata or
  groups of strata no satisfactory estimates can yet be formed. Certain
  general conclusions may indeed be drawn, and comparisons may be made
  between different series of rocks. Sandstones full of false-bedding
  were probably accumulated more rapidly than finely-laminated shales or
  clays. It is not uncommon in certain Carboniferous formations to find
  coniferous and other trunks embedded in sandstone. Some of these trees
  seem to have been carried along and to have sunk, their heavier or
  root end touching the bottom and their upper end slanting upward in
  the direction of the current, exactly as in the case of the snags of
  the Mississippi. In other cases the trees have been submerged while
  still in their positions of growth. The continuous deposit of sand at
  last rose above the level of the trunks and buried them. It is clear
  then that the rate of deposit must have been sometimes sufficiently
  rapid to allow sand to accumulate to a depth of 30 ft. or more before
  the decay of the wood. Modern instances are known where, under certain
  circumstances, submerged trees may last for some centuries, but even
  the most durable must decay in what, after all, is a brief space of
  geological time. Since continuous layers of the same kind of deposit
  suggest a persistence of geological conditions, while numerous
  alternations of different kinds of sedimentary matter point to
  vicissitudes or alternations of conditions, it may be supposed that
  the time represented by a given thickness of similar strata was less
  than that shown by the same thickness of dissimilar strata, because
  the changes needed to bring new varieties of sediment into the area of
  deposit would usually require the lapse of some time for their
  completion. But this conclusion may often be erroneous. It will be
  best supported when, from the very nature of the rocks, wide
  variations in the character of the water-bottom can be established.
  Thus a group of shales followed by a fossiliferous limestone would
  almost always mark the lapse of a much longer period than an equal
  depth of sandy strata. A thick mass of limestone, made up of organic
  remains which lived and died upon the spot, and whose remains are
  crowded together generation above generation, must have demanded many
  years or centuries for its formation.

  But in all speculations of this kind we must bear in mind that the
  length of time represented by a given depth of strata is not to be
  estimated merely from their thickness or lithological character. The
  interval between the deposit of two successive laminae of shale may
  have been as long as, or even longer than, that required for the
  formation of one of the laminae. In like manner the interval needed
  for the transition from one stratum or kind of strata to another may
  often have been more than equal to the time required for the formation
  of the strata on either side. But the relative chronological
  importance of the bars or lines in the geological record can seldom be
  satisfactorily discussed merely on lithological grounds. This must
  mainly be decided on the evidence of organic remains, as shown in
  Part VI., where the grouping of the stratified rocks into formations
  and systems is described.

(b) _Igneous Rocks._--As part of the earth's crust these rocks present
characters by which they are strongly differentiated from the stratified
series. While the broad petrographical distinctions of their several
varieties remain persistent, they present sufficient local variations of
type to point to the existence of what have been called petrographic
provinces, in each of which the eruptive masses are connected by a
general family relationship, differing more or less from that of a
neighbouring province. In each region presenting a long chronological
series of eruptive rocks a petrographical sequence can be traced, which
is observed to be not absolutely the same everywhere, though its general
features may be persistent. The earliest manifestations of eruptive
material in any district appear to have been most frequently of an
intermediate type between acid and basic, passing thence into a
thoroughly acid series and concluding with an effusion of basic
material.

Considered as part of the architecture of the crust of the earth,
igneous rocks are conveniently divisible into two great series: (1)
those bodies of material which have been injected into the crust and
have solidified there, and (2) those which have reached the surface and
have been ejected there, either in a molten state as lava or in a
fragmental form as dust, ashes and scoriae. The first of these divisions
represents the plutonic, intrusive or subsequent phase of eruptivity;
the second marks the volcanic, interstratified or contemporaneous phase.

  1. The plutonic or intrusive rocks, which have been forced into the
  crust and have consolidated there, present a wide range of texture
  from the most coarse-grained granites to the most perfect natural
  glass. Seeing that they have usually cooled with extreme slowness
  underground, they are as a general rule more largely crystalline than
  the volcanic series. The form assumed by each individual body of
  intrusive material has depended upon the shape of the space into which
  it has been injected, and where it has cooled and become solid. This
  shape has been determined by the local structure of the earth's crust
  on the one hand and by the energy of the eruptive force on the other.
  It offers a convenient basis for the classification of the intrusive
  rocks, which, as part of the framework of the crust, may thus be
  grouped according to the shape of the cavity which received them, as
  bosses, sills, dikes and necks.

  Bosses, or stocks, are the largest and most shapeless extravasations
  of erupted material. They include the great bodies of granite which,
  in most countries of the world, have risen for many miles through the
  stratified formations and have altered the rocks around them by
  contact-metamorphism. Sills, or intrusive sheets, are bed-like masses
  which have been thrust between the planes of sedimentary or even of
  igneous rocks. The term laccolite has been applied to sills which are
  connected with bosses. Intrusive sheets are distinguishable from true
  contemporaneously intercalated lavas by not keeping always to the same
  platform, but breaking across and altering the contiguous strata, and
  by the closeness of their texture where they come in contact with the
  contiguous rocks, which, being cold, chilled the molten material and
  caused it to consolidate on its outer margins more rapidly than in its
  interior. Dikes or veins are vertical walls or ramifying branches of
  intrusive material which has consolidated in fissures or irregular
  clefts of the crust. Necks are volcanic chimneys which have been
  filled up with erupted material, and have now been exposed at the
  surface after prolonged denudation has removed not only the
  superficial volcanic masses originally associated with them, but also
  more or less of the upper part of the vents. Plutonic rocks do not
  present evidence of their precise geological age. All that can be
  certainly affirmed from them is that they must be younger than the
  rocks into which they have been intruded. From their internal
  structure, however, and from the evidence of the rocks associated with
  them, some more or less definite conjectures may be made as to the
  limits of time within which they were probably injected.

  2. The interstratified or volcanic series is of special importance in
  geology, inasmuch as it contains the records of volcanic action during
  the past history of the globe. It was pointed out in Part I. that
  while towards the end of the 18th and in the beginning of the 19th
  century much attention was paid by Hutton and his followers to the
  proofs of intrusion afforded by what they called the "unerupted lavas"
  within the earth's crust, these observers lost sight of the
  possibility that some of these rocks might have been erupted at the
  surface, and might thus be chronicles of volcanic action in former
  geological periods. It is not always possible to satisfactorily
  discriminate between the two types of contemporaneously intercalated
  and subsequently injected material. But rocks of the former type have
  not broken into or involved the overlying strata, and they are usually
  marked by the characteristic structures of superficial lavas and by
  their association with volcanic tuffs. By means of the evidence which
  they supply, it has been ascertained that volcanic action has been
  manifested in the globe since the earliest geological periods. In the
  British Isles, for example, the volcanic record is remarkably full for
  the long series of ages from Cambrian to Permian time, and again for
  the older Tertiary period.


2. SUBSEQUENTLY INDUCED STRUCTURES

After their accumulation, whether as stratified or eruptive masses, all
kinds of rocks have been subject to various changes, and have acquired
in consequence a variety of superinduced structures. It has been pointed
out in the part of this article dealing with dynamical geology that one
of the most important forms of energy in the evolution of geological
processes is to be found in the movements that take place within the
crust of the earth. Some of these movements are so slight as to be only
recognizable by means of delicate instruments; but from this inferior
limit they range up to gigantic convulsions by which mountain-chains are
upheaved. The crust must be regarded as in a perpetual state of strain,
and its component materials are therefore subject to all the effects
which flow from that condition. It is the one great object of the
geotectonic division of geology to study the structures which have been
developed in consequence of earth-movements, and to discover from this
investigation the nature of the processes whereby the rocks of the crust
have been brought into the condition and the positions in which we now
find them. The details of this subject will be found in separate
articles descriptive of each of the technical terms applied to the
several kinds of superinduced structures. All that need be offered here
is a general outline connecting the several portions of the subject
together.

  One of the most universal of these later structures is to be seen in
  the divisional planes, usually vertical or highly inclined, by which
  rocks are split into quadrangular or irregularly shaped blocks. To
  these planes the name of joints has been given. They are of prime
  importance from an industrial point of view, seeing that the art of
  quarrying consists mainly in detecting and making proper use of them.
  Their abundance in all kinds of rocks, from those of recent date up to
  those of the highest antiquity, affords a remarkable testimony to the
  strains which the terrestrial crust has suffered. They have arisen
  sometimes from tension, such as that caused by contraction from the
  drying and consolidation of an aqueous sediment or from the cooling of
  a molten mass; sometimes from torsion during movements of the crust.

  Although the stratified rocks were originally deposited in a more or
  less nearly horizontal position on the floor of the sea, where now
  visible on the dry land they are seldom found to have retained their
  flatness. On the contrary, they are seen to have been generally tilted
  up at various angles, sometimes even placed on end (crop, dip,
  strike). When a sufficiently large area of ground is examined, the
  inclination into which the strata have been thrown may be observed not
  to continue far in the same direction, but to turn over to the
  opposite or another quarter. It can then be seen that in reality the
  rocks have been thrown into undulations. From the lowest and flattest
  arches where the departure from horizontality may be only trifling,
  every step may be followed up to intense curvature, where the strata
  have been compressed and plicated as if they had been piles of soft
  carpets (anticline, syncline, monocline, geo-anticline, geo-syncline,
  isoclinal, plication, curvature, quaquaversal). It has further
  happened abundantly all over the surface of the globe that relief from
  internal strain in the crust has been obtained by fracture, and the
  consequent subsidence or elevation of one or both sides of the
  fissure. The differential movement between the two sides may be
  scarcely perceptible in the feeblest dislocation, but in the extreme
  cases it may amount to many thousand feet (fault, fissure,
  dislocation, hade, slickensides). The great faults in a country are
  among its most important structural features, and as they not
  infrequently continue to be lines of weakness in the crust along which
  sudden slipping may from time to time take place, they become the
  lines of origin of earthquakes. The San Francisco earthquake of 1906,
  already cited, affords a memorable illustration of this connexion.

  It is in a great mountain-chain that the extraordinary complication of
  plicated and faulted structures in the crust of the earth can be most
  impressively beheld. The combination of overturned folds with rupture
  has been already referred to as a characteristic feature in the Alps
  (Part IV.). The gigantic folds have in many places been pushed over
  each other so as to lie almost flat, while the upper limb has not
  infrequently been driven for many miles beyond the lower by a rupture
  along the axis. In this way successive slices of a thick series of
  formations have been carried northwards on the northern slope of the
  Alps, and have been piled so abnormally above each other that some of
  their oldest members recur several times on different thrust-planes,
  the whole being underlain by Tertiary strata (see ALPS). Further
  proof of the colossal compression to which the rocks have been
  subjected is afforded by their intense crumpling and corrugation, and
  by the abundantly faulted and crushed condition to which they have
  been reduced. Similar evidence as to stresses in the terrestrial crust
  and the important changes which they produce among the rocks may also
  be obtained on a smaller scale in many non-mountainous countries.

  Another marked result of the compression of the terrestrial crust has
  been induced in some rocks by the production of the fissile structure
  which is typically shown in roofing-slate (cleavage). Closely
  connected with this internal rearrangement has been the development of
  microscopic microlites or crystals (rutile, mica, &c.) in argillaceous
  slates which were undoubtedly originally fine marine mud and silt.
  From this incipient form of metamorphism successive stages may be
  traced through the various kinds of argillite and phyllite into
  mica-schist, and thence into more crystalline gneissoid varieties
  (foliation, slate, mica-schist, gneiss). The Alps afford excellent
  illustrations of these transformations.

  The fissures produced in the crust are sometimes clean, sharply
  defined divisional planes, like cracks across a pane of glass. Much
  more usually, however, the rocks on either side have been broken up by
  the friction of movement, and the fault is marked by a variable
  breadth of this broken material. Sometimes the walls have separated
  and molten rock has risen from below and solidified between them as a
  dike. Occasionally the fissures have opened to the surface, and have
  been filled in from above with detritus, as in the sandstone-dikes of
  Colorado and California. In mineral districts the fissures have been
  filled with various spars and ores, forming what are known as mineral
  veins.

  Where one series of rocks is covered by another without any break or
  discordance in the stratification they are said to be conformable. But
  where the older series has been tilted up or visibly denuded before
  being overlain by the younger, the latter is termed unconformable.
  This relation is one of the greatest value in structural geology, for
  it marks a gap in the geological record, which may represent a vast
  lapse of time not there recorded by strata.


PART VI.--PALEONTOLOGICAL GEOLOGY

This division of the science deals with fossils, or the traces of plants
and animals preserved in the rocks of the earth's crust, and endeavours
to gather from them information as to the history of the globe and its
inhabitants. The term "fossil" (Lat. _fossilis_, from _fodere_, to dig
up), meaning literally anything "dug up," was formerly applied
indiscriminately to any mineral substance taken out of the earth's
crust, whether organized or not. Since the time of Lamarck, however, the
meaning of the word has been restricted, so as to include only the
remains or traces of plants and animals preserved in any natural
formation whether hard rock or superficial deposit. It includes not
merely the petrified structures of organisms, but whatever was directly
connected with or produced by these organisms. Thus the resin which was
exuded from trees of long-perished forests is as much a fossil as any
portion of the stem, leaves, flowers or fruit, and in some respects is
even more valuable to the geologist than more determinable remains of
its parent trees, because it has often preserved in admirable perfection
the insects which flitted about in the woodlands. The burrows and trails
of a worm preserved in sandstone and shale claim recognition as fossils,
and indeed are commonly the only indications to be met with of the
existence of annelid life among old geological formations. The droppings
of fishes and reptiles, called coprolites, are excellent fossils, and
tell their tale as to the presence and food of vertebrate life in
ancient waters. The little agglutinated cases of the caddis-worm remain
as fossils in formations from which, perchance, most other traces of
life may have passed away. Nay, the very handiwork of man, when
preserved in any natural manner, is entitled to rank among fossils; as
where his flint-implements have been dropped into the pre-historic
gravels of river-valleys or where his canoes have been buried in the
silt of lake-bottoms.

  A study of the land-surfaces and sea-floors of the present time shows
  that there are so many chances against the conservation of the remains
  of either terrestrial or marine animals and plants that if, as is
  probable, the same conditions existed in former geological periods, we
  should regard the occurrence of organic remains among the stratified
  formations of the earth's crust as generally the result of various
  fortunate accidents.

  Let us consider, in the first place, the chances for the preservation
  of remains of the present fauna and flora of a country. The surface of
  the land may be densely clothed with forest and abundantly peopled
  with animal life. But the trees die and moulder into soil. The
  animals, too, disappear, generation after generation, and leave few or
  no perceptible traces of their existence. If we were not aware from
  authentic records that central and northern Europe were covered with
  vast forests at the beginning of our era, how could we know this fact?
  What has become of the herds of wild oxen, the bears, wolves and other
  denizens of primeval Europe? How could we prove from the examination
  of the surface soil of any country that those creatures had once
  abounded there? The conditions for the preservation of any relics of
  the plant and animal life of a terrestrial surface must obviously be
  always exceptional. They are supplied only where the organic remains
  can be protected from the air and superficial decay. Hence they may be
  observed in (1) the deposits on the floors of lakes; (2) in
  peat-mosses; (3) in deltas at river-mouths; and (4) under the
  stalagmite of caverns in limestone districts. But in these and other
  favourable places a mere infinitesimal fraction of the fauna or flora
  of a land-surface is likely to be entombed or preserved.

  In the second place, although in the sea the conditions for the
  preservation of organic remains are in many respects more favourable
  than on land, they are apt to be frustrated by many adverse
  circumstances. While the level of the land remains stationary, there
  can be but little effective entombment of marine organisms in littoral
  deposits; for only a limited accumulation of sediment will be formed
  until subsidence of the sea-floor takes place. In the trifling beds of
  sand or gravel thrown up on a stationary shore, only the harder and
  more durable forms of life, such as gastropods and lamellibranchs,
  which can withstand the triturating effects of the beach waves, are
  likely to remain uneffaced.

  Below tide-marks, along the margin of the land where sediment is
  gradually deposited, the conditions are more favourable for the
  preservation of marine organisms. In the sheets of sand and mud there
  laid down the harder parts of many forms of life may be entombed and
  protected from decay. But only a small proportion of the total marine
  fauna may be expected to appear in such deposits. At the best, merely
  littoral and shallow-water forms will occur, and, even under the most
  favourable conditions, they will represent but a fraction of the whole
  assemblage of life in these juxta-terrestrial parts of the ocean. As
  we recede from the land the rate of deposition of sediment on the
  sea-floor must become feebler, until, in the remote central abysses,
  it reaches a hardly appreciable minimum. Except, therefore, where some
  kind of ooze or other deposit is accumulating in these more pelagic
  regions, the conditions must be on the whole unfavourable for the
  preservation of any adequate representation of the deep-sea fauna.
  Hard durable objects, such as teeth and bones, may slowly accumulate,
  and be protected by a coating of peroxide of manganese, or of some of
  the silicates now forming here and there over the deep-sea bottom; or
  the rate of growth of the abysmal deposit may be so tardy that most of
  the remains of at least the larger animals will disappear, owing to
  decay, before they can be covered up and preserved. Any such deep-sea
  formation, if raised into land, would supply but a meagre picture of
  the whole life of the sea.

  It would thus appear that the portion of the sea-floor best suited for
  receiving and preserving the most varied assemblage of marine organic
  remains is the area in front of the land, to which rivers and currents
  bring continual supplies of sediment. The most favourable conditions
  for the accumulation of a thick mass of marine fossiliferous strata
  will arise when the area of deposit is undergoing a gradual
  subsidence. If the rate of depression and that of deposit were equal,
  or nearly so, the movement might proceed for a vast period without
  producing any great apparent change in marine geography, and even
  without seriously affecting the distribution of life over the
  sea-floor within the area of subsidence. Hundreds or thousands of feet
  of sedimentary strata might in this way be heaped up round the
  continents, containing a fragmentary series of organic remains
  belonging to those forms of comparatively shallow-water life which had
  hard parts capable of preservation. There can be little doubt that
  such has, in fact, been the history of the main mass of stratified
  formations in the earth's crust. By far the largest proportion of
  these piles of marine strata has unquestionably been laid down in
  water of no great depth within the area of deposit of terrestrial
  sediment. The enormous thickness to which they attain seems only
  explicable by prolonged and repeated movements of subsidence,
  interrupted, however, as we know, by other movements of a contrary
  kind.

  Since the conditions for the preservation of organic remains exist
  more favourably under the sea than on land, marine organisms must be
  far more abundantly conserved than those of the land. This is true
  to-day, and has, as far as known, been true in all past geological
  time. Hence for the purposes of the geologist the fossil remains of
  marine forms of life far surpass all others in value. Among them there
  will necessarily be a gradation of importance, regulated chiefly by
  their relative abundance. Now, of all the marine tribes which live
  within the juxta-terrestrial belt of sedimentation, unquestionably the
  Mollusca stand in the place of pre-eminence as regards their aptitude
  for becoming fossils. They almost all possess a hard, durable shell,
  capable of resisting considerable abrasion and readily passing into a
  mineralized condition. They are extremely abundant both as to
  individuals and genera. They occur on the shore within tide mark, and
  range thence down into the abysses. Moreover, they appear to have
  possessed these qualifications from early geological times. In the
  marine Mollusca, therefore, we have a common ground of comparison
  between the stratified formations of different periods. They have been
  styled the alphabet of palaeontological inquiry.

There are two main purposes to which fossils may be put in geological
research: (1) to throw light upon former conditions of physical
geography, such as the presence of land, rivers, lakes and seas, in
places where they do not now exist, changes of climate, and the former
distribution of plants and animals; and (2) to furnish a guide in
geological chronology whereby rocks may be classified according to
relative date, and the facts of geological history may be arranged and
interpreted as a connected record of the earth's progress.

  1. As examples of the first of these two directions of inquiry
  reference may be made to (a) former land-surfaces revealed by the
  occurrence of layers of soil with tree-stumps and roots still in the
  position of growth (see PURBECKIAN); (b) ancient lakes proved by beds
  of marl or limestone full of lacustrine shells; (c) old sea-bottoms
  marked by the occurrence of marine organisms; (d) variations in the
  quality of the water, such as freshness or saltness, indicated by
  changes in the size and shape of the fossils; (e) proximity to former
  land, suggested by the occurrence of abundant drift-wood in the
  strata; (f) former conditions of climate, different from the present,
  as evidenced by such organisms as tropical types of plants and animals
  intercalated among the strata of temperate or northern countries.

  2. In applying fossils to the determination of geological chronology
  it is first necessary to ascertain the order of superposition of the
  rocks. Obviously, in a continuous series of undisturbed sedimentary
  deposits the lowest must necessarily be the oldest, and the plants or
  animals which they contain must have lived and died before any of the
  organisms that occur in the overlying strata. This order of
  superposition having been settled in a series of formations, it is
  found that the fossils at the bottom are not quite the same as those
  at the top of the series. Tracing the beds upward, we discover that
  species after species of the lowest platforms disappears, until
  perhaps not one of them is found. With the cessation of these older
  species others make their entrance. These, in turn, are found to die
  out, and to be replaced by newer forms. After patient examination of
  the rocks, it has been ascertained that every well-marked "formation,"
  or group of strata, is characterized by its own species or genera, or
  by a general assemblage, or _facies_, of organic forms. Such a
  generalization can only, of course, be determined by actual practical
  experience over an area of some size. When the typical fossils of a
  formation are known, they serve to identify that formation in its
  progress across a country. Thus, in tracts where the true order of
  superposition cannot be determined, owing to the want of sections or
  to the disturbed condition of the rocks, fossils serve as a means of
  identification and furnish a guide to the succession of the rocks.
  They even demonstrate that in some mountainous ground the beds have
  been turned completely upside down, where it can be shown that the
  fossils in what are now the uppermost strata ought properly to lie
  underneath those in the beds below them.

  It is by their characteristic fossils that the stratified rocks of the
  earth's crust can be most satisfactorily subdivided into convenient
  groups of strata and classed in chronological order. Each "formation"
  is distinguished by its own peculiar assemblage of organic remains, by
  means of which it can be followed and recognized, even amid the
  crumplings and dislocations of a disturbed region. The same general
  succession of organic types can be observed over a large part of the
  world, though, of course, with important modifications in different
  countries. This similarity of succession has been termed _homotaxis_,
  a term which expresses the fact that the order in which the leading
  types of organized existence have appeared upon the earth has been
  similar even in widely separated regions. It is evident that, in this
  way, a reliable method of comparison is furnished, whereby the
  stratified formations of different parts of the earth's crust can be
  brought into relation with each other. Had the geologist continued to
  remain, as in the days of Werner, hampered by the limitations imposed
  by a reliance on mere lithological characters, he would have made
  little or no progress in deciphering the record of the successive
  phases of the history of the globe chronicled in the crust. Just as,
  at the present time, sheets of gravel in one place are contemporaneous
  with sheets of mud at another, so in the past all kinds of
  sedimentation have been in progress simultaneously, and those of one
  period may not be distinguishable in themselves from those of another.
  Little or no reliance can be placed upon lithological resemblances or
  differences in comparing the sedimentary formations of different
  countries.

  In making use of fossil evidence for the purpose of subdividing the
  stratified rocks of the earth's crust, it is found to be applicable to
  the smaller details of stratigraphy as well as to the definition of
  large groups of strata. Thus a particular stratum may be marked by the
  occurrence in it of various fossils, one or more of which may be
  distinctive, either from occurring in no other bed above and below or
  from special abundance in that stratum. One or more of these species
  is therefore used as a guide to the occurrence of the bed in
  question, which is called by the name of the most abundant species. In
  this way what is called a "geological horizon," or "zone," is marked
  off, and its exact position in the series of formations is fixed.

  Perhaps the most distinctive feature in the progress of
  palaeontological geology during the last half century has been the
  recognition and wide application of this method of zonal stratigraphy,
  which, in itself, was only a further development of William Smith's
  famous idea, "Strata identified by Organized Fossils." It was first
  carried out in detail by various palaeontologists in reference to the
  Jurassic formations, notably by F.A. von Quenstedt and C.A. Oppel in
  Germany and A.D. d'Orbigny in France. The publication of Oppel's
  classic work _Die Juraformation Englands, Frankreichs und des
  südwestlichen Deutschlands_ (1856-1858) marked an epoch in the
  development of stratigraphical geology. Combining what had been done
  by various observers with his own laborious researches in France,
  England, Württemberg and Bavaria, he drew up a classification of the
  Jurassic system, grouping its several formations into zones, each
  characterized by some distinctly predominant fossil after which it was
  named (see LIAS). The same method of classification was afterwards
  extended to the Cretaceous series by A.D. d'Orbigny, E. Hébert and
  others, until the whole Mesozoic rocks from the Trias to the top of
  the Chalk has now been partitioned into zones, each named after some
  characteristic species or genus of fossils. More recently the
  principle has been extended to the Palaeozoic formations, though as
  yet less fully than to the younger parts of the geological record. It
  has been successfully applied by Professor C. Lapworth to the
  investigation of the Silurian series (see SILURIAN; ORDOVICIAN
  SYSTEM). He found that the species of graptolites have each a
  comparatively narrow vertical range, and they may consequently be used
  for stratigraphical purposes. Applying the method, in the first
  instance, to the highly plicated Silurian rocks of the south of
  Scotland, he found that by means of graptolites he was able to work
  out the structure of the ground. Each great group of strata was seen
  to possess its own graptolitic zones, and by their means could be
  identified not only in the original complex Scottish area, but in
  England and Wales and in Ireland. It was eventually ascertained that
  the succession of zones in Great Britain could be recognized on the
  Continent, in North America and even in Australia. The brachiopods and
  trilobites have likewise been made use of for zonal purposes among the
  oldest sedimentary formations. The most ancient of the Palaeozoic
  systems has as its fitting base the _Olenellus_ zone.

  Within undefined and no doubt variable geographical limits
  palaeontological zones have been found to be remarkably persistent.
  They follow each other in the same general order, but not always with
  equal definiteness. The type fossil may appear in some districts on a
  higher or a lower platform than it does in others. Only to a limited
  degree is there any coincidence between lithological variations in the
  strata and the sequence of the zones. In the Jurassic formations,
  indeed, where frequent alternations of different sedimentary materials
  are to be met with, it is in some cases possible to trace a definite
  upward or downward limit for a zone by some abrupt change in the
  sedimentation, such as from limestone to shale. But such a precise
  demarcation is impossible where no distinct bands of different
  sediments are to be seen. The zones can then only be vaguely
  determined by finding their characteristic fossils, and noting where
  these begin to appear in the strata and where they cease. It would
  seem, therefore, that the sequence of palaeontological zones, or
  life-horizons, has not depended merely upon changes in the nature of
  the conditions under which the organisms lived. We should naturally
  expect that these changes would have had a marked influence; that, for
  instance, a difference should be perceptible between the character of
  the fossils in a limestone and that of those in a shale or a
  sandstone. The environment, when a limestone was in course of
  deposition, would generally be one of clear water, favourable for a
  more vigorous and more varied fauna than where a shale series was
  accumulating, when the water would be discoloured, and only such
  animals would continue to live in it, or on the bottom, as could
  maintain themselves in the midst of mud. But no such lithological
  reason, betokening geographical changes that would affect living
  creatures, can be adduced as a universally applicable explanation of
  the occurrence and limitation of palaeontological zones. One of these
  zones may be only a few inches, or feet or yards in vertical extent,
  and no obvious lithological or other cause can be seen why its
  specially characteristic fossils should not be found just as
  frequently in the similar strata above and below. There is often
  little or no evidence of any serious change in the conditions of
  sedimentation, still less of any widespread physical disturbance, such
  as the catastrophes by which the older geologists explained the
  extinction of successive types of life.

  It has been suggested that, where the life-zones are well defined,
  sedimentation has been extremely slow, and that though these zones
  follow each other with no break in the sedimentation, they were really
  separated by prolonged intervals of time during which organic
  evolution could come effectively into play. But it is not easy to
  explain how, for example in the Lower Lias, there could have been a
  succession of prodigious intervals, when practically no sediment was
  laid down, and yet that the strata should show no sign of
  contemporaneous disturbance or denudation, but succeed each other as
  if they had been accumulated by one continuous process of deposit. It
  must be admitted that the problem of life-zones in stratigraphical
  geology has not yet been solved.

  As Darwin first cogently showed, the history of life has been very
  imperfectly registered in the stratified parts of the earth's crust.
  Apart from the fact that, even under the most favourable conditions,
  only a small proportion of the total flora and fauna of any period
  would be preserved in the fossil state, enormous gaps occur where no
  record has survived at all. It is as if whole chapters and books were
  missing from a historical work. Some of these lacunae are sufficiently
  obvious. Thus, in some cases, powerful dislocations have thrown
  considerable portions of the rocks out of sight. Sometimes extensive
  metamorphism has so affected them that their original characters,
  including their organic contents, have been destroyed. Oftenest of
  all, denudation has come into play, and vast masses of fossiliferous
  rock have been entirely worn away, as is demonstrated by the abundant
  unconformabilities in the structure of the earth's crust.

  While the mere fact that one series of rocks lies unconformably on
  another proves the lapse of a considerable interval between their
  respective dates, the relative length of this interval may sometimes
  be proved by means of fossil evidence, and by this alone. Let us
  suppose, for example, that a certain group of formations has been
  disturbed, upraised, denuded and covered unconformably by a second
  group. In lithological characters the two may closely resemble each
  other, and there may be nothing to show that the gap represented by
  their unconformability is of an important character. In many cases,
  indeed, it would be quite impossible to pronounce any well-grounded
  judgment as to the amount of interval, even measured by the vague
  relative standards of geological chronology. But if each group
  contains a well-preserved suite of organic remains, it may not only be
  possible, but easy, to say exactly how much of the geological record
  has been left out between the two sets of formations. By comparing the
  fossils with those obtained from regions where the geological record
  is more complete, it may be ascertained, perhaps, that the lower rocks
  belong to a certain platform or stage in geological history which for
  our present purpose we may call D, and that the upper rocks can in
  like manner be paralleled with stage H. It would be then apparent that
  at this locality the chronicles of three great geological periods E,
  F, and G were wanting, which are elsewhere found to be intercalated
  between D and H. The lapse of time represented by this
  unconformability would thus be equivalent to that required for the
  accumulation of the three missing formations in those regions where
  sedimentation was more continuous.

  Fossil evidence may be made to prove the existence of gaps which are
  not otherwise apparent. As has been already remarked, changes in
  organic forms must, on the whole, have been extremely slow in the
  geological past. The whole species of a sea-floor could not pass
  entirely away, and be replaced by other forms, without the lapse of
  long periods of time. If then among the conformable stratified
  formations of former ages we encounter sudden and abrupt changes in
  the _facies_ of the fossils, we may be certain that these must mark
  omissions in the record, which we may hope to fill in from a more
  perfect series elsewhere. The complete biological contrasts between
  the fossil contents of unconformable strata are sufficiently
  explicable. It is not so easy to give a satisfactory account of those
  which occur where the beds are strictly conformable, and where no
  evidence can be observed of any considerable change of physical
  conditions at the time of deposit. A group of strata having the same
  general lithological characters throughout may be marked by a great
  discrepance between the fossils above and below a certain line. A few
  species may pass from the one into the other, or perhaps every species
  may be different. In cases of this kind, when proved to be not merely
  local but persistent over wide areas, we must admit, notwithstanding
  the apparently undisturbed and continuous character of the original
  deposition of the strata, that the abrupt transition from the one
  _facies_ of fossils to the other represents a long interval of time
  which has not been recorded by the deposit of strata. A.C. Ramsay, who
  called attention to these gaps, termed them "breaks in the succession
  of organic remains." He showed that they occur abundantly among the
  Palaeozoic and Secondary rocks of England. It is obvious, of course,
  that such breaks, even though traceable over wide regions, were not
  general over the whole globe. There have never been any universal
  interruptions in the continuity of the chain of being, so far as
  geological evidence can show. But the physical changes which caused
  the breaks may have been general over a zoological district or minor
  region. They no doubt often caused the complete extinction of genera
  and species which had a small geographical range.

  From all these facts it is clear that the geological record, as it now
  exists, is at the best but an imperfect chronicle of geological
  history. In no country is it complete. The lacunae of one region must
  be supplied from another. Yet in proportion to the geographical
  distance between the localities where the gaps occur and those whence
  the missing intervals are supplied, the element of uncertainty in our
  reading of the record is increased. The most desirable method of
  research is to exhaust the evidence for each area or province, and to
  compare the general order of its succession as a whole with that which
  can be established for other provinces.


PART VII.--STRATIGRAPHICAL GEOLOGY

This branch of the science arranges the rocks of the earth's crust in
the order of their appearance, and interprets the sequence of events of
which they form the records. Its province is to cull from the other
departments of geology the facts which may be needed to show what has
been the progress of our planet, and of each continent and country, from
the earliest times of which the rocks have preserved any memorial. Thus
from mineralogy and petrography it contains information regarding the
origin and subsequent mutations of minerals and rocks. From dynamical
geology it learns by what agencies the materials of the earth's crust
have been formed, altered, broken, upheaved and melted. From geotectonic
geology it understands the various processes whereby these materials
were put together so as to build up the complicated crust of the earth.
From palaeontological geology it receives in well-determined fossil
remains a clue by which to discriminate the different stratified
formations, and to trace the grand onward march of organized existence
upon this planet. Stratigraphical geology thus gathers up the sum of all
that is made known by the other departments of the science, and makes it
subservient to the interpretation of the geological history of the
earth.

The leading principles of stratigraphy may be summed up as follows:

1. In every stratigraphical research the fundamental requisite is to
establish the order of superposition of the strata. Until this is
accomplished it is impossible to arrange the dates, and make out the
sequence of geological history.

2. The stratified portion of the earth's crust, or what has been called
the "geological record," can be subdivided into natural groups, or
series of strata, characterized by distinctive organic remains and
recognizable by these remains, in spite of great changes in lithological
character from place to place. A bed, or a number of beds, linked
together by containing one or more distinctive species or genera of
fossils is termed a _zone_ or _horizon_, and usually bears the name of
one of its more characteristic fossils, as the _Planorbis_-zone of the
Lower Lias, which is so called from the prevalence in it of the ammonite
_Psiloceras planorbis_. Two or more such zones related to each other by
the possession of a number of the same characteristic species or genera
have been designated _beds_ or an _assise_. Two or more sets of beds or
assises similarly related form a _group_ or _stage_; a number of groups
or stages make a _series_, _formation_ or _section_, and a succession of
formations may be united into a _system_.

3. Some living species of plants and animals can be traced downwards
through the more recent geological formations; but the number which can
be so followed grows smaller as the examination is pursued into more
ancient deposits. With their disappearance other species or genera
present themselves which are no longer living. These in turn may be
traced backward into earlier formations, till they too cease and their
places are taken by yet older forms. It is thus shown that the
stratified rocks contain the records of a gradual progression of organic
forms. A species which has once died out does not seem ever to have
reappeared.

4. When the order of succession of organic remains among the stratified
rocks has been determined, they become an invaluable guide in the
investigation of the relative age of rocks and the structure of the
land. Each zone and formation, being characterized by its own species or
genera, may be recognized by their means, and the true succession of
strata may thus be confidently established even in a country wherein the
rocks have been shattered by dislocation, folded, inverted or
metamorphosed.

5. Though local differences exist in regard to the precise zone in which
a given species of organism may make its first appearance, the general
order of succession of the organic forms found in the rocks is never
inverted. The record is nowhere complete in any region, but the portions
represented, even though extremely imperfect, always follow each other
in their proper chronological order, unless where disturbance of the
crust has intervened to destroy the original sequence.

6. The relative chronological value of the divisions of the geological
record is not to be measured by mere depth of strata. While it may be
reasonably assumed that, in general, a great thickness of stratified
rock must mark the passage of a long period of time, it cannot safely be
affirmed that a much less thickness elsewhere must represent a
correspondingly diminished period. The need for this caution may
sometimes be made evident by an unconformability between two sets of
rocks, as has already been explained. The total depth of both groups
together may be, say 1000 ft. Elsewhere we may find a single unbroken
formation reaching a depth of 10,000 ft.; but it would be unwarrantable
to assume that the latter represents ten times the length of time
indicated by the former two. So far from this being the case, it might
not be difficult to show that the minor thickness of rock really denotes
by far the longer geological interval. If, for instance, it could be
proved that the upper part of both the sections lies on one and the same
geological platform, but that the lower unconformable series in the one
locality belongs to a far lower and older system of rocks than the base
of the thick conformable series in the other, then it would be clear
that the gap marked by the unconformability really indicates a longer
period than the massive succession of deposits.

7. Fossil evidence furnishes the chief means of comparing the relative
value of formations and groups of rock. A "break in the succession of
organic remains," as already explained, marks an interval of time often
unrepresented by strata at the place where the break is found. The
relative importance of these breaks, and therefore, probably, the
comparative intervals of time which they mark, may be estimated by the
difference of the _facies_ or general character of the fossils on each
side. If, for example, in one case we find every species to be
dissimilar above and below a certain horizon, while in another locality
only half of the species on each side are peculiar, we naturally infer,
if the total number of species seems large enough to warrant the
inference, that the interval marked by the former break was much longer
than that marked by the second. But we may go further and compare by
means of fossil evidence the relation between breaks in the succession
of organic remains and the depth of strata between them.

  Three formations of fossiliferous strata, A, C, and H, may occur
  conformably above each other. By a comparison of the fossil contents
  of all parts of A, it may be ascertained that, while some species are
  peculiar to its lower, others to its higher portions, yet the majority
  extend throughout the formation. If now it is found that of the total
  number of species in the upper portion of A only one-third passes up
  into C, it may be inferred with some plausibility that the time
  represented by the break between A and C was really longer than that
  required for the accumulation of the whole of the formation A. It
  might even be possible to discover elsewhere a thick intermediate
  formation B filling up the gap between A and C. In like manner were it
  to be discovered that, while the whole of the formation C is
  characterized by a common suite of fossils, not one of the species and
  only one half of the genera pass up into H, the inference could hardly
  be resisted that the gap between the two formations marks the passage
  of a far longer interval than was needed for the deposition of the
  whole of C. And thus we reach the remarkable conclusion that, thick
  though the stratified formations of a country may be, in some cases
  they may not represent so long a total period of time as do the gaps
  in their succession,--in other words, that non-deposition was more
  frequent and prolonged than deposition, or that the intervals of time
  which have been recorded by strata have not been so long as those
  which have not been so recorded.

In all speculations of this nature, however, it is necessary to reason
from as wide a basis of observation as possible, seeing that so much of
the evidence is negative. Especially needful is it to bear in mind that
the cessation of one or more species at a certain line among the rocks
of a particular district may mean nothing more than that, onward from
the time marked by that line, these species, owing to some change in the
conditions of life, were compelled to migrate or became locally extinct
or, from some alteration in the conditions of fossilization, were no
longer imbedded and preserved as fossils. They may have continued to
flourish abundantly in neighbouring districts for a long period
afterward. Many examples of this obvious truth might be cited. Thus in a
great succession of mingled marine, brackish-water and terrestrial
strata, like that of the Carboniferous Limestone series of Scotland,
corals, crinoids and brachiopods abound in the limestones and
accompanying shales, but disappear as the sandstones, ironstones, clays,
coals and bituminous shales supervene. An observer meeting for the first
time with an instance of this disappearance, and remembering what he had
read about breaks in succession, might be tempted to speculate about the
extinction of these organisms, and their replacement by other and later
forms of life, such as the ferns, lycopods, estuarine or fresh-water
shells, ganoid fishes and other fossils so abundant in the overlying
strata. But further research would show him that high above the
plant-bearing sandstones and coals other limestones and shales might be
observed, once more charged with the same marine fossils as before, and
still farther overlying groups of sandstones, coals and carbonaceous
beds followed by yet higher marine limestones. He would thus learn that
the same organisms, after being locally exterminated, returned again and
again to the same area. After such a lesson he would probably pause
before too confidently asserting that the highest bed in which we can
detect certain fossils marks their final appearance in the history of
life. Some breaks in the succession may thus be extremely local, one set
of organisms having been driven to a different part of the same region,
while another set occupied their place until the first was enabled to
return.

8. The geological record is at the best but an imperfect chronicle of
the geological history of the earth. It abounds in gaps, some of which
have been caused by the destruction of strata owing to metamorphism,
denudation or otherwise, others by original non-deposition, as above
explained. Nevertheless from this record alone can the progress of the
earth be traced. It contains the registers of the appearance and
disappearance of tribes of plants and animals which have from time to
time flourished on the earth. Only a small proportion of the total
number of species which have lived in past time have been thus
chronicled, yet by collecting the broken fragments of the record an
outline at least of the history of life upon the earth can be
deciphered.

It cannot be too frequently stated, nor too prominently kept in view,
that, although gaps occur in the succession of organic remains as
recorded in the rocks, they do not warrant the conclusion that any such
blank intervals ever interrupted the progress of plant and animal life
upon the globe. There is every reason to believe that the march of life
has been unbroken, onward and upward. Geological history, therefore, if
its records in the stratified formations were perfect, ought to show a
blending and gradation of epoch with epoch. But the progress has been
constantly interrupted, now by upheaval, now by volcanic outbursts, now
by depression. These interruptions serve as natural divisions in the
chronicle, and enable the geologist to arrange his history into periods.
As the order of succession among stratified rocks was first made out in
Europe, and as many of the gaps in that succession were found to be
widespread over the European area, the divisions which experience
established for that portion of the globe came to be regarded as
typical, and the names adopted for them were applied to the rocks of
other and far distant regions. This application has brought out the fact
that some of the most marked breaks in the European series do not exist
elsewhere, and, on the other hand, that some portions of that series are
much more complete than the corresponding sections in other regions.
Hence, while the general similarity of succession may remain, different
subdivisions and nomenclature are required as we pass from continent to
continent.

The nomenclature adopted for the subdivisions of the geological record
bears witness to the rapid growth of geology. It is a patch-work in
which no system nor language has been adhered to, but where the
influences by which the progress of the science has been moulded may be
distinctly traced. Some of the earliest names are lithological, and
remind us of the fact that mineralogy and petrography preceded geology
in the order of birth--Chalk, Oolite, Greensand, Millstone Grit. Others
are topographical, and often recall the labours of the early geologists
of England--London Clay, Oxford Clay, Purbeck, Portland, Kimmeridge
beds. Others are taken from local English provincial names, and remind
us of the debt we owe to William Smith, by whom so many of them were
first used--Lias, Gault, Crag, Cornbrash. Others of later date recognize
an order of superposition as already established among formations--Old
Red Sandstone, New Red Sandstone. By common consent it is admitted that
names taken from the region where a formation or group of rocks is
typically developed are best adapted for general use. Cambrian,
Silurian, Devonian, Permian, Jurassic are of this class, and have been
adopted all over the globe.

But whatever be the name chosen to designate a particular group of
strata, it soon comes to be used as a chronological or homotaxial term,
apart altogether from the stratigraphical character of the strata to
which it is applied. Thus we speak of the Chalk or Cretaceous system,
and embrace under that term formations which may contain no chalk; and
we may describe as Silurian a series of strata utterly unlike in
lithological characters to the formations in the typical Silurian
country. In using these terms we unconsciously allow the idea of
relative date to arise prominently before us. Hence such a word as
"chalk" or "cretaceous" does not suggest so much to us the group of
strata so called as the interval of geological history which these
strata represent. We speak of the Cretaceous, Jurassic, and Cambrian
periods, and of the Cretaceous fauna, the Jurassic flora, the Cambrian
trilobites, as if these adjectives denoted simply epochs of geological
time.

The stratified formations of the earth's crust, or geological record,
are classified into five main divisions, which in their order of
antiquity are as follows: (1) Archean or Pre-Cambrian, called also
sometimes Azoic (lifeless) or Eozoic (dawn of life); (2) Palaeozoic
(ancient life) or Primary; (3) Mesozoic (middle life) or Secondary; (4)
Cainozoic (recent life) or Tertiary; (5) Quaternary or Post-Tertiary.
These divisions are further ranged into systems, formations, groups or
stages, assises and zones. Accounts of the various subdivisions named
are given in separate articles under their own headings. In order,
however, that the sequence of the formations and their parallelism in
Europe and North America may be presented together a stratigraphical
table is given on next page.


PART VIII.--PHYSIOGRAPHICAL GEOLOGY

This department of geological inquiry investigates the origin and
history of the present topographical features of the land. As these
features must obviously be related to those of earlier time which are
recorded in the rocks of the earth's crust, they cannot be
satisfactorily studied until at least the main outlines of the history
of these rocks have been traced. Hence physiographical research comes
appropriately after the other branches of the science have been
considered.

From the stratigraphy of the terrestrial crust we learn that by far the
largest part of the area of dry land is built up of marine formations;
and therefore that the present land is not an aboriginal portion of the
earth's surface, but has been overspread by the sea in which its rocks
were mainly accumulated. We further discover that this submergence of
the land did not happen once only, but again and again in past ages and
in all parts of the world. Yet although the terrestrial areas varied
much from age to age in their extent and in their distribution, being at
one time more continental, at another more insular, there is reason to
believe that these successive diminutions and expansions have on the
whole been effected within, or not far outside, the limits of the
existing continents. There is no evidence that any portion of the
present land ever lay under the deeper parts of the ocean. The abysmal
deposits of the ocean-floor have no true representatives among the
sedimentary formations anywhere visible on the land. Nor, on the other
hand, can it be shown that any part of the existing ocean abysses ever
rose above sea-level into dry land. Hence geologists have drawn the
inference that the ocean basins have probably been always where they now
are; and that although the continental areas have often been narrowed by
submergence and by denudation, there has probably seldom or never been a
complete disappearance of land. The fact that the sedimentary formations
of each successive geological period consist to so large an extent of
mechanically formed terrigenous detritus, affords good evidence of the
coexistence of tracts of land as well as of extensive denudation.


  _The Geological Record or Order of Succession of the Stratified
  Formations of the Earth's Crust._

  +---+---+-------------------------------------------+----------------------------------+
  |   |   |                  Europe.                  |          North America.          |
  +---+---+-------------------------------------------+----------------------------------+
  | Q |    \      Historic, up to the present time.   | Similar to the European          |
  | u |     \     Prehistoric, comprising deposits of |   development, but with scantier |
  | a |      \     the Iron, Bronze, and later        |   traces of the presence of man. |
  | t |       \     Stone Ages.                       |                                  |
  | e |        \    Neolithic--alluvium, peat, lake-  |                                  |
  | r | Recent, \   dwellings, loess, &c.             |                                  |
  | n | Post-   | Palaeolithic--river-gravels, cave-  |                                  |
  | a | glacial |   deposits, &c.                     |                                  |
  | r | or      |                                     |                                  |
  | y | Human.  |                                     |                                  |
  |   |         |                                     |                                  |
  | o |         |                                     |                                  |
  | r +---------+-------------------------------------+----------------------------------+
  |   | Pleist- | Older Loess and valley-gravels;     | As in Europe, it is hardly       |
  | P | ocene   |   cave-deposits.                    |   possible to assign a definite  |
  | o | or      | Strand-lines or raised beaches;     |   chronological place to each of |
  | s | Glacial.|   youngest moraines.                |   the various deposits of this   |
  | t |         | Upper Boulder-clays; eskers; marine |   period, terrestrial and marine.|
  | | |         |   sands and clays.                  |   They generally resemble the    |
  | T |         | Interglacial deposits.              |   European series. The           |
  | e |         | Lower boulder-clay or Till, with    |   characteristic marine,         |
  | r |         |   striated rock-surfaces below.     |   fluviatile and lacustrine      |
  | t |         |                                     |   terraces, which overlie the    |
  | i |        /                                      |   older drifts, have been        |
  | a |       /                                       |   classed as the Champlain Group.|
  | r |      /                                        |                                  |
  | y |     /                                         |                                  |
  | . |    /                                          |                                  |
  +---+---+-------------------------------------------+----------------------------------+
  |   | P | Newer:--English Forest-Bed Group; Red and | On the Atlantic border           |
  |   | l |   Norwich Crag; Amstelian and Scaldesian  |   represented by the marine      |
  |   | i |   groups of Belgium and Holland; Sicilian |   Floridian series; in the       |
  |   | o |   and Astian of France and Italy.         |   interior by a subaerial and    |
  |   | c | Older:--English Coralline Crag; Diestian  |   lacustrine series; and on the  |
  |   | e |   of Belgium; Plaisancian of southern     |   Pacific border by the thick    |
  |   | n |   France and Italy.                       |   marine series of San Francisco.|
  |   | e |                                           |                                  |
  |   | . |                                           |                                  |
  |   +---+-------------------------------------------+----------------------------------+
  |   | M | Wanting in Britain; well developed in     | Represented in the Eastern States|
  |   | i |   France, S. E. Europe and Italy;         |   by a marine series (Yorktown or|
  | C | o |   divisible into the following groups in  |   Chesapeake, Chipola and        |
  | a | c |   descending order: (1) Pontian; (2)      |   Chattahoochee groups), and in  |
  | i | e |   Sarmatian; (3) Tortonian; (4) Helvetian;|   the interior by the lacustrine |
  | n | n |   (5) Langhian (Burdigalian).             |   Loup Fork (Nebraska), Deep     |
  | o | e |                                           |   River, and John Day groups.    |
  | z | . |                                           |                                  |
  | o +---+-------------------------------------------+----------------------------------+
  | i |   | In Britain the "fluvio-marine series" of  | On the Atlantic border no        |
  | c | O |   the Isle of Wight; also the volcanic    |   equivalents have been          |
  |   | l |   plateaux of Antrim and Inner Hebrides   |   satisfactorily recognised, but |
  | o | i |   and those of the Faeroe Isles and       |   on the Pacific side there are  |
  | r | g |   Iceland. In continental Europe the      |   marine deposits in N. W.       |
  |   | o |   following subdivisions have been        |   Oregon, which may represent    |
  | T | c |   established in descending order: (1)    |   this division. In the interior |
  | e | e |   Aquitanian, (2) Stampian (Rupelian),    |   the equivalent is believed to  |
  | r | n |   (3) Tongrain (Sannoisian).              |   be the fresh-water White River |
  | t | e |                                           |   series, including (1)          |
  | i | . |                                           |   _Protoceras_ beds, (2)         |
  | a |   |                                           |   _Oreodon_ beds, and (3)        |
  | r |   |                                           |   _Titanothervum_ beds.          |
  | y +---+-------------------------------------------+----------------------------------+
  | . |   | Barton sands and clays; Ludian series of  | Woodstock and Aquia Creek groups |
  |   |   |   France.                                 |   of Potomac River; Vicksburg,   |
  |   |   | Bracklesham Beds; Lutetian (Calcaire      |   Jackson, Claiborne, Buhrstone, |
  |   | E |   grossier and Caillasses) of Paris       |   and Lignitic groups of         |
  |   | o |   basin.                                  |   Mississippi.                   |
  |   | c | London clay, Woolwich and Reading Beds;   | In the interior a thick series of|
  |   | e |   Thanet sands; Ypresian or Londinian of  |   fresh-water formations,        |
  |   | n |   N. France and Belgium; Sparnacian and   |   comprising, in descending      |
  |   | e |   Thanetian groups.                       |   order, the Uinta, Bridger,     |
  |   | . |                                           |   Wind River, Wasatch, Torrejon, |
  |   |   |                                           |   and Puerco groups.             |
  |   |   |                                           | On the Pacific side the marine   |
  |   |   |                                           |   Tejon series of Oregon and     |
  |   |   |                                           |   California.                    |
  |---+---+-------------------------------------------+----------------------------------|
  |   |   |     Upper                                 | On the Atlantic border both      |
  |   |   |     =====                                 |   marine strata and others       |
  |   |   | Danian--wanting in Britain; uppermost     |   containing a terrestrial flora |
  |   |   |   limestone of Denmark.                   |   represent the Cretaceous series|
  |   |   | Senonian--Upper Chalk with Flints of      |   of formations.                 |
  |   |   |   England; Aturian and Emscherian stages  | In the interior there is also a  |
  |   |   |   on the European continent.              |   commingling of marine with     |
  |   |   | Turonian--Middle Chalk with few flints,   |   lacustrine deposits. At the top|
  |   |   |   and comprising the Angoumian and stages.|   lies the Laramie or Lignitic   |
  |   | C | Cenomanian--Lower Chalk and Chalk Marl.   |   series with an abundant        |
  |   | r |                                           |   terrestrial flora, passing down|
  |   | e |     Lower                                 |   into the lacustrine and        |
  |   | t |     =====                                 |   brackish-water Montana series. |
  |   | a | Albian--Upper Greensand and Gault.        |   Of older date, the Colorado    |
  |   | c | Aptian--Lower Greensand; Marls and        |   series contains an abundant    |
  |   | e |   limestones of Provence, &c.             |   marine fauna, yet includes also|
  |   | o | Urgonian (Barremian)--Atherfield clay;    |   some Niobrara marls and        |
  |   | u |   massive Hippurite limestones of         |   limestones are likewise of     |
  |   | s |   southern France.                        |   marine origin, but the lower   |
  |   | . | Neocomian--Weald clay and Hastings sand;  |   members of the series (Benton  |
  |   |   |   Hauterivian and Valanginian sub-stages  |   and Dakota) show another great |
  |   |   |   of Switzerland and France.              |   representation of fresh-water  |
  | M |   |                                           |   sedimentation with lignites and|
  | e |   |                                           |   coals.                         |
  | s |   |                                           | In California a vast succession  |
  | o |   |                                           |   of marine deposits (Shasta-    |
  | z |   |                                           |   Chico) represents the          |
  | o |   |                                           |   Cretaceous system; and in      |
  | i |   |                                           |   western British N. America     |
  | c |   |                                           |   coal-seams also occur.         |
  |   +---+-------------------------------------------+----------------------------------+
  | o |   | Purbeckian--Purbeck beds; Münder Mergel;  | Representatives of the Middle and|
  | r |   |   largely present in Westphalia.          |   lower Jurassic formations have |
  |   |   | Portlandian--Portland group of England,   |   been found in California and   |
  | S |   |   represented in S. France by the thick   |   Oregon, and farther north among|
  | e |   |   Tithonian limestones.                   |   the Arctic islands.            |
  | c |   | Kimmeridgian--Kimmeridge Clay of England; | Strata containing Lower Jurassic |
  | o |   |   Virgulian and Pterocerian groups of N.  |   marine fossils appear in       |
  | n | J |   France; represented by thick limestones |   Wyoming and Dakota; and above  |
  | d | u |   in the Mediterranean basin.             |   them come the _Atlantosaurus_  |
  | a | r | Corallian--Coral Rag, Coralline Oolite;   |   and _Baptanodon_ beds, which   |
  | r | a |   Sequanian stages of the Continent,      |   have yielded so large a        |
  | y | s |   comprising the sub-stages of Astartian  |   variety of deinosaurs and other|
  | . | s |   and Rauracian.                          |   vertebrates, and especially the|
  |   | i | Oxfordian--Oxford Clay; Axgovian and      |   remains of a number of genera  |
  |   | c |   Neuvizyan stages.                       |   of small mammals.              |
  |   | . | Callovian--Kellaways Rock, Divesian       |                                  |
  |   |   |   sub-stage of N. France.                 |                                  |
  |   |   | Bathonian--series of English strata from  |                                  |
  |   |   |   Cornbrash down to Fuller's Earth.       |                                  |
  |   |   | Bajocian--Inferior Oolite of England.     |                                  |
  |   |   | Lassic--divisible into (1) Upper Lias     |                                  |
  |   |   |   or Toarcian, (2) Middle Lias, Marlstone |                                  |
  |   |   |   or Charmouthian, (3) Lower Lias of      |                                  |
  |   |   |   Sinemurian and Hettangian.              |                                  |
  |   +---+-------------------------------------------+----------------------------------+
  |   |   | In Germany and western Europe this        | In New York, Connecticut, New    |
  |   | T |   division represents the deposits of     |   Brunswick, and Nova Scotia     |
  |   | r |   inland seas or lagoons, and is divisible|   a series of red sandstone      |
  |   | i |   into the following stages in descending |   (Newark series) contains land- |
  |   | a |   order: (1) Rhaetic, (2) Keuper, (3)     |   plants and labyrinthodonts     |
  |   | s |   Muschelkalk, (4) Bunter.  In the        |   like the lagoon type of central|
  |   | s |   eastern Alps and the Mediterranean      |   and western Europe. On the     |
  |   | i |   basin the contemporaneous sedimentary   |   Pacific slope, however, marine |
  |   | c |   formations are those of open clear      |   equivalents occur, representing|
  |   | . |   sea, in which a thickness of many       |   the pelagic type of south-     |
  |   |   |   thousand feet of strata was accumulated.|   eastern Europe.                |
  +---+---+-------------------------------------------+----------------------------------+
  |   | P | Thuringian--Zechstein, Magnesian          | To this division of the geologi- |
  |   | e |   Limestone; named from its development   |   cal record the Upper Barren    |
  |   | r |   in Thuringia; well represented          |   Measures of the coal-fields of |
  |   | m |   also in Saxony, Bavaria and Bohemia.    |   Pennsylvania, Prince Edward    |
  |   | i | Saxonian--Rothliegendes Group; Red        |   Island, Nova Scotia and        |
  |   | a |   Sandstones, &c.                         |   New Brunswick have been        |
  |   | n | Autunian--where the strata present the    |   assigned.                      |
  |   | . |   lagoon facies, well displayed at Autun  | Farther south in Kansas, Texas,  |
  |   |   |   in France; where the marine type is     |   and Nebraska the representa-   |
  |   |   |   predominant, as in Russia, the group    |   tives of the division have an  |
  |   |   |   has been termed Artinskian.             |   abundant marine fauna.         |
  |   +---+-------------------------------------------+----------------------------------+
  |   | C | Stephanian or Uralian--represented in     | Upper productive Coal-measures.  |
  |   | a |   Russia by marine formations, and in     | Lower Barren measures.           |
  |   | r |   central and western Europe by numerous  | Lower productive Coal-measures.  |
  |   | b |   small basins containing a peculiar      | Pottsville conglomerate.         |
  |   | o |   flora and in some places a great variety| Mauch Chunk shales; limestones   |
  |   | n |   of insects.                             |   of Chester, St Louis, &c.      |
  |   | i | Westphalian or Moscovian--Coal-measures,  | Pocono series; Kinderhook        |
  |   | f |   Millstone Grit.                         |   limestone.                     |
  |   | e | Culm or Dinantian--Carboniferous Limestone|                                  |
  |   | r |   and Calciferous Sandstone series.       |                                  |
  |   | o |                                           |                                  |
  |   | u |                                           |                                  |
  |   | s |                                           |                                  |
  |   | . |                                           |                                  |
  |   +---+-------------------------------------------+----------------------------------+
  |   |                            Devonian and Old Red Sandstone.                       |
  | P +----------------------+------------------------+----------------------------------+
  | a |     Devonian type.   |  Old Red Sandstone     |                                  |
  | l |                      |       type.            |                                  |
  | a +----------------------+------------------------+ / Catskill red sandstone; Old    |
  | e |         / Famennian. | Yellow and red         | |   Red Sandstone type: the      |
  | o | Upper  <             |   sandstone with       |<    strata below show the        |
  | z |         \ Frasnian.  |   _Holoptychius_,      | |   Devonian type.               |
  | o |                      |   _Bothriolepis_,&c.   | | Chemung Group.                 |
  | i |                      |                        | \ Genesee   "                    |
  | c |                      |                        |                                  |
  |   |         / Givetian.  | Caithness Flagstones   |                                  |
  | o | Middle <             |   with _Osteolepus_,   | / Hamilton Group.                |
  | r |         \ Eifelian.  |   _Dipterus_,          | \ Marcellus  "                   |
  |   |                      |   _Homosteus_, &c.     |                                  |
  | P |                      |                        |                                  |
  | r |                      | Red and purple         | / Corniferous Limestone. / Upper |
  | i |        /Coblentizian.|   sandstones and       | |                        | Held- |
  | m | Lower <              |   conglomerates with   |<  Onondaga Limestone.   <  erberg|
  | a |        \Gedinnian.   |   _Cephalaspis_,       | |                        \ Group.|
  | r |                      |   _Pteraspis_, &c.     | \ Oriskany Sandstone.            |
  | y +---+------------------+------------------------+----------------------------------+
  | . |   |                                           |  / Lower Helderberg Group.       |
  |   | S |               / Ludlow  Group.            |  | Water-Lime.                   |
  |   | i |   Upper      <  Wenlock   "               | <  Niagara Shale and Limestone.  |
  |   | l |               \ Llandovery"               |  | Clinton Group.                |
  |   | u |                                           |  \ Medina    "                   |
  |   | r |                                           |                                  |
  |   | i |                                           |  / Cincinnati Group.             |
  |   | a |   Lower       / Caradoc or Bala Group.    |  | Utica        "                |
  |   | n | (Ordovician) <  Llandeilo         "       | <  Trenton      "                |
  |   | . |               \ Arenig            "       |  | Chazy        "                |
  |   |   |                                           |  \ Calciferous  "                |
  |   +---+-------------------------------------------+----------------------------------+
  |   | C | Upper or _Olenus_ series--Tremadoc        | Upper or Potsdam series with     |
  |   | a |   slates and _Lingula_ Flags.             |   _Olenus_ and _Dicelocephalus_  |
  |   | m | Middle or _Pardoxides_ series--Menevian   |   fauna.                         |
  |   | b |   Group.                                  | Middle or Acadian series with    |
  |   | r | Lower or _Olenellus_ series--Llanberis    |   _Paradoxides_ fauna.           |
  |   | i |   and Harlech Group, and _Olenellus_-     | Lower or Georgian series with    |
  |   | a |   zone.                                   |   _Olenellus_ fauna.             |
  |   | n |                                           |                                  |
  |   | . |                                           |                                  |
  +---+---+-------------------------------------------+----------------------------------+
  |   |   |                       Archean, Pre-Cambrian, Eozoic.                         |
  +---+---+-------------------------------------------+----------------------------------+
  |   |   | In Scotland, underneath the Cambrian      | In Canada and the Lake Superior  |
  |   |   |   Olenellus group, lies unconformably     |   region of the United States    |
  |   |   |   a mass of red sandstone and con-        |   a vast succession of rocks of  |
  |   |   |   glomerate (Torridonian) 8000 or 10,000  |   Pre-Cambrian age has been      |
  |   |   |   ft. thick, which rests with a strong    |   grouped into the following     |
  |   |   |   gneisses and schists (Lewisian).  A     |   subdivisions in descending     |
  |   |   |   thick series of slates and phyllites    |   order: (1) Keweenwan, lying    |
  |   |   |   lies below the oldest Palaeozoic rocks  |   unconformably on (2) Animikie, |
  |   |   |   in central Europe, with coarse          |   separated by a strong          |
  |   |   |   gneisses below.                         |   unconformability from (3) Upper|
  |   |   |                                           |   Huronian, (4) Lower Huronian   |
  |   |   |                                           |   with an unconformable base, (5)|
  |   |   |                                           |   Goutchiching, (6) Laurentian.  |
  |   |   |                                           |   In the eastern part of Canada, |
  |   |   |                                           |   Newfoundland, &c., and also in |
  |   |   |                                           |   Montana, sedimentary formations|
  |   |   |                                           |   of great thickness below the   |
  |   |   |                                           |   lowest Cambrian zone have been |
  |   |   |                                           |   found to contain some obscure  |
  |   |   |                                           |   organisms.                     |
  +---+---+-------------------------------------------+----------------------------------+

From these general considerations we proceed to inquire how the existing
topographical features of the land arose. Obviously the co-operation of
the two great geological agencies of hypogene and epigene energy, which
have been at work from the beginning of our globe's decipherable
history, must have been the cause to which these features are to be
assigned; and the task of the geologist is to ascertain, if possible,
the part that has been taken by each. There is a natural tendency to see
in a stupendous piece of scenery, such as a deep ravine, a range of
hills, a line of precipice or a chain of mountains, evidence only of
subterranean convulsion; and before the subject was taken up as a matter
of strict scientific induction, an appeal to former cataclysms was
considered a sufficient solution of the problems presented by such
features of landscape. The rise of the modern Huttonian school, however,
led to a more careful examination of these problems. The important share
taken by erosion in the determination of the present features of
landscape was then recognized, while a fuller appreciation of the
relative parts played by the hypogene and epigene causes has gradually
been reached.

1. The study of the progress of denudation at the present time has led
to the conclusion that even if the rate of waste were not more rapid
than it is to-day, it would yet suffice in a comparatively brief
geological period to reduce the dry land to below the sea-level. But not
only would the area of the land be diminished by denudation, it could
hardly fail to be more or less involved in those widespread movements of
subsidence, during which the thick sedimentary formations of the crust
appear to have been accumulated. It is thus manifest that there must
have been from time to time during the history of our globe upward
movements of the crust, whereby the balance between land and sea was
redressed. Proofs of such movements have been abundantly preserved among
the stratified formations. We there learn that the uplifts have usually
followed each other at long intervals between which subsidence
prevailed, and thus that there has been a prolonged oscillation of the
crust over the great continental areas of the earth's surface.

An examination of that surface leads to the recognition of two great
types of upheaval. In the one, the sea-floor, with all its thick
accumulations of sediment, has been carried upwards, sometimes for
several thousand feet, so equably that the strata retain their original
flatness with hardly any sensible disturbance for hundreds of square
miles. In the other type the solid crust has been plicated, corrugated
and dislocated, especially along particular lines, and has attained its
most stupendous disruption in lofty chains of mountains. Between these
two phases of uplift many intermediate stages have been developed,
according to the direction and intensity of the subterranean force and
the varying nature and disposition of the rocks Of the crust.

(a) Where the uplift has extended over wide spaces, without appreciable
deformation of the crust, the flat strata have given rise to low plains,
or if the amount of uprise has been great enough, to high plains,
plateaux or tablelands. The plains of Russia, for example, lie for the
most part on such tracts of equably uplifted strata. The great plains of
the western interior of the United States form a great plateau or
tableland, 5000 or 6000 ft. above the sea, and many thousands of square
miles in extent, on which the Rocky Mountains have been ridged up.

(b) It is in a great mountain-chain that the complicated structures
developed during disturbances of the earth's crust can best be studied
(see Parts IV. and V. of this article), and where the influence of these
structures on the topography of the surface is most effectively
displayed. Such a chain may be the result of one colossal disturbance;
but those of high geological antiquity usually furnish proofs of
successive uplifts with more or less intervening denudation. Formed
along lines of continental displacement in the crust, they have again
and again given relief from the strain of compression by fresh
crumpling, fracture and uprise. The chief guide in tracing these
successive stages of growth is supplied by unconformability. If, for
example, a mountain-range consists of upraised Silurian rocks, upon the
upturned and denuded edges of which the Carboniferous Limestone lies
transgressively, it is clear that its original upheaval must have taken
place in the period of geological time represented by the interval
between the Silurian and the Carboniferous Limestone formations. If, as
the range is followed along its course, the Carboniferous Limestone is
found to be also highly inclined and covered unconformably by the Upper
Coal-measures, a second uplift of that portion of the ground can be
proved to have taken place between the time of the Limestone and that of
the Upper Coal-measures. By this simple and obvious kind of evidence the
relative ages of different mountain-chains may be compared. In most
great chains, however, the rocks have been so intensely crumpled, and
even inverted, that much labour may be required before their true
relations can be determined.

The Alps furnish an instructive example of the long series of
revolutions through which a great mountain-system may have passed before
reaching its present development. The first beginnings of the chain may
have been upraised before the oldest Palaeozoic formations were laid
down. There are at least traces of land and shore-lines in the
Carboniferous period. Subsequent submergences and uplifts appear to have
occurred during the Mesozoic periods. There is evidence that thereafter
the whole region sank deep under the sea, in which the older Tertiary
sediments were accumulated, and which seems to have spread right across
the heart of the Old World. But after the deposition of the Eocene
formations came the gigantic disruptions whereby all the rocks of the
Alpine region were folded over each other, crushed, corrugated,
fractured and displaced, some of their older portions, including the
fundamental gneisses and schists, being squeezed up, torn off, and
pushed horizontally for many miles over the younger rocks. But this
upheaval, though the most momentous, was not the last which the chain
has undergone, for at a later epoch in Tertiary time renewed disturbance
gave rise to a further series of ruptures and plications. The chain thus
successively upheaved has been continuously exposed to denudation and
has consequently lost much of its original height. That it has been left
in a state of instability is indicated by the frequent earthquakes of
the Alpine region, which doubtless arise from the sudden snapping of
rocks under intense strain.

A distinct type of mountain due to direct hypogene action is to be seen
in a volcano. It has been already pointed out (Part IV. sect. 1) that at
the vents which maintain a communication between the molten magma of the
earth's interior and the surface, eruptions take place whereby
quantities of lava and fragmentary materials are heaped round each
orifice of discharge. A typical volcanic mountain takes the form of a
perfect cone, but as it grows in size and its main vent is choked, while
the sides of the cone are unable to withstand the force of the
explosions or the pressure of the ascending column of lava, eruptions
take place laterally, and numerous parasitic cones arise on the flanks
of the parent mountain. Where lava flows out from long fissures, it may
pile up vast sheets of rock, and bury the surrounding country under
several thousand feet of solid stone, covering many hundreds of square
miles. In this way volcanic tablelands have been formed which, attacked
by the denuding forces, are gradually trenched by valleys and ravines,
until the original level surface of the lava-field may be almost or
wholly lost. As striking examples of this physiographical type reference
may be made to the plateau of Abyssinia, the Ghats of India, the
plateaux of Antrim, the Inner Hebrides and Iceland, and the great
lava-plains of the western territories of the United States.

2. But while the subterranean movements have upraised portions of the
surface of the lithosphere above the level of the ocean, and have thus
been instrumental in producing the existing tracts of land, the detailed
topographical features of a landscape are not solely, nor in general
even chiefly, attributable to these movements. From the time that any
portion of the sea-floor appears above sea-level, it undergoes erosion
by the various epigene agents. Each climate and geological region has
its own development of these agents, which include air, aridity, rapid
and frequent alternations of wetness and dryness or of heat and cold,
rain, springs, frosts, rivers, glaciers, the sea, plant and animal life.
In a dry climate subject to great extremes of temperature the character
and rate of decay will differ from those of a moist or an arctic
climate. But it must be remembered that, however much they may vary in
activity and in the results which they effect, the epigene forces work
without intermission, while the hypogene forces bring about the upheaval
of land only after long intervals. Hence, trifling as the results during
a human life may appear, if we realize the multiplying influence of time
we are led to perceive that the apparently feeble superficial agents
can, in the course of ages, achieve stupendous transformations in the
aspect of the land. If this efficacy may be deduced from what can be
seen to be in progress now, it may not less convincingly be shown, from
the nature of the sedimentary rocks of the earth's crust, to have been
in progress from the early beginnings of geological history. Side by
side with the various upheavals and subsidences, there has been a
continuous removal of materials from the land, and an equally persistent
deposit of these materials under water, with the consequent growth of
new rocks. Denudation has been aptly compared to a process of
sculpturing wherein, while each of the implements employed by nature,
like a special kind of graving tool, produces its own characteristic
impress on the land, they all combine harmoniously towards the
achievement of their one common task. Hence the present contours of the
land depend partly on the original configuration of the ground, and the
influence it may have had in guiding the operations of the erosive
agents, partly on the vigour with which these agents perform their work,
and partly on the varying structure and powers of resistance possessed
by the rocks on which the erosion is carried on.

Where a new tract of land has been raised out of the sea by such an
energetic movement as broke up the crust and produced the complicated
structure and tumultuous external forms of a great mountain chain, the
influence of the hypogene forces on the topography attains its highest
development. But even the youngest existing chain has suffered so
greatly from denudation that the aspect which it presented at the time
of its uplift can only be dimly perceived. No more striking illustration
of this feature can be found than that supplied by the Alps, nor one
where the geotectonic structures have been so fully studied in detail.
On the outer flanks of these mountains the longitudinal ridges and
valleys of the Jura correspond with lines of anticline and syncline. Yet
though the dominant topographical elements of the region have obviously
been produced by the plication of the stratified formations, each ridge
has suffered so large an amount of erosion that the younger rocks have
been removed from its crest where the older members of the series are
now exposed to view, while on every slope proofs may be seen of
extensive denudation. If from these long wave-like undulations of the
ground, where the relations between the disposition of the rocks below
and the forms of the surface are so clearly traceable, the observer
proceeds inwards to the main chain, he finds that the plications and
displacements of the various formations assume an increasingly
complicated character; and that although proofs of great denudation
continue to abound, it becomes increasingly difficult to form any
satisfactory conjecture as to the shape of the ground when the upheaval
ended or any reliable estimate of the amount of material which has since
then been removed. Along the central heights the mountains lift
themselves towards the sky like the storm-swept crests of vast
earth-billows. The whole aspect of the ground suggests intense
commotion, and the impression thus given is often much intensified by
the twisted and crumpled strata, visible from a long distance, on the
crags and crests. On this broken-up surface the various agents of
denudation have been ceaselessly engaged since it emerged from the sea.
They have excavated valleys, sometimes along depressions provided for
them by the subterranean disturbances, sometimes down the slopes of the
disrupted blocks of ground. So powerful has been this erosion that
valleys cut out along lines of anticline, which were natural ridges,
have sometimes become more important than those in lines of syncline,
which were structurally depressions. The same subaerial forces have
eroded lake-basins, dug out corries or cirques, notched the ridges,
splintered the crests and furrowed the slopes, leaving no part of the
original surface of the uplifted chain unmodified.

It has often been noted with surprise that features of underground
structure which, it might have been confidently anticipated, should have
exercised a marked influence on the topography of the surface have not
been able to resist the levelling action of the denuding agents, and do
not now affect the surface at all. This result is conspicuously seen in
coal-fields where the strata are abundantly traversed by faults. These
dislocations, having sometimes a displacement of several hundred feet,
might have been expected to break up the surface into a network of
cliffs and plains; yet in general they do not modify the level character
of the ground above. One of the most remarkable faults in Europe is the
great thrust which bounds the southern edge of the Belgian coal-field
and brings the Devonian rocks above the Coal-measures. It can be traced
across Belgium into the Boulonnais, and may not improbably run beneath
the Secondary and Tertiary rocks of the south of England. It is crossed
by the valleys of the Meuse and other northerly-flowing streams. Yet so
indistinctly is it marked in the Meuse valley that no one would suspect
its existence from any peculiarity in the general form of the ground,
and even an experienced geologist, until he had learned the structure of
the district, would scarcely detect any fault at all.

Where faults have influenced the superficial topography, it is usually
by giving rise to a hollow along which the subaerial agents and
especially running water can act effectively. Such a hollow may be
eventually widened and deepened into a valley. On bare crags and crests,
lines of fault are apt to be marked by notches or clefts, and they thus
help to produce the pinnacles and serrated outlines of these exposed
uplands.

It was cogently enforced by Hutton and Playfair, and independently by
Lamarck, that no co-operation of underground agency is needed to produce
such topography as may be seen in a great part of the world, but that if
a tract of sea-floor were upraised into a wide plain, the fall of rain
and the circulation of water over its surface would in the end carve out
such a system of hills and valleys as may be seen on the dry land now.
No such plain would be a dead-level. It would have inequalities on its
surface which would serve as channels to guide the drainage from the
first showers of rain. And these channels would be slowly widened and
deepened until they would become ravines and valleys, while the ground
between them would be left projecting as ridges and hills. Nor would the
erosion of such a system of water-courses require a long series of
geological periods for its accomplishment. From measurements and
estimates of the amount of erosion now taking place in the basin of the
Mississippi river it has been computed that valleys 800 ft. deep might
be carved out in less than a million years. In the vast tablelands of
Colorado and other western regions of the United States an impressive
picture is presented of the results of mere subaerial erosion on
undisturbed and nearly level strata. Systems of stream-courses and
valleys, river gorges unexampled elsewhere in the world for depth and
length, vast winding lines of escarpment, like ranges of sea-cliffs,
terraced slopes rising from plateau to plateau, huge buttresses and
solitary stacks standing like islands out of the plains, great
mountain-masses towering into picturesque peaks and pinnacles cleft by
innumerable gullies, yet everywhere marked by the parallel bars of the
horizontal strata out of which they have been carved--these are the
orderly symmetrical characteristics of a country where the scenery is
due entirely to the action of subaerial agents on the one hand and the
varying resistance of perfectly regular stratified rocks on the other.

The details of the sculpture of the land have mainly depended on the
nature of the materials on which nature's erosive tools have been
employed. The joints by which all rocks are traversed have been
especially serviceable as dominant lines down which the rain has
filtered, up which the springs have risen and into which the frost
wedges have been driven. On the high bare scarps of a lofty mountain the
inner structure of the mass is laid open, and there the system of joints
even more than faults is seen to have determined the lines of crest, the
vertical walls of cliff and precipice, the forms of buttress and recess,
the position of cleft and chasm, the outline of spire and pinnacle. On
the lower slopes, even under the tapestry of verdure which nature
delights to hang where she can over her naked rocks, we may detect the
same pervading influence of the joints upon the forms assumed by ravines
and crags. Each kind of stone, too, gives rise to its own characteristic
form of scenery. Massive crystalline rocks, such as granite, break up
along their joints and often decay into sand or earth along their
exposed surfaces, giving rise to rugged crags with long talus slopes at
their base. The stratified rocks besides splitting at their joints are
especially distinguished by parallel ledges, cornices and recesses,
produced by the irregular decay of their component strata, so that they
often assume curiously architectural types of scenery. But besides this
family feature they display many minor varieties of aspect according to
their lithological composition. A range of sandstone hills, for example,
presents a marked contrast to one of limestone, and a line of chalk
downs to the escarpments formed by alternating bands of harder and
softer clays and shales.

It may suffice here merely to allude to a few of the more important
parts of the topography of the land in their relation to physiographical
geology. A true mountain-chain, viewed from the geological side, is a
mass of high ground which owes its prominence to a ridging-up of the
earth's crust, and the intense plication and rupture of the rocks of
which it is composed. But ranges of hills almost mountainous in their
bulk may be formed by the gradual erosion of valleys out of a mass of
original high ground, such as a high plateau or tableland. Eminences
which have been isolated by denudation from the main mass of the
formations of which they originally formed part are known as "outliers"
or "hills of circumdenudation."

Tablelands, as already pointed out, may be produced either by the
upheaval of tracts of horizontal strata from the sea-floor into land; or
by the uprise of plains of denudation, where rocks of various
composition, structure and age have been levelled down to near or below
the level of the sea by the co-operation of the various erosive agents.
Most of the great tablelands of the globe are platforms of
little-disturbed strata which have been upraised bodily to a
considerable elevation. No sooner, however, are they placed in that
position than they are attacked by running water, and begin to be
hollowed out into systems of valleys. As the valleys sink, the platforms
between them grow into narrower and more definite ridges, until
eventually the level tableland is converted into a complicated network
of hills and valleys, wherein, nevertheless, the key to the whole
arrangement is furnished by a knowledge of the disposition and effects
of the flow of water. The examples of this process brought to light in
Colorado, Wyoming, Nevada and the other western regions by Newberry,
King, Hayden, Powell and other explorers, are among the most striking
monuments of geological operations in the world.

Examples of ancient and much decayed tablelands formed by the denudation
of much disturbed rocks are furnished by the Highlands of Scotland and
of Norway. Each of these tracts of high ground consists of some of the
oldest and most dislocated formations of Europe, which at a remote
period were worn down into a plain, and in that condition may have lain
long submerged under the sea and may possibly have been overspread there
with younger formations. Having at a much later time been raised several
thousand feet above sea-level the ancient platforms of Britain and
Scandinavia have been since exposed to denudation, whereby each of them
has been so deeply channeled into glens and fjords that it presents
to-day a surface of rugged hills, either isolated or connected along the
flanks, while only fragments of the general surface of the tableland can
here and there be recognized amidst the general destruction.

Valleys have in general been hollowed out by the greater erosive action
of running water along the channels of drainage. Their direction has
been probably determined in the great majority of cases by
irregularities of the surface along which the drainage flowed on the
first emergence of the land. Sometimes these irregularities have been
produced by folds of the terrestrial crust, sometimes by faults,
sometimes by the irregularities on the surface of an uplifted platform
of deposition or of denudation. Two dominant trends may be observed
among them. Some are longitudinal and run along the line of flexures in
the upraised tract of land, others are transverse where the drainage has
flowed down the slopes of the ridges into the longitudinal valleys or
into the sea. The forms of valleys have been governed partly by the
structure and composition of the rocks, and partly by the relative
potency of the different denuding agents. Where the influence of rain
and frost has been slight, and the streams, supplied from distant
sources, have had sufficient declivity, deep, narrow, precipitous
ravines or gorges have been excavated. The canyons of the arid region of
the Colorado are a magnificent example of this result. Where, on the
other hand, ordinary atmospheric action has been more rapid, the sides
of the river channels have been attacked, and open sloping glens and
valleys have been hollowed out. A gorge or defile is usually due to the
action of a waterfall, which, beginning with some abrupt declivity or
precipice in the course of the river when it first commenced to flow, or
caused by some hard rock crossing the channel, has eaten its way
backward.

Lakes have been already referred to, and their modes of origin have been
mentioned. As they are continually being filled up with the detritus
washed into them from the surrounding regions they cannot be of any
great geological antiquity, unless where by some unknown process their
basins are from time to time widened and deepened.

In the general subaerial denudation of a country, innumerable minor
features are worked out as the structure of the rocks controls the
operations of the eroding agents. Thus, among comparatively undisturbed
strata, a hard bed resting upon others of a softer kind is apt to form
along its outcrop a line of cliff or escarpment. Though a long range of
such cliffs resembles a coast that has been worn by the sea, it may be
entirely due to mere atmospheric waste. Again, the more resisting
portions of a rock may be seen projecting as crags or knolls. An igneous
mass will stand out as a bold hill from amidst the more decomposable
strata through which it has risen. These features, often so marked on
the lower grounds, attain their most conspicuous development among the
higher and barer parts of the mountains, where subaerial disintegration
is most rapid. The torrents tear out deep gullies from the sides of the
declivities. Corries or cirques are scooped out on the one hand and
naked precipices are left on the other. The harder bands of rock project
as massive ribs down the slopes, shoot up into prominent _aiguilles_, or
help to give to the summits the notched saw-like outlines they so often
present.

The materials worn from the surface of the higher are spread out over
the lower grounds. The streams as they descend begin to drop their
freight of sediment when, by the lessening of their declivity, their
carrying power is diminished. The great plains of the earth's surface
are due to this deposit of gravel, sand and loam. They are thus
monuments at once of the destructive and reproductive processes which
have been in progress unceasingly since the first land rose above the
sea and the first shower of rain fell. Every pebble and particle of
their soil, once part of the distant mountains, has travelled slowly and
fitfully to lower levels. Again and again have these materials been
shifted, ever moving downward and sea-ward. For centuries, perhaps, they
have taken their share in the fertility of the plains and have
ministered to the nurture of flower and tree, of the bird of the air,
the beast of the field and of man himself. But their destiny is still
the great ocean. In that bourne alone can they find undisturbed repose,
and there, slowly accumulating in massive beds, they will remain until,
in the course of ages, renewed upheaval shall raise them into future
land, there once more to pass through the same cycle of change. (A. Ge.)

  LITERATURE.--_Historical_: The standard work is Karl A. von Zittel's
  _Geschichte der Geologie und Paläontologie_ (1899), of which there is
  an abbreviated, but still valuable, English translation; D'Archiac,
  _Histoire des progrès de la géologie_, deals especially with the
  period 1834-1850; Keferstein, _Geschichte und Literatur der
  Geognosie_, gives a summary up to 1840; while Sir A. Geikie's
  _Founders of Geology_ (1897; 2nd ed., 1906) deals more particularly
  with the period 1750-1820. General treatises: Sir Charles Lyell's
  _Principles of Geology_ is a classic. Of modern English works, Sir A.
  Geikie's _Text Book of Geology_ (4th ed., 1903) occupies the first
  place; the work of T.C. Chamberlin and R.D. Salisbury, _Geology; Earth
  History_ (3 vols., 1905-1906), is especially valuable for American
  geology. A. de Lapparent's _Traité de géologie_ (5th ed., 1906), is
  the standard French work. H. Credner's _Elemente der Geologie_ has
  gone through several editions in Germany. Dynamical and
  physiographical geology are elaborately treated by E. Suess, _Das
  Antlitz der Erde_, translated into English, with the title _The Face
  of the Earth_. The practical study of the science is treated of by F.
  von Richthofen, _Führer für Forschungsreisende_ (1886); G.A. Cole,
  _Aids in Practical Geology_ (5th ed., 1906); A. Geikie, _Outlines of
  Field Geology_ (5th ed., 1900). The practical applications of Geology
  are discussed by J.V. Elsden, _Applied Geology_ (1898-1899). The
  relations of Geology to scenery are dealt with by Sir A. Geikie,
  _Scenery of Scotland_ (3rd ed., 1901); J.E. Marr, _The Scientific
  Study of Scenery_ (1900); Lord Avebury, _The Scenery of Switzerland_
  (1896); _The Scenery of England_ (1902); and J. Geikie, _Earth
  Sculpture_ (1898). A detailed bibliography is given in Sir A. Geikie's
  _Text Book of Geology_. See also the separate articles on geological
  subjects for special references to authorities.


FOOTNOTES:

  [1] In De Luc's _Lettres physiques et morales sur les montagnes_
    (1778), the word "cosmology" is used for our science, the author
    stating that "geology" is more appropriate, but it "was not a word in
    use." In a completed edition, published in 1779, the same statement
    is made, but "geology" occurs in the text; in the same year De
    Saussure used the word without any explanation, as if it were well
    known.

  [2] The subject of the age of the earth has also been discussed by
    Professor J. Joly and Professor W.J. Sollas. The former geologist,
    approaching the question from a novel point of view, has estimated
    the total quantity of sodium in the water of the ocean and the
    quantity of that element received annually by the ocean from the
    denudation of the land. Dividing the one sum by the other, he arrives
    at the result that the probable age of the earth is between 90 and
    100 millions of years (_Trans. Roy. Dublin Soc._ ser. ii. vol. vii.,
    1899, p. 23: _Geol. Mag._, 1900, p. 220). Professor Sollas believes
    that this limit exceeds what is required for the evolution of
    geological history, that the lower limit assigned by Lord Kelvin
    falls short of what the facts demand, and that geological time will
    probably be found to have been comprised within some indeterminate
    period between these limits. (Address to Section C, _Brit. Assoc.
    Report_, 1900; _Age of the Earth_, London, 1905.)




GEOMETRICAL CONTINUITY. In a report of the Institute prefixed to Jean
Victor Poncelet's _Traité des propriétés projectives des figures_
(Paris, 1822), it is said that he employed "ce qu'il appelle le principe
de continuité." The law or principle thus named by him had, he tells us,
been tacitly assumed as axiomatic by "les plus savans géomètres." It had
in fact been enunciated as "lex continuationis," and "la loi de la
continuité," by Gottfried Wilhelm Leibnitz (Oxf. N.E.D.), and previously
under another name by Johann Kepler in cap. iv. 4 of his _Ad Vitellionem
paralipomena quibus astronomiae pars optica traditur_ (Francofurti,
1604). Of sections of the cone, he says, there are five species from the
"recta linea" or line-pair to the circle. From the line-pair we pass
through an infinity of hyperbolas to the parabola, and thence through an
infinity of ellipses to the circle. Related to the sections are certain
remarkable points which have no name. Kepler calls them foci. The circle
has one focus at the centre, an ellipse or hyperbola two foci
equidistant from the centre. The parabola has one focus within it, and
another, the "caecus focus," which may be imagined to be _at infinity_
on the axis _within or without the curve_. The line from it to any point
of the section is parallel to the axis. To carry out the analogy we must
speak paradoxically, and say that the line-pair likewise has foci, which
in this case coalesce as in the circle and fall upon the lines
themselves; for our geometrical terms should be subject to analogy.
Kepler dearly loves analogies, his most trusty teachers, acquainted with
all the secrets of nature, "_omnium naturae arcanorum conscios_." And
they are to be especially regarded in geometry as, by the use of
"however absurd expressions," classing extreme limiting forms with an
infinity of intermediate cases, and placing the whole essence of a thing
clearly before the eyes.

Here, then, we find formulated by Kepler the doctrine of the concurrence
of parallels at a single point at infinity and the principle of
continuity (under the name analogy) in relation to the infinitely great.
Such conceptions so strikingly propounded in a famous work could not
escape the notice of contemporary mathematicians. Henry Briggs, in a
letter to Kepler from Merton College, Oxford, dated "10 Cal. Martiis
1625," suggests improvements in the _Ad Vitellionem paralipomena_, and
gives the following construction: Draw a line CBADC, and let an ellipse,
a parabola, and a hyperbola have B and A for focus and vertex. Let CC
be the other foci of the ellipse and the hyperbola. Make AD equal to AB,
and with centres CC and radius in each case equal to CD describe
circles. Then any point of the ellipse is equidistant from the focus B
and one circle, and any point of the hyperbola from the focus B and the
other circle. Any point P of the parabola, in which the second focus is
missing or infinitely distant, is equidistant from the focus B and the
line through D which we call the directrix, this taking the place of
either circle when its centre C is at infinity, and every line CP being
then parallel to the axis. Thus Briggs, and we know not how many "savans
géomètres" who have left no record, had already taken up the new
doctrine in geometry in its author's lifetime. Six years after Kepler's
death in 1630 Girard Desargues, "the Monge of his age," brought out the
first of his remarkable works founded on the same principles, a short
tract entitled _Méthode universelle de mettre en perspective les objets
donnés réellement ou en devis_ (Paris, 1636); but "Le privilége étoit de
1630." (Poudra, _[OE]uvres de Des._, i. 55). Kepler as a modern geometer
is best known by his _New Stereometry of Wine Casks_ (Lincii, 1615), in
which he replaces the circuitous Archimedean method of exhaustion by a
direct "royal road" of infinitesimals, treating a vanishing arc as a
straight line and regarding a curve as made up of a succession of short
chords. Some 2000 years previously one Antipho, probably the well-known
opponent of Socrates, has regarded a circle in like manner as the
limiting form of a many-sided inscribed rectilinear figure. Antipho's
notion was rejected by the men of his day as unsound, and when
reproduced by Kepler it was again stoutly opposed as incapable of any
sort of geometrical demonstration--not altogether without reason, for it
rested on an assumed law of continuity rather than on palpable proof.

To complete the theory of continuity, the one thing needful was the idea
of imaginary points implied in the algebraical geometry of René
Descartes, in which equations between variables representing
co-ordinates were found often to have imaginary roots. Newton, in his
two sections on "Inventio orbium" (_Principia_ i. 4, 5), shows in his
brief way that he is familiar with the principles of modern geometry. In
two propositions he uses an auxiliary line which is supposed to cut the
conic in X and Y, but, as he remarks at the end of the second (prop.
24), it may not cut it at all. For the sake of brevity he passes on at
once with the observation that the required constructions are evident
from the case in which the line cuts the trajectory. In the scholium
appended to prop. 27, after saying that an asymptote is a tangent at
infinity, he gives an unexplained general construction for the axes of a
conic, which seems to imply that it has asymptotes. In all such cases,
having equations to his loci in the background, he may have thought of
elements of the figure as passing into the imaginary state in such
manner as not to vitiate conclusions arrived at on the hypothesis of
their reality.

Roger Joseph Boscovich, a careful student of Newton's works, has a full
and thorough discussion of geometrical continuity in the third and last
volume of his _Elementa universae matheseos_ (ed. prim. Venet, 1757),
which contains _Sectionum conicarum elementa nova quadam methodo
concinnata et dissertationem de transformatione locorum geometricorum,
ubi de continuitatis lege, et de quibusdam infiniti mysteriis_. His
first principle is that all varieties of a defined locus have the same
properties, so that what is demonstrable of one should be demonstrable
in like manner of all, although some artifice may be required to bring
out the underlying analogy between them. The opposite extremities of an
infinite straight line, he says, are to be regarded as joined, as if the
line were a circle having its centre at the infinity on either side of
it. This leads up to the idea of a _veluti plus quam infinita extensio_,
a line-circle containing, as we say, the line infinity. Change from the
real to the imaginary state is contingent upon the passage of some
element of a figure through zero or infinity and never takes place _per
saltum_. Lines being some positive and some negative, there must be
negative rectangles and negative squares, such as those of the exterior
diameters of a hyperbola. Boscovich's first principle was that of
Kepler, by whose _quantumvis absurdis locutionibus_ the boldest
applications of it are covered, as when we say with Poncelet that all
concentric circles in a plane touch one another in two imaginary fixed
points at infinity. In G.K. Ch. von Staudt's _Geometrie der Lage and
Beiträge zur G. der L._ (Nürnberg, 1847, 1856-1860) the geometry of
position, including the extension of the field of pure geometry to the
infinite and the imaginary, is presented as an independent science,
"welche des Messens nicht bedarf." (See GEOMETRY: _Projective_.)

Ocular illusions due to distance, such as Roger Bacon notices in the
_Opus majus_ (i. 126, ii. 108, 497; Oxford, 1897), lead up to or
illustrate the mathematical uses of the infinite and its reciprocal the
infinitesimal. Specious objections can, of course, be made to the
anomalies of the law of continuity, but they are inherent in the higher
geometry, which has taught us so much of the "secrets of nature."
Kepler's excursus on the "analogy" between the conic sections
hereinbefore referred to is given at length in an article on "The
Geometry of Kepler and Newton" in vol. xviii. of the _Transactions of
the Cambridge Philosophical Society_ (1900). It had been generally
overlooked, until attention was called to it by the present writer in a
note read in 1880 (_Proc. C.P.S._ iv. 14-17), and shortly afterwards in
_The Ancient and Modern Geometry of Conics, with Historical Notes and
Prolegomena_ (Cambridge 1881).     (C. T.*)




GEOMETRY, the general term for the branch of mathematics which has for
its province the study of the properties of space. From experience, or
possibly intuitively, we characterize existent space by certain
fundamental qualities, termed axioms, which are insusceptible of proof;
and these axioms, in conjunction with the mathematical entities of the
point, straight line, curve, surface and solid, appropriately defined,
are the premises from which the geometer draws conclusions. The
geometrical axioms are merely conventions; on the one hand, the system
may be based upon inductions from experience, in which case the deduced
geometry may be regarded as a branch of physical science; or, on the
other hand, the system may be formed by purely logical methods, in which
case the geometry is a phase of pure mathematics. Obviously the geometry
with which we are most familiar is that of existent space--the
three-dimensional space of experience; this geometry may be termed
Euclidean, after its most famous expositor. But other geometries exist,
for it is possible to frame systems of axioms which definitely
characterize some other kind of space, and from these axioms to deduce a
series of non-contradictory propositions; such geometries are called
non-Euclidean.

It is convenient to discuss the subject-matter of geometry under the
following headings:

I. _Euclidean Geometry_: a discussion of the axioms of existent space
and of the geometrical entities, followed by a synoptical account of
Euclid's Elements.

II. _Projective Geometry_: primarily Euclidean, but differing from I. in
employing the notion of geometrical continuity (q.v.)--points and lines
at infinity.

III. _Descriptive Geometry_: the methods for representing upon planes
figures placed in space of three dimensions.

IV. _Analytical Geometry_: the representation of geometrical figures and
their relations by algebraic equations.

V. _Line Geometry_: an analytical treatment of the line regarded as the
space element.

VI. _Non-Euclidean Geometry_: a discussion of geometries other than that
of the space of experience.

VII. _Axioms of Geometry_: a critical analysis of the foundations of
geometry.

  Special subjects are treated under their own headings: e.g.
  PROJECTION, PERSPECTIVE; CURVE, SURFACE; CIRCLE, CONIC SECTION;
  TRIANGLE, POLYGON, POLYHEDRON; there are also articles on special
  curves and figures, e.g. ELLIPSE, PARABOLA, HYPERBOLA; TETRAHEDRON,
  CUBE, OCTAHEDRON, DODECAHEDRON, ICOSAHEDRON; CARDIOID, CATENARY,
  CISSOID, CONCHOID, CYCLOID, EPICYCLOID, LIMAÇON, OVAL, QUADRATRIX,
  SPIRAL, &c.

_History._--The origin of geometry (Gr. [Greek: gê], earth, [Greek:
metron], a measure) is, according to Herodotus, to be found in the
etymology of the word. Its birthplace was Egypt, and it arose from the
need of surveying the lands inundated by the Nile floods. In its
infancy it therefore consisted of a few rules, very rough and
approximate, for computing the areas of triangles and quadrilaterals;
and, with the Egyptians, it proceeded no further, the geometrical
entities--the point, line, surface and solid--being only discussed in so
far as they were involved in practical affairs. The point was realized
as a mark or position, a straight line as a stretched string or the
tracing of a pole, a surface as an area; but these units were not
abstracted; and for the Egyptians geometry was only an art--an auxiliary
to surveying.[1] The first step towards its elevation to the rank of a
science was made by Thales (q.v.) of Miletus, who transplanted the
elementary Egyptian mensuration to Greece. Thales clearly abstracted the
notions of points and lines, founding the geometry of the latter unit,
and discovering _per saltum_ many propositions concerning areas, the
circle, &c. The empirical rules of the Egyptians were corrected and
developed by the Ionic School which he founded, especially by
Anaximander and Anaxagoras, and in the 6th century B.C. passed into the
care of the Pythagoreans. From this time geometry exercised a powerful
influence on Greek thought. Pythagoras (q.v.), seeking the key of the
universe in arithmetic and geometry, investigated logically the
principles underlying the known propositions; and this resulted in the
formulation of definitions, axioms and postulates which, in addition to
founding a _science_ of geometry, permitted a crystallization,
fractional, it is true, of the amorphous collection of material at hand.
Pythagorean geometry was essentially a geometry of areas and solids; its
goal was the regular solids--the tetrahedron, cube, octahedron,
dodecahedron and icosahedron--which symbolized the five elements of
Greek cosmology. The geometry of the circle, previously studied in Egypt
and much more seriously by Thales, was somewhat neglected, although this
curve was regarded as the most perfect of all plane figures and the
sphere the most perfect of all solids. The circle, however, was taken up
by the Sophists, who made most of their discoveries in attempts to solve
the classical problems of squaring the circle, doubling the cube and
trisecting an angle. These problems, besides stimulating pure geometry,
i.e. the geometry of constructions made by the ruler and compasses,
exercised considerable influence in other directions. The first problem
led to the discovery of the method of _exhaustion_ for determining
areas. Antiphon inscribed a square in a circle, and on each side an
isosceles triangle having its vertex on the circle; on the sides of the
octagon so obtained, isosceles triangles were again constructed, the
process leading to inscribed polygons of 8, 16 and 32 sides; and the
areas of these polygons, which are easily determined, are successive
approximations to the area of the circle. Bryson of Heraclea took an
important step when he circumscribed, in addition to inscribing,
polygons to a circle, but he committed an error in treating the circle
as the mean of the two polygons. The method of Antiphon, in assuming
that by continued division a polygon can be constructed coincident with
the circle, demanded that magnitudes are not infinitely divisible. Much
controversy ranged about this point; Aristotle supported the doctrine of
infinite divisibility; Zeno attempted to show its absurdity. The
mechanical tracing of loci, a principle initiated by Archytas of
Tarentum to solve the last two problems, was a frequent subject for
study, and several mechanical curves were thus discovered at subsequent
dates (cissoid, conchoid, quadratrix). Mention may be made of
Hippocrates, who, besides developing the known methods, made a study of
similar figures, and, as a consequence, of proportion. This step is
important as bringing into line discontinuous number and continuous
magnitude.

A fresh stimulus was given by the succeeding Platonists, who, accepting
in part the Pythagorean cosmology, made the study of geometry
preliminary to that of philosophy. The many discoveries made by this
school were facilitated in no small measure by the clarification of the
axioms and definitions, the logical sequence of propositions which was
adopted, and, more especially, by the formulation of the analytic
method, i.e. of assuming the truth of a proposition and then reasoning
to a known truth. The main strength of the Platonist geometers lies in
stereometry or the geometry of solids. The Pythagoreans had dealt with
the sphere and regular solids, but the pyramid, prism, cone and cylinder
were but little known until the Platonists took them in hand. Eudoxus
established their mensuration, proving the pyramid and cone to have
one-third the content of a prism and cylinder on the same base and of
the same height, and was probably the discoverer of a proof that the
volumes of spheres are as the cubes of their radii. The discussion of
sections of the cone and cylinder led to the discovery of the three
curves named the parabola, ellipse and hyperbola (see CONIC SECTION); it
is difficult to over-estimate the importance of this discovery; its
investigation marks the crowning achievement of Greek geometry, and led
in later years to the fundamental theorems and methods of modern
geometry.

The presentation of the subject-matter of geometry as a connected and
logical series of propositions, prefaced by [Greek: Horoi] or
foundations, had been attempted by many; but it is to Euclid that we owe
a complete exposition. Little indeed in the _Elements_ is probably
original except the arrangement; but in this Euclid surpassed such
predecessors as Hippocrates, Leon, pupil of Neocleides, and Theudius of
Magnesia, devising an apt logical model, although when scrutinized in
the light of modern mathematical conceptions the proofs are riddled with
fallacies. According to the commentator Proclus, the _Elements_ were
written with a twofold object, first, to introduce the novice to
geometry, and secondly, to lead him to the regular solids; conic
sections found no place therein. What Euclid did for the line and
circle, Apollonius did for the conic sections, but there we have a
discoverer as well as editor. These two works, which contain the
greatest contributions to ancient geometry, are treated in detail in
Section I. _Euclidean Geometry_ and the articles EUCLID; CONIC SECTION;
APPOLONIUS. Between Euclid and Apollonius there flourished the
illustrious Archimedes, whose geometrical discoveries are mainly
concerned with the mensuration of the circle and conic sections, and of
the sphere, cone and cylinder, and whose greatest contribution to
geometrical method is the elevation of the method of exhaustion to the
dignity of an instrument of research. Apollonius was followed by
Nicomedes, the inventor of the conchoid; Diocles, the inventor of the
cissoid; Zenodorus, the founder of the study of isoperimetrical figures;
Hipparchus, the founder of trigonometry; and Heron the elder, who wrote
after the manner of the Egyptians, and primarily directed attention to
problems of practical surveying.

Of the many isolated discoveries made by the later Alexandrian
mathematicians, those of Menelaus are of importance. He showed how to
treat spherical triangles, establishing their properties and determining
their congruence; his theorem on the products of the segments in which
the sides of a triangle are cut by a line was the foundation on which
Carnot erected his theory of transversals. These propositions, and also
those of Hipparchus, were utilized and developed by Ptolemy (q.v.), the
expositor of trigonometry and discoverer of many isolated propositions.
Mention may be made of the commentator Pappus, whose _Mathematical
Collections_ is valuable for its wealth of historical matter; of Theon,
an editor of Euclid's _Elements_ and commentator of Ptolemy's
_Almagest_; of Proclus, a commentator of Euclid; and of Eutocius, a
commentator of Apollonius and Archimedes.

The Romans, essentially practical and having no inclination to study
science _qua_ science, only had a geometry which sufficed for surveying;
and even here there were abundant inaccuracies, the empirical rules
employed being akin to those of the Egyptians and Heron. The Hindus,
likewise, gave more attention to computation, and their geometry was
either of Greek origin or in the form presented in trigonometry, more
particularly connected with arithmetic. It had no logical foundations;
each proposition stood alone; and the results were empirical. The Arabs
more closely followed the Greeks, a plan adopted as a sequel to the
translation of the works of Euclid, Apollonius, Archimedes and many
others into Arabic. Their chief contribution to geometry is exhibited in
their solution of algebraic equations by intersecting conics, a step
already taken by the Greeks in isolated cases, but only elevated into a
_method_ by Omar al Hayyami, who flourished in the 11th century. During
the middle ages little was added to Greek and Arabic geometry. Leonardo
of Pisa wrote a _Practica geometriae_ (1220), wherein Euclidean methods
are employed; but it was not until the 14th century that geometry,
generally Euclid's _Elements_, became an essential item in university
curricula. There was, however, no sign of original development, other
branches of mathematics, mainly algebra and trigonometry, exercising a
greater fascination until the 16th century, when the subject again came
into favour.

The extraordinary mathematical talent which came into being in the 16th
and 17th centuries reacted on geometry and gave rise to all those
characters which distinguish modern from ancient geometry. The first
innovation of moment was the formulation of the principle of geometrical
continuity by Kepler. The notion of infinity which it involved permitted
generalizations and systematizations hitherto unthought of (see
GEOMETRICAL CONTINUITY); and the method of indefinite division applied
to rectification, and quadrature and cubature problems avoided the
cumbrous method of exhaustion and provided more accurate results.
Further progress was made by Bonaventura Cavalieri, who, in his
_Geometria indivisibilibus continuorum_ (1620), devised a method
intermediate between that of exhaustion and the infinitesimal calculus
of Leibnitz and Newton. The logical basis of his system was corrected by
Roberval and Pascal; and their discoveries, taken in conjunction with
those of Leibnitz, Newton, and many others in the fluxional calculus,
culminated in the branch of our subject known as differential geometry
(see INFINITESIMAL CALCULUS; CURVE; SURFACE).

A second important advance followed the recognition that conics could be
regarded as projections of a circle, a conception which led at the hands
of Desargues and Pascal to modern _projective geometry_ and
_perspective_. A third, and perhaps the most important, advance attended
the application of algebra to geometry by Descartes, who thereby founded
_analytical geometry_. The new fields thus opened up were diligently
explored, but the calculus exercised the greatest attraction and
relatively little progress was made in geometry until the beginning of
the 19th century, when a new era opened.

Gaspard Monge was the first important contributor, stimulating
analytical and differential geometry and founding _descriptive geometry_
in a series of papers and especially in his lectures at the École
polytechnique. Projective geometry, founded by Desargues, Pascal, Monge
and L.N.M. Carnot, was crystallized by J.V. Poncelet, the creator of the
modern methods. In his _Traité des propriétés des figures_ (1822) the
line and circular points at infinity, imaginaries, polar reciprocation,
homology, cross-ratio and projection are systematically employed. In
Germany, A.F. Möbius, J. Plücker and J. Steiner were making far-reaching
contributions. Möbius, in his _Barycentrische Calcul_ (1827), introduced
homogeneous co-ordinates, and also the powerful notion of geometrical
transformation, including the special cases of collineation and duality;
Plücker, in his _Analytisch-geometrische Entwickelungen_ (1828-1831),
and his _System der analytischen Geometrie_ (1835), introduced the
abridged notation, line and plane co-ordinates, and the conception of
generalized space elements; while Steiner, besides enriching geometry in
numerous directions, was the first to systematically generate figures by
projective pencils. We may also notice M. Chasles, whose _Aperçu
historique_ (1837) is a classic. Synthetic geometry, characterized by
its fruitfulness and beauty, attracted most attention, and it so
happened that its originally weak logical foundations became replaced by
a more substantial set of axioms. These were found in the anharmonic
ratio, a device leading to the liberation of synthetic geometry from
metrical relations, and in involution, which yielded rigorous
definitions of imaginaries. These innovations were made by K.J.C. von
Staudt. Analytical geometry was stimulated by the algebra of invariants,
a subject much developed by A. Cayley, G. Salmon, S.H. Aronhold, L.O.
Hesse, and more particularly by R.F.A. Clebsch.

The introduction of the line as a space element, initiated by H.
Grassmann (1844) and Cayley (1859), yielded at the hands of Plücker a
new geometry, termed _line geometry_, a subject developed more notably
by F. Klein, Clebsch, C.T. Reye and F.O.R. Sturm (see Section V., _Line
Geometry_).

_Non-euclidean geometries_, having primarily their origin in the
discussion of Euclidean parallels, and treated by Wallis, Saccheri and
Lambert, have been especially developed during the 19th century. Four
lines of investigation may be distinguished:--the naïve-synthetic,
associated with Lobatschewski, Bolyai, Gauss; the metric differential,
studied by Riemann, Helmholtz, Beltrami; the projective, developed by
Cayley, Klein, Clifford; and the critical-synthetic, promoted chiefly by
the Italian mathematicians Peano, Veronese, Burali-Forte, Levi Civittà,
and the Germans Pasch and Hilbert.     (C. E.*)


I. EUCLIDEAN GEOMETRY

This branch of the science of geometry is so named since its methods and
arrangement are those laid down in Euclid's _Elements_.

§ 1. _Axioms._--The object of geometry is to investigate the properties
of space. The first step must consist in establishing those fundamental
properties from which all others follow by processes of deductive
reasoning. They are laid down in the Axioms, and these ought to form
such a system that nothing need be added to them in order fully to
characterize space, and that nothing may be omitted without making the
system incomplete. They must, in fact, completely "define" space.

§ 2. _Definitions._--The axioms of Euclidean Geometry are obtained from
inspection of existent space and of solids in existent space,--hence
from experience. The same source gives us the notions of the geometrical
entities to which the axioms relate, viz. solids, surfaces, lines or
curves, and points. A solid is directly given by experience; we have
only to abstract all material from it in order to gain the notion of a
geometrical solid. This has shape, size, position, and may be moved. Its
boundary or boundaries are called surfaces. They separate one part of
space from another, and are said to have no thickness. Their boundaries
are curves or lines, and these have length only. Their boundaries,
again, are points, which have no magnitude but only position. We thus
come in three steps from solids to points which have no magnitude; in
each step we lose one extension. Hence we say a solid has three
dimensions, a surface two, a line one, and a point none. Space itself,
of which a solid forms only a part, is also said to be of three
dimensions. The same thing is intended to be expressed by saying that a
solid has length, breadth and thickness, a surface length and breadth, a
line length only, and a point no extension whatsoever.

Euclid gives the essence of these statements as definitions:--

  Def. 1, I. _A point is that which has no parts, or which has no
  magnitude._

  Def. 2, I. _A line is length without breadth._

  Def. 5, I. _A superficies is that which has only length and breadth._

  Def. 1, XI. _A solid is that which has length, breadth and thickness._

It is to be noted that the synthetic method is adopted by Euclid; the
analytical derivation of the successive ideas of "surface," "line," and
"point" from the experimental realization of a "solid" does not find a
place in his system, although possessing more advantages.

If we allow motion in geometry, we may generate these entities by moving
a point, a line, or a surface, thus:--

  The path of a moving point is a line.

  The path of a moving line is, in general, a surface.

  The path of a moving surface is, in general, a solid.

And we may then assume that the lines, surfaces and solids, as defined
before, can all be generated in this manner. From this generation of the
entities it follows again that the boundaries--the first and last
position of the moving element--of a line are points, and so on; and
thus we come back to the considerations with which we started.

Euclid points this out in his definitions,--Def. 3, I., Def. 6, I., and
Def. 2, XI. He does not, however, show the connexion which these
definitions have with those mentioned before. When points and lines have
been defined, a statement like Def. 3, I., "The extremities of a line
are points," is a proposition which either has to be proved, and then it
is a theorem, or which has to be taken for granted, in which case it is
an axiom. And so with Def. 6, I., and Def. 2, XI.

§ 3. Euclid's definitions mentioned above are attempts to describe, in a
few words, notions which we have obtained by inspection of and
abstraction from solids. A few more notions have to be added to these,
principally those of the simplest line--the straight line, and of the
simplest surface--the flat surface or plane. These notions we possess,
but to define them accurately is difficult. Euclid's Definition 4, I.,
"A straight line is that which lies evenly between its extreme points,"
must be meaningless to any one who has not the notion of straightness in
his mind. Neither does it state a property of the straight line which
can be used in any further investigation. Such a property is given in
Axiom 10, I. It is really this axiom, together with Postulates 2 and 3,
which characterizes the straight line.

Whilst for the straight line the verbal definition and axiom are kept
apart, Euclid mixes them up in the case of the plane. Here the
Definition 7, I., includes an axiom. It defines a plane as a surface
which has the property that every straight line which joins any two
points in it lies altogether in the surface. But if we take a straight
line and a point in such a surface, and draw all straight lines which
join the latter to all points in the first line, the surface will be
fully determined. This construction is therefore sufficient as a
definition. That every other straight line which joins any two points in
this surface lies altogether in it is a further property, and to assume
it gives another axiom.

Thus a number of Euclid's axioms are hidden among his first definitions.
A still greater confusion exists in the present editions of Euclid
between the postulates and axioms so called, but this is due to later
editors and not to Euclid himself. The latter had the last three axioms
put together with the postulates [Greek: (aitêmata)], so that these were
meant to include all assumptions relating to space. The remaining
assumptions, which relate to magnitudes in general, viz. the first eight
"axioms" in modern editions, were called "common notions" [Greek:
(koivai ennoiai)]. Of the latter a few may be said to be definitions.
Thus the eighth might be taken as a definition of "equal," and the
seventh of "halves." If we wish to collect the axioms used in Euclid's
_Elements_, we have therefore to take the three postulates, the last
three axioms as generally given, a few axioms hidden in the definitions,
and an axiom used by Euclid in the proof of Prop. 4, I, and on a few
other occasions, viz. that figures may be moved in space without change
of shape or size.

  § 4. _Postulates._--The assumptions actually made by Euclid may be
  stated as follows:--

  (1) Straight lines exist which have the property that any one of them
  may be produced both ways without limit, that through any two points
  in space such a line may be drawn, and that any two of them coincide
  throughout their indefinite extensions as soon as two points in the
  one coincide with two points in the other. (This gives the contents of
  Def. 4, part of Def. 35, the first two Postulates, and Axiom 10.)

  (2) Plane surfaces or planes exist having the property laid down in
  Def. 7, that every straight line joining any two points in such a
  surface lies altogether in it.

  (3) Right angles, as defined in Def. 10, are possible, and all right
  angles are equal; that is to say, wherever in space we take a plane,
  and wherever in that plane we construct a right angle, all angles thus
  constructed will be equal, so that any one of them may be made to
  coincide with any other. (Axiom 11.)

  (4) The 12th Axiom of Euclid. This we shall not state now, but only
  introduce it when we cannot proceed any further without it.

  (5) Figures maybe freely moved in space without change of shape or
  size. This is assumed by Euclid, but not stated as an axiom.

  (6) In any plane a circle may be described, having any point in that
  plane as centre, and its distance from any other point in that plane
  as radius. (Postulate 3.)

The definitions which have not been mentioned are all "nominal
definitions," that is to say, they fix a name for a thing described.
Many of them overdetermine a figure.

§ 5. Euclid's _Elements_ (see EUCLID) are contained in thirteen books.
Of these the first four and the sixth are devoted to "plane geometry,"
as the investigation of figures in a plane is generally called. The 5th
book contains the theory of proportion which is used in Book VI. The
7th, 8th and 9th books are purely arithmetical, whilst the 10th contains
a most ingenious treatment of geometrical irrational quantities. These
four books will be excluded from our survey. The remaining three books
relate to figures in space, or, as it is generally called, to "solid
geometry." The 7th, 8th, 9th, 10th, 13th and part of the 11th and 12th
books are now generally omitted from the school editions of the
_Elements_. In the first four and in the 6th book it is to be understood
that all figures are drawn in a plane.


  BOOK I. OF EUCLID'S "ELEMENTS."

  § 6. According to the third postulate it is possible to draw in any
  plane a circle which has its centre at any given point, and its radius
  equal to the distance of this point from any other point given in the
  plane. This makes it possible (Prop. 1) to construct on a given line
  AB an equilateral triangle, by drawing first a circle with A as centre
  and AB as radius, and then a circle with B as centre and BA as radius.
  The point where these circles intersect--that they intersect Euclid
  quietly assumes--is the vertex of the required triangle. Euclid does
  not suppose, however, that a circle may be drawn which has its radius
  equal to the distance between any two points unless one of the points
  be the centre. This implies also that we are not supposed to be able
  to make any straight line equal to any other straight line, or to
  carry a distance about in space. Euclid therefore next solves the
  problem: It is required along a given straight line from a point in it
  to set off a distance equal to the length of another straight line
  given anywhere in the plane. This is done in two steps. It is shown in
  Prop. 2 how a straight line may be drawn from a given point equal in
  length to another given straight line not drawn from that point. And
  then the problem itself is solved in Prop. 3, by drawing first through
  the given point some straight line of the required length, and then
  about the same point as centre a circle having this length as radius.
  This circle will cut off from the given straight line a length equal
  to the required one. Nowadays, instead of going through this long
  process, we take a pair of compasses and set off the given length by
  its aid. This assumes that we may move a length about without changing
  it. But Euclid has not assumed it, and this proceeding would be fully
  justified by his desire not to take for granted more than was
  necessary, if he were not obliged at his very next step actually to
  make this assumption, though without stating it.

  § 7. We now come (in Prop. 4) to the first theorem. It is the
  fundamental theorem of Euclid's whole system, there being only a very
  few propositions (like Props. 13, 14, 15, I.), except those in the 5th
  book and the first half of the 11th, which do not depend upon it. It
  is stated very accurately, though somewhat clumsily, as follows:--

  _If two triangles have two sides of the one equal to two sides of the
  other, each to each, and have also the angles contained by those sides
  equal to one another, they shall also have their bases or third sides
  equal; and the two triangles shall be equal; and their other angles
  shall be equal, each to each, namely, those to which the equal sides
  are opposite._

  That is to say, the triangles are "identically" equal, and one may be
  considered as a copy of the other. The proof is very simple. The first
  triangle is taken up and placed on the second, so that the parts of
  the triangles which are known to be equal fall upon each other. It is
  then easily seen that also the remaining parts of one coincide with
  those of the other, and that they are therefore equal. This process of
  applying one figure to another Euclid scarcely uses again, though many
  proofs would be simplified by doing so. The process introduces motion
  into geometry, and includes, as already stated, the axiom that figures
  may be moved without change of shape or size.

  If the last proposition be applied to an isosceles triangle, which has
  two sides equal, we obtain the theorem (Prop. 5), _if two sides of a
  triangle are equal, then the angles opposite these sides are equal_.

  Euclid's proof is somewhat complicated, and a stumbling-block to many
  schoolboys. The proof becomes much simpler if we consider the
  isosceles triangle ABC (AB = AC) twice over, once as a triangle BAC,
  and once as a triangle CAB; and now remember that AB, AC in the first
  are equal respectively to AC, AB in the second, and the angles
  included by these sides are equal. Hence the triangles are equal, and
  the angles in the one are equal to those in the other, viz. those
  which are opposite equal sides, i.e. angle ABC in the first equals
  angle ACB in the second, as they are opposite the equal sides AC and
  AB in the two triangles.

  There follows the converse theorem (Prop. 6). _If two angles in a
  triangle are equal, then the sides opposite them are equal_,--i.e. the
  triangle is isosceles. The proof given consists in what is called a
  _reductio ad absurdum_, a kind of proof often used by Euclid, and
  principally in proving the converse of a previous theorem. It assumes
  that the theorem to be proved is wrong, and then shows that this
  assumption leads to an absurdity, i.e. to a conclusion which is in
  contradiction to a proposition proved before--that therefore the
  assumption made cannot be true, and hence that the theorem is true. It
  is often stated that Euclid invented this kind of proof, but the
  method is most likely much older.

  § 8. It is next proved that _two triangles which have the three sides
  of the one equal respectively to those of the other are identically
  equal, hence that the angles of the one are equal respectively to
  those of the other, those being equal which are opposite equal sides_.
  This is Prop. 8, Prop. 7 containing only a first step towards its
  proof.

  These theorems allow now of the solution of a number of problems,
  viz.:--

  _To bisect a given angle_ (Prop. 9).

  _To bisect a given finite straight line_ (Prop. 10).

  _To draw a straight line perpendicularly to a given straight line
  through a given point in it_ (Prop. 11), _and also through a given
  point not in it_ (Prop. 12).

  The solutions all depend upon properties of isosceles triangles.

  § 9. The next three theorems relate to angles only, and might have
  been proved before Prop. 4, or even at the very beginning. The first
  (Prop. 13) says, _The angles which one straight line makes with
  another straight line on one side of it either are two right angles or
  are together equal to two right angles_. This theorem would have been
  unnecessary if Euclid had admitted the notion of an angle such that
  its two limits are in the same straight line, and had besides defined
  the sum of two angles.

  Its converse (Prop. 14) is of great use, inasmuch as it enables us in
  many cases to prove that two straight lines drawn from the same point
  are one the continuation of the other. So also is

  Prop. 15. _If two straight lines cut one another, the vertical or
  opposite angles shall be equal._

  § 10. Euclid returns now to properties of triangles. Of great
  importance for the next steps (though afterwards superseded by a more
  complete theorem) is

  Prop. 16. _If one side of a triangle be produced, the exterior angle
  shall be greater than either of the interior opposite angles._

  Prop. 17. _Any two angles of a triangle are together less than two
  right angles, is an immediate consequence of it._ By the aid of these
  two, the following fundamental properties of triangles are easily
  proved:--

  Prop. 18. _The greater side of every triangle has the greater angle
  opposite to it_;

  Its converse, Prop. 19. _The greater angle of every triangle is
  subtended by the greater side, or has the greater side opposite to
  it_;

  Prop. 20. _Any two sides of a triangle are together greater than the
  third side_;

  And also Prop. 21. _If from the ends of the side of a triangle there
  be drawn two straight lines to a point within the triangle, these
  shall be less than the other two sides of the triangle, but shall
  contain a greater angle._

  § 11. Having solved two problems (Props. 22, 23), he returns to two
  triangles which have two sides of the one equal respectively to two
  sides of the other. It is known (Prop. 4) that if the included angles
  are equal then the third sides are equal; and conversely (Prop. 8), if
  the third sides are equal, then the angles included by the first sides
  are equal. From this it follows that if the included angles are not
  equal, the third sides are not equal; and conversely, that if the
  third sides are not equal, the included angles are not equal. Euclid
  now completes this knowledge by proving, that "_if the included angles
  are not equal, then the third side in that triangle is the greater
  which contains the greater angle_"; and conversely, that "_if the
  third sides are unequal, that triangle contains the greater angle
  which contains the greater side_." These are Prop. 24 and Prop. 25.

  § 12. The next theorem (Prop. 26) says that _if two triangles have one
  side and two angles of the one equal respectively to one side and two
  angles of the other, viz. in both triangles either the angles adjacent
  to the equal side, or one angle adjacent and one angle opposite it,
  then the two triangles are identically equal_.

  This theorem belongs to a group with Prop. 4 and Prop. 8. Its first
  case might have been given immediately after Prop. 4, but the second
  case requires Prop. 16 for its proof.

  § 13. We come now to the investigation of parallel straight lines,
  i.e. of straight lines which lie in the same plane, and cannot be made
  to meet however far they be produced either way. The investigation
  which starts from Prop. 16, will become clearer if a few names be
  explained which are not all used by Euclid. If two straight lines be
  cut by a third, the latter is now generally called a "transversal" of
  the figure. It forms at the two points where it cuts the given lines
  four angles with each. Those of the angles which lie between the given
  lines are called interior angles, and of these, again, any two which
  lie on opposite sides of the transversal but one at each of the two
  points are called "alternate angles."

  We may now state Prop. 16 thus:--_If two straight lines which meet are
  cut by a transversal, their alternate angles are unequal_. For the
  lines will form a triangle, and one of the alternate angles will be an
  exterior angle to the triangle, the other interior and opposite to it.

  From this follows at once the theorem contained in Prop. 27. _If two
  straight lines which are cut by a transversal make alternate angles
  equal, the lines cannot meet, however far they be produced, hence they
  are parallel._ This proves the existence of parallel lines.

  Prop. 28 states the same fact in different forms. _If a straight line,
  falling on two other straight lines, make the exterior angle equal to
  the interior and opposite angle on the same side of the line, or make_
  _the interior angles on the same side together equal to two right
  angles, the two straight lines shall be parallel to one another_.

  Hence we know that, "if two straight lines which are cut by a
  transversal meet, their alternate angles are not equal"; and hence
  that, "if alternate angles are equal, then the lines are parallel."

  The question now arises, Are the propositions converse to these true
  or not? That is to say, "If alternate angles are unequal, do the lines
  meet?" And "if the lines are parallel, are alternate angles
  necessarily equal?"

  The answer to either of these two questions implies the answer to the
  other. But it has been found impossible to prove that the negation or
  the affirmation of either is true.

  The difficulty which thus arises is overcome by Euclid assuming that
  the first question has to be answered in the affirmative. This gives
  his last axiom (12), which we quote in his own words.

  Axiom 12.--_If a straight line meet two straight lines, so as to make
  the two interior angles on the same side of it taken together less
  than two right angles, these straight lines, being continually
  produced, shall at length meet on that side on which are the angles
  which are less than two right angles._

  The answer to the second of the above questions follows from this, and
  gives the theorem Prop. 29:--_If a straight line fall on two parallel
  straight lines, it makes the alternate angles equal to one another,
  and the exterior angle equal to the interior and opposite angle on the
  same side, and also the two interior angles on the same side together
  equal to two right angles_.

  § 14. With this a new part of elementary geometry begins. The earlier
  propositions are independent of this axiom, and would be true even if
  a wrong assumption had been made in it. They all relate to figures in
  a plane. But a plane is only one among an infinite number of
  conceivable surfaces. We may draw figures on any one of them and study
  their properties. We may, for instance, take a sphere instead of the
  plane, and obtain "spherical" in the place of "plane" geometry. If on
  one of these surfaces lines and figures could be drawn, answering to
  all the definitions of our plane figures, and if the axioms with the
  exception of the last all hold, then all propositions up to the 28th
  will be true for these figures. This is the case in spherical geometry
  if we substitute "shortest line" or "great circle" for "straight
  line," "small circle" for "circle," and if, besides, we limit all
  figures to a part of the sphere which is less than a hemisphere, so
  that two points on it cannot be opposite ends of a diameter, and
  therefore determine always one and only one great circle.

  For spherical triangles, therefore, all the important propositions 4,
  8, 26; 5 and 6; and 18, 19 and 20 will hold good.

  This remark will be sufficient to show the impossibility of proving
  Euclid's last axiom, which would mean proving that this axiom is a
  consequence of the others, and hence that the theory of parallels
  would hold on a spherical surface, where the other axioms do hold,
  whilst parallels do not even exist.

  It follows that the axiom in question states an inherent difference
  between the plane and other surfaces, and that the plane is only fully
  characterized when this axiom is added to the other assumptions.

  § 15. The introduction of the new axiom and of parallel lines leads to
  a new class of propositions.

  After proving (Prop. 30) that "_two lines which are each parallel to a
  third are parallel to each other_," we obtain the new properties of
  triangles contained in Prop. 32. Of these the second part is the most
  important, viz. the theorem, _The three interior angles of every
  triangle are together equal to two right angles_.

  As easy deductions not given by Euclid but added by Simson follow the
  propositions about the angles in polygons, they are given in English
  editions as corollaries to Prop. 32.

  These theorems do not hold for spherical figures. The sum of the
  interior angles of a spherical triangle is always greater than two
  right angles, and increases with the area.

  § 16. The theory of parallels as such may be said to be finished with
  Props. 33 and 34, which state properties of the parallelogram, i.e. of
  a quadrilateral formed by two pairs of parallels. They are--

  Prop. 33. _The straight lines which join the extremities of two equal
  and parallel straight lines towards the same parts are themselves
  equal and parallel_; and

  Prop. 34. _The opposite sides and angles of a parallelogram are equal
  to one another, and the diameter (diagonal) bisects the parallelogram,
  that is, divides it into two equal parts._

  § 17. The rest of the first book relates to areas of figures.

  The theory is made to depend upon the theorems--

  Prop. 35. _Parallelograms on the same base and between the same
  parallels are equal to one another_; and

  Prop. 36. _Parallelograms on equal bases and between the same
  parallels are equal to one another_.

  As each parallelogram is bisected by a diagonal, the last theorems
  hold also if the word parallelogram be replaced by "triangle," as is
  done in Props. 37 and 38.

  It is to be remarked that Euclid proves these propositions only in the
  case when the parallelograms or triangles have their bases in the same
  straight line.

  The theorems converse to the last form the contents of the next three
  propositions, viz.: Props, 40 and 41.--_Equal triangles, on the same
  or on equal bases, in the same straight line, and on the same side of
  it, are between the same parallels_.

  That the two cases here stated are given by Euclid in two separate
  propositions proved separately is characteristic of his method.

  § 18. To compare areas of other figures, Euclid shows first, in Prop.
  42, how _to draw a parallelogram which is equal in area to a given
  triangle, and has one of its angles equal to a given angle_. If the
  given angle is right, then the problem is solved _to draw a
  "rectangle" equal in area to a given triangle_.

  Next this parallelogram is transformed into another parallelogram,
  _which has one of its sides equal to a given straight line_, whilst
  its angles remain unaltered. This may be done by aid of the theorem in

  Prop. 43. _The complements of the parallelograms which are about the
  diameter of any parallelogram are equal to one another._

  Thus the problem (Prop. 44) is solved to _construct a parallelogram on
  a given line, which is equal in area to a given triangle, and which
  has one angle equal to a given angle_ (generally a right angle).

  As every polygon can be divided into a number of triangles, we can now
  construct a parallelogram having a given angle, say a right angle, and
  being equal in area to a given polygon. For each of the triangles into
  which the polygon has been divided, a parallelogram may be
  constructed, having one side equal to a given straight line and one
  angle equal to a given angle. If these parallelograms be placed side
  by side, they may be added together to form a single parallelogram,
  having still one side of the given length. This is done in Prop. 45.

  Herewith a means is found to compare areas of different polygons. We
  need only construct two rectangles equal in area to the given
  polygons, and having each one side of given length. By comparing the
  unequal sides we are enabled to judge whether the areas are equal, or
  which is the greater. Euclid does not state this consequence, but the
  problem is taken up again at the end of the second book, where it is
  shown how to construct a square equal in area to a given polygon.

  Prop. 46 is: _To describe a square on a given straight line_.

  § 19. The first book concludes with one of the most important theorems
  in the whole of geometry, and one which has been celebrated since the
  earliest times. It is stated, but on doubtful authority, that
  Pythagoras discovered it, and it has been called by his name. If we
  call that side in a right-angled triangle which is opposite the right
  angle the hypotenuse, we may state it as follows:--

  Theorem of Pythagoras (Prop. 47).--_In every right-angled triangle the
  square on the hypotenuse is equal to the sum of the squares of the
  other sides._

  And conversely--

  Prop. 48. _If the square described on one of the sides of a triangle
  be equal to the squares described on the other sides, then the angle
  contained by these two sides is a right angle._

  On this theorem (Prop. 47) almost all geometrical measurement depends,
  which cannot be directly obtained.


  BOOK II.

  § 20. The propositions in the second book are very different in
  character from those in the first; they all relate to areas of
  rectangles and squares. Their true significance is best seen by
  stating them in an algebraic form. This is often done by expressing
  the lengths of lines by aid of numbers, which tell how many times a
  chosen unit is contained in the lines. If there is a unit to be found
  which is contained an exact number of times in each side of a
  rectangle, it is easily seen, and generally shown in the teaching of
  arithmetic, that the rectangle contains a number of unit squares equal
  to the product of the numbers which measure the sides, a unit square
  being the square on the unit line. If, however, no such unit can be
  found, this process requires that connexion between lines and numbers
  which is only established by aid of ratios of lines, and which is
  therefore at this stage altogether inadmissible. But there exists
  another way of connecting these propositions with algebra, based on
  modern notions which seem destined greatly to change and to simplify
  mathematics. We shall introduce here as much of it as is required for
  our present purpose.

  At the beginning of the second book we find a definition according to
  which "a rectangle is said to be 'contained' by the two sides which
  contain one of its right angles"; in the text this phraseology is
  extended by speaking of rectangles contained by any two straight
  lines, meaning the rectangle which has two adjacent sides equal to the
  two straight lines.

  We shall denote a finite straight line by a single small letter, a, b,
  c, ... x, and the area of the rectangle contained by two lines a and b
  by ab, and this we shall call the product of the two lines a and b. It
  will be understood that this definition has nothing to do with the
  definition of a product of numbers.

  We define as follows:--

  The _sum_ of two straight lines a and b means a straight line c which
  may be divided in two parts equal respectively to a and b. This sum is
  denoted by a + b.

  The _difference_ of two lines a and b (in symbols, a-b) means a line c
  which when added to b gives a; that is,

    a - b = c if b + c = a.

  The _product_ of two lines a and b (in symbols, ab) means the area of
  the rectangle contained by the lines a and b. For aa, which means the
  square on the line a, we write a².

  § 21. The first ten of the fourteen propositions of the second book
  may then be written in the form of formulae as follows:--

    Prop. 1. a(b + c + d + ... ) = ab + ac + ad + ...

      "   2. ab + ac = a² if b + c = a.

      "   3. a(a + b) = a² + ab.

      "   4. (a + b)² = a² + 2ab + b².

      "   5. (a + b)(a - b) + b² = a².

      "   6. (a + b)(a - b) + b² = a².

      "   7. a² + (a - b)² = 2a(a - b) + b².

      "   8. 4(a + b)a + b² = (2a + b)².

      "   9. (a + b)² + (a - b)² = 2a² + 2b².

      "  10. (a + b)² + (a - b)² = 2a² + 2b².

  It will be seen that 5 and 6, and also 9 and 10, are identical. In
  Euclid's statement they do not look the same, the figures being
  arranged differently.

  If the letters a, b, c, ... denoted numbers, it follows from algebra
  that each of these formulae is true. But this does not prove them in
  our case, where the letters denote lines, and their products areas
  without any reference to numbers. To prove them we have to discover
  the laws which rule the operations introduced, viz. addition and
  multiplication of segments. This we shall do now; and we shall find
  that these laws are the same with those which hold in algebraical
  addition and multiplication.

  § 22. In a sum of numbers we may change the order in which the numbers
  are added, and we may also add the numbers together in groups and then
  add these groups. But this also holds for the sum of segments and for
  the sum of rectangles, as a little consideration shows. That the sum
  of rectangles has always a meaning follows from the Props. 43-45 in
  the first book. These laws about addition are reducible to the two--

    a + b = b + a            (1),

    a + (b + c) = a + b + c  (2);

  or, when expressed for rectangles,

    ab + ed = ed + ab              (3),

    ab + (cd + ef) = ab + cd + ef  (4).

  The brackets mean that the terms in the bracket have been added
  together before they are added to another term. The more general cases
  for more terms may be deduced from the above.

  For the product of two numbers we have the law that it remains
  unaltered if the factors be interchanged. This also holds for our
  geometrical product. For if ab denotes the area of the rectangle which
  has a as base and b as altitude, then ba will denote the area of the
  rectangle which has b as base and a as altitude. But in a rectangle we
  may take either of the two lines which contain it as base, and then
  the other will be the altitude. This gives

    ab = ba       (5).

  In order further to multiply a sum by a number, we have in algebra the
  rule:--Multiply each term of the sum, and add the products thus
  obtained. That this holds for our geometrical products is shown by
  Euclid in his first proposition of the second book, where he proves
  that the area of a rectangle whose base is the sum of a number of
  segments is equal to the sum of rectangles which have these segments
  separately as bases. In symbols this gives, in the simplest case,

     a(b + c) = ab + ac   \
                           >  (6).
  and (b + c)a = ba + ca  /

  To these laws, which have been investigated by Sir William Hamilton
  and by Hermann Grassmann, the former has given special names. He calls
  the laws expressed in

    (1) and (3) the commutative law for addition;

            (5)          "       "   "  multiplication;

    (2) and (4) the associative laws for addition;

            (6) the distributive law.

  § 23. Having proved that these six laws hold, we can at once prove
  every one of the above propositions in their algebraical form.

  The first is proved geometrically, it being one of the fundamental
  laws. The next two propositions are only special cases of the first.
  Of the others we shall prove one, viz. the fourth:--

         (a + b)² = (a + b)(a + b) = (a + b)a + (a + b)b   by (6).

  But                     (a + b)a = aa + ba       by (6),
                                   = aa + ab       by (5);

  and                     (a + b)b = ab + bb       by (6).

  Therefore     (a + b)² = aa + ab + (ab + bb) \
                         = aa + (ab + ab) + bb  >  by (4).
                         = aa + 2ab + bb       /

  This gives the theorem in question.

  In the same manner every one of the first ten propositions is proved.

  It will be seen that the operations performed are exactly the same as
  if the letters denoted numbers.

  Props. 5 and 6 may also be written thus--

    (a + b)(a - b) = a² - b².

  Prop. 7, which is an easy consequence of Prop. 4, may be transformed.
  If we denote by c the line a + b, so that

    c = a + b, a = c - b,

  we get

    c² + (c - b)² = 2c(c - b) + b²
                  = 2c² - 2bc + b².

  Subtracting c² from both sides, and writing a for c, we get

    (a - b)² = a² - 2ab + b².

  In Euclid's _Elements_ this form of the theorem does not appear, all
  propositions being so stated that the notion of subtraction does not
  enter into them.

  § 24. The remaining two theorems (Props. 12 and 13) connect the square
  on one side of a triangle with the sum of the squares on the other
  sides, in case that the angle between the latter is acute or obtuse.
  They are important theorems in trigonometry, where it is possible to
  include them in a single theorem.

  § 25. There are in the second book two problems, Props. 11 and 14.

  If written in the above symbolic language, the former requires to find
  a line x such that a(a - x) = x². Prop. 11 contains, therefore, the
  solution of a quadratic equation, which we may write x² + ax = a². The
  solution is required later on in the construction of a regular
  decagon.

  More important is the problem in the last proposition (Prop. 14). It
  requires the construction of a square equal in area to a given
  rectangle, hence a solution of the equation

    x² = ab.

  In Book I., 42-45, it has been shown how a rectangle may be
  constructed equal in area to a given figure bounded by straight lines.
  By aid of the new proposition we may therefore now determine a line
  such that the square on that line is equal in area to any given
  rectilinear figure, or we can _square_ any such figure.

  As of two squares that is the greater which has the greater side, it
  follows that now the comparison of two areas has been reduced to the
  comparison of two lines.

  The problem of reducing other areas to squares is frequently met with
  among Greek mathematicians. We need only mention the problem of
  squaring the circle (see CIRCLE).

  In the present day the comparison of areas is performed in a simpler
  way by reducing all areas to rectangles having a common base. Their
  altitudes give then a measure of their areas.

  The construction of a rectangle having the base u, and being equal in
  area to a given rectangle, depends upon Prop. 43, I. This therefore
  gives a solution of the equation

    ab = ux,

  where x denotes the unknown altitude.


  BOOK III.

  § 26. The third book of the _Elements_ relates exclusively to
  properties of the circle. A circle and its circumference have been
  defined in Book I., Def. 15. We restate it here in slightly different
  words:--

  _Definition_.--The circumference of a circle is a plane curve such
  that all points in it have the same distance from a fixed point in the
  plane. This point is called the "centre" of the circle.

  Of the new definitions, of which eleven are given at the beginning of
  the third book, a few only require special mention. The first, which
  says that circles with equal radii are equal, is in part a theorem,
  but easily proved by applying the one circle to the other. Or it may
  be considered proved by aid of Prop. 24, equal circles not being used
  till after this theorem.

  In the second definition is explained what is meant by a line which
  "touches" a circle. Such a line is now generally called a tangent to
  the circle. The introduction of this name allows us to state many of
  Euclid's propositions in a much shorter form.

  For the same reason we shall call a straight line joining two points
  on the circumference of a circle a "chord."

  Definitions 4 and 5 may be replaced with a slight generalization by
  the following:--

  _Definition_.--By the distance of a point from a line is meant the
  length of the perpendicular drawn from the point to the line.

  § 27. From the definition of a circle it follows that every circle has
  a centre. Prop. 1 requires to find it when the circle is given, i.e.
  when its circumference is drawn.

  To solve this problem a chord is drawn (that is, any two points in the
  circumference are joined), and through the point where this is
  bisected a perpendicular to it is erected. Euclid then proves, first,
  that no point off this perpendicular can be the centre, hence that the
  centre must lie in this line; and, secondly, that of the points on the
  perpendicular one only can be the centre, viz. the one which bisects
  the parts of the perpendicular bounded by the circle. In the second
  part Euclid silently assumes that the perpendicular there used does
  cut the circumference in two, and only in two points. The proof
  therefore is incomplete. The proof of the first part, however, is
  exact. By drawing two non-parallel chords, and the perpendiculars
  which bisect them, the centre will be found as the point where these
  perpendiculars intersect.

  § 28. In Prop. 2 it is proved that a chord of a circle lies altogether
  within the circle.

  What we have called the first part of Euclid's solution of Prop. 1 may
  be stated as a theorem:--

  _Every straight line which bisects a chord, and is at right angles to
  it, passes through the centre of the circle._

  The converse to this gives Prop. 3, which may be stated thus:--

  _If a straight line through the centre of a circle bisect a chord,
  then it is perpendicular to the chord, and if it be perpendicular to
  the chord it bisects it._

  An easy consequence of this is the following theorem, which is
  essentially the same as Prop. 4:--

  _Two chords of a circle, of which neither passes through the centre,
  cannot bisect each other._

  These last three theorems are fundamental for the theory of the
  circle. It is to be remarked that Euclid never proves that a straight
  line cannot have more than two points in common with a circumference.

  § 29. The next two propositions (5 and 6) might be replaced by a
  single and a simpler theorem, viz:--

  _Two circles which have a common centre, and whose circumferences have
  one point in common, coincide._

  Or, more in agreement with Euclid's form:--

  _Two different circles, whose circumferences have a point in common,
  cannot have the same centre._

  That Euclid treats of two cases is characteristic of Greek
  mathematics.

  The next two propositions (7 and 8) again belong together. They may be
  combined thus:--

  _If from a point in a plane of a circle, which is not the centre,
  straight lines be drawn to the different points of the circumference,
  then of all these lines one is the shortest, and one the longest, and
  these lie both in that straight line which joins the given point to
  the centre. Of all the remaining lines each is equal to one and only
  one other, and these equal lines lie on opposite sides of the shortest
  or longest, and make equal angles with them._

  Euclid distinguishes the two cases where the given point lies within
  or without the circle, omitting the case where it lies in the
  circumference.

  From the last proposition it follows that if from a point more than
  two equal straight lines can be drawn to the circumference, this point
  must be the centre. This is Prop. 9.

  As a consequence of this we get

  _If the circumferences of the two circles have three points in common
  they coincide._

  For in this case the two circles have a common centre, because from
  the centre of the one three equal lines can be drawn to points on the
  circumference of the other. But two circles which have a common
  centre, and whose circumferences have a point in common, coincide.
  (Compare above statement of Props. 5 and 6.)

  This theorem may also be stated thus:--

  _Through three points only one circumference may be drawn; or, Three
  points determine a circle._

  Euclid does not give the theorem in this form. He proves, however,
  _that the two circles cannot cut another in more than two points_
  (Prop. 10), and _that two circles cannot touch one another in more
  points than one_ (Prop. 13).

  § 30. Propositions 11 and 12 assert that _if two circles touch, then
  the point of contact lies on the line joining their centres_. This
  gives two propositions, because the circles may touch either
  internally or externally.

  § 31. Propositions 14 and 15 relate to the length of chords. The first
  says _that equal chords are equidistant from the centre, and that
  chords which are equidistant from the centre are equal_;

  Whilst Prop. 15 compares unequal chords, viz. _Of all chords the
  diameter is the greatest, and of other chords that is the greater
  which is nearer to the centre_; and conversely, _the greater chord is
  nearer to the centre_.

  § 32. In Prop. 16 the tangent to a circle is for the first time
  introduced. The proposition is meant to show that the straight line at
  the end point of the diameter and at right angles to it is a tangent.
  The proposition itself does not state this. It runs thus:--

  Prop. 16. _The straight line drawn at right angles to the diameter of
  a circle, from the extremity of it, falls without the circle; and no
  straight line can be drawn from the extremity, between that straight
  line and the circumference, so as not to cut the circle._

  _Corollary_.--The straight line at right angles to a diameter drawn
  through the end point of it touches the circle.

  The statement of the proposition and its whole treatment show the
  difficulties which the tangents presented to Euclid.

  Prop. 17 solves the problem _through a given point, either in the
  circumference or without it, to draw a tangent to a given circle_.

  Closely connected with Prop. 16 are Props. 18 and 19, which state
  (Prop. 18), _that the line joining the centre of a circle to the point
  of contact of a tangent is perpendicular to the tangent_; and
  conversely (Prop. 19), _that the straight line through the point of
  contact of, and perpendicular to, a tangent to a circle passes through
  the centre of the circle_.

  § 33. The rest of the book relates to angles connected with a circle,
  viz. angles which have the vertex either at the centre or on the
  circumference, and which are called respectively angles at the centre
  and angles at the circumference. Between these two kinds of angles
  exists the important relation expressed as follows:--

  Prop. 20. _The angle at the centre of a circle is double of the angle
  at the circumference on the same base, that is, on the same arc._

  This is of great importance for its consequences, of which the two
  following are the principal:--

  Prop. 21. _The angles in the same segment of a circle are equal to one
  another_;

  Prop. 22. _The opposite angles of any quadrilateral figure inscribed
  in a circle are together equal to two right angles._

  Further consequences are:--

  Prop. 23. _On the same straight line, and on the same side of it,
  there cannot be two similar segments of circles, not coinciding with
  one another_;

  Prop. 24. _Similar segments of circles on equal straight lines are
  equal to one another._

  The problem Prop. 25. _A segment of a circle being given to describe
  the circle of which it is a segment_, may be solved much more easily
  by aid of the construction described in relation to Prop. 1, III., in
  § 27.

  § 34. There follow four theorems connecting the angles at the centre,
  the arcs into which they divide the circumference, and the chords
  subtending these arcs. They are expressed for angles, arcs and chords
  in equal circles, but they hold also for angles, arcs and chords in
  the same circle.

  The theorems are:--

  Prop. 26. _In equal circles equal angles stand on equal arcs, whether
  they be at the centres or circumferences_;

  Prop. 27. (converse to Prop. 26). _In equal circles the angles which
  stand on equal arcs are equal to one another, whether they be at the
  centres or the circumferences_;

  Prop. 28. _In equal circles equal straight lines_ (equal chords) _cut
  off equal arcs, the greater equal to the greater, and the less equal
  to the less_;

  Prop. 29 (converse to Prop. 28). _In equal circles equal arcs are
  subtended by equal straight lines._

  § 35. Other important consequences of Props. 20-22 are:--

  Prop. 31. _In a circle the angle in a semicircle is a right angle; but
  the angle in a segment greater than a semicircle is less than a right
  angle; and the angle in a segment less than a semicircle is greater
  than a right angle_;

  Prop. 32. _If a straight line touch a circle, and from the point of
  contact a straight line be drawn cutting the circle, the angles which
  this line makes with the line touching the circle shall be equal to
  the angles which are in the alternate segments of the circle._

  § 36. Propositions 30, 33, 34, contain problems which are solved by
  aid of the propositions preceding them:--

  Prop. 30. _To bisect a given arc, that is, to divide it into two equal
  parts_;

  Prop. 33. _On a given straight line to describe a segment of a circle
  containing an angle equal to a given rectilineal angle_;

  Prop. 34. _From a given circle to cut off a segment containing an
  angle equal to a given rectilineal angle_.

  § 37. If we draw chords through a point A within a circle, they will
  each be divided by A into two segments. Between these segments the law
  holds that the rectangle contained by them has the same area on
  whatever chord through A the segments are taken. The value of this
  rectangle changes, of course, with the position of A.

  A similar theorem holds if the point A be taken without the circle. On
  every straight line through A, which cuts the circle in two points B
  and C, we have two segments AB and AC, and the rectangles contained by
  them are again equal to one another, and equal to the square on a
  tangent drawn from A to the circle.

  The first of these theorems gives Prop. 35, and the second Prop. 36,
  with its corollary, whilst Prop. 37, the last of Book III., gives the
  converse to Prop. 36. The first two theorems may be combined in one:--

  _If through a point A in the plane of a circle a straight line be
  drawn cutting the circle in B and C, then the rectangle AB·AC has a
  constant value so long as the point A be fixed; and if from A a
  tangent AD can be drawn to the circle, touching at D, then the above
  rectangle equals the square on AD._

  Prop. 37 may be stated thus:--

  _If from a point A without a circle a line be drawn cutting the circle
  in B and C, and another line to a point D on the circle, and AB·AC =
  AD², then the line AD touches the circle at D._

  It is not difficult to prove also the converse to the general
  proposition as above stated. This proposition and its converse may be
  expressed as follows:--

  _If four points ABCD be taken on the circumference of a circle, and if
  the lines AB, CD, produced if necessary, meet at E, then_

    EA·EB = EC·ED;

  _and conversely, if this relation holds then the four points lie on a
  circle, that is, the circle drawn through three of them passes through
  the fourth._

  That a circle may always be drawn through three points, provided that
  they do not lie in a straight line, is proved only later on in Book
  IV.


  BOOK IV.

  § 38. The fourth book contains only problems, all relating to the
  construction of triangles and polygons inscribed in and circumscribed
  about circles, and of circles inscribed in or circumscribed about
  triangles and polygons. They are nearly all given for their own sake,
  and not for future use in the construction of figures, as are most of
  those in the former books. In seven definitions at the beginning of
  the book it is explained what is understood by figures inscribed in or
  described about other figures, with special reference to the case
  where one figure is a circle. Instead, however, of saying that one
  figure is described about another, it is now generally said that the
  one figure is circumscribed about the other. We may then state the
  definitions 3 or 4 thus:--

  _Definition._--A polygon is said to be inscribed in a circle, and the
  circle is said to be circumscribed about the polygon, if the vertices
  of the polygon lie in the circumference of the circle.

  And definitions 5 and 6 thus:--

  _Definition._--A polygon is said to be circumscribed about a circle,
  and a circle is said to be inscribed in a polygon, if the sides of the
  polygon are tangents to the circle.

  § 39. The first problem is merely constructive. It requires to draw in
  a given circle a chord equal to a given straight line, which is not
  greater than the diameter of the circle. The problem is not a
  determinate one, inasmuch as the chord may be drawn from any point in
  the circumference. This may be said of almost all problems in this
  book, especially of the next two. They are:--

  Prop. 2. _In a given circle to inscribe a triangle equiangular to a
  given triangle;_

  Prop. 3. _About a given circle to circumscribe a triangle equiangular
  to a given triangle._

  § 40. Of somewhat greater interest are the next problems, where the
  triangles are given and the circles to be found.

  Prop. 4. _To inscribe a circle in a given triangle._

  The result is that the problem has always a solution, viz. the centre
  of the circle is the point where the bisectors of two of the interior
  angles of the triangle meet. The solution shows, though Euclid does
  not state this, that the problem has but one solution; and also,

  _The three bisectors of the interior angles of any triangle meet in a
  point, and this is the centre of the circle inscribed in the
  triangle._

  The solutions of most of the other problems contain also theorems. Of
  these we shall state those which are of special interest; Euclid does
  not state any one of them.

  § 41. Prop. 5. _To circumscribe a circle about a given triangle._

  The one solution which always exists contains the following:--

  _The three straight lines which bisect the sides of a triangle at
  right angles meet in a point, and this point is the centre of the
  circle circumscribed about the triangle._

  Euclid adds in a corollary the following property:--

  The centre of the circle circumscribed about a triangle lies within,
  on a side of, or without the triangle, according as the triangle is
  acute-angled, right-angled or obtuse-angled.

  § 42. Whilst it is always possible to draw a circle which is inscribed
  in or circumscribed about a given triangle, this is not the case with
  quadrilaterals or polygons of more sides. Of those for which this is
  possible the regular polygons, i.e. polygons which have all their
  sides and angles equal, are the most interesting. In each of them a
  circle may be inscribed, and another may be circumscribed about it.

  Euclid does not use the word regular, but he describes the polygons in
  question as _equiangular_ and _equilateral_. We shall use the name
  regular polygon. The regular triangle is equilateral, the regular
  quadrilateral is the square.

  Euclid considers the regular polygons of 4, 5, 6 and 15 sides. For
  each of the first three he solves the problems--(1) to inscribe such a
  polygon in a given circle; (2) to circumscribe it about a given
  circle; (3) to inscribe a circle in, and (4) to circumscribe a circle
  about, such a polygon.

  For the regular triangle the problems are not repeated, because more
  general problems have been solved.

  Props. 6, 7, 8 and 9 solve these problems for the square.

  The general problem of inscribing in a given circle a regular polygon
  of n sides depends upon the problem of dividing the circumference of a
  circle into n equal parts, or what comes to the same thing, of drawing
  from the centre of the circle n radii such that the angles between
  consecutive radii are equal, that is, to divide the space about the
  centre into n equal angles. Thus, if it is required to inscribe a
  square in a circle, we have to draw four lines from the centre, making
  the four angles equal. This is done by drawing two diameters at right
  angles to one another. The ends of these diameters are the vertices of
  the required square. If, on the other hand, tangents be drawn at these
  ends, we obtain a square circumscribed about the circle.

  § 43. To construct a _regular pentagon_, we find it convenient first
  to construct a _regular decagon_. This requires to divide the space
  about the centre into ten equal angles. Each will be 1/10th of a right
  angle, or 1/5th of two right angles. If we suppose the decagon
  constructed, and if we join the centre to the end of one side, we get
  an isosceles triangle, where the angle at the centre equals 1/5th of
  two right angles; hence each of the angles at the base will be 2/5ths
  of two right angles, as all three angles together equal two right
  angles. Thus we have to construct an isosceles triangle, having the
  angle at the vertex equal to half an angle at the base. This is solved
  in Prop. 10, by aid of the problem in Prop. 11 of the second book. If
  we make the sides of this triangle equal to the radius of the given
  circle, then the base will be the side of the regular decagon
  inscribed in the circle. This side being known the decagon can be
  constructed, and if the vertices are joined alternately, leaving out
  half their number, we obtain the regular pentagon. (Prop. 11.)

  Euclid does not proceed thus. He wants the pentagon before the
  decagon. This, however, does not change the real nature of his
  solution, nor does his solution become simpler by not mentioning the
  decagon.

  Once the regular pentagon is inscribed, it is easy to circumscribe
  another by drawing tangents at the vertices of the inscribed pentagon.
  This is shown in Prop. 12.

  Props. 13 and 14 teach how a circle may be inscribed in or
  circumscribed about any given regular pentagon.

  § 44. The _regular hexagon_ is more easily constructed, as shown in
  Prop. 15. The result is that the side of the regular hexagon inscribed
  in a circle is equal to the radius of the circle.

  For this polygon the other three problems mentioned are not solved.

  § 45. The book closes with Prop. 16. To inscribe a regular quindecagon
  in a given circle. If we inscribe a regular pentagon and a regular
  hexagon in the circle, having one vertex in common, then the arc from
  the common vertex to the next vertex of the pentagon is 1/5th of the
  circumference, and to the next vertex of the hexagon is 1/6th of the
  circumference. The difference between these arcs is, therefore, 1/5 -
  1/6 = 1/30th of the circumference. The latter may, therefore, be
  divided into thirty, and hence also in fifteen equal parts, and the
  regular quindecagon be described.

  § 46. We conclude with a few theorems about regular polygons which are
  not given by Euclid.

  _The straight lines perpendicular to and bisecting the sides of any
  regular polygon meet in a point. The straight lines bisecting the
  angles in the regular polygon meet in the same point. This point is
  the centre of the circles circumscribed about and inscribed in the
  regular polygon._

  We can bisect any given arc (Prop. 30, III.). Hence we can divide a
  circumference into 2n equal parts as soon as it has been divided into
  n equal parts, or as soon as a regular polygon of n sides has been
  constructed. Hence--

  _If a regular polygon of n sides has been constructed, then a regular
  polygon of 2n sides, of 4n, of 8n sides, &c., may also be
  constructed._ Euclid shows how to construct regular polygons of 3, 4,
  5 and 15 sides. It follows that we can construct regular polygons of

     3,  6, 12,  24  sides
     4,  8, 16,  32    "
     5, 10, 20,  40    "
    15, 30, 60, 120    "

  The construction of any new regular polygon not included in one of
  these series will give rise to a new series. Till the beginning of the
  19th century nothing was added to the knowledge of regular polygons as
  given by Euclid. Then Gauss, in his celebrated _Arithmetic_, proved
  that every regular polygon of 2^n + 1 sides may be constructed if this
  number 2^n + 1 be prime, and that no others except those with 2^m(2^n
  + 1) sides can be constructed by elementary methods. This shows that
  regular polygons of 7, 9, 13 sides cannot thus be constructed, but
  that a regular polygon of 17 sides is possible; for 17 = 2^4 + 1. The
  next polygon is one of 257 sides. The construction becomes already
  rather complicated for 17 sides.


  BOOK V.

  § 47. The fifth book of the _Elements_ is not exclusively geometrical.
  It contains the theory of ratios and proportion of quantities in
  general. The treatment, as here given, is admirable, and in every
  respect superior to the algebraical method by which Euclid's theory is
  now generally replaced. We shall treat the subject in order to show
  why the usual algebraical treatment of proportion is not really sound.
  We begin by quoting those definitions at the beginning of Book V.
  which are most important. These definitions have given rise to much
  discussion.

  The only definitions which are essential for the fifth book are Defs.
  1, 2, 4, 5, 6 and 7. Of the remainder 3, 8 and 9 are more than
  useless, and probably not Euclid's, but additions of later editors, of
  whom Theon of Alexandria was the most prominent. Defs. 10 and 11
  belong rather to the sixth book, whilst all the others are merely
  nominal. The really important ones are 4, 5, 6 and 7.

  § 48. To define a magnitude is not attempted by Euclid. The first two
  definitions state what is meant by a "part," that is, a submultiple or
  measure, and by a "multiple" of a given magnitude. The meaning of Def.
  4 is that two given quantities can have a ratio to one another only in
  case that they are comparable as to their magnitude, that is, if they
  are of the same kind.

  Def. 3, which is probably due to Theon, professes to define a ratio,
  but is as meaningless as it is uncalled for, for all that is wanted is
  given in Defs. 5 and 7.

  In Def. 5 it is explained what is meant by saying that two magnitudes
  have the same ratio to one another as two other magnitudes, and in
  Def. 7 what we have to understand by a greater or a less ratio. The
  6th definition is only nominal, explaining the meaning of the word
  _proportional_.

  Euclid represents magnitudes by lines, and often denotes them either
  by single letters or, like lines, by two letters. We shall use only
  single letters for the purpose. If a and b denote two magnitudes of
  the same kind, their ratio will be denoted by a : b; if c and d are
  two other magnitudes of the same kind, but possibly of a different
  kind from a and b, then if c and d have the same ratio to one another
  as a and b, this will be expressed by writing--

    a : b :: c : d.

  Further, if m is a (whole) number, ma shall denote the multiple of a
  which is obtained by taking it m times.

  § 49. The whole theory of ratios is based on Def. 5.

  Def. 5. _The first of four magnitudes is said to have the same ratio
  to the second that the third has to the fourth when, any equimultiples
  whatever of the first and the third being taken, and any equimultiples
  whatever of the second and the fourth, if the multiple of the first be
  less than that of the second, the multiple of the third is also less
  than that of the fourth; and if the multiple of the first is equal to
  that of the second, the multiple of the third is also equal to that of
  the fourth; and if the multiple of the first is greater than that of
  the second, the multiple of the third is also greater than that of the
  fourth._

  It will be well to show at once in an example how this definition can
  be used, by proving the first part of the first proposition in the
  sixth book. _Triangles of the same altitude are to one another as
  their bases_, or if a and b are the bases, and [alpha] and ß the
  areas, of two triangles which have the same altitude, then a : b ::
  [alpha] : ß.

  To prove this, we have, according to Definition 5, to show--

    if ma > nb, then m[alpha] > nß,
    if ma = nb, then m[alpha] = nß,
    if ma < nb, then m[alpha] < nß.

  That this is true is in our case easily seen. We may suppose that the
  triangles have a common vertex, and their bases in the same line. We
  set off the base a along the line containing the bases m times; we
  then join the different parts of division to the vertex, and get m
  triangles all equal to [alpha]. The triangle on ma as base equals,
  therefore, m[alpha]. If we proceed in the same manner with the base b,
  setting it off n times, we find that the area of the triangle on the
  base nb equals nß, the vertex of all triangles being the same. But if
  two triangles have the same altitude, then their areas are equal if
  the bases are equal; hence m[alpha] = nß if ma = nb, and if their
  bases are unequal, then that has the greater area which is on the
  greater base; in other words, m[alpha] is greater than, equal to, or
  less than nß, according as ma is greater than, equal to, or less than
  nb, which was to be proved.

  § 50. It will be seen that even in this example it does not become
  evident what a ratio really is. It is still an open question whether
  ratios are magnitudes which we can compare. We do not know whether the
  ratio of two lines is a magnitude of the same kind as the ratio of two
  areas. Though we might say that Def. 5 defines _equal _ratios, still
  we do not know whether they are equal in the sense of the axiom, that
  two things which are equal to a third are equal to one another. That
  this is the case requires a proof, and until this proof is given we
  shall use the :: instead of the sign = , which, however, we shall
  afterwards introduce.

  As soon as it has been established that all ratios are like
  magnitudes, it becomes easy to show that, in some cases at least, they
  are numbers. This step was never made by Greek mathematicians. They
  distinguished always most carefully between continuous magnitudes and
  the discrete series of numbers. In modern times it has become the
  custom to ignore this difference.

  If, in determining the ratio of two lines, a common measure can be
  found, which is contained m times in the first, and n times in the
  second, then the ratio of the two lines equals the ratio of the two
  numbers m : n. This is shown by Euclid in Prop. 5, X. But the ratio of
  two numbers is, as a rule, a fraction, and the Greeks did not, as we
  do, consider fractions as numbers. Far less had they any notion of
  introducing irrational numbers, which are neither whole nor
  fractional, as we are obliged to do if we wish to say that all ratios
  are numbers. The incommensurable numbers which are thus introduced as
  ratios of incommensurable quantities are nowadays as familiar to us as
  fractions; but a proof is generally omitted that we may apply to them
  the rules which have been established for rational numbers only.
  Euclid's treatment of ratios avoids this difficulty. His definitions
  hold for commensurable as well as for incommensurable quantities. Even
  the notion of incommensurable quantities is avoided in Book V. But he
  proves that the more elementary rules of algebra hold for ratios. We
  shall state all his propositions in that algebraical form to which we
  are now accustomed. This may, of course, be done without changing the
  character of Euclid's method.

  §. 51. Using the notation explained above we express the first
  propositions as follows:--

  Prop. 1. If   a = ma', b = mb', c = mc',
  then          a + b + c = m(a' + b' + c').

  Prop. 2. If   a = mb, and c = md,
                e = nb, and f = nd,

  then a + e is the same multiple of b as c + f is of d, viz.:--

    a + e = (m + n)b, and c + f = (m + n)d.

  Prop. 3. If a = mb, c = md, then is na the same multiple of b that nc
  is of d, viz. na = nmb, nc = nmd.

  Prop. 4. If   a : b :: c : d,
  then          ma : nb :: mc : nd.

  Prop. 5. If   a = mb, and c = md,
  then          a - c = m(b - d).

  Prop. 6. If   a = mb, c = md,

  then are a - nb and c - nd either equal to, or equimultiples of, b and
  d, viz. a - nb = (m - n)b and c - nd = (m - n)d, where m - n may be
  unity.

  All these propositions relate to _equimultiples_. Now follow
  propositions about ratios which are compared as to their magnitude.

  § 52. Prop. 7. If a = b, then a : c :: b : c and c : a :: c : b.

  The proof is simply this. As a = b we know that ma = mb; therefore

    if   ma > nc, then mb > nc,
    if   ma = nc, then mb = nc,
    if   ma < nc, then mb < nc,

  therefore the first proportion holds by Definition 5.

  Prop. 8. If a > b, then a : c > b : c,
  and                     c : a < c : b.

  The proof depends on Definition 7.

  Prop. 9 (converse to Prop. 7). If
             a : c :: b : c,
  or if      c : a :: c : b, then a = b.

  Prop. 10 (converse to Prop. 8). If
             a : c > b : c, then a > b,
  and if     c : a < c : b, then a < b.

  Prop. 11. If  a : b :: c : d,
  and           a : b :: e : f,
  then          c : d :: e : f.

  In words, _if too ratios are equal to a third, they are equal to one
  another_. After these propositions have been proved, we have a right
  to consider a ratio as a _magnitude_, for only now can we consider a
  ratio as something for which the axiom about magnitudes holds: things
  which are equal to a third are equal to one another.

  We shall indicate this by writing in future the sign = instead of ::.
  The remaining propositions, which explain themselves, may then be
  stated as follows:

  § 53. Prop. 12. If  a : b = c : d = e : f,
  then                a + c + e : b + d + f = a : b.

  Prop. 13. If    a : b = c : d and c : d > e : f,
  then            a : b > e : f.

  Prop. 14. If    a : b = c : d, and a > c, then b > d.

  Prop. 15. Magnitudes have the same ratio to one another that their
  equimultiples have--

  ma : mb = a : b.

  Prop. 16. If a, b, c, d are magnitudes of the same kind, and if
                  a : b = c : d,
  then            a : c = b : d.

  Prop. 17. If    a + b : b = c + d : d,
  then            a : b = c : d.

  Prop. 18 (converse to 17). If
                  a : b = c : d
  then            a + b : b = c + d : d.

  Prop. 19. If a, b, c, d are quantities of the same kind, and if
                  a : b = c : d,
  then            a - c : b - d = a : b.

  § 54. Prop. 20. _If there be three magnitudes, and another three,
  which have the same ratio, taken two and two, then if the first be
  greater than the third, the fourth shall be greater than the sixth:
  and if equal, equal; and if less, less._

  If we understand by

    a : b : c : d : e : ... = a' : b' : c' : d' : e' : ...

  that the ratio of any two consecutive magnitudes on the first side
  equals that of the corresponding magnitudes on the second side, we may
  write this theorem in symbols, thus:--

  If a, b, c be quantities of one, and d, e, f magnitudes of the same or
  any other kind, such that

           a : b : c = d : e : f,
  and if   a > c, then d > f,
  but if   a = c, then d = f,
  and if   a < c, then d < f.

  Prop. 21. If  a : b = e : f and b : c = d : e,
  or if         a : b : c = 1/f : 1/e : 1/d,
  and if        a > c, then d > f,
  but if        a = c, then d = f,
  and if        a < c, then d < f.

  By aid of these two propositions the following two are proved.

  § 55. Prop. 22. _If there be any number of magnitudes, and as many
  others, which have the same ratio, taken two and two in order, the
  first shall have to the last of the first magnitudes the same ratio
  which the first of the others has to the last._

  We may state it more generally, thus:

  If   a : b : c : d : e: ... = a' : b' : c' : d' : e' : ... ,

  then not only have two consecutive, but any two magnitudes on the
  first side, the same ratio as the corresponding magnitudes on the
  other. For instance--

    a : c = a' : c'; b : e = b' : e', &c.

  Prop. 23 we state only in symbols, viz.:--

  If  a : b : c : d : e : ... = 1/a' : 1/b' : 1/c' : 1/d' : 1/e' ...,

  then  a : c = c' : a',
        b : e = e' : b',

  and so on.

  Prop. 24 comes to this: If a : b = c : d and e : b = f : d, then

    a + e : b = c + f : d.

  Some of the proportions which are considered in the above propositions
  have special names. These we have omitted, as being of no use, since
  algebra has enabled us to bring the different operations contained in
  the propositions under a common point of view.

  § 56. The last proposition in the fifth book is of a different
  character.

  Prop. 25. _If four magnitudes of the same kind be proportional, the
  greatest and least of them together shall be greater than the other
  two together._ In symbols--

  If a, b, c, d be magnitudes of the same kind, and if a : b = c : d,
  and if a is the greatest, hence d the least, then a + d > b + c.

  § 57. We return once again to the question. What is a ratio? We have
  seen that we may treat ratios as magnitudes, and that all ratios are
  magnitudes of the same kind, for we may compare any two as to their
  magnitude. It will presently be shown that ratios of lines may be
  considered as _quotients_ of lines, so that a ratio appears as answer
  to the question, How often is one line contained in another? But the
  answer to this question is given by a number, at least in some cases,
  and in all cases if we admit incommensurable numbers. Considered from
  this point of view, we may say the fifth book of the _Elements_ shows
  that some of the simpler algebraical operations hold for
  incommensurable numbers. In the ordinary algebraical treatment of
  numbers this proof is altogether omitted, or given by a process of
  limits which does not seem to be natural to the subject.


  BOOK VI.

  § 58. The sixth book contains the theory of similar figures. After a
  few definitions explaining terms, the first proposition gives the
  first application of the theory of proportion.

  Prop. 1. _Triangles and parallelograms of the same altitude are to one
  another as their bases._

  The proof has already been considered in § 49.

  From this follows easily the important theorem

  Prop. 2. _If a straight line be drawn parallel to one of the sides of
  a triangle it shall cut the other sides, or those sides produced,
  proportionally; and if the sides or the sides produced be cut
  proportionally, the straight line which joins the points of section
  shall be parallel to the remaining side of the triangle._

  § 59. The next proposition, together with one added by Simson as Prop.
  A, may be expressed more conveniently if we introduce a modern
  phraseology, viz. if in a line AB we assume a point C between A and B,
  we shall say that C divides AB internally in the ratio AC : CB; but if
  C be taken in the line AB produced, we shall say that AB is divided
  externally in the ratio AC : CB.

  The two propositions then come to this:

  Prop. 3. _The bisector of an angle in a triangle divides the opposite
  side internally in a ratio equal to the ratio of the two sides
  including that angle;_ and conversely, _if a line through the vertex
  of a triangle divide the base internally in the ratio of the two other
  sides, then that line bisects the angle at the vertex_.

  Simson's Prop. A. _The line which bisects an exterior angle of a
  triangle divides the opposite side externally in the ratio of the
  other sides;_ and conversely, _if a line through the vertex of a
  triangle divide the base externally in the ratio of the sides, then it
  bisects an exterior angle at the vertex of the triangle_.

  If we combine both we have--

  _The two lines which bisect the interior and exterior angles at one
  vertex of a triangle divide the opposite side internally and
  externally in the same ratio, viz. in the ratio of the other two
  sides._

  § 60. The next four propositions contain the theory of similar
  triangles, of which four cases are considered. They may be stated
  together.

  _Two triangles are similar_,--

  1. (Prop. 4). _If the triangles are equiangular:_

  2. (Prop. 5). _If the sides of the one are proportional to those of
  the other_;

  3. (Prop. 6). _If two sides in one are proportional to two sides in
  the other, and if the angles contained by these sides are equal_;

  4. (Prop. 7). _If two sides in one are proportional to two sides in
  the other, if the angles opposite homologous sides are equal, and if
  the angles opposite the other homologous sides are both acute, both
  right or both obtuse; homologous sides being in each case those which
  are opposite equal angles_.

  An important application of these theorems is at once made to a
  right-angled triangle, viz.:--

  Prop. 8. _In a right-angled triangle, if a perpendicular be drawn from
  the right angle to the base, the triangles on each side of it are
  similar to the whole triangle, and to one another_.

  _Corollary._--From this it is manifest that the perpendicular drawn
  from the right angle of a right-angled triangle to the base is a mean
  proportional between the segments of the base, and also that each of
  the sides is a mean proportional between the base and the segment of
  the base adjacent to that side.

  § 61. There follow four propositions containing problems, in language
  slightly different from Euclid's, viz.:--

  Prop. 9. _To divide a straight line into a given number of equal
  parts_.

  Prop. 10. _To divide a straight line in a given ratio_.

  Prop. 11. _To find a third proportional to two given straight lines_.

  Prop. 12. _To find a fourth proportional to three given straight
  lines_.

  Prop. 13. _To find a mean proportional between two given straight
  lines_.

  The last three may be written as equations with one unknown
  quantity--viz. if we call the given straight lines a, b, c, and the
  required line x, we have to find a line x so that

  Prop. 11.  a : b = b : x;

  Prop. 12.  a : b = c : x;

  Prop. 13.  a : x = x : b.

  We shall see presently how these may be written without the signs of
  ratios.

  § 62. Euclid considers next proportions connected with parallelograms
  and triangles which are equal in area.

  Prop. 14. _Equal parallelograms which have one angle of the one equal
  to one angle of the other have their sides about the equal angles
  reciprocally proportional; and parallelograms which have one angle of
  the one equal to one angle of the other, and their sides about the
  equal angles reciprocally proportional, are equal to one another_.

  Prop. 15. _Equal triangles which have one angle of the one equal to
  one angle of the other, have their sides about the equal angles
  reciprocally proportional; and triangles which have one angle of the
  one equal to one angle of the other, and their sides about the equal
  angles reciprocally proportional, are equal to one another_.

  [Illustration]

  The latter proposition is really the same as the former, for if, as in
  the accompanying diagram, in the figure belonging to the former the
  two equal parallelograms AB and BC be bisected by the lines DF and EG,
  and if EF be drawn, we get the figure belonging to the latter.

  It is worth noticing that the lines FE and DG are parallel. We may
  state therefore the theorem--

  _If two triangles are equal in area, and have one angle in the one
  vertically opposite to one angle in the other, then the two straight
  lines which join the remaining two vertices of the one to those of the
  other triangle are parallel_.

  § 63. A most important theorem is

  _Prop. 16. If four straight lines be proportionals, the rectangle
  contained by the extremes is equal to the rectangle contained by the
  means; and if the rectangle contained by the extremes be equal to the
  rectangle contained by the means, the four straight lines are
  proportionals_.

  In symbols, if a, b, c, d are the four lines, and
  if                  a : b = c : d,
  then                ad = bc;
  and conversely, if  ad = bc,
  then                a : b = c : d,

  where ad and bc denote (as in § 20), the areas of the rectangles
  contained by a and d and by b and c respectively.

  This allows us to transform every proportion between four lines into
  an equation between two products.

  It shows further that the operation of forming a product of two lines,
  and the operation of forming their ratio are each the inverse of the
  other.

  If we now define a quotient a/b of two lines as the _number_ which
  multiplied into b gives a, so that

    a
    -- b = a,
    b

  we see that from the equality of two quotients

    a    c
    -- = --
    b    d

  follows, if we multiply both sides by bd,

    a        c
    -- b·d = -- d·b,
    b        d

    ad = cb.

  But from this it follows, according to the last theorem, that

    a : b = c : d.

  Hence we conclude that the quotient a/b and the ratio a : b are
  different forms of the same magnitude, only with this important
  difference that the quotient a/b would have a meaning only if a and b
  have a common measure, until we introduce incommensurable numbers,
  while the ratio a : b has always a meaning, and thus gives rise to the
  introduction of incommensurable numbers.

  Thus it is really the theory of ratios in the fifth book which enables
  us to extend the geometrical calculus given before in connexion with
  Book II. It will also be seen that if we write the ratios in Book V.
  as quotients, or rather as fractions, then most of the theorems state
  properties of quotients or of fractions.

  § 64. Prop. 17. _If three straight lines are proportional the
  rectangle contained by the extremes is equal to the square on the
  mean;_ and conversely, is only a special case of 16. After the
  problem, Prop. 18, _On a given straight line to describe a rectilineal
  figure similar and similarly situated to a given rectilineal figure_,
  there follows another fundamental theorem:

  Prop. 19. _Similar triangles are to one another in the duplicate ratio
  of their homologous sides._ In other words, the areas of similar
  triangles are to one another as the squares on homologous sides. This
  is generalized in:

  Prop. 20. _Similar polygons may be divided into the same number of
  similar triangles, having the same ratio to one another that the
  polygons have; and the polygons are to one another in the duplicate
  ratio of their homologous sides._

  § 65. Prop. 21. _Rectilineal figures which are similar to the same
  rectilineal figure are also similar to each other_, is an immediate
  consequence of the definition of similar figures. As similar figures
  may be said to be equal in "shape" but not in "size," we may state it
  also thus:

  "Figures which are equal in shape to a third are equal in shape to
  each other."

  Prop. 22. _If four straight lines be proportionals, the similar
  rectilineal figures similarly described on them shall also be
  proportionals; and if the similar rectilineal figures similarly
  described on four straight lines be proportionals, those straight
  lines shall be proportionals._

  This is essentially the same as the following:--

  _If_    a : b = c : d,
  _then_  a² : b² = c² : d².

  § 66. Now follows a proposition which has been much discussed with
  regard to Euclid's exact meaning in saying that a ratio is
  _compounded_ of two other ratios, viz.:

  Prop. 23. _Parallelograms which are equiangular to one another, have
  to one another the ratio which is compounded of the ratios of their
  sides._

  The proof of the proposition makes its meaning clear. In symbols the
  ratio a : c is compounded of the two ratios a : b and b : c, and if a
  : b = a' : b', b : c = b" : c", then a : c is compounded of a' : b'
  and b" : c".

  If we consider the ratios as numbers, we may say that the one ratio is
  the product of those of which it is compounded, or in symbols,

    a    a    b    a'   b"     a    a'     b    b"
    -- = -- · -- = -- · --, if -- = -- and -- = --.
    c    b    c    b'   c"     b    b'     c    c"

  The theorem in Prop. 23 is the foundation of all mensuration of areas.
  From it we see at once that two rectangles have the ratio of their
  areas compounded of the ratios of their sides.

  If A is the area of a rectangle contained by a and b, and B that of a
  rectangle contained by c and d, so that A = ab, B = cd, then A : B =
  ab : cd, and this is, the theorem says, compounded of the ratios a : c
  and b : d. In forms of quotients,

    a    b    ab
    -- · -- = --.
    c    d    cd

  This shows how to multiply quotients in our geometrical calculus.

  Further, _Two triangles have the ratios of their areas compounded of
  the ratios of their bases and their altitude._ For a triangle is equal
  in area to half a parallelogram which has the same base and the same
  altitude.

  § 67. To bring these theorems to the form in which they are usually
  given, we assume a straight line u as our unit of length (generally an
  inch, a foot, a mile, &c.), and determine the number [alpha] which
  expresses how often u is contained in a line a, so that [alpha]
  denotes the ratio a : u whether commensurable or not, and that a =
  [alpha]u. We call this number [alpha] the numerical value of a. If in
  the same manner ß be the numerical value of a line b we have

    a : b = [alpha] : ß;

  in words: _The ratio of two lines (and of two like quantities in
  general) is equal to that of their numerical values._

  This is easily proved by observing that a = [alpha]u, b = ßu,
  therefore a : b = [alpha]u : ßu, and this may without difficulty be
  shown to equal [alpha] : ß.

  If now a, b be base and altitude of one, a', b' those of another
  parallelogram, [alpha], ß and [alpha]', ß' their numerical values
  respectively, and A, A' their areas, then

    A    a    b    [alpha]    ß     [alpha]ß
    -- = -- · -- = -------- · -- = ----------.
    A'   a'   b'   [alpha]'   ß'   [alpha]'ß'

  In words: _The areas of two parallelograms are to each other as the
  products of the numerical values of their bases and altitudes._

  If especially the second parallelogram is the unit square, i.e. a
  square on the unit of length, then [alpha]' = ß' = 1, A' = u², and we
  have

    A
    -- = [alpha]ß or A = [alpha]ß · u².
    A'

  This gives the theorem: The number of unit squares contained in a
  parallelogram equals the product of the numerical values of base and
  altitude, and similarly the number of unit squares contained in a
  triangle equals half the product of the numerical values of base and
  altitude.

  This is often stated by saying that the area of a parallelogram is
  equal to the product of the base and the altitude, meaning by this
  product the product of the numerical values, and not the product as
  defined above in § 20.

  § 68. Propositions 24 and 26 relate to parallelograms about diagonals,
  such as are considered in Book I., 43. They are--

  Prop. 24. _Parallelograms about the diameter of any parallelogram are
  similar to the whole parallelogram and to one another_; and its
  converse (Prop. 26), _If two similar parallelograms have a common
  angle, and be similarly situated, they are about the same diameter._

  Between these is inserted a problem.

  Prop. 25. _To describe a rectilineal figure which shall be similar to
  one given rectilinear figure, and equal to another given rectilineal
  figure_.

  § 69. Prop. 27 contains a theorem relating to the theory of maxima and
  minima. We may state it thus:

  Prop. 27. _If a parallelogram be divided into two by a straight line
  cutting the base, and if on half the base another parallelogram be
  constructed similar to one of those parts, then this third
  parallelogram is greater than the other part._

  Of far greater interest than this general theorem is a special case of
  it, where the parallelograms are changed into rectangles, and where
  one of the parts into which the parallelogram is divided is made a
  square; for then the theorem changes into one which is easily
  recognized to be identical with the following:--

  _Of all rectangles which have the same perimeter the square has the
  greatest area._

  This may also be stated thus:--

  _Of all rectangles which have the same area the square has the least
  perimeter._

  § 70. The next three propositions contain problems which may be said
  to be solutions of quadratic equations. The first two are, like the
  last, involved in somewhat obscure language. We transcribe them as
  follows:

  _Problem_.--To describe on a given base a parallelogram, and to divide
  it either internally (Prop. 28) or externally (Prop. 29) from a point
  on the base into two parallelograms, of which the one has a given size
  (is equal in area to a given figure), whilst the other has a given
  shape (is similar to a given parallelogram).

  If we express this again in symbols, calling the given base a, the one
  part x, and the altitude y, we have to determine x and y in the first
  case from the equations

    (a - x)y = k²,

    x    p
    -- = --,
    y    q

  k² being the given size of the first, and p and q the base and
  altitude of the parallelogram which determine the shape of the second
  of the required parallelograms.

  If we substitute the value of y, we get

               pk²
    (a - x)x = ---,
                q

  or,

    ax - x² = b²,

  where a and b are known quantities, taking b² = pk²/q.

  The second case (Prop. 29) gives rise, in the same manner, to the
  quadratic

    ax + x² = b².

  The next problem--

  Prop. 30. _To cut a given straight line in extreme and mean ratio_,
  leads to the equation

    ax + x² = a².

  This is, therefore, only a special case of the last, and is, besides,
  an old acquaintance, being essentially the same problem as that
  proposed in II. 11.

  Prop. 30 may therefore be solved in two ways, either by aid of Prop.
  29 or by aid of II. 11. Euclid gives both solutions.

  § 71. Prop. 31 (Theorem). _In any right-angled triangle, any
  rectilineal figure described on the side subtending the right angle is
  equal to the similar and similarly-described figures on the sides
  containing the right angle_,--is a pretty generalization of the
  theorem of Pythagoras (I. 47).

  Leaving out the next proposition, which is of little interest, we come
  to the last in this book.

  Prop. 33. _In equal circles angles, whether at the centres or the
  circumferences, have the same ratio which the arcs on which they stand
  have to one another; so also have the sectors_.

  Of this, the part relating to angles at the centre is of special
  importance; it enables us to measure angles by arcs.

  With this closes that part of the _Elements_ which is devoted to the
  study of figures in a plane.


  BOOK XI.

  § 72. In this book figures are considered which are not confined to a
  plane, viz. first relations between lines and planes in space, and
  afterwards properties of solids.

  Of new definitions we mention those which relate to the
  perpendicularity and the inclination of lines and planes.

  Def. 3. _A straight line is perpendicular, or at right angles, to a
  plane when it makes right angles with every straight line meeting it
  in that plane_.

  The definition of perpendicular planes (Def. 4) offers no difficulty.
  Euclid defines the inclination of lines to planes and of planes to
  planes (Defs. 5 and 6) by aid of plane angles, included by straight
  lines, with which we have been made familiar in the first books.

  The other important definitions are those of parallel planes, which
  never meet (Def. 8), and of solid angles formed by three or more
  planes meeting in a point (Def. 9).

  To these we add the definition of a line parallel to a plane as a line
  which does not meet the plane.

  § 73. Before we investigate the contents of Book XI., it will be well
  to recapitulate shortly what we know of planes and lines from the
  definitions and axioms of the first book. There a plane has been
  defined as a surface which has the property that every straight line
  which joins two points in it lies altogether in it. This is equivalent
  to saying that a straight line which has two points in a plane has all
  points in the plane. Hence, a straight line which does not lie in the
  plane cannot have more than one point in common with the plane. This
  is virtually the same as Euclid's Prop. 1, viz.:--

  Prop. 1. _One part of a straight line cannot be in a plane and another
  part without it_.

  It also follows, as was pointed out in § 3, in discussing the
  definitions of Book I., that a plane is determined already by one
  straight line and a point without it, viz. if all lines be drawn
  through the point, and cutting the line, they will form a plane.

  This may be stated thus:--

  _A plane is determined_--

  1st, _By a straight line and a point which does not lie on it;_

  2nd, _By three points which do not lie in a straight line_; for if two
  of these points be joined by a straight line we have case 1;

  3rd, _By two intersecting straight lines_; for the point of
  intersection and two other points, one in each line, give case 2;

  4th, _By two parallel lines_ (Def. 35, I.).

  The third case of this theorem is Euclid's

  Prop. 2. _Two straight lines which cut one another are in one plane,
  and three straight lines which meet one another are in one plane_.

  And the fourth is Euclid's

  Prop. 7. _If two straight lines be parallel, the straight line drawn
  from any point in one to any point in the other is in the same plane
  with the parallels_. From the definition of a plane further follows

  Prop. 3. _If two planes cut one another, their common section is a
  straight line_.

  § 74. Whilst these propositions are virtually contained in the
  definition of a plane, the next gives us a new and fundamental
  property of space, showing at the same time that it is possible to
  have a straight line perpendicular to a plane, according to Def. 3. It
  states--

  Prop. 4. _If a straight line is perpendicular to two straight lines in
  a plane which it meets, then it is perpendicular to all lines in the
  plane which it meets, and hence it is perpendicular to the plane_.

  Def. 3 may be stated thus: If a straight line is perpendicular to a
  plane, then it is perpendicular to every line in the plane which it
  meets. The converse to this would be

  _All straight lines which meet a given straight line in the same
  point, and are perpendicular to it, lie in a plane which is
  perpendicular to that line_.

  This Euclid states thus:

  Prop. 5. _If three straight lines meet all at one point, and a
  straight line stands at right angles to each of them at that point,
  the three straight lines shall be in one and the same plane_.

  § 75. There follow theorems relating to the theory of parallel lines
  in space, viz.:--

  Prop. 6. _Any two lines which are perpendicular to the same plane are
  parallel to each other;_ and conversely

  Prop. 8. _If of two parallel straight lines one is perpendicular to a
  plane, the other is so also._

  Prop. 7. _If two straight lines are parallel, the straight line which
  joins any point in one to any point in the other is in the same plane
  as the parallels._ (See above, § 73.)

  Prop. 9. _Two straight lines which are each of them parallel to the
  same straight line, and not in the same plane with it, are parallel to
  one another;_ where the words, "and not in the same plane with it,"
  may be omitted, for they exclude the case of three parallels in a
  plane, which has been proved before; and

  Prop. 10. _If two angles in different planes have the two limits of
  the one parallel to those of the other, then the angles are equal._
  That their planes are parallel is shown later on in Prop. 15.

  This theorem is not necessarily true, for the angles in question may
  be supplementary; but then the one angle will be equal to that which
  is adjacent and supplementary to the other, and this latter angle will
  also have its limits parallel to those of the first.

  From this theorem it follows that if we take any two straight lines in
  space which do not meet, and if we draw through any point P in space
  two lines parallel to them, then the angle included by these lines
  will always be the same, whatever the position of the point P may be.
  This angle has in modern times been called the angle between the given
  lines:--

  _By the angles between two not intersecting lines we understand the
  angles which two intersecting lines include that are parallel
  respectively to the two given lines._

  § 76. It is now possible to solve the following two problems:--

  _To draw a straight line perpendicular to a given plane from a given
  point which lies_

  1. _Not in the plane_ (Prop. 11).

  2. _In the plane_ (Prop. 12).

  The second case is easily reduced to the first--viz. if by aid of the
  first we have drawn any perpendicular to the plane from some point
  without it, we need only draw through the given point in the plane a
  line parallel to it, in order to have the required perpendicular
  given. The solution of the first part is of interest in itself. It
  depends upon a construction which may be expressed as a theorem.

  _If from a point A without a plane a perpendicular AB be drawn to the
  plane, and if from the foot B of this perpendicular another
  perpendicular BC be drawn to any straight line in the plane, then the
  straight line joining A to the foot C of this second perpendicular
  will also be perpendicular to the line in the plane._

  The theory of perpendiculars to a plane is concluded by the theorem--

  Prop. 13. _Through any point in space, whether in or without a plane,
  only one straight line can be drawn perpendicular to the plane._

  § 77. The next four propositions treat of parallel planes. It is shown
  _that planes which have a common perpendicular are parallel_ (Prop.
  14); _that two planes are parallel if two intersecting straight lines
  in the one are parallel respectively to two straight lines in the
  other plane_ (Prop. 15); _that parallel planes are cut by any plane in
  parallel straight lines_ (Prop. 16); and lastly, _that any two
  straight lines are cut proportionally by a series of parallel planes_
  (Prop. 17).

  This theory is made more complete by adding the following theorems,
  which are easy deductions from the last: _Two parallel planes have
  common perpendiculars_ (converse to 14); and _Two planes which are
  parallel to a third plane are parallel to each other._

  It will be noted that Prop. 15 at once allows of the solution of the
  problem: "Through a given point to draw a plane parallel to a given
  plane." And it is also easily proved that this problem allows always
  of one, and only of one, solution.

  § 78. We come now to planes which are perpendicular to one another.
  Two theorems relate to them.

  Prop. 18. _If a straight line be at right angles to a plane, every
  plane which passes through it shall be at right angles to that plane._

  Prop. 19. _If two planes which cut one another be each of them
  perpendicular to a third plane, their common section shall be
  perpendicular to the same plane._

  § 79. If three planes pass through a common point, and if they bound
  each other, a solid angle of three faces, or a _trihedral_ angle, is
  formed, and similarly by more planes a solid angle of more faces, or a
  _polyhedral_ angle. These have many properties which are quite
  analogous to those of triangles and polygons in a plane. Euclid states
  some, viz.:--

  Prop. 20. _If a solid angle be contained by three plane angles, any
  two of them are together greater than the third._

  But the next--

  Prop. 21. _Every solid angle is contained by plane angles, which are
  together less than four right angles_--has no analogous theorem in the
  plane.

  We may mention, however, that the theorems about triangles contained
  in the propositions of Book I., which do not depend upon the theory of
  parallels (that is all up to Prop. 27), have their corresponding
  theorems about trihedral angles. The latter are formed, if for "side
  of a triangle" we write "plane angle" or "face" of trihedral angle,
  and for "angle of triangle" we substitute "angle between two faces"
  where the planes containing the solid angle are called its _faces_. We
  get, for instance, from I. 4, the theorem, _If two trihedral angles
  have the angles of two faces in the one equal to the angles of two
  faces in the other, and have likewise the angles included by these
  faces equal, then the angles in the remaining faces are equal, and the
  angles between the other faces are equal each to each, viz. those
  which are opposite equal faces._ The solid angles themselves are not
  necessarily equal, for they may be only symmetrical like the right
  hand and the left.

  The connexion indicated between triangles and trihedral angles will
  also be recognized in

  Prop. 22. _If every two of three plane angles be greater than the
  third, and if the straight lines which contain them be all equal, a
  triangle may be made of the straight lines that join the extremities
  of those equal straight lines._

  And Prop. 23 solves the problem, _To construct a trihedral angle
  having the angles of its faces equal to three given plane angles, any
  two of them being greater than the third._ It is, of course, analogous
  to the problem of constructing a triangle having its sides of given
  length.

  Two other theorems of this kind are added by Simson in his edition of
  Euclid's _Elements_.

  § 80. These are the principal properties of lines and planes in space,
  but before we go on to their applications it will be well to define
  the word _distance_. In geometry distance means always "shortest
  distance"; viz. the distance of a point from a straight line, or from
  a plane, is the length of the perpendicular from the point to the line
  or plane. The distance between two non-intersecting lines is the
  length of their common perpendicular, there being but one. The
  distance between two parallel lines or between two parallel planes is
  the length of the common perpendicular between the lines or the
  planes.

  § 81. _Parallelepipeds_.--The rest of the book is devoted to the study
  of the parallelepiped. In Prop. 24 the possibility of such a solid is
  proved, viz.:--

  Prop. 24. _If a solid be contained by six planes two and two of which
  are parallel, the opposite planes are similar and equal
  parallelograms._

  Euclid calls this solid henceforth a parallelepiped, though he never
  defines the word. Either face of it may be taken as _base_, and its
  distance from the opposite face as _altitude_.

  Prop. 25. _If a solid parallelepiped be cut by a plane parallel to two
  of its opposite planes, it divides the whole into two solids, the base
  of one of which shall be to the base of the other as the one solid is
  to the other_.

  This theorem corresponds to the theorem (VI. 1) that parallelograms
  between the same parallels are to one another as their bases. A
  similar analogy is to be observed among a number of the remaining
  propositions.

  § 82. After solving a few problems we come to

  Prop. 28. _If a solid parallelepiped be cut by a plane passing through
  the diagonals of two of the opposite planes, it shall be cut in two
  equal parts._

  In the proof of this, as of several other propositions, Euclid
  neglects the difference between solids which are symmetrical like the
  right hand and the left.

  Prop. 31. _Solid parallelepipeds, which are upon equal bases, and of
  the same altitude, are equal to one another._

  Props. 29 and 30 contain special cases of this theorem leading up to
  the proof of the general theorem.

  As consequences of this fundamental theorem we get

  Prop. 32. _Solid parallelepipeds, which have the same altitude, are to
  one another as their bases;_ and

  Prop. 33. _Similar solid parallelepipeds are to one another in the
  triplicate ratio of their homologous sides._

  If we consider, as in § 67, the ratios of lines as numbers, we may
  also say--

  _The ratio of the volumes of similar parallelepipeds is equal to the
  ratio of the third powers of homologous sides._

  Parallelepipeds which are not similar but equal are compared by aid of
  the theorem

  Prop. 34. _The bases and altitudes of equal solid parallelepipeds are
  reciprocally proportional; and if the bases and altitudes be
  reciprocally proportional, the solid parallelepipeds are equal._

  § 83. Of the following propositions the 37th and 40th are of special
  interest.

  Prop. 37. _If four straight lines be proportionals, the similar solid
  parallelepipeds, similarly described from them, shall also be
  proportionals; and if the similar parallelepipeds similarly described
  from four straight lines be proportionals, the straight lines shall be
  proportionals._

  In symbols it says--

    If a : b = c : d, then a³ : b³ = c³ : d³.

  Prop. 40 teaches how to compare the volumes of triangular prisms with
  those of parallelepipeds, by proving _that a triangular prism is equal
  in volume to a parallelepiped, which has its altitude and its base
  equal to the altitude and the base of the triangular prism._

  § 84. From these propositions follow all results relating to the
  mensuration of volumes. We shall state these as we did in the case of
  areas. The starting-point is the "rectangular" parallelepiped, which
  has every edge perpendicular to the planes it meets, and which takes
  the place of the rectangle in the plane. If this has all its edges
  equal we obtain the "cube."

  If we take a certain line u as unit length, then the square on u is
  the unit of area, and the cube on u the unit of volume, that is to
  say, if we wish to measure a volume we have to determine how many unit
  cubes it contains.

  A rectangular parallelepiped has, as a rule, the three edges unequal,
  which meet at a point. Every other edge is equal to one of them. If a,
  b, c be the three edges meeting at a point, then we may take the
  rectangle contained by two of them, say by b and c, as base and the
  third as altitude. Let V be its volume, V' that of another rectangular
  parallelepiped which has the edges a', b, c, hence the same base as
  the first. It follows then easily, from Prop. 25 or 32, that V : V' =
  a : a'; or in words,

  _Rectangular parallelepipeds on equal bases are proportional to their
  altitudes._

  If we have two rectangular parallelepipeds, of which the first has the
  volume V and the edges a, b, c, and the second, the volume V' and the
  edges a', b', c', we may compare them by aid of two new ones which
  have respectively the edges a', b, c and a', b', c, and the volumes V1
  and V2. We then have

    V : V1 = a : a'; V1 : V2 = b : b', V2 : V' = c : c'.

  Compounding these, we have

    V : V' = (a : a')(b : b')(c : c'),

  or

    V    a    b    c
    -- = -- · -- · --.
    V'   a'   b'   c'

  Hence, as a special case, making V' equal to the unit cube U on u we
  get

    V    a    b    c
    -- = -- · -- · -- = [alpha]·ß·[gamma],
    U    u    u    u

  where [alpha], ß, [gamma] are the numerical values of a, b, c; that
  is, _The number of unit cubes in a rectangular parallelepiped_ is
  equal to the product of the numerical values of its three edges. This
  is generally expressed by saying the volume of a rectangular
  parallelepiped is measured by the product of its sides, or by the
  product of its base into its altitude, which in this case is the same.

  Prop. 31 allows us to extend this to any parallelepipeds, and Props.
  28 or 40, to triangular prisms.

  _The volume of any parallelepiped, or of any triangular prism, is
  measured by the product of base and altitude._

  The consideration that any polygonal prism may be divided into a
  number of triangular prisms, which have the same altitude and the sum
  of their bases equal to the base of the polygonal prism, shows further
  that the same holds for any prism whatever.


  BOOK XII.

  § 85. In the last part of Book XI. we have learnt how to compare the
  volumes of parallelepipeds and of prisms. In order to determine the
  volume of any solid bounded by plane faces we must determine the
  volume of pyramids, for every such solid may be decomposed into a
  number of pyramids.

  As every pyramid may again be decomposed into triangular pyramids, it
  becomes only necessary to determine their volume. This is done by the

  _Theorem._--Every triangular pyramid is equal in volume to one third
  of a triangular prism having the same base and the same altitude as
  the pyramid.

  This is an immediate consequence of Euclid's

  Prop. 7. _Every prism having a triangular base may be divided into
  three pyramids that have triangular bases, and are equal to one
  another._

  The proof of this theorem is difficult, because the three triangular
  pyramids into which the prism is divided are by no means equal in
  shape, and cannot be made to coincide. It has first to be proved that
  two triangular pyramids have equal volumes, if they have equal bases
  and equal altitudes. This Euclid does in the following manner. He
  first shows (Prop. 3) that a triangular pyramid may be divided into
  four parts, of which two are equal triangular pyramids similar to the
  whole pyramid, whilst the other two are equal triangular prisms, and
  further, that these two prisms together are greater than the two
  pyramids, hence more than half the given pyramid. He next shows (Prop.
  4) that if two triangular pyramids are given, having equal bases and
  equal altitudes, and if each be divided as above, then the two
  triangular prisms in the one are equal to those in the other, and each
  of the remaining pyramids in the one has its base and altitude equal
  to the base and altitude of the remaining pyramids in the other. Hence
  to these pyramids the same process is again applicable. We are thus
  enabled to cut out of the two given pyramids equal parts, each greater
  than half the original pyramid. Of the remainder we can again cut out
  equal parts greater than half these remainders, and so on as far as we
  like. This process may be continued till the last remainder is smaller
  than any assignable quantity, however small. It follows, so we should
  conclude at present, that the two volumes must be equal, for they
  cannot differ by any assignable quantity.

  To Greek mathematicians this conclusion offers far greater
  difficulties. They prove elaborately, by a _reductio ad absurdum_,
  that the volumes cannot be unequal. This proof must be read in the
  _Elements._ We must, however, state that we have in the above not
  proved Euclid's Prop. 5, but only a special case of it. Euclid does
  not suppose that the bases of the two pyramids to be compared are
  equal, and hence he proves that the volumes are as the bases. The
  reasoning of the proof becomes clearer in the special case, from which
  the general one may be easily deduced.

  § 86. Prop. 6 extends the result to pyramids with polygonal bases.
  From these results follow again the rules at present given for the
  mensuration of solids, viz. a pyramid is the third part of a
  triangular prism having the same base and the same altitude. But a
  triangular prism is equal in volume to a parallelepiped which has the
  same base and altitude. Hence if B is the base and h the altitude, we
  have

    Volume of prism   = Bh,
    Volume of pyramid = 1/3Bh,

  statements which have to be taken in the sense that B means the number
  of square units in the base, h the number of units of length in the
  altitude, or that B and h denote the numerical values of base and
  altitude.

  § 87. A method similar to that used in proving Prop. 5 leads to the
  following results relating to solids bounded by simple curved
  surfaces:--

  Prop. 10. _Every cone is the third part of a cylinder which has the
  same base, and is of an equal altitude with it._

  Prop. 11. _Cones or cylinders of the same altitude are to one another
  as their bases._

  Prop. 12. _Similar cones or cylinders have to one another the
  triplicate ratio of that which the diameters of their bases have._

  Prop. 13. _If a cylinder be cut by a plane parallel to its opposite
  planes or bases, it divides the cylinder into two cylinders, one of
  which is to the other as the axis of the first to the axis of the
  other;_ which may also be stated thus:--

  _Cylinders on the same base are proportional to their altitudes._

  Prop. 14. _Cones or cylinders upon equal bases are to one another as
  their altitudes._

  Prop. 15. _The bases and altitudes of equal cones or cylinders are
  reciprocally proportional, and if the bases and altitudes be
  reciprocally proportional, the cones or cylinders are equal to one
  another._

  These theorems again lead to formulae in mensuration, if we compare a
  cylinder with a prism having its base and altitude equal to the base
  and altitude of the cylinder. This may be done by the method of
  exhaustion. We get, then, the result that their bases are equal, and
  have, if B denotes the numerical value of the base, and h that of the
  altitude,

    Volume of cylinder = Bh,
    Volume of cone     = 1/3Bh.

  § 88. The remaining propositions relate to circles and spheres. Of the
  sphere only one property is proved, viz.:--

  Prop. 18. _Spheres have to one another the triplicate ratio of that
  which their diameters have._ The mensuration of the sphere, like that
  of the circle, the cylinder and the cone, had not been settled in the
  time of Euclid. It was done by Archimedes.


  BOOK XIII.

  § 89. The 13th and last book of Euclid's _Elements_ is devoted to the
  regular solids (see POLYHEDRON). It is shown that there are five of
  them, viz.:--

  1. The regular _tetrahedron_, with 4 triangular faces and 4 vertices;

  2. The _cube_, with 8 vertices and 6 square faces;

  3. The _octahedron_, with 6 vertices and 8 triangular faces;

  4. The _dodecahedron_, with 12 pentagonal faces, 3 at each of the
  20 vertices;

  5. The _icosahedron_, with 20 triangular faces, 5 at each of the
  12 vertices.

  It is shown how to inscribe these solids in a given sphere, and how to
  determine the lengths of their edges.

  § 90. The 13th book, and therefore the _Elements_, conclude with the
  scholium, "that no other regular solid exists besides the five ones
  enumerated."

  The proof is very simple. Each face is a regular polygon, hence the
  angles of the faces at any vertex must be angles in equal regular
  polygons, must be together less than four right angles (XI. 21), and
  must be three or more in number. Each angle in a regular triangle
  equals two-thirds of one right angle. Hence it is possible to form a
  solid angle with three, four or five regular triangles or faces. These
  give the solid angles of the tetrahedron, the octahedron and the
  icosahedron. The angle in a square (the regular quadrilateral) equals
  one right angle. Hence three will form a solid angle, that of the
  cube, and four will not. The angle in the regular pentagon equals 6/5
  of a right angle. Hence three of them equal 18/5 (i.e. less than 4)
  right angles, and form the solid angle of the dodecahedron. Three
  regular polygons of six or more sides cannot form a solid angle.
  Therefore no other regular solids are possible.     (O. H.)


II. PROJECTIVE GEOMETRY

It is difficult, at the outset, to characterize projective geometry as
compared with Euclidean. But a few examples will at least indicate the
practical differences between the two.

In Euclid's _Elements_ almost all propositions refer to the _magnitude_
of lines, angles, areas or volumes, and therefore to measurement. The
statement that an angle is right, or that two straight lines are
parallel, refers to measurement. On the other hand, the fact that a
straight line does or does not cut a circle is independent of
measurement, it being dependent only upon the mutual "position" of the
line and the circle. This difference becomes clearer if we project any
figure from one plane to another (see PROJECTION). By this the length of
lines, the magnitude of angles and areas, is altered, so that the
projection, or shadow, of a square on a plane will not be a square; it
will, however, be some quadrilateral. Again, the projection of a circle
will not be a circle, but some other curve more or less resembling a
circle. But one property may be stated at once--no straight line can cut
the projection of a circle in more than two points, because no straight
line can cut a circle in more than two points. There are, then, some
properties of figures which do not alter by projection, whilst others
do. To the latter belong nearly all properties relating to measurement,
at least in the form in which they are generally given. The others are
said to be projective properties, and their investigation forms the
subject of projective geometry.

Different as are the kinds of properties investigated in the old and the
new sciences, the methods followed differ in a still greater degree. In
Euclid each proposition stands by itself; its connexion with others is
never indicated; the leading ideas contained in its proof are not
stated; general principles do not exist. In the modern methods, on the
other hand, the greatest importance is attached to the leading thoughts
which pervade the whole; and general principles, which bring whole
groups of theorems under one aspect, are given rather than separate
propositions. The whole tendency is towards generalization. A straight
line is considered as given in its entirety, extending both ways to
infinity, while Euclid never admits anything but finite quantities. The
treatment of the infinite is in fact another fundamental difference
between the two methods: Euclid avoids it; in modern geometry it is
systematically introduced.

Of the different modern methods of geometry, we shall treat principally
of the methods of projection and correspondence which have proved to be
the most powerful. These have become independent of Euclidean Geometry,
especially through the _Geometrie der Lage_ of V. Staudt and the
_Ausdehnungslehre_ of Grassmann.

For the sake of brevity we shall presuppose a knowledge of Euclid's
_Elements_, although we shall use only a few of his propositions.

  § 1. _Geometrical Elements._ We consider space as filled with points,
  lines and planes, and these we call the elements out of which our
  figures are to be formed, calling any combination of these elements a
  "figure."

  By a line we mean a straight line in its entirety, extending both ways
  to infinity; and by a plane, a plane surface, extending in all
  directions to infinity.

  We accept the three-dimensional space of experience--the space assumed
  by Euclid--which has for its properties (among others):--

  Through any two points in space one and only one line may be drawn;

  Through any three points which are not in a line, one and only one
  plane may be placed;

  The intersection of two planes is a line;

  A line which has two points in common with a plane lies in the plane,
  hence the intersection of a line and a plane is a single point; and

  Three planes which do not meet in a line have one single point in
  common.

  These results may be stated differently in the following form:--

  I. A plane is determined--           A point is determined--
       1. By three points which do         1. By three planes which do
           not lie in a line;                  not pass through a line;
       2. By two intersecting lines;       2. By two intersecting lines;
       3. By a line and a point            3. By a plane and a line
           which does not lie in it.           which does not lie in it.
  II. A line is determined--
       1. By two points;                   2. By two planes.

  It will be observed that not only are planes determined by points, but
  also points by planes; that therefore the planes may be considered as
  elements, like points; and also that in any one of the above
  statements we may interchange the words point and plane, and we obtain
  again a correct statement, provided that these statements themselves
  are true. As they stand, we ought, in several cases, to add "if they
  are not parallel," or some such words, parallel lines and planes being
  evidently left altogether out of consideration. To correct this we
  have to reconsider the theory of parallels.

  [Illustration: FIG. 1.]

  § 2. _Parallels. Point at Infinity._--Let us take in a plane a line p
  (fig. 1), a point S not in this line, and a line q drawn through S.
  Then this line q will meet the line p in a point A. If we turn the
  line q about S towards q', its point of intersection with p will move
  along p towards B, passing, on continued turning, to a greater and
  greater distance, until it is moved out of our reach. If we turn q
  still farther, its continuation will meet p, but now at the other side
  of A. The point of intersection has disappeared to the right and
  reappeared to the left. There is one intermediate position where q is
  parallel to p--that is where it does not cut p. In every other
  position it cuts p in some finite point. If, on the other hand, we
  move the point A to an infinite distance in p, then the line q which
  passes through A will be a line which does not cut p at any finite
  point. Thus we are led to say: _Every_ line through S which joins it
  to any point at an infinite distance in p is parallel to p. But by
  Euclid's 12th axiom there is but one line parallel to p through S. The
  difficulty in which we are thus involved is due to the fact that we
  try to reason about infinity as if we, with our finite capabilities,
  could comprehend the infinite. To overcome this difficulty, we may say
  that all points at infinity in a line _appear_ to us as one, and may
  be replaced by a single "ideal" point.

  We may therefore now give the following definitions and axiom:--

  _Definition._--Lines which meet at infinity are called parallel.

  _Axiom._--All points at an infinite distance in a line may be
  considered as one single point.

  _Definition._--This ideal point is called the _point at infinity_ in
  the line.

  The axiom is equivalent to Euclid's Axiom 12, for it follows from
  either that through any point only one line may be drawn parallel to a
  given line.

  This point at infinity in a line is reached whether we move a point in
  the one or in the opposite direction of a line to infinity. A line
  thus appears closed by this point, and we speak as if we could move a
  point along the line from one position A to another B in two ways,
  either through the point at infinity or through finite points only.

  It must never be forgotten that this point at infinity is ideal; in
  fact, the whole notion of "infinity" is only a mathematical
  conception, and owes its introduction (as a method of research) to the
  working generalizations which it permits.

  § 3. _Line and Plane at Infinity._--Having arrived at the notion of
  replacing all points at infinity in a line by one ideal point, there
  is no difficulty in replacing all points at infinity in a plane by one
  ideal line.

  To make this clear, let us suppose that a line p, which cuts two fixed
  lines a and b in the points A and B, moves parallel to itself to a
  greater and greater distance. It will at last cut both a and b at
  their points at infinity, so that a line which joins the two points at
  infinity in two intersecting lines lies altogether at infinity. Every
  other line in the plane will meet it therefore at infinity, and thus
  it contains all points at infinity in the plane.

  _All points at infinity in a plane lie in a line, which is called the_
  line at infinity _in the plane._

  It follows that parallel planes must be considered as planes having a
  common line at infinity, for any other plane cuts them in parallel
  lines which have a point at infinity in common.

  If we next take two intersecting planes, then the point at infinity in
  their line of intersection lies in both planes, so that their lines at
  infinity meet. Hence every line at infinity meets every other line at
  infinity, and they are therefore all in one plane.

  _All points at infinity in space may be considered as lying in one
  ideal plane, which is called the_ plane at infinity.

  § 4. _Parallelism._--We have now the following definitions:--

  Parallel lines are lines which meet at infinity;

  Parallel planes are planes which meet at infinity;

  A line is parallel to a plane if it meets it at infinity.

  Theorems like this--Lines (or planes) which are parallel to a third
  are parallel to each other--follow at once.

  This view of parallels leads therefore to no contradiction of Euclid's
  _Elements._

  As immediate consequences we get the propositions:--

  Every line meets a plane in one point, or it lies in it;

  Every plane meets every other plane in a line;

  Any two lines in the same plane meet.

  § 5. _Aggregates of Geometrical Elements._--We have called points,
  lines and planes the elements of geometrical figures. We also say that
  an element of one kind contains one of the other if it lies in it or
  passes through it.

  All the elements of one kind which are contained in one or two
  elements of a different kind form aggregates which have to be
  enumerated. They are the following:--

  I. Of one dimension.

     1. The _row_, or range, _of points_ formed by all points in a line,
          which is called its base.

     2. The _flat pencil_ formed by all the lines through a point in a
          plane. Its base is the point in the plane.

     3. The _axial pencil_ formed by all planes through a line which is
          called its base or axis.

  II. Of two dimensions.

     1. The field of points and lines--that is, a plane with all its
          points and all its lines.

     2. The pencil of lines and planes--that is, a point in space with
          all lines and all planes through it.

  III. Of three dimensions.

     The space of points--that is, all points in space.

     The space of planes--that is, all planes in space.

  IV. Of four dimensions.

     The space of lines, or all lines in space.

  § 6. _Meaning of "Dimensions."_--The word dimension in the above needs
  explanation. If in a plane we take a row p and a pencil with centre Q,
  then through every point in p one line in the pencil will pass, and
  every ray in Q will cut p in one point, so that we are entitled to say
  a row contains as many points as a flat pencil lines, and, we may add,
  as an axial pencil planes, because an axial pencil is cut by a plane
  in a flat pencil.

  The number of elements in the row, in the flat pencil, and in the
  axial pencil is, of course, infinite and indefinite too, but the same
  in all. This number may be denoted by [infinity]. Then a plane
  contains [infinity]² points and as many lines. To see this, take a
  flat pencil in a plane. It contains [infinity] lines, and each line
  contains [infinity] points, whilst each point in the plane lies on one
  of these lines. Similarly, in a plane each line cuts a fixed line in a
  point. But this line is cut at each point by [infinity] lines and
  contains [infinity] points; hence there are [infinity]² lines in a
  plane.

  A pencil in space contains as many lines as a plane contains points
  and as many planes as a plane contains lines, for any plane cuts the
  pencil in a field of points and lines. Hence a pencil contains
  [infinity]² lines and [infinity]² planes. _The field and the pencil
  are of two dimensions._

  To count the number of points in space we observe that each point lies
  on some line in a pencil. But the pencil contains [infinity]² lines,
  and each line [infinity] points; hence space contains [infinity]³
  points. Each plane cuts any fixed plane in a line. But a plane
  contains [infinity]² lines, and through each pass [infinity] planes;
  therefore space contains [infinity]³ planes.

  Hence space contains as many planes as points, but it contains an
  infinite number of times more lines than points or planes. To count
  them, notice that every line cuts a fixed plane in one point. But
  [infinity]² lines pass through each point, and there are [infinity]²
  points in the plane. Hence there are [infinity]^4 lines in space. _The
  space of points and planes is of three dimensions, but the space of
  lines is of four dimensions._

  A field of points or lines contains an infinite number of rows and
  flat pencils; a pencil contains an infinite number of flat pencils and
  of axial pencils; space contains a triple infinite number of pencils
  and of fields, [infinity]^4 rows and axial pencils and [infinity]^5
  flat pencils--or, in other words, each point is a centre of
  [infinity]² flat pencils.

  § 7. The above enumeration allows a classification of figures. Figures
  in a row consist of groups of points only, and figures in the flat or
  axial pencil consist of groups of lines or planes. In the plane we may
  draw polygons; and in the pencil or in the point, solid angles, and so
  on.

  We may also distinguish the different measurements We have--

    In the row, length of segment;
    In the flat pencil, angles;
    In the axial pencil, dihedral angles between two planes;
    In the plane, areas;
    In the pencil, solid angles;
    In the space of points or planes, volumes.


  SEGMENTS OF A LINE

  § 8. Any two points A and B in space determine on the line through
  them a finite part, which may be considered as being described by a
  point moving from A to B. This we shall denote by AB, and distinguish
  it from BA, which is supposed as being described by a point moving
  from B to A, and hence in a direction or in a "sense" opposite to AB.
  Such a finite line, which has a definite sense, we shall call a
  "segment," so that AB and BA denote different segments, which are said
  to be equal in length but of opposite sense. The one sense is often
  called positive and the other negative.

  In introducing the word "sense" for direction in a line, we have the
  word direction reserved for direction of the line itself, so that
  different lines have different directions, unless they be parallel,
  whilst in each line we have a positive and negative sense.

  We may also say, with Clifford, that AB denotes the "step" of going
  from A to B.

  [Illustration: FIG. 2.]

  § 9. If we have three points A, B, C in a line (fig. 2), the step AB
  will bring us from A to B, and the step BC from B to C. Hence both
  steps are equivalent to the one step AC. This is expressed by saying
  that AC is the "sum" of AB and BC; in symbols--

    AB + BC = AC,

  where account is to be taken of the sense.

  This equation is true whatever be the position of the three points on
  the line. As a special case we have

    AB + BA = 0, (1)

  and similarly

    AB + BC + CA = 0, (2)

  which again is true for any three points in a line.

  We further write

    AB = -BA.

  where - denotes negative sense.

  We can then, just as in algebra, change subtraction of segments into
  addition by changing the sense, so that AB - CB is the same as AB +
  (-CB) or AB + BC. A figure will at once show the truth of this. The
  sense is, in fact, in every respect equivalent to the "sign" of a
  number in algebra.

  § 10. Of the many formulae which exist between points in a line we
  shall have to use only one more, which connects the segments between
  any four points A, B, C, D in a line. We have

    BC = BD + DC, CA = CD + DA, AB = AD + DB;

  or multiplying these by AD, BD, CD respectively, we get

    BC·AD = BD·AD + DC·AD = BD·AD - CD·AD

    CA·BD = CD·BD + DA·BD = CD·BD - AD·BD

    AB·CD = AD·CD + DB·CD = AD·CD - BD·CD.

  It will be seen that the sum of the right-hand sides vanishes, hence
  that

    BC·AD + CA·BD + AB·CD = 0 (3)

  for any four points on a line.

  [Illustration: FIG. 3.]

  § 11. If C is any point in the line AB, then we say that C divides the
  segment AB in the ratio AC/CB, account being taken of the sense of the
  two segments AC and CB. If C lies between A and B the ratio is
  positive, as AC and CB have the same sense. But if C lies without the
  segment AB, i.e. if C divides AB externally, then the ratio is
  negative. To see how the value of this ratio changes with C, we will
  move C along the whole line (fig. 3), whilst A and B remain fixed. If
  C lies at the point A, then AC = 0, hence the ratio AC : CB vanishes.
  As C moves towards B, AC increases and CB decreases, so that our ratio
  increases. At the middle point M of AB it assumes the value +1, and
  then increases till it reaches an infinitely large value, when C
  arrives at B. On passing beyond B the ratio becomes negative. If C is
  at P we have AC = AP = AB + BP, hence

    AC   AB   BP     AB
    -- = -- + -- = - -- - 1.
    CB   PB   PB     BP

  In the last expression the ratio AB : BP is positive, has its greatest
  value [infinity] when C coincides with B, and vanishes when BC becomes
  infinite. Hence, as C moves from B to the right to the point at
  infinity, the ratio AC : CB varies from -[infinity] to -1.

  If, on the other hand, C is to the left of A, say at Q, we have AC =
  AQ = AB + BQ = AB - QB, hence AC/CB = AB/QB - 1.

  Here AB < QB, hence the ratio AB : QB is positive and always less than
  one, so that the whole is negative and < 1. If C is at the point at
  infinity it is -1, and then increases as C moves to the right, till
  for C at A we get the ratio = 0. Hence--

  "As C moves along the line from an infinite distance to the left to an
  infinite distance at the right, the ratio always increases; it starts
  with the value -1, reaches 0 at A, +1 at M, [infinity] at B, now
  changes sign to -[infinity], and increases till at an infinite
  distance it reaches again the value -1. _It assumes therefore all
  possible values from -[infinity] to +[infinity], and each value only
  once, so that not only does every position of C determine a definite
  value of the ratio AC : CB, but also, conversely, to every positive or
  negative value of this ratio belongs one single point in the line AB._

  [Relations between segments of lines are interesting as showing an
  application of algebra to geometry. The genesis of such relations
  from algebraic identities is very simple. For example, if a, b, c, x
  be any four quantities, then

              a                       b
    --------------------- + --------------------- +
    (a - b)(a - c)(x - a)   (b - c)(b - a)(x - b)

                c                       x
      --------------------- = ---------------------;
      (c - a)(c - b)(x - c)   (x - a)(x - b)(x - c)

  this may be proved, cumbrously, by multiplying up, or, simply, by
  decomposing the right-hand member of the identity into partial
  fractions. Now take a line ABCDX, and let AB = a, AC = b, AD = c, AX =
  x. Then obviously (a - b) = AB - AC = -BC, paying regard to signs; (a
  - c) = AB - AD = DB, and so on. Substituting these values in the
  identity we obtain the following relation connecting the segments
  formed by five points on a line:--

       AB         AC         AD         AX
    -------- + -------- + -------- = --------.
    BC·BD·BX   CD·CB·CX   DB·DC·DX   BX·CX·DX

  Conversely, if a metrical relation be given, its validity may be
  tested by reducing to an algebraic equation, which is an identity if
  the relation be true. For example, if ABCDX be five collinear points,
  prove

    AD·AX   BD·BX   CD·CX
    ----- + ----- + ----- = 1.
    AB·AC   BC·BA   CA·CB

  Clearing of fractions by multiplying throughout by AB·BC·CA, we have
  to prove

    -AD·AX·BC - BD·BX·CA - CD·CX·AB = AB·BC·CA.

  Take A as origin and let AB = a, AC = b, AD = c, AX = x. Substituting
  for the segments in terms of a, b, c, x, we obtain on simplification

    a²b - ab² = -ab² + a²b, an obvious identity.

  An alternative method of testing a relation is illustrated in the
  following example:-- If A, B, C, D, E, F be six collinear points, then

      AE·AF      BE·BF      CE·CF      DE·DF
    -------- + -------- + -------- + -------- = 0.
    AB·AC·AD   BC·BD·BA   CD·CA·CB   DA·DB·DC

  Clearing of fractions by multiplying throughout by AB·BC·CD·DA, and
  reducing to a common origin O (calling OA = a, OB = b, &c.), an
  equation containing the second and lower powers of OA (= a), &c., is
  obtained. Calling OA = x, it is found that x = b, x = c, x = d are
  solutions. Hence the quadratic has three roots; consequently it is an
  identity.

  The relations connecting five points which we have instanced above may
  be readily deduced from the six-point relation; the first by taking D
  at infinity, and the second by taking F at infinity, and then making
  the obvious permutations of the points.]


  PROJECTION AND CROSS-RATIOS

  § 12. If we join a point A to a point S, then the point where the line
  SA cuts a fixed plane [pi] is called the projection of A on the plane
  [pi] from S as centre of projection. If we have two planes [pi] and
  [pi]' and a point S, we may project every point A in [pi] to the other
  plane. If A' is the projection of A, then A is also the projection of
  A', so that the relations are reciprocal. To every figure in [pi] we
  get as its projection a corresponding figure in [pi]'.

  We shall determine such properties of figures as remain true for the
  projection, and which are called projective properties. For this
  purpose it will be sufficient to consider at first only constructions
  in one plane.

  [Illustration: FIG. 4.]

  [Illustration: FIG. 5.]

  Let us suppose we have given in a plane two lines p and p' and a
  centre S (fig. 4); we may then project the points in p from S to p'.
  Let A', B' ... be the projections of A, B ..., the point at infinity
  in p which we shall denote by I will be projected into a finite point
  I' in p', viz. into the point where the parallel to p through S cuts
  p'. Similarly one point J in p will be projected into the point J' at
  infinity in p'. This point J is of course the point where the parallel
  to p' through S cuts p. We thus see that every point in p is projected
  into a single point in p'.

  Fig. 5 shows that a segment AB will be projected into a segment A'B'
  which is not equal to it, at least not as a rule; and also that the
  ratio AC : CB is not equal to the ratio A'C' : C'B' formed by the
  projections. These ratios will become equal only if p and p' are
  parallel, for in this case the triangle SAB is similar to the triangle
  SA'B'. Between three points in a line and their projections there
  exists therefore in general no relation. But between four points a
  relation does exist.

  § 13. Let A, B, C, D be four points in p, A', B', C, D' their
  projections in p', then the ratio of the two ratios AC : CB and AD :
  DB into which C and D divide the segment AB is equal to the
  corresponding expression between A', B', C', D'. In symbols we have

    AC   AD   A'C'   A'D'
    -- : -- = ---- : ----.
    CB   DB   C'B'   D'B'

  This is easily proved by aid of similar triangles.

  [Illustration: FIG. 6.]

  Through the points A and B on p draw parallels to p', which cut the
  projecting rays in C2, D2, B2 and A1, C1, D1, as indicated in fig. 6.
  The two triangles ACC2 and BCC1 will be similar, as will also be the
  triangles ADD2 and BDD1.

  The proof is left to the reader.

  This result is of fundamental importance.

  The expression AC/CB : AD/DB has been called by Chasles the
  "anharmonic ratio of the four points A, B, C, D." Professor Clifford
  proposed the shorter name of "cross-ratio." We shall adopt the latter.
  We have then the

  FUNDAMENTAL THEOREM.--_The cross-ratio of four points in a line is
  equal to the cross-ratio of their projections on any other line which
  lies in the same plane with it._

  § 14. Before we draw conclusions from this result, we must investigate
  the meaning of a cross-ratio somewhat more fully.

  If four points A, B, C, D are given, and we wish to form their
  cross-ratio, we have first to divide them into two groups of two, the
  points in each group being taken in a definite order. Thus, let A, B
  be the first, C, D the second pair, A and C being the first points in
  each pair. The cross-ratio is then the ratio AC : CB divided by AD :
  DB. This will be denoted by (AB, CD), so that

               AC   AD
    (AB, CD) = -- : --.
               CB   DB

  This is easily remembered. In order to write it out, make first the
  two lines for the fractions, and put above and below these the letters
  A and B in their places, thus, A*/B : A*/B; and then fill up,
  crosswise, the first by C and the other by D.

  § 15. If we take the points in a different order, the value of the
  cross-ratio will change. We can do this in twenty-four different ways
  by forming all permutations of the letters. But of these twenty-four
  cross-ratios groups of four are equal, so that there are really only
  six different ones, and these six are reciprocals in pairs.

  We have the following rules:--

  I. If in a cross-ratio the two groups be interchanged, its value
  remains unaltered, i.e.

    (AB, CD) = (CD, AB) = (BA, DC) = (DC, BA).

  II. If in a cross-ratio the two points belonging to one of the two
  groups be interchanged, the cross-ratio changes into its reciprocal,
  i.e.

    (AB, CD) = 1/(AB, DC) = 1/(BA, CD) = 1/(CD, BA) = 1/(DC, AB).

  From I. and II. we see that eight cross-ratios are associated with
  (AB, CD).

  III. If in a cross-ratio the two middle letters be interchanged, the
  cross-ratio [alpha] changes into its complement 1 - [alpha], i.e. (AB,
  CD) = 1 - (AC, BD).

  [§ 16. If [lambda] = (AB, CD), µ = (AC, DB), [nu] = (AD, BC), then
  [lambda], µ, [nu] and their reciprocals 1/[lambda], 1/µ, 1/[nu] are
  the values of the total number of twenty-four cross-ratios. Moreover,
  [lambda], µ, [nu] are connected by the relations

    [lambda] + 1/µ = µ + 1/[nu] = [nu] + 1/[lambda] = -[lambda]µ[nu] = 1;

  this proposition may be proved by substituting for [lambda], µ, [nu]
  and reducing to a common origin. There are therefore four equations
  between three unknowns; hence if one cross-ratio be given, the
  remaining twenty-three are determinate. Moreover, two of the
  quantities [lambda], µ, [nu] are positive, and the remaining one
  negative.

  The following scheme shows the twenty-four cross-ratios expressed in
  terms of [lambda], µ, [nu].]

    +---------+-----------------------+---------------+---------------+
    |(AB, CD) |                       |               |               |
    |(BA, DC) |       [lambda]        |     1 - µ     |  1/(1 - [nu]) |
    |(CD, AB) |                       |               |               |
    |(DC, BA) |                       |               |               |
    +---------+-----------------------+---------------+---------------+
    |(AC, DB) |                       |               |               |
    |(BD, CA) |   1/(1 - [lambda])    |      1/µ      |([nu] - 1)/[nu]|
    |(CA, BD) |                       |               |               |
    |(DB, AC) |                       |               |               |
    +---------+-----------------------+---------------+---------------+
    |(AB, DC) |                       |               |               |
    |(BA, CD) |      1/[lambda]       |   1/(1 - µ)   |   1 - [nu]    |
    |(CD, BA) |                       |               |               |
    |(DC, AB) |                       |               |               |
    +---------+-----------------------+---------------+---------------+
    |(AD, BC) |                       |               |               |
    |(BC, AD) |([lambda] - 1)/[lambda]|   µ/(µ - 1)   |     [nu]      |
    |(CB, DA) |                       |               |               |
    |(DA, CB) |                       |               |               |
    +---------+-----------------------+---------------+---------------+
    |(AC, BD) |                       |               |               |
    |(BD, AC) |     1 - [lambda]      |       µ       |[nu]/([nu] - 1)|
    |(CA, DB) |                       |               |               |
    |(DB, CA) |                       |               |               |
    +---------+-----------------------+---------------+---------------+
    |(AD, CB) |                       |               |               |
    |(BC, DA) |[lambda]/([lambda] - 1)|   (µ - 1)/µ   |    1/[nu]     |
    |(CB, AD) |                       |               |               |
    |(DA, BC) |                       |               |               |
    +---------+-----------------------+---------------+---------------+

  § 17. If one of the points of which a cross-ratio is formed is the
  point at infinity in the line, the cross-ratio changes into a simple
  ratio. It is convenient to let the point at infinity occupy the last
  place in the symbolic expression for the cross-ratio. Thus if I is a
  point at infinity, we have (AB, CI) = -AC/CB, because AI : IB = -1.

  Every common ratio of three points in a line may thus be expressed as
  a cross-ratio, by adding the point at infinity to the group of points.


  HARMONIC RANGES

  § 18. If the points have special positions, the cross-ratios may have
  such a value that, of the six different ones, two and two become
  equal. If the first two shall be equal, we get [lambda] = 1/[lambda],
  or [lambda]² = 1, [lambda] = ±1.

  If we take [lambda] = +1, we have (AB, CD) = 1, or AC/CB = AD/DB; that
  is, the points C and D coincide, provided that A and B are different.

  If we take [lambda] = -1, so that (AB, CD) = -1, we have AC/CB =
  -AD/DB. _Hence C and D divide AB internally and externally in the same
  ratio._

  The four points are in this case said to be _harmonic points_, and _C
  and D are said to be harmonic conjugates with regard to A and B._

  But we have also (CD, AB) = -1, so that A and B are harmonic
  conjugates with regard to C and D.

  The principal property of harmonic points is that their cross-ratio
  remains unaltered if we interchange the two points belonging to one
  pair, viz.

    (AB, CD) = (AB, DC) = (BA, CD).

  For four harmonic points the six cross-ratios become equal two and
  two:

                                       [lambda]
    [lambda] = -1, 1 - [lambda] = 2, ------------ = ½,
                                     [lambda] - 1

           1                1            [lambda] - 1
      = -------- = -1, ------------ = ½, ------------ = 2.
        [lambda]       1 - [lambda]        [lambda]

  Hence if we get four points whose cross-ratio is 2 or ½, then they are
  harmonic, but not arranged so that conjugates are paired. If this is
  the case the cross-ratio = -1.

  § 19. If we equate any two of the above six values of the
  cross-ratios, we get either [lambda] = 1, 0, [infinity], or [lambda] =
  -1, 2, ½, or else [lambda] becomes a root of the equation [lambda]² -
  [lambda] + 1 = 0, that is, an imaginary cube root of -1. In this case
  the six values become three and three equal, so that only two
  different values remain. This case, though important in the theory of
  cubic curves, is for our purposes of no interest, whilst harmonic
  points are all-important.

  § 20. From the definition of harmonic points, and by aid of § 11, the
  following properties are easily deduced.

  If C and D are harmonic conjugates with regard to A and B, then one of
  them lies in, the other without AB; it is impossible to move from A to
  B without passing either through C or through D; the one blocks the
  finite way, the other the way through infinity. This is expressed by
  saying A and B are "separated" by C and D.

  For every position of C there will be one and only one point D which
  is its harmonic conjugate with regard to any point pair A, B.

  If A and B are different points, and if C coincides with A or B, D
  does. But if A and B coincide, one of the points C or D, lying between
  them, coincides with them, and the other may be anywhere in the line.
  It follows that, "_if of four harmonic conjugates two coincide, then a
  third coincides with them, and the fourth may be any point in the
  line_."

  If C is the middle point between A and B, then D is the point at
  infinity; for AC : CB = +1, hence AD : DB must be equal to -1. _The
  harmonic conjugate of the point at infinity in a line with regard to
  two points A, B is the middle point of AB._

  This important property gives a first example how metric properties
  are connected with projective ones.

  [§ 21. _Harmonic properties of the complete quadrilateral and
  quadrangle._

  [Illustration: FIG. 7.]

  [Illustration: FIG. 8.]

  A figure formed by four lines in a plane is called a _complete
  quadrilateral_, or, shorter, a _four-side_. The four sides meet in six
  points, named the "vertices," which may be joined by three lines
  (other than the sides), named the "diagonals" or "harmonic lines." The
  diagonals enclose the "harmonic triangle of the quadrilateral." In
  fig. 7, A'B'C', B'AC, C'AB, CBA' are the sides, A, A', B, B', C, C'
  the vertices, AA', BB', CC' the harmonic lines, and [alpha]ß[gamma]
  the harmonic triangle of the quadrilateral. A figure formed by four
  coplanar points is named a _complete quadrangle_, or, shorter, a
  _four-point_. The four points may be joined by six lines, named the
  "sides," which intersect in three other points, termed the "diagonal
  or harmonic points." The harmonic points are the vertices of the
  "harmonic triangle of the complete quadrangle." In fig. 8, AA', BB'
  are the points, AA', BB', A'B', B'A, AB, BA' are the sides, L, M, N
  are the diagonal points, and LMN is the harmonic triangle of the
  quadrangle.

  The harmonic property of the complete quadrilateral is: Any diagonal
  or harmonic line is harmonically divided by the other two; and of a
  complete quadrangle: The angle at any harmonic point is divided
  harmonically by the joins to the other harmonic points. To prove the
  first theorem, we have to prove (AA', ß[gamma]), (BB',
  [gamma][alpha]), (CC', ß[alpha]) are harmonic. Consider the
  cross-ratio (CC', [alpha]ß). Then projecting from A on BB' we have
  A(CC', [alpha]ß) = A(B'B, [alpha][gamma]). Projecting from A' on BB',
  A'(CC', [alpha]ß) = A'(BB', [alpha][gamma]). Hence (B'B,
  [alpha][gamma]) = (BB', [alpha][gamma]), i.e. the cross-ratio (BB',
  [alpha][gamma]) equals that of its reciprocal; hence the range is
  harmonic.

  The second theorem states that the pencils L(BA, NM), M(B'A, LN),
  N(BA, LM) are harmonic. Deferring the subject of harmonic pencils to
  the next section, it will suffice to state here that any transversal
  intersects an harmonic pencil in an harmonic range. Consider the
  pencil L(BA, NM), then it is sufficient to prove (BA', NM') is
  harmonic. This follows from the previous theorem by considering A'B as
  a diagonal of the quadrilateral ALB'M.]

  This property of the complete quadrilateral allows the solution of the
  problem:

  _To construct the harmonic conjugate D to a point C with regard to two
  given points A and B._

  Through A draw any two lines, and through C one cutting the former two
  in G and H. Join these points to B, cutting the former two lines in E
  and F. The point D where EF cuts AB will be the harmonic conjugate
  required.

  This remarkable construction requires nothing but the drawing of
  lines, and is therefore independent of measurement. In a similar
  manner the harmonic conjugate of the line VA for two lines VC, VD is
  constructed with the aid of the property of the complete quadrangle.

  § 22. _Harmonic Pencils._--The theory of cross-ratios may be extended
  from points in a row to lines in a flat pencil and to planes in an
  axial pencil. We have seen (§ 13) that if the lines which join four
  points A, B, C, D to any point S be cut by any other line in A', B',
  C', D', then (AB, CD) = (A'B', C'D'). In other words, four lines in a
  flat pencil are cut by every other line in four points whose
  cross-ratio is constant.

  _Definition._--By the cross-ratio of four rays in a flat pencil is
  meant the cross-ratio of the four points in which the rays are cut by
  any line. If a, b, c, d be the lines, then this cross-ratio is denoted
  by (ab, cd).

  _Definition._--By the cross-ratio of four planes in an axial pencil is
  understood the cross-ratio of the four points in which any line cuts
  the planes, or, what is the same thing, the cross-ratio of the four
  rays in which any plane cuts the four planes.

  In order that this definition may have a meaning, it has to be proved
  that all lines cut the pencil in points which have the same
  cross-ratio. This is seen at once for two intersecting lines, as their
  plane cuts the axial pencil in a flat pencil, which is itself cut by
  the two lines. The cross-ratio of the four points on one line is
  therefore equal to that on the other, and equal to that of the four
  rays in the flat pencil.

  If two non-intersecting lines p and q cut the four planes in A, B, C,
  D and A', B', C', D', draw a line r to meet both p and q, and let this
  line cut the planes in A", B", C", D". Then (AB, CD) = (A'B', C'D'),
  for each is equal to (A"B", C"D").

  § 23. We may now also extend the notion of harmonic elements, viz.

  _Definition._--Four rays in a flat pencil and four planes in an axial
  pencil are said to be harmonic if their cross-ratio equals -1, that
  is, if they are cut by a line in four harmonic points.

  If we understand by a "median line" of a triangle a line which joins a
  vertex to the middle point of the opposite side, and by a "median
  line" of a parallelogram a line joining middle points of opposite
  sides, we get as special cases of the last theorem:

  _The diagonals and median lines of a parallelogram form an harmonic
  pencil_; and

  _At a vertex of any triangle, the two sides, the median line, and the
  line parallel to the base form an harmonic pencil._

  Taking the parallelogram a rectangle, or the triangle isosceles, we
  get:

  _Any two lines and the bisections of their angles form an harmonic
  pencil._ Or:

  _In an harmonic pencil, if two conjugate rays are perpendicular, then
  the other two are equally inclined to them_; and, conversely, _if one
  ray bisects the angle between conjugate rays, it is perpendicular to
  its conjugate_.

  This connects perpendicularity and bisection of angles with projective
  properties.

  § 24. We add a few theorems and problems which are easily proved or
  solved by aid of harmonics.

  An harmonic pencil is cut by a line parallel to one of its rays in
  three equidistant points.

  Through a given point to draw a line such that the segment determined
  on it by a given angle is bisected at that point.

  Having given two parallel lines, to bisect on either any given segment
  without using a pair of compasses.

  Having given in a line a segment and its middle point, to draw through
  any given point in the plane a line parallel to the given line.

  To draw a line which joins a given point to the intersection of two
  given lines which meet off the drawing paper (by aid of § 21).


  CORRESPONDENCE. HOMOGRAPHIC AND PERSPECTIVE RANGES

  § 25. Two rows, p and p', which are one the projection of the other
  (as in fig. 5), stand in a definite relation to each other,
  characterized by the following properties.

  1. _To each point in either corresponds one point in the other_; that
  is, those points are said to correspond which are projections of one
  another.

  2. _The cross-ratio of any four points in one equals that of the
  corresponding points in the other._

  3. _The lines joining corresponding points all pass through the same
  point._

  If we suppose corresponding points marked, and the rows brought into
  any other position, then the lines joining corresponding points will
  no longer meet in a common point, and hence the third of the above
  properties will not hold any longer; but we have still a
  correspondence between the points in the two rows possessing the first
  two properties. Such a correspondence has been called a _one-one
  correspondence_, whilst the two rows between which such correspondence
  has been established are said to be _projective_ or _homographic_. Two
  rows which are each the projection of the other are therefore
  _projective_. We shall presently see, also, that any two projective
  rows may always be placed in such a position that one appears as the
  projection of the other. If they are in such a position the rows are
  said to be in _perspective position_, or simply to be in
  _perspective_.

  § 26. The notion of a one-one correspondence between rows may be
  extended to flat and axial pencils, viz. a flat pencil will be said to
  be projective to a flat pencil if to each ray in the first corresponds
  one ray in the second, and if the cross-ratio of four rays in one
  equals that of the corresponding rays in the second.

  Similarly an axial pencil may be projective to an axial pencil. But a
  flat pencil may also be projective to an axial pencil, or either
  pencil may be projective to a row. The definition is the same in each
  case: there is a one-one correspondence between the elements, and four
  elements have the same cross-ratio as the corresponding ones.

  § 27. There is also in each case a special position which is called
  _perspective_, viz.

  1. Two projective rows are perspective if they lie in the same plane,
  and if the one row is a projection of the other.

  2. Two projective flat pencils are perspective--(1) if they lie in the
  same plane, and have a row as a common section; (2) if they lie in the
  same pencil (in space), and are both sections of the same axial
  pencil; (3) if they are in space and have a row as common section, or
  are both sections of the same axial pencil, one of the conditions
  involving the other.

  3. Two projective axial pencils, if their axes meet, and if they have
  a flat pencil as a common section.

  4. A row and a projective flat pencil, if the row is a section of the
  pencil, each point lying in its corresponding line.

  5. A row and a projective axial pencil, if the row is a section of the
  pencil, each point lying in its corresponding line.

  6. A flat and a projective axial pencil, if the former is a section of
  the other, each ray lying in its corresponding plane.

  That in each case the correspondence established by the position
  indicated is such as has been called projective follows at once from
  the definition. It is not so evident that the perspective position may
  always be obtained. We shall show in § 30 this for the first three
  cases. First, however, we shall give a few theorems which relate to
  the general correspondence, not to the perspective position.

  § 28. _Two rows or pencils, flat or axial, which are projective to a
  third are projective to each other_; this follows at once from the
  definitions.

  § 29. _If two rows, or two pencils, either flat or axial, or a row and
  a pencil, be projective, we may assume to any three elements in the
  one the three corresponding elements in the other, and then the
  correspondence is uniquely determined._

  For if in two projective rows we assume that the points A, B, C in the
  first correspond to the given points A', B', C' in the second, then to
  any fourth point D in the first will correspond a point D' in the
  second, so that

    (AB, CD) = (A'B', C'D').

  But there is only one point, D', which makes the cross-ratio (A'B',
  C'D') equal to the given number (AB, CD).

  The same reasoning holds in the other cases.

  § 30. If two rows are perspective, then the lines joining
  corresponding points all meet in a point, the centre of projection;
  and the point in which the two bases of the rows intersect as a point
  in the first row coincides with its corresponding point in the second.

  This follows from the definition. The converse also holds, viz.

  _If two projective rows have such a position that one point in the one
  coincides with its corresponding point in the other, then they are
  perspective, that is, the lines joining corresponding points all pass
  through a common point, and form a flat pencil._

  For let A, B, C, D ... be points in the one, and A', B', C', D' ...
  the corresponding points in the other row, and let A be made to
  coincide with its corresponding point A'. Let S be the point where the
  lines BB' and CC' meet, and let us join S to the point D in the first
  row. This line will cut the second row in a point D", so that A, B, C,
  D are projected from S into the points A, B', C', D". The cross-ratio
  (AB, CD) is therefore equal to (AB', C'D"), and by hypothesis it is
  equal to (A'B', C'D'). Hence (A'B', C'D") = (A'B', C'D'), that is, D"
  is the same point as D'.

  § 31. If two projected flat pencils in the same plane are in
  perspective, then the intersections of corresponding lines form a row,
  and the line joining the two centres as a line in the first pencil
  corresponds to the same line as a line in the second. And conversely,

  _If two projective pencils in the same plane, but with different
  centres, have one line in the one coincident with its corresponding
  line in the other, then the two pencils are perspective, that is, the
  intersection of corresponding lines lie in a line._

  The proof is the same as in § 30.

  § 32. If two projective flat pencils in the same point (pencil in
  space), but not in the same plane, are perspective, then the planes
  joining corresponding rays all pass through a line (they form an axial
  pencil), and the line common to the two pencils (in which their planes
  intersect) corresponds to itself. And conversely:--

  If two flat pencils which have a common centre, but do not lie in a
  common plane, are placed so that one ray in the one coincides with its
  corresponding ray in the other, then they are perspective, that is,
  the planes joining corresponding lines all pass through a line.

  § 33. If two projective axial pencils are perspective, then the
  intersection of corresponding planes lie in a plane, and the plane
  common to the two pencils (in which the two axes lie) corresponds to
  itself. And conversely:--

  If two projective axial pencils are placed in such a position that a
  plane in the one coincides with its corresponding plane, then the two
  pencils are perspective, that is, corresponding planes meet in lines
  which lie in a plane.

  The proof again is the same as in § 30.

  § 34. These theorems relating to perspective position become illusory
  if the projective rows of pencils have a common base. We then have:--

  In two projective rows on the same line--and also in two projective
  and concentric flat pencils in the same plane, or in two projective
  axial pencils with a common axis--every element in the one coincides
  with its corresponding element in the other as soon as three elements
  in the one coincide with their corresponding elements in the other.

  _Proof_ (in case of two rows).--Between four elements A, B, C, D and
  their corresponding elements A', B', C', D' exists the relation (ABCD)
  = (A'B'C'D'). If now A', B', C' coincide respectively with A, B, C, we
  get (AB, CD) = (AB, CD'), hence D and D' coincide.

  The last theorem may also be stated thus:--

  In two projective rows or pencils, which have a common base but are
  not identical, not more than two elements in the one can coincide with
  their corresponding elements in the other.

  Thus two projective rows on the same line cannot have more than two
  pairs of coincident points unless every point coincides with its
  corresponding point.

  It is easy to construct two projective rows on the same line, which
  have two pairs of corresponding points coincident. Let the points A,
  B, C as points belonging to the one row correspond to A, B, and C' as
  points in the second. Then A and B coincide with their corresponding
  points, but C does not. It is, however, not necessary that two such
  rows have twice a point coincident with its corresponding point; it is
  possible that this happens only once or not at all. Of this we shall
  see examples later.

  [Illustration: FIG. 9.]

  § 35. If two projective rows or pencils are in perspective position,
  we know at once which element in one corresponds to any given element
  in the other. If p and q (fig. 9) are two projective rows, so that K
  corresponds to itself, and if we know that to A and B in p correspond
  A' and B' in q, then the point S, where AA' meets BB', is the centre
  of projection, and hence, in order to find the point C' corresponding
  to C, we have only to join C to S; the point C', where this line cuts
  q, is the point required.

  [Illustration: FIG. 10.]

  If two flat pencils, S1 and S2, in a plane are perspective (fig. 10),
  we need only to know two pairs, a, a' and b, b', of corresponding rays
  in order to find the axis s of projection. This being known, a ray c'
  in S2, corresponding to a given ray c in S1, is found by joining S2 to
  the point where c cuts the axis s.

  A similar construction holds in the other cases of perspective
  figures.

  On this depends the solution of the following general problem.

  § 36. Three pairs of corresponding elements in two projective rows or
  pencils being given, to determine for any element in one the
  corresponding element in the other.

  We solve this in the two cases of two projective rows and of two
  projective flat pencils in a plane.

  _Problem_ I.--Let A, B, C be          _Problem_ II.--Let a, b, c be
  three points in a row s, A', B',      three rays in a pencil S, a',
  C' the corresponding points in a      b', c' the corresponding rays in
  projective row s', both being in      a projective pencil S', both
  a plane; it is required to find       being in the same plane; it is
  for any point D in s the              required to find for any ray d
  corresponding point D' in s'.         in S the corresponding ray d' in
                                        S'.

  The solution is made to depend on the construction of an auxiliary row
  or pencil which is perspective to both the given ones. This is found
  as follows:--

  [Illustration: FIG. 11.]

  _Solution of Problem_ I.--On the line joining two corresponding
  points, say AA' (fig. 11), take any two points, S and S', as centres
  of auxiliary pencils. Join the intersection B1 of SB and S'B' to the
  intersection C1 of SC and S'C' by the line s1. Then a row on s1 will
  be perspective to s with S as centre of projection, and to s' with S'
  as centre. To find now the point D' on s' corresponding to a point D
  on s we have only to determine the point D1, where the line SD cuts
  s1, and to draw S'D1; the point where this line cuts s' will be the
  required point D'.

  _Proof._--The rows s and s' are both perspective to the row s1, hence
  they are projective to one another. To A, B, C, D on s correspond A1,
  B1, C1, D1 on s1, and to these correspond A', B', C', D' on s'; so
  that D and D' are corresponding points as required.

  [Illustration: FIG. 12.]

  _Solution of Problem_ II.--Through the intersection A of two
  corresponding rays a and a' (fig. 12), take two lines, s and s', as
  bases of auxiliary rows. Let S1 be the point where the line b1, which
  joins B and B', cuts the line c1, which joins C and C'. Then a pencil
  S1 will be perspective to S with s as axis of projection. To find the
  ray d' in S' corresponding to a given ray d in S, cut d by s at D;
  project this point from S1 to D' on s' and join D' to S'. This will be
  the required ray.

  _Proof._--That the pencil S1 is perspective to S and also to S'
  follows from construction. To the lines a1, b1, c1, d1 in S1
  correspond the lines a, b, c, d in S and the lines a', b', c', d' in
  S', so that d and d' are corresponding rays.

  In the first solution the two centres, S, S', are _any_ two points on
  a line joining any two corresponding points, so that the solution of
  the problem allows of a great many different constructions. _But
  whatever construction be used, the point D', corresponding to D, must
  be always the same_, according to the theorem in § 29. This gives rise
  to a number of theorems, into which, however, we shall not enter. The
  same remarks hold for the second problem.

  § 37. _Homological Triangles._--As a further application of the
  theorems about perspective rows and pencils we shall prove the
  following important theorem.

  _Theorem._--If ABC and A'B'C' (fig. 13) be two triangles, such that
  the lines AA', BB', CC' meet in a point S, then the intersections of
  BC and B'C', of CA and C'A', and of AB and A'B' will lie in a line.
  Such triangles are said to be homological, or in perspective. The
  triangles are "co-axial" in virtue of the property that the meets of
  corresponding sides are collinear and copolar, since the lines joining
  corresponding vertices are concurrent.

  _Proof._--Let a, b, c denote the lines AA', BB', CC', which meet at S.
  Then these may be taken as bases of projective rows, so that A, A', S
  on a correspond to B, B', S on b, and to C, C', S on c. As the point S
  is common to all, any two of these rows will be perspective.

    If S1 be the centre of projection of rows b and c,
       S2      "          "          "        c and a,
       S3      "          "          "        a and b,

  and if the line S1S2 cuts a in A1, and b in B1, and c in C1, then A1,
  B1 will be corresponding points in a and b, both corresponding to C1
  in c. But a and b are perspective, therefore the line A1B1, that is
  S1S2, joining corresponding points must pass through the centre of
  projection S3 of a and b. In other words, S1, S2, S3 lie in a line.
  This is Desargues' celebrated theorem if we state it thus:--

  [Illustration: FIG. 13.]

  _Theorem of Desargues._--If each of two triangles has one vertex on
  each of three concurrent lines, then the intersections of
  corresponding sides lie in a line, those sides being called
  corresponding which are opposite to vertices on the same line.

  The converse theorem holds also, viz.

  _Theorem._--If the sides of one triangle meet those of another in
  three points which lie in a line, then the vertices lie on three lines
  which meet in a point.

  The proof is almost the same as before.

  § 38. _Metrical Relations between Projective Rows._--Every row
  contains one point which is distinguished from all others, viz. the
  point at infinity. In two projective rows, to the point I at infinity
  in one corresponds a point I' in the other, and to the point J' at
  infinity in the second corresponds a point J in the first. The points
  I' and J are in general finite. If now A and B are any two points in
  the one, A', B' the corresponding points in the other row, then

    (AB, JI) = (A'B', J'I'),

  or

    AJ/JB : AI/IB = A'J'/J'B' : A'I'/I'B'.

  But, by § 17,

    AI/IB = A'J'/J'B' = -1;

  therefore the last equation changes into

    AJ·A'I' = BJ·B'I',

  that is to say--

  _Theorem._--The product of the distances of any two corresponding
  points in two projective rows from the points which correspond to the
  points at infinity in the other is constant, viz. AJ·A'I' = k.
  Steiner has called this number k the _Power of the correspondence_.

  [The relation AJ . A'I' = k shows that if J, I' be given then the
  point A' corresponding to a specified point A is readily found; hence
  A, A' generate homographic ranges of which I and J' correspond to the
  points at infinity on the ranges. If we take any two origins O, O', on
  the ranges and reduce the expression AJ . A'I' = k to its algebraic
  equivalent, we derive an equation of the form [alpha]xx' + ßx +
  [gamma]x' + [delta] = 0. Conversely, if a relation of this nature
  holds, then points corresponding to solutions in x, x' form
  homographic ranges.]

  § 39. _Similar Rows._--If the points at infinity in two projective
  rows correspond so that I' and J are at infinity, this result loses
  its meaning. But if A, B, C be any three points in one, A', B', C' the
  corresponding ones on the other row, we have

    (AB, CI) = (A'B', C'I'),

  which reduces to

    AC/CB = A'C'/C'B' or AC/A'C' = BC/B'C',

  that is, corresponding segments are proportional. Conversely, if
  corresponding segments are proportional, then to the point at infinity
  in one corresponds the point at infinity in the other. If we call such
  rows _similar_, we may state the result thus--

  _Theorem._--Two projective rows are similar if to the point at
  infinity in one corresponds the point at infinity in the other, and
  conversely, if two rows are similar then they are projective, and the
  points at infinity are corresponding points.

  From this the well-known propositions follow:--

  Two lines are cut proportionally (in similar rows) by a series of
  parallels. The rows are perspective, with centre of projection at
  infinity.

  If two similar rows are placed parallel, then the lines joining
  homologous points pass through a common point.

  § 40. If two flat pencils be projective, then there exists in either,
  one single pair of lines at right angles to one another, such that the
  corresponding lines in the other pencil are again at right angles.

  [Illustration: FIG. 14.]

  To prove this, we place the pencils in perspective position (fig. 14)
  by making one ray coincident with its corresponding ray. Corresponding
  rays meet then on a line p. And now we draw the circle which has its
  centre O on p, and which passes through the centres S and S' of the
  two pencils. This circle cuts p in two points H and K. The two pairs
  of rays, h, k, and h', k', joining these points to S and S' will be
  pairs of corresponding rays at right angles. The construction gives in
  general but one circle, but if the line p is the perpendicular
  bisector of SS', there exists an infinite number, and _to every right
  angle in the one pencil corresponds a right angle in the other_.


  PRINCIPLE OF DUALITY

  § 41. It has been stated in § 1 that not only points, but also planes
  and lines, are taken as elements out of which figures are built up. We
  shall now see that the construction of one figure which possesses
  certain properties gives rise in many cases to the construction of
  another figure, by replacing, according to definite rules, elements of
  one kind by those of another. The new figure thus obtained will then
  possess properties which may be stated as soon as those of the
  original figure are known.

  We obtain thus a principle, known as the _principle of duality_ or of
  _reciprocity_, which enables us to construct to any figure not
  containing any measurement in its construction a _reciprocal_ figure,
  as it is called, and to deduce from any theorem a _reciprocal_
  theorem, for which no further proof is needed.

  It is convenient to print reciprocal propositions on opposite sides of
  a page broken into two columns, and this plan will occasionally be
  adopted.

  We begin by repeating in this form a few of our former statements:--

  Two points determine a line.        Two planes determine a line.

  Three points which are not in a     Three planes which do not pass
  line determine a plane.             through a line determine a point.

  A line and a point without it       A line and a plane not through
  determine a plane.                  it determine a point.

  Two lines in a plane determine      Two lines through a point
  a point.                            determine a plane.

  These propositions show that it will be possible, when any figure is
  given, to construct a second figure by taking planes instead of
  points, and points instead of planes, but lines where we had lines.

  For instance, if in the first figure we take a plane and three points
  in it, we have to take in the second figure a point and three planes
  through it. The three points in the first, together with the three
  lines joining them two and two, form a triangle; the three planes in
  the second and their three lines of intersection form a trihedral
  angle. A triangle and a trihedral angle are therefore reciprocal
  figures.

  Similarly, to any figure in a plane consisting of points and lines
  will correspond a figure consisting of planes and lines passing
  through a point S, and hence belonging to the pencil which has S as
  centre.

  The figure reciprocal to four points in space which do not lie in a
  plane will consist of four planes which do not meet in a point. In
  this case each figure forms a tetrahedron.

  § 42. As other examples we have the following:--

  To a row                   is reciprocal  an axial pencil,

  " a flat pencil                  "        a flat pencil,

  " a field of points and lines    "        a pencil of planes and lines,

  " the space of points            "        the space of planes.

  For the row consists of a line and all the points in it, reciprocal to
  it therefore will be a line with all planes through it, that is, an
  axial pencil; and so for the other cases.

  This correspondence of reciprocity breaks down, however, if we take
  figures which contain measurement in their construction. For instance,
  there is no figure reciprocal to two planes at _right angles_, because
  there is no segment in a row which has a magnitude as definite as a
  right angle.

  We add a few examples of reciprocal propositions which are easily
  proved.

  _Theorem._--If A, B, C, D are        _Theorem._--If [alpha], ß,
  any four points in space, and if     [gamma], [delta] are four planes
  the lines AB and CD meet, then       in space, and if the lines
  all four points lie in a plane,      [alpha]ß and [gamma][delta] meet,
  hence also AC and BD, as well        then all four planes lie in a
  as AD and BC, meet.                  point (pencil), hence also
                                       [alpha][gamma] and ß[delta], as
                                       well as [alpha][delta] and
                                       ß[gamma], meet.

  Theorem.--_If of any number of lines every one meets every other,
  whilst all do not_

  _lie in a point, then all lie in     _lie in a plane, then all lie in
  a plane._                            a point (pencil)._

  § 43. Reciprocal figures as explained lie both in space of three
  dimensions. If the one is confined to a plane (is formed of elements
  which lie in a plane), then the reciprocal figure is confined to a
  pencil (is formed of elements which pass through a point).

  But there is also a more special principle of duality, according to
  which figures are reciprocal which lie both in a plane or both in a
  pencil. In the plane we take points and lines as reciprocal elements,
  for they have this fundamental property in common, that two elements
  of one kind determine one of the other. In the pencil, on the other
  hand, lines and planes have to be taken as reciprocal, and here it
  holds again that two lines or planes determine one plane or line.

  Thus, to one plane figure we can construct one reciprocal figure in
  the plane, and to each one reciprocal figure in a pencil. We mention a
  few of these. At first we explain a few names:--

  A figure consisting of n points     A figure consisting of n lines
  in a plane will be called an        in a plane will be called an
  n-point.                            n-side.

  A figure consisting of n planes     A figure consisting of n lines
  in a pencil will be called an       in a pencil will be called an
  n-flat.                             n-edge.

  It will be understood that an n-side is different from a polygon of n
  sides. The latter has sides of finite length and n vertices, the
  former has sides all of infinite extension, and every point where two
  of the sides meet will be a vertex. A similar difference exists
  between a solid angle and an n-edge or an n-flat. We notice
  particularly--

  A four-point has six sides, of      A four-side has six vertices, of
  which two and two are opposite,     which two and two are opposite,
  and three diagonal points, which    and three diagonals, which join
  are intersections of opposite       opposite vertices.
  sides.

  A four-flat has six edges, of       A four-edge has six faces, of
  which two and two are opposite,     which two and two are opposite,
  and three diagonal planes, which    and three diagonal edges, which
  pass through opposite edges.        are intersections of opposite
                                      faces.

  A four-side is usually called a complete quadrilateral, and a
  four-point a complete quadrangle. The above notation, however, seems
  better adapted for the statement of reciprocal propositions.

  § 44.

  If a point moves in a plane it      If a line moves in a plane it
  describes a plane curve.            envelopes a plane curve (fig. 15).

  If a plane moves in a pencil it     If a line moves in a pencil it
  envelopes a cone.                   describes a cone.

  A curve thus appears as generated either by points, and then we call
  it a "locus," or by lines, and then we call it an "envelope." In the
  same manner a cone, which means here a surface, appears either as the
  locus of lines passing through a fixed point, the "vertex" of the
  cone, or as the envelope of planes passing through the same point.

  [Illustration: FIG. 15.]

  To a surface as locus of points corresponds, in the same manner, a
  surface as envelope of planes; and to a curve in space as locus of
  points corresponds a developable surface as envelope of planes.

  It will be seen from the above that we may, by aid of the principle of
  duality, construct for every figure a reciprocal figure, and that to
  any property of the one a reciprocal property of the other will exist,
  as long as we consider only properties which depend upon nothing but
  the positions and intersections of the different elements and not upon
  measurement.

  For such propositions it will therefore be unnecessary to prove more
  than one of two reciprocal theorems.


  GENERATION OF CURVES AND CONES OF SECOND ORDER OR SECOND CLASS

  § 45. _Conics._--If we have two projective pencils in a plane,
  corresponding rays will meet, and their point of intersection will
  constitute some locus which we have to investigate. Reciprocally, if
  two projective rows in a plane are given, then the lines which join
  corresponding points will envelope some curve. We prove first:--

  _Theorem._--If two projective       _Theorem._--If two projective
  flat pencils lie in a plane, but    rows lie in a plane, but are
  are neither in perspective nor      neither in perspective nor on a
  concentric, then the locus of       common base, then the envelope
  intersections of corresponding      of lines joining corresponding
  rays is a curve of the second       points is a curve of the second
  order, that is, no line contains    class, that is, through no point
  more than two points of the         pass more than two of the
  locus.                              enveloping lines.

  Proof.--We draw any line t.         _Proof._--We take any point T
  This cuts each of the pencils in    and join it to all points in each
  a row, so that we have on t two     row. This gives two concentric
  rows, and these are projective      pencils, which are projective
  because the pencils are             because the rows are projective.
  projective. If corresponding rays   If a line joining corresponding
  of the two pencils meet on the      points in the two rows passes
  line t, their intersection will     through T, it will be a line in
  be a point in the one row which     the one pencil which coincides
  coincides with its corresponding    with its corresponding line in
  point in the other. But two         the other. But two projective
  projective rows on the same base    concentric flat pencils in the
  cannot have more than two           same plane cannot have more than
  points of one coincident with       two lines of one coincident with
  their corresponding points in       their corresponding line in the
  the other (§ 34).                   other (§ 34).

  It will be seen that the proofs are reciprocal, so that the one may be
  copied from the other by simply interchanging the words point and
  line, locus and envelope, row and pencil, and so on. We shall
  therefore in future prove seldom more than one of two reciprocal
  theorems, and often state one theorem only, the reader being
  recommended to go through the reciprocal proof by himself, and to
  supply the reciprocal theorems when not given.

  § 46. We state the theorems in the pencil reciprocal to the last,
  without proving them:--

  _Theorem._--If two projective       _Theorem._--If two projective
  flat pencils are concentric, but    axial pencils lie in the same
  are neither perspective nor         pencil (their axes meet in a
  coplanar, then the envelope of      point), but are neither perspective
  the planes joining corresponding    nor co-axial, then the locus
  rays is a cone of the second        of lines joining corresponding
  class; that is, no line through     planes is a cone of the second
  the common centre contains more     order; that is, no plane in the
  than two of the enveloping          pencil contains more than two
  planes.

  § 47. Of theorems about cones of second order and cones of second
  class we shall state only very few. We point out, however, the
  following connexion between the curves and cones under consideration:

  The lines which join any point      Every plane section of a cone
  in space to the points on a curve   of the second order is a curve of
  of the second order form a cone     the second order.
  of the second order.

  The planes which join any           Every plane section of a cone
  point in space to the lines         of the second class is a curve of
  enveloping a curve of the           the second class.
  second class envelope themselves
  a cone of the second class.

  By its aid, or by the principle of duality, it will be easy to obtain
  theorems about them from the theorems about the curves.

  We prove the first. A curve of the second order is generated by two
  projective pencils. These pencils, when joined to the point in space,
  give rise to two projective axial pencils, which generate the cone in
  question as the locus of the lines where corresponding planes meet.

  §48.

  _Theorem._--The curve of second    _Theorem._--The envelope of
  order which is generated by two    second class which is generated
  projective flat pencils passes     by two projective rows contains
  through the centres of the two     the bases of these rows as
  pencils.                           enveloping lines or tangents.

  _Proof._--If S and S' are the      _Proof._--If s and s' are the
  two pencils, then to the ray SS'   two rows, then to the point ss'
  or p' in the pencil S'             or P' as a point in s'
  corresponds in the pencil S a      corresponds in s a point P,
  ray p, which is different from     which is not coincident with P',
  p', for the pencils are not        for the rows are not
  perspective. But p and p' meet     perspective. But P and P' are
  at S, so that S is a point on      joined by s, so that s is one of
  the curve, and similarly S'.       the enveloping lines, and
                                     similarly s'.

  It follows that every line in one of the two pencils cuts the curve in
  two points, viz. once at the centre S of the pencil, and once where it
  cuts its corresponding ray in the other pencil. These two points,
  however, coincide, if the line is cut by its corresponding line at S
  itself. The line p in S, which corresponds to the line SS' in S', is
  therefore the only line through S which has but one point in common
  with the curve, or which cuts the curve in two coincident points. Such
  a line is called a _tangent_ to the curve, touching the latter at the
  point S, which is called the "point of contact."

  In the same manner we get in the reciprocal investigation the result
  that through every point in one of the rows, say in s, two tangents
  may be drawn to the curve, the one being s, the other the line joining
  the point to its corresponding point in s'. There is, however, one
  point P in s for which these two lines coincide. Such a point in one
  of the tangents is called the "point of contact" of the tangent. We
  thus get--

  _Theorem._--To the line joining    _Theorem._--To the point of
  the centres of the projective      intersection of the bases of two
  pencils as a line in one pencil    projective rows as a point in
  corresponds in the other the       one row corresponds in the other
  tangent at its centre.             the _point of contact_ of its
                                     base.

  § 49. Two projective pencils are determined if three pairs of
  corresponding lines are given. Hence if a1, b1, c1 are three lines in
  a pencil S1, and a2, b2, c2 the corresponding lines in a projective
  pencil S2, the correspondence and therefore the curve of the second
  order generated by the points of intersection of corresponding rays is
  determined. Of this curve we know the two centres S1 and S2, and the
  three points a1a2, b1b2, c1c2, hence five points in all. This and the
  reciprocal considerations enable us to solve the following two
  problems:

  _Problem._--To construct a curve   _Problem._--To construct a curve
  of the second order, of which      of the second class, of which
  five points S1, S2, A, B, C are    five tangents u1, u2, a, b, c
  given.                             are given.

  In order to solve the left-hand problem, we take two of the given
  points, say S1 and S2, as centres of pencils. These we make projective
  by taking the rays a1, b1, c1, which join S1 to A, B, C respectively,
  as corresponding to the rays a2, b2, c2, which join S2 to A, B, C
  respectively, so that three rays meet their corresponding rays at the
  given points A, B, C. This determines the correspondence of the
  pencils which will generate a curve of the second order passing
  through A, B, C and through the centres S1 and S2, hence through the
  five given points. To find more points on the curve we have to
  construct for any ray in S1 the corresponding ray in S2. This has been
  done in § 36. But we repeat the construction in order to deduce
  further properties from it. We also solve the right-hand problem. Here
  we select two, viz. u1, u2 of the five given lines, u1, u2, a, b, c,
  as bases of two rows, and the points A1, B1, C1 where a, b, c cut u1
  as corresponding to the points A2, B2, C2 where a, b, c cut u2.

  We get then the following solutions of the two problems:

  _Solution._--Through the point A   _Solution._--In the line a take
  draw any two lines, u1 and u2      any two points S1 and S2 as
  (fig. 16), the first u1 to cut     centres of pencils (fig. 17),
  the pencil S1 in a row AB1C1,      the first S1 (A1B1C1) to project
  the other u2 to cut the pencil     the row u1, the other S2
  S2 in a row AB2C2. These two       (A2B2C2) to project the row u2.
  rows will be perspective, as the   These two pencils will be
  point A corresponds to itself,     perspective, the line S1A1 being
  and the centre of projection       the same as the corresponding
  will be the point S, where the     line S2A2, and the axis of
  lines B1B2 and C1C2 meet. To       projection will be the line u,
  find now for any ray d1 in S1      which joins the intersection B
  its corresponding ray d2 in S2,    of S1B1 and S2B2 to the
  we determine the point D1 where    intersection C of S1C1 and S2C2.
  d1 cuts u1, project this point     To find now for any point D1 in
  from S to D2 on u2 and join S2     u1 the corresponding point D2 in
  to D2. This will be the required   u2, we draw S1D1 and project the
  ray d2 which cuts d1 at some       point D where this line cuts u
  point D on the curve.              from S2 to u2. This will give
                                     the required point D2, and the
                                     line d joining D1 to D2 will be
                                     a new tangent to the curve.

  § 50. These constructions prove, when rightly interpreted, very
  important properties of the curves in question.

  [Illustration: FIG. 16.]

  If in fig. 16 we draw in the pencil S1 the ray k1 which passes through
  the auxiliary centre S, it will be found that the corresponding ray k2
  cuts it on u2. Hence--

  _Theorem._--In the above           _Theorem._--In the above
  construction the bases of the      construction (fig. 17) the
  auxiliary rows u1 and u2 cut the   tangents to the curve from the
  curve where they cut the rays      centres of the auxiliary pencils
  S2S and S1S respectively.          S1 and S2 are the lines which
                                     pass through u2u and u1u
                                     respectively.

  As A is any given point on the curve, and u1 any line through it, we
  have solved the problems:

  _Problem._--To find the second     _Problem._--To find the second
  point in which any line through    tangent which can be drawn from
  a known point on the curve cuts    any point in a given tangent to
  the curve.                         the curve.

  If we determine in S1 (fig. 16) the ray corresponding to the ray S2S1
  in S2, we get the tangent at S1. Similarly, we can determine the point
  of contact of the tangents u1 or u2 in fig. 17.

  [Illustration: FIG. 17.]

  § 51. If five points are given, of which not three are in a line, then
  we can, as has just been shown, always draw a curve of the second
  order through them; we select two of the points as centres of
  projective pencils, and then one such curve is determined. It will be
  presently shown that we get always the same curve if two other points
  are taken as centres of pencils, that therefore five points
  _determine_ one curve of the second order, and reciprocally, that five
  tangents determine one curve of the second class. Six points taken at
  random will therefore not lie on a curve of the second order. In order
  that this may be the case a certain condition has to be satisfied, and
  this condition is easily obtained from the construction in § 49, fig.
  16. If we consider the conic determined by the five points A, S1, S2,
  K, L, then the point D will be on the curve if, and only if, the
  points on D1, S, D2 be in a line.

  [Illustration: FIG. 18.]

  This may be stated differently if we take AKS1DS2L (figs. 16 and 18)
  as a hexagon inscribed in the conic, then AK and DS2 will be opposite
  sides, so will be KS1 and S2L, as well as S1D and LA. The first two
  meet in D2, the others in S and D1 respectively. We may therefore
  state the required condition, together with the reciprocal one, as
  follows:--

  _Pascal's Theorem._--If a hexagon  _Brianchon's Theorem._--If a
  be inscribed in a curve of the     hexagon be circumscribed about
  second order, then the             a curve of the second class, then
  intersectionsof opposite sides     the lines joining opposite vertices
  are three points in a line.        are three lines meeting in a point.

  These celebrated theorems, which are known by the names of their
  discoverers, are perhaps the most fruitful in the whole theory of
  conics. Before we go over to their applications we have to show that
  we obtain the same curve if we take, instead of S1, S2, any two other
  points on the curve as centres of projective pencils.

  § 52. We know that the curve depends only upon the correspondence
  between the pencils S1 and S2, and not upon the special construction
  used for finding new points on the curve. The point A (fig. 16 or 18),
  through which the two auxiliary rows u1, u2 were drawn, may therefore
  be changed to any other point on the curve. Let us now suppose the
  curve drawn, and keep the points S1, S2, K, L and D, and hence also
  the point S fixed, whilst we move A along the curve. Then the line AL
  will describe a pencil about L as centre, and the point D1 a row on
  S1D perspective to the pencil L. At the same time AK describes a
  pencil about K and D2 a row perspective to it on S2D. But by Pascal's
  theorem D1 and D2 will always lie in a line with S, so that the rows
  described by D1 and D2 are perspective. It follows that the pencils K
  and L will themselves be projective, corresponding rays meeting on the
  curve. This proves that we get the same curve whatever pair of the
  five given points we take as centres of projective pencils. Hence--

  Only one curve of the second       Only one curve of the second
  order can be drawn which passes    class can be drawn which touches
  through five given points.         five given lines.

  We have seen that if on a curve of the second order two points
  coincide at A, the line joining them becomes the tangent at A. If,
  therefore, a point on the curve and its tangent are given, this will
  be equivalent to having given two points on the curve. Similarly, if
  on the curve of second class a tangent and its point of contact are
  given, this will be equivalent to two given tangents.

  We may therefore extend the last theorem:

  Only one curve of the second       Only one curve of the second
  order can be drawn, of which       class can be drawn, of which four
  four points and the tangent at     tangents and the point of contact
  oneof them, or three points        at one of them, or three tangents
  and the tangents at two of         and the points of contact at two
  them, are given.                   of them, are given.

  § 53. At the same time it has been proved:

  If all points on a curve of the    All tangents to a curve of second
  second order be joined to any      class are cut by any two of
  two of them, then the two          them in projective rows, those
  pencils thus formed are            being corresponding points which
  projective, those rays being       lie on the same tangent. Hence--
  corresponding which meet on the
  curve. Hence--

  The cross-ratio of four rays       The cross-ratio of the four
  joining a point S on a curve of    points in which any tangent u is
  second order to four fixed         cut by four fixed tangents a, b, c,
  points A, B, C, D in the curve     d is independent of the position of
  is independent of the position     u, and is called the cross-ratio of
  of S, and is called the cross-     the four tangents a, b, c, d.
  ratio of the four points A, B,
  C, D.

  If this cross-ratio equals -1     If this cross-ratio equals -1
  the four points are said to be    the four tangents are said to be
  four harmonic points.             four harmonic tangents.

  We have seen that a curve of second order, as generated by projective
  pencils, has at the centre of each pencil one tangent; and further,
  that any point on the curve may be taken as centre of such pencil.
  Hence--

  A curve of second order has       A curve of second class has on
  at every point one tangent.       every tangent a point of contact.

  § 54. We return to Pascal's and Brianchon's theorems and their
  applications, and shall, as before, state the results both for curves
  of the second order and curves of the second class, but prove them
  only for the former.

  Pascal's theorem may be used when five points are given to find more
  points on the curve, viz. it enables us to find the point where any
  line through one of the given points cuts the curve again. It is
  convenient, in making use of Pascal's theorem, to number the points,
  to indicate the order in which they are to be taken in forming a
  hexagon, which, by the way, may be done in 60 different ways. It will
  be seen that 1 2 (leaving out 3) 4 5 are opposite sides, so are 2 3
  and (leaving out 4) 5 6, and also 3 4 and (leaving out 5) 6 1.

  If the points 1 2 3 4 5 are given, and we want a 6th point on a line
  drawn through 1, we know all the sides of the hexagon with the
  exception of 5 6, and this is found by Pascal's theorem.

  If this line should happen to pass through 1, then 6 and 1 coincide,
  or the line 6 1 is the tangent at 1. And always if two consecutive
  vertices of the hexagon approach nearer and nearer, then the side
  joining them will ultimately become a tangent.

  We may therefore consider a pentagon inscribed in a curve of second
  order and the tangent at one of its vertices as a hexagon, and thus
  get the theorem:

  Every pentagon inscribed in a      Every pentagon circumscribed
  curve of second order has the      about a curve of the second class
  property that the intersections    has the property that the lines
  of two pairs of non-consecutive    which join two pairs of non-
  sides lie in a line with the       consecutive vertices meet on that
  point where the fifth side cuts    line which joins the fifth vertex
  the tangent at the opposite        to the point of contact of the
  vertex.                            opposite side.

  This enables us also to solve the following problems.

  Given five points on a curve of    Given five tangents to a curve
  second order to construct the      of second class to construct the
  tangent at any one of them.        point of contact of any one of
                                     them.

  [Illustration: FIG. 19.]

  If two pairs of adjacent vertices coincide, the hexagon becomes a
  quadrilateral, with tangents at two vertices. These we take to be
  opposite, and get the following theorems:

  If a quadrilateral be inscribed    If a quadrilateral be circumscribed
  in a curve of second order, the    about a curve of second
  intersections of opposite sides,   class, the lines joining opposite
  and also the intersections of      vertices, and also the lines joining
  the tangents at opposite           points of contact of opposite
  vertices, lie in a line (fig.      sides, meet in a point.
  19).

  [Illustration: FIG. 20.]

  If we consider the hexagon made up of a triangle and the tangents at
  its vertices, we get--

  If a triangle is inscribed in a    If a triangle be circumscribed
  curve of the second order, the     about a curve of second class,
  points in which the sides are      the lines which join the vertices
  cut by the tangents at the         to the points of contact of the
  opposite vertices meet in a        opposite sides meet in a point
  point.                             (fig. 20).

  § 55. Of these theorems, those about the quadrilateral give rise to a
  number of others. Four points A, B, C, D may in three different ways
  be formed into a quadrilateral, for we may take them in the order
  ABCD, or ACBD, or ACDB, so that either of the points B, C, D may be
  taken as the vertex opposite to A. Accordingly we may apply the
  theorem in three different ways.

  Let A, B, C, D be four points on a curve of second order (fig. 21),
  and let us take them as forming a quadrilateral by taking the points
  in the order ABCD, so that A, C and also B, D are pairs of opposite
  vertices. Then P, Q will be the points where opposite sides meet, and
  E, F the intersections of tangents at opposite vertices. The four
  points P, Q, E, F lie therefore in a line. The quadrilateral ACBD
  gives us in the same way the four points Q, R, G, H in a line, and the
  quadrilateral ABDC a line containing the four points R, P, I, K. These
  three lines form a triangle PQR.

  The relation between the points and lines in this figure may be
  expressed more clearly if we consider ABCD as a four-point inscribed
  in a conic, and the tangents at these points as a four-side
  circumscribed about it,--viz. it will be seen that P, Q, R are the
  diagonal points of the four-point ABCD, whilst the sides of the
  triangle PQR are the diagonals of the circumscribing four-side. Hence
  the theorem--

  _Any four-point on a curve of the second order and the four-side
  formed by the tangents at these points stand in this relation that the
  diagonal points of the four-point lie in the diagonals of the
  four-side._ And conversely,

  _If a four-point and a circumscribed four-side stand in the above
  relation, then a curve of the second order may be described which
  passes through the four points and touches there the four sides of
  these figures._

  That the last part of the theorem is true follows from the fact that
  the four points A, B, C, D and the line a, as tangent at A, determine
  a curve of the second order, and the tangents to this curve at the
  other points B, C, D are given by the construction which leads to fig.
  21.

  [Illustration: FIG. 21.]

  The theorem reciprocal to the last is--

  _Any four-side circumscribed about a curve of second class and the
  four-point formed by the points of contact stand in this relation that
  the diagonals of the four-side pass through the diagonal points of the
  four-point._ And conversely,

  _If a four-side and an inscribed four-point stand in the above
  relation, then a curve of the second class may be described which
  touches the sides of the four-side at the points of the four-point._

  § 56. The four-point and the four-side in the two reciprocal theorems
  are alike. Hence if we have a four-point ABCD and a four-side abcd
  related in the manner described, then not only may a curve of the
  second order be drawn, but also a curve of the second class, which
  both touch the lines a, b, c, d at the points A, B, C, D.

  The curve of second order is already more than determined by the
  points A, B, C and the tangents a, b, c at A, B and C. The point D may
  therefore be _any_ point on this curve, and d any tangent to the
  curve. On the other hand the curve of the second class is more than
  determined by the three tangents a, b, c and their points of contact
  A, B, C, so that d is any tangent to this curve. It follows that every
  tangent to the curve of second order is a tangent of a curve of the
  second class having the same point of contact. In other words, the
  curve of second order is a curve of second class, and _vice versa_.
  Hence the important theorems--

  _Every curve of second order is    _Every curve of second class is a
  a curve of second class._          curve of second order._

  The curves of second order and of second class, having thus been
  proved to be identical, shall henceforth be called by the common name
  of _Conics_.

  For these curves hold, therefore, all properties which have been
  proved for curves of second order or of second class. We may therefore
  now state Pascal's and Brianchon's theorem thus--

  _Pascal's Theorem._--If a hexagon be inscribed in a conic, then the
  intersections of opposite sides lie in a line.

  _Brianchon's Theorem._--If a hexagon be circumscribed about a conic,
  then the diagonals forming opposite centres meet in a point.

  § 57. If we suppose in fig. 21 that the point D together with the
  tangent d moves along the curve, whilst A, B, C and their tangents a,
  b, c remain fixed, then the ray DA will describe a pencil about A, the
  point Q a projective row on the fixed line BC, the point F the row b,
  and the ray EF a pencil about E. But EF passes always through Q. Hence
  the pencil described by AD is projective to the pencil described by
  EF, and therefore to the row described by F on b. At the same time the
  line BD describes a pencil about B projective to that described by AD
  (§ 53). Therefore the pencil BD and the row F on b are projective.
  Hence--

  _If on a conic a point A be taken and the tangent a at this point,
  then the cross-ratio of the four rays which join A to any four points
  on the curve is equal to the cross-ratio of the points in which the
  tangents at these points cut the tangent at A._

  § 58. There are theorems about cones of second order and second class
  in a pencil which are reciprocal to the above, according to § 43. We
  mention only a few of the more important ones.

  The locus of intersections of corresponding planes in two projective
  axial pencils whose axes meet is a cone of the second order.

  The envelope of planes which join corresponding lines in two
  projective flat pencils, not in the same plane, is a cone of the
  second class.

  Cones of second order and cones of second class are identical.

  Every plane cuts a cone of the second order in a conic.

  _A cone of second order is uniquely determined by five of its edges or
  by five of its tangent planes, or by four edges and the tangent plane
  at one of them, &c. &c._

  _Pascal's Theorem._--If a solid angle of six faces be inscribed in a
  cone of the second order, then the intersections of opposite faces are
  three lines in a plane.

  _Brianchon's Theorem._--If a solid angle of six edges be circumscribed
  about a cone of the second order, then the planes through opposite
  edges meet in a line.

  Each of the other theorems about conics may be stated for cones of the
  second order.

  § 59. _Projective Definitions of the Conics._--We now consider the
  shape of the conics. We know that any line in the plane of the conic,
  and hence that the line at infinity, either has no point in common
  with the curve, or one (counting for two coincident points) or two
  distinct points. If the line at infinity has no point on the curve the
  latter is altogether finite, and is called an _Ellipse_ (fig. 21). If
  the line at infinity has only one point in common with the conic, the
  latter extends to infinity, and has the line at infinity a tangent. It
  is called a _Parabola_ (fig. 22). If, lastly, the line at infinity
  cuts the curve in two points, it consists of two separate parts which
  each extend in two branches to the points at infinity where they meet.
  The curve is in this case called an _Hyperbola_ (see fig. 20). The
  tangents at the two points at infinity are finite because the line at
  infinity is not a tangent. They are called _Asymptotes_. The branches
  of the hyperbola approach these lines indefinitely as a point on the
  curves moves to infinity.

  [Illustration: FIG. 22.]

  § 60. That the circle belongs to the curves of the second order is
  seen at once if we state in a slightly different form the theorem that
  in a circle all angles at the circumference standing upon the same arc
  are equal. If two points S1, S2 on a circle be joined to any other two
  points A and B on the circle, then the angle included by the rays S1A
  and S1B is equal to that between the rays S2A and S2B, so that as A
  moves along the circumference the rays S1A and S2A describe equal and
  therefore projective pencils. The circle can thus be generated by two
  projective pencils, and is a curve of the second order.

  If we join a point in space to all points on a circle, we get a
  (circular) cone of the second order (§ 43). Every plane section of
  this cone is a conic. This conic will be an ellipse, a parabola, or an
  hyperbola, according as the line at infinity in the plane has no, one
  or two points in common with the conic in which the plane at infinity
  cuts the cone. It follows that our curves of second order may be
  obtained as sections of a circular cone, and that they are identical
  with the "Conic Sections" of the Greek mathematicians.

  § 61. Any two tangents to a parabola are cut by all others in
  projective rows; but the line at infinity being one of the tangents,
  the points at infinity on the rows are corresponding points, and the
  rows therefore similar. Hence the theorem--

  _The tangents to a parabola cut each other proportionally._


  POLE AND POLAR

  § 62. We return once again to fig. 21, which we obtained in § 55.

  If a four-side be circumscribed about and a four-point inscribed in a
  conic, so that the vertices of the second are the points of contact of
  the sides of the first, then the triangle formed by the diagonals of
  the first is the same as that formed by the diagonal points of the
  other.

  Such a triangle will be called a _polar-triangle_ of the conic, so
  that PQR in fig. 21 is a polar-triangle. It has the property that on
  the side p opposite P meet the tangents at A and B, and also those at
  C and D. From the harmonic properties of four-points and four-sides it
  follows further that the points L, M, where it cuts the lines AB and
  CD, are harmonic conjugates with regard to AB and CD respectively.

  If the point P is given, and we draw a line through it, cutting the
  conic in A and B, then the point Q harmonic conjugate to P with regard
  to AB, and the point H where the tangents at A and B meet, are
  determined. But they lie both on p, and therefore this line is
  determined. If we now draw a second line through P, cutting the conic
  in C and D, then the point M harmonic conjugate to P with regard to
  CD, and the point G where the tangents at C and D meet, must also lie
  on p. As the first line through P already determines p, the second may
  be any line through P. Now every two lines through P determine a
  four-point ABCD on the conic, and therefore a polar-triangle which has
  one vertex at P and its opposite side at p. This result, together with
  its reciprocal, gives the theorems--

  _All polar-triangles which have one vertex in common have also the
  opposite side in common._

  _All polar-triangles which have one side in common have also the
  opposite vertex in common._

  § 63. To any point P in the plane of, but not on, a conic corresponds
  thus one line p as the side opposite to P in all polar-triangles which
  have one vertex at P, and reciprocally to every line p corresponds one
  point P as the vertex opposite to p in all triangles which have p as
  one side.

  We call the line p the _polar_ of P, and the point P the _pole_ of the
  line p with regard to the conic.

  If a point lies on the conic, we call the tangent at that point its
  polar; and reciprocally we call the point of contact the pole of
  tangent.

  § 64. From these definitions and former results follow--

  The polar of any point P not       The pole of any line p not a
  on the conic is a line p, which    tangent to the conic is a point
  has the following properties:--    P, which has the following
                                     properties:--

  1. On every line through P         1. Of all lines through a point
  which cuts the conic, the polar    on p from which two tangents
  of P contains the harmonic         may be drawn to the conic, the
  conjugate  of P with regard to     pole P contains the line which is
  those points on the conic.         harmonic conjugate to p, with
                                     regard to the two tangents.

  2. If tangents can be drawn        2. If p cuts the conic, the
  from P, their points of contact    tangents at the intersections
  lie on p.                          meet at P.

  3. Tangents drawn at the           3. The point of contact of
  points where any line through P    tangents drawn from any point
  cuts the conic meet on p; and      on p to the conic lie in a line
  conversely,                        with P; and conversely,

  4. If from any point on p,         4. Tangents drawn at points
  tangents be drawn, their points    where any line through P cuts the
  of contact will lie in a line      conic meet on p.
  with P.

  5. Any four-point on the conic     5. Any four-side circumscribed
  which has one diagonal point at    about a conic which has one
  P has the other two lying on p.    diagonal on p has the other two
                                     meeting at P.

  The truth of 2 follows from 1. If T be a point where p cuts the conic,
  then one of the points where PT cuts the conic, and which are harmonic
  conjugates with regard to PT, coincides with T; hence the other
  does--that is, PT touches the curve at T.

  That 4 is true follows thus: If we draw from a point H on the polar
  one tangent a to the conic, join its point of contact A to the pole P,
  determine the second point of intersection B of this line with the
  conic, and draw the tangent at B, it will pass through H, and will
  therefore be the second tangent which may be drawn from H to the
  curve.

  § 65. The second property of the polar or pole gives rise to the
  theorem--

  From a point in the plane of a     A line in the plane of a conic
  conic, two, one or no tangents     has two, one or no points in
  may be drawn to the conic,         common with the conic, according
  as its polar has two,              as two, one or no tangents
  one, or no points in common        can be drawn from its pole to the
  with the curve.                    conic.

  Of any point in the plane of a conic we say that it was _without_, on
  or _within_ the curve according as two, one or no tangents to the
  curve pass through it. The points on the conic separate those within
  the conic from those without. That this is true for a circle is known
  from elementary geometry. That it also holds for other conics follows
  from the fact that every conic may be considered as the projection of
  a circle, which will be proved later on.

  The fifth property of pole and polar stated in § 64 shows how to find
  the polar of any point and the pole of any line by aid of the
  straight-edge only. Practically it is often convenient to draw three
  secants through the pole, and to determine only one of the diagonal
  points for two of the four-points formed by pairs of these lines and
  the conic (fig. 22).

  These constructions also solve the problem--

  From a point without a conic, to draw the two tangents to the conic by
  aid of the straight-edge only.

  For we need only draw the polar of the point in order to find the
  points of contact.

  § 66. The property of a polar-triangle may now be stated thus--

  In a polar-triangle each side is the polar of the opposite vertex, and
  each vertex is the pole of the opposite side.

  [Illustration: FIG. 23.]

  If P is one vertex of a polar-triangle, then the other vertices, Q and
  R, lie on the polar p of P. One of these vertices we may choose
  arbitrarily. For if from any point Q on the polar a secant be drawn
  cutting the conic in A and D (fig. 23), and if the lines joining these
  points to P cut the conic again at B and C, then the line BC will pass
  through Q. Hence P and Q are two of the vertices on the polar-triangle
  which is determined by the four-point ABCD. The third vertex R lies
  also on the line p. It follows, therefore, also--

  _If Q is a point on the polar of P, then P is a point on the polar of
  Q_; and reciprocally,

  _If q is a line through the pole of p, then p is a line through the
  pole of q._

  This is a very important theorem. It may also be stated thus--

  _If a point moves along a line describing a row, its polar turns about
  the pole of the line describing a pencil._

  _This pencil is projective to the row, so that the cross-ratio of four
  poles in a row equals the cross-ratio of its four polars, which pass
  through the pole of the row._

  To prove the last part, let us suppose that P, A and B in fig. 23
  remain fixed, whilst Q moves along the polar p of P. This will make CD
  turn about P and move R along p, whilst QD and RD describe projective
  pencils about A and B. Hence Q and R describe projective rows, and
  hence PR, which is the polar of Q, describes a pencil projective to
  either.

  § 67. Two points, of which one, and therefore each, lies on the polar
  of the other, are said to be _conjugate with regard to the conic_; and
  two lines, of which one, and therefore each, passes through the pole
  of the other, are said to be _conjugate with regard to the conic_.
  Hence all points conjugate to a point P lie on the polar of P; all
  lines conjugate to a line p pass through the pole of p.

  If the line joining two conjugate poles cuts the conic, then the poles
  are harmonic conjugates with regard to the points of intersection;
  hence one lies within the other without the conic, and all points
  conjugate to a point within a conic lie without it.

  Of a polar-triangle any two vertices are conjugate poles, any two
  sides conjugate lines. If, therefore, one side cuts a conic, then one
  of the two vertices which lie on this side is within and the other
  without the conic. The vertex opposite this side lies also without,
  for it is the pole of a line which cuts the curve. In this case
  therefore one vertex lies within, the other two without. If, on the
  other hand, we begin with a side which does not cut the conic, then
  its pole lies within and the other vertices without. Hence--

  Every polar-triangle has one and only one vertex within the conic.

  We add, without a proof, the theorem--

  The four points in which a conic is cut by two conjugate polars are
  four harmonic points in the conic.

  § 68. If two conics intersect in four points (they cannot have more
  points in common, § 52), there exists one and only one four-point
  which is inscribed in both, and therefore one polar-triangle common to
  both.

  _Theorem._--Two conics which intersect in four points have always one
  and only one common polar-triangle; and reciprocally,

  Two conics which have four common tangents have always one and only
  one common polar-triangle.


  DIAMETERS AND AXES OF CONICS

  § 69. _Diameters._--The theorems about the harmonic properties of
  poles and polars contain, as special cases, a number of important
  metrical properties of conics. These are obtained if either the pole
  or the polar is moved to infinity,--it being remembered that the
  harmonic conjugate to a point at infinity, with regard to two points
  A, B, is the middle point of the segment AB. The most important
  properties are stated in the following theorems:--

  _The middle points of parallel chords of a conic lie in a line--viz.
  on the polar to the point at infinity on the parallel chords._

  This line is called a _diameter_.

  _The polar of every point at infinity is a diameter._

  _The tangents at the end points of a diameter are parallel, and are
  parallel to the chords bisected by the diameter._

  _All diameters pass through a common point, the pole of the line at
  infinity._

  _All diameters of a parabola are parallel_, the pole to the line at
  infinity being the point where the curve touches the line at
  infinity.

  In case of the ellipse and hyperbola, the pole to the line at infinity
  is a finite point called the _centre_ of the curve.

  _A centre of a conic bisects every chord through it._

  _The centre of an ellipse is within the curve_, for the line at
  infinity does not cut the ellipse.

  _The centre of an hyperbola is without the curve_, because the line at
  infinity cuts the curve. Hence also--

  _From the centre of an hyperbola two tangents can be drawn to the
  curve which have their point of contact at infinity._ These are called
  _Asymptotes_ (§ 59).

  _To construct a diameter_ of a conic, draw two parallel chords and
  join their middle points.

  _To find the centre_ of a conic, draw two diameters; their
  intersection will be the centre.

  § 70. _Conjugate Diameters._--A polar-triangle with one vertex at the
  centre will have the opposite side at infinity. The other two sides
  pass through the centre, and are called _conjugate diameters_, each
  being the polar of the point at infinity on the other.

  _Of two conjugate diameters each bisects the chords parallel to the
  other, and if one cuts the curve, the tangents at its ends are
  parallel to the other diameter._

  Further--

  _Every parallelogram inscribed in a conic has its sides parallel to
  two conjugate diameters_; and

  _Every parallelogram circumscribed about a conic has as diagonals two
  conjugate diameters._

  This will be seen by considering the parallelogram in the first case
  as an inscribed four-point, in the other as a circumscribed four-side,
  and determining in each case the corresponding polar-triangle. The
  first may also be enunciated thus--

  _The lines which join any point on an ellipse or an hyperbola to the
  ends of a diameter are parallel to two conjugate diameters._

  § 71. _If every diameter is perpendicular to its conjugate the conic
  is a circle._

  For the lines which join the ends of a diameter to any point on the
  curve include a right angle.

  _A conic which has more than one pair of conjugate diameters at right
  angles to each other is a circle._

  [Illustration: FIG. 24.]

  Let AA' and BB' (fig. 24) be one pair of conjugate diameters at right
  angles to each other, CC and DD' a second pair. If we draw through the
  end point A of one diameter a chord AP parallel to DD', and join P to
  A', then PA and PA' are, according to § 70, parallel to two conjugate
  diameters. But PA is parallel to DD', hence PA' is parallel to CC, and
  therefore PA and PA' are perpendicular. If we further draw the
  tangents to the conic at A and A', these will be perpendicular to AA',
  they being parallel to the conjugate diameter BB'. We know thus five
  points on the conic, viz. the points A and A' with their tangents, and
  the point P. Through these a circle may be drawn having AA' as
  diameter; and as through five points one conic only can be drawn, this
  circle must coincide with the given conic.

  § 72. _Axes._--Conjugate diameters perpendicular to each other are
  called _axes_, and the points where they cut the curve _vertices_ of
  the conic.

  In a circle every diameter is an axis, every point on it is a vertex;
  and any two lines at right angles to each other may be taken as a pair
  of axes of any circle which has its centre at their intersection.

  [Illustration: FIG. 25.]

  If we describe on a diameter AB of an ellipse or hyperbola a circle
  concentric to the conic, it will cut the latter in A and B (fig. 25).
  Each of the semicircles in which it is divided by AB will be partly
  within, partly without the curve, and must cut the latter therefore
  again in a point. The circle and the conic have thus four points A, B,
  C, D, and therefore one polar-triangle, in common (§ 68). Of this the
  centre is one vertex, for the line at infinity is the polar to this
  point, both with regard to the circle and the other conic. The other
  two sides are conjugate diameters of both, hence perpendicular to each
  other. This gives--

  An ellipse as well as an hyperbola has one pair of axes.

  This reasoning shows at the same time _how to construct the axis of an
  ellipse or of an hyperbola_.

  _A parabola has one axis_, if we define an axis as a diameter
  perpendicular to the chords which it bisects. It is easily
  constructed. The line which bisects any two parallel chords is a
  diameter. Chords perpendicular to it will be bisected by a parallel
  diameter, and this is the axis.

  § 73. The first part of the right-hand theorem in § 64 may be stated
  thus: any two conjugate lines through a point P without a conic are
  harmonic conjugates with regard to the two tangents that may be drawn
  from P to the conic.

  If we take instead of P the centre C of an hyperbola, then the
  conjugate lines become conjugate diameters, and the tangents
  asymptotes. Hence--

  _Any two conjugate diameters of an hyperbola are harmonic conjugates
  with regard to the asymptotes._

  As the axes are conjugate diameters at right angles to one another, it
  follows (§ 23)--

  _The axes of an hyperbola bisect the angles between the asymptotes._

  [Illustration: FIG. 26.]

  Let O be the centre of the hyperbola (fig. 26), t any secant which
  cuts the hyperbola in C, D and the asymptotes in E, F, then the line
  OM which bisects the chord CD is a diameter conjugate to the diameter
  OK which is parallel to the secant t, so that OK and OM are harmonic
  with regard to the asymptotes. The point M therefore bisects EF. But
  by construction M bisects CD. It follows that DF = EC, and ED = CF; or

  _On any secant of an hyperbola the segments between the curve and the
  asymptotes are equal._

  If the chord is changed into a tangent, this gives--

  _The segment between the asymptotes on any tangent to an hyperbola is
  bisected by the point of contact._

  The first part allows a simple solution of the problem to find any
  number of points on an hyperbola, of which the asymptotes and one
  point are given. This is equivalent to three points and the tangents
  at two of them. This construction requires measurement.

  § 74. For the parabola, too, follow some metrical properties. A
  diameter PM (fig. 27) bisects every chord conjugate to it, and the
  pole P of such a chord BC lies on the diameter. But a diameter cuts
  the parabola once at infinity. Hence--

  _The segment PM which joins the middle point M of a chord of a
  parabola to the pole P of the chord is bisected by the parabola at A._

  § 75. Two asymptotes and any two tangents to an hyperbola may be
  considered as a quadrilateral circumscribed about the hyperbola. But
  in such a quadrilateral the intersections of the diagonals and the
  points of contact of opposite sides lie in a line (§ 54). If therefore
  DEFG (fig. 28) is such a quadrilateral, then the diagonals DF and GE
  will meet on the line which joins the points of contact of the
  asymptotes, that is, on the line at infinity; hence they are parallel.
  From this the following theorem is a simple deduction:

  _All triangles formed by a tangent and the asymptotes of an hyperbola
  are equal in area._

  [Illustration: FIG. 27.]

  [Illustration: FIG. 28.]

  If we draw at a point P (fig. 28) on an hyperbola a tangent, the part
  HK between the asymptotes is bisected at P. The parallelogram PQOQ'
  formed by the asymptotes and lines parallel to them through P will be
  half the triangle OHK, and will therefore be constant. If we now take
  the asymptotes OX and OY as oblique axes of co-ordinates, the lines OQ
  and QP will be the co-ordinates of P, and will satisfy the equation xy
  = const. = a².

  _For the asymptotes as axes of co-ordinates the equation of the
  hyperbola is xy = const._


  INVOLUTION

  [Illustration: FIG. 29.]

  § 76. If we have two projective rows, ABC on u and A'B'C' on u', and
  place their bases on the same line, then each point in this line
  counts twice, once as a point in the row u and once as a point in the
  row u'. In fig. 29 we denote the points as points in the one row by
  letters above the line A, B, C ..., and as points in the second row by
  A', B', C' ... below the line. Let now A and B' be the same point,
  then to A will correspond a point A' in the second, and to B' a point
  B in the first row. In general these points A' and B will be
  different. It may, however, happen that they coincide. Then the
  correspondence is a peculiar one, as the following theorem shows:

  _If two projective rows lie on the same base, and if it happens that
  to one point in the base the same point corresponds, whether we
  consider the point as belonging to the first or to the second row,
  then the same will happen for every point in the base--that is to say,
  to every point in the line corresponds the same point in the first as
  in the second row._

  [Illustration: FIG. 30.]

  In order to determine the correspondence, we may assume three pairs of
  corresponding points in two projective rows. Let then A', B', C', in
  fig. 30, correspond to A, B, C, so that A and B', and also B and A',
  denote the same point. Let us further denote the point C' when
  considered as a point in the first row by D; then it is to be proved
  that the point D', which corresponds to D, is the same point as C. We
  know that the cross-ratio of four points is equal to that of the
  corresponding row. Hence

    (AB, CD) = (A'B', C'D')

  but replacing the dashed letters by those undashed ones which denote
  the same points, the second cross-ratio equals (BA, DD'), which,
  according to § 15, equals (AB, D'D); so that the equation becomes

    (AB, CD) = (AB, D'D).

  This requires that C and D' coincide.

  § 77. Two projective rows on the same base, which have the above
  property, that to every point, whether it be considered as a point in
  the one or in the other row, corresponds the same point, are said to
  be in _involution_, or to form an _involution_ of points on the line.

  We mention, but without proving it, that any two projective rows may
  be placed so as to form an involution.

  An involution may be said to consist of a row of pairs of points, to
  every point A corresponding a point A', and to A' again the point A.
  These points are said to be conjugate, or, better, one point is termed
  the "mate" of the other.

  From the definition, according to which an involution may be
  considered as made up of two projective rows, follow at once the
  following important properties:

  1. The cross-ratio of four points equals that of the four conjugate
  points.

  2. If we call a point which coincides with its mate a "focus" or
  "double point" of the involution, we may say: An involution has either
  two foci, or one, or none, and is called respectively a hyperbolic,
  parabolic or elliptic involution (§ 34).

  3. In an hyperbolic involution any two conjugate points are harmonic
  conjugates with regard to the two foci.

  For if A, A' be two conjugate points, F1, F2 the two foci, then to the
  points F1, F2, A, A' in the one row correspond the points F1, F2, A',
  A in the other, each focus corresponding to itself. Hence (F1F2, AA')
  = (F1F2, A'A)--that is, we may interchange the two points AA' without
  altering the value of the cross-ratio, which is the characteristic
  property of harmonic conjugates (§ 18).

  4. The point conjugate to the point at infinity is called the "centre"
  of the involution. Every involution has a centre, unless the point at
  infinity be a focus, in which case we may say that the centre is at
  infinity.

  In an hyperbolic involution the centre is the middle point between the
  foci.

  5. The product of the distances of two conjugate points A, A' from the
  centre O is constant: OA . OA' = c.

  For let A, A' and B, B' be two pairs of conjugate points, the centre,
  I the point at infinity, then

    (AB, OI) = (A'B', IO),

  or

    OA . OA' = OB . OB'.

  In order to determine the distances of the foci from the centre, we
  write F for A and A' and get

  OF² = c; OF = ±[root]c.

  Hence if c is positive OF is real, and has two values, equal and
  opposite. The involution is hyperbolic.

  If c = 0, OF = 0, and the two foci both coincide with the centre. If c
  is negative, [root]c becomes imaginary, and there are no foci. Hence
  we may write--

    In an hyperbolic involution, OA·OA' = k²,
    In a parabolic involution, OA·OA' = 0,
    In an elliptic involution, OA·OA' = -k².

  From these expressions it follows that conjugate points A, A' in an
  hyperbolic involution lie on the same side of the centre, and in an
  elliptic involution on opposite sides of the centre, and that in a
  parabolic involution one coincides with the centre.

  In the first case, for instance, OA·OA' is positive; hence OA and OA'
  have the same sign.

  It also follows that two segments, AA' and BB', between pairs of
  conjugate points have the following positions: in an hyperbolic
  involution they lie either one altogether within or altogether without
  each other; in a parabolic involution they have one point in common;
  and in an elliptic involution they overlap, each being partly within
  and partly without the other.

  _Proof._--We have OA·OA' = OB·OB' = k² in case of an hyperbolic
  involution. Let A and B be the points in each pair which are nearer to
  the centre O. If now A, A' and B, B' lie on the same side of O, and if
  B is nearer to O than A, so that OB < OA, then OB' > OA'; hence B' lies
  farther away from O than A', or the segment AA' lies within BB'. And so
  on for the other cases.

  6. An involution is determined--

    ([alpha]) By two pairs of conjugate points. Hence also
    (ß) By one pair of conjugate points and the centre;
    ([gamma]) By the two foci;
    ([delta]) By one focus and one pair of conjugate points;
    ([epsilon]) By one focus and the centre.

  7. The condition that A, B, C and A', B', C' may form an involution
  may be written in one of the forms--

     (AB, CC') = (A'B', C'C),

  or (AB, CA') = (A'B', C'A),

  or (AB, C'A') = (A'B', CA),

  for each expresses that in the two projective rows in which A, B, C
  and A', B', C' are conjugate points two conjugate elements may be
  interchanged.

  8. Any three pairs. A, A', B, B', C, C', of conjugate points are
  connected by the relations:

    AB'·BC'·CA'   AB'·BC·C'A'   AB·B'C'·CA'   AB·B'C·C'A'
    ----------- = ----------- = ----------- = ----------- = -1.
    A'B·B'C·C'A   A'B·B'C'·CA   A'B'·BC·C'A   A'B'·BC'·CA

  These relations readily follow by working out the relations in (7)
  (above).

  § 78. _Involution of a quadrangle.--The sides of any four-point are
  cut by any line in six points in involution, opposite sides being cut
  in conjugate points._

  Let A1B1C1D1 (fig. 31) be the four-point. If its sides be cut by the
  line p in the points A, A', B, B', C, C', if further, C1D1 cuts the
  line A1B1 in C2, and if we project the row A1B1C2C to p once from D1
  and once from C1, we get (A'B', C'C) = (BA, C'C).

  Interchanging in the last cross-ratio the letters in each pair we get
  (A'B', C'C) = (AB, CC'). Hence by § 77 (7) the points are in
  involution.

  The theorem may also be stated thus:

  _The three points in which any line cuts the sides of a triangle and
  the projections, from any point in the plane, of the vertices of the
  triangle on to the same line are six points in involution._

  [Illustration: FIG. 31.]

  Or again--

  The projections from any point on to any line of the six vertices of a
  four-side are six points in involution, the projections of opposite
  vertices being conjugate points.

  This property gives a simple means to construct, by aid of the
  straight edge only, in an involution of which two pairs of conjugate
  points are given, to any point its conjugate.

  § 79. _Pencils in Involution._--The theory of involution may at once
  be extended from the row to the flat and the axial pencil--viz. we say
  that there is an involution in a flat or in an axial pencil if any
  line cuts the pencil in an involution of points. An involution in a
  pencil consists of pairs of conjugate rays or planes; it has two, one
  or no _focal rays_ (double lines) or _planes_, but nothing
  corresponding to a centre.

  An involution in a flat pencil contains always one, and in general
  only one, pair of conjugate rays which are perpendicular to one
  another. For in two projective flat pencils exist always two
  corresponding right angles (§ 40).

  Each involution in an axial pencil contains in the same manner one
  pair of conjugate planes at right angles to one another.

  As a rule, there exists but one pair of conjugate lines or planes at
  right angles to each other. But it is possible that there are more,
  and then there is an infinite number of such pairs. An involution in a
  flat pencil, in which every ray is perpendicular to its conjugate ray,
  is said to be _circular_. That such involution is possible is easily
  seen thus: if in two concentric flat pencils each ray on one is made
  to correspond to that ray on the other which is perpendicular to it,
  then the two pencils are projective, for if we turn the one pencil
  through a right angle each ray in one coincides with its corresponding
  ray in the other. But these two projective pencils are in involution.

  A circular involution has no focal rays, because no ray in a pencil
  coincides with the ray perpendicular to it.

  § 80. _Every elliptical involution in a row may be considered as a
  section of a circular involution._

  In an elliptical involution any two segments AA' and BB' lie partly
  within and partly without each other (fig. 32). Hence two circles
  described on AA' and BB' as diameters will intersect in two points E
  and E'. The line EE' cuts the base of the involution at a point O,
  which has the property that OA·OA' = OB . OB', for each is equal to
  OE . OE'. The point O is therefore the centre of the involution. If we
  wish to construct to any point C the conjugate point C', we may draw
  the circle through CEE'. This will cut the base in the required point
  C' for OC·OC' = OA·OA'. But EC and EC' are at right angles. Hence the
  involution which is obtained by joining E or E' to the points in the
  given involution is circular. This may also be expressed thus:

  [Illustration: FIG. 32.]

  _Every elliptical involution has the property that there are two
  definite points in the plane from which any two conjugate points are
  seen under a right angle._

  At the same time the following problem has been solved:

  To determine the centre and also the point corresponding to any given
  point in an elliptical involution of which two pairs of conjugate
  points are given.

  § 81. _Involution Range on a Conic._--By the aid of § 53, the points
  on a conic may be made to correspond to those on a line, so that the
  row of points on the conic is projective to a row of points on a line.
  We may also have two projective rows on the same conic, and these will
  be in involution as soon as one point on the conic has the same point
  corresponding to it all the same to whatever row it belongs. An
  involution of points on a conic will have the property (as follows
  from its definition, and from § 53) that the lines which join
  conjugate points of the involution to any point on the conic are
  conjugate lines of an involution in a pencil, and that a fixed tangent
  is cut by the tangents at conjugate points on the conic in points
  which are again conjugate points of an involution on the fixed
  tangent. For such involution on a conic the following theorem holds:

  _The lines which join corresponding points in an involution on a conic
  all pass through a fixed point; and reciprocally, the points of
  intersection of conjugate lines in an involution among tangents to a
  conic lie on a line._

  [Illustration: FIG. 33]

  We prove the first part only. The involution is determined by two
  pairs of conjugate points, say by A, A' and B, B' (fig. 33). Let AA'
  and BB' meet in P. If we join the points in involution to any point on
  the conic, and the conjugate points to another point on the conic, we
  obtain two projective pencils. We take A and A' as centres of these
  pencils, so that the pencils A(A'BB') and A'(AB'B) are projective, and
  in perspective position, because AA' corresponds to A'A. Hence
  corresponding rays meet in a line, of which two points are found by
  joining AB' to A'B and AB to A'B'. It follows that the _axis_ of
  perspective is the polar of the point P, where AA' and BB' meet. If we
  now wish to construct to any other point C on the conic the
  corresponding point C', we join C to A' and the point where this line
  cuts p to A. The latter line cuts the conic again in C'. But we know
  from the theory of pole and polar that the line CC' passes through P.
  The point of concurrence is called the "pole of the involution," and
  the line of collinearity of the meets is called the "axis of the
  involution."


  INVOLUTION DETERMINED BY A CONIC ON A LINE.--FOCI

  § 82. The polars, with regard to a conic, of points in a row p form a
  pencil P projective to the row (§ 66). This pencil cuts the base of
  the row p in a projective row.

  If A is a point in the given row, A' the point where the polar of A
  cuts p, then A and A' will be corresponding points. If we take A' a
  point in the first row, then the polar of A' will pass through A, so
  that A corresponds to A'--in other words, the rows are in involution.
  The conjugate points in this involution are conjugate points with
  regard to the conic. Conjugate points coincide only if the polar of a
  point A passes through A--that is, if A lies on the conic. Hence--

  _A conic determines on every line in its plane an involution, in which
  those points are conjugate which are also conjugate with regard to the
  conic._

  _If the line cuts the conic the involution is hyperbolic, the points
  of intersection being the foci._

  _If the line touches the conic the involution is parabolic, the two
  foci coinciding at the point of contact._

  _If the line does not cut the conic the involution is elliptic, having
  no foci._

  If, on the other hand, we take a point P in the plane of a conic, we
  get to each line a through P one conjugate line which joins P to the
  pole of a. These pairs of conjugate lines through P form an involution
  in the pencil at P. The focal rays of this involution are the tangents
  drawn from P to the conic. This gives the theorem reciprocal to the
  last, viz:--

  _A conic determines in every pencil in its plane an involution,
  corresponding lines being conjugate lines with regard to the conic._

  _If the point is without the conic the involution is hyperbolic, the
  tangents from the points being the focal rays._

  _If the point lies on the conic the involution is parabolic, the
  tangent at the point counting for coincident focal rays._

  _If the point is within the conic the involution is elliptic, having
  no focal rays._

  It will further be seen that the involution determined by a conic on
  any line p is a section of the involution, which is determined by the
  conic at the pole P of p.

  § 83. _Foci._--The centre of a pencil in which the conic determines a
  circular involution is called a "focus" of the conic.

  In other words, a focus is such a point that every line through it is
  perpendicular to its conjugate line. The polar to a focus is called a
  _directrix_ of the conic.

  From the definition it follows that _every focus lies on an axis_, for
  the line joining a focus to the centre of the conic is a diameter to
  which the conjugate lines are perpendicular; and _every line joining
  two foci is an axis_, for the perpendiculars to this line through the
  foci are conjugate to it. These conjugate lines pass through the pole
  of the line, the pole lies therefore at infinity, and the line is a
  diameter, hence by the last property an axis.

  It follows that all _foci lie on one axis_, for no line joining a
  point in one axis to a point in the other can be an axis.

  As the conic determines in the pencil which has its centre at a focus
  a circular involution, no tangents can be drawn from the focus to the
  conic. Hence _each focus lies within a conic_; and _a directrix does
  not cut the conic_.

  Further properties are found by the following considerations:

  § 84. Through a point P one line p can be drawn, which is with regard
  to a given conic conjugate to a given line q, viz. that line which
  joins the point P to the pole of the line q. If the line q is made to
  describe a pencil about a point Q, then the line p will describe a
  pencil about P. These two pencils will be projective, for the line p
  passes through the pole of q, and whilst q describes the pencil Q, its
  pole describes a projective row, and this row is perspective to the
  pencil P.

  We now take the point P on an axis of the conic, draw any line p
  through it, and from the pole of p draw a perpendicular q to p. Let q
  cut the axis in Q. Then, in the pencils of conjugate lines, which have
  their centres at P and Q, the lines p and q are conjugate lines at
  right angles to one another. Besides, to the axis as a ray in either
  pencil will correspond in the other the perpendicular to the axis (§
  72). The conic generated by the intersection of corresponding lines in
  the two pencils is therefore the circle on PQ as diameter, _so that
  every line in P is perpendicular to its corresponding line in Q_.

  To every point P on an axis of a conic corresponds thus a point Q,
  such that conjugate lines through P and Q are perpendicular.

  We shall show that these _point-pairs_ P, Q _form an involution_. To
  do this let us move P along the axis, and with it the line p, keeping
  the latter parallel to itself. Then P describes a row, p a perspective
  pencil (of parallels), and the pole of p a projective row. At the same
  time the line q describes a pencil of parallels perpendicular to p,
  and perspective to the row formed by the pole of p. The point Q,
  therefore, where q cuts the axis, describes a row projective to the
  row of points P. The two points P and Q describe thus two projective
  rows on the axis; and not only does P as a point in the first row
  correspond to Q, but also Q as a point in the first corresponds to P.
  The two rows therefore form an involution. _The centre of this
  involution, it is easily seen, is the centre of the conic._

  _A focus of this involution has the property that any two conjugate
  lines through it are perpendicular; hence, it is a focus to the
  conic._

  Such involution exists on each axis. But only one of these can have
  foci, because all foci lie on the same axis. The involution on one of
  the axes is elliptic, and appears (§ 80) therefore as the section of
  two circular involutions in two pencils whose centres lie in the other
  axis. These centres are foci, hence the one axis contains two foci,
  the other axis none; _or every central conic has two foci which lie on
  one axis equidistant from the centre_.

  The axis which contains the foci is called the _principal axis_; in
  case of an hyperbola it is the axis which cuts the curve, because the
  foci lie within the conic.

  In case of the parabola there is but one axis. The involution on this
  axis has its centre at infinity. One focus is therefore at infinity,
  the one focus only is finite. _A parabola has only one focus._

  [Illustration: FIG. 34.]

  § 85. If through any point P (fig. 34) on a conic the tangent PT and
  the normal PN (i.e. the perpendicular to the tangent through the point
  of contact) be drawn, these will be conjugate lines with regard to the
  conic, and at right angles to each other. They will therefore cut the
  principal axis in two points, which are conjugate in the involution
  considered in § 84; hence they are harmonic conjugates with regard to
  the foci. If therefore the two foci F1 and F2 be joined to P, these
  lines will be harmonic with regard to the tangent and normal. As the
  latter are perpendicular, they will bisect the angles between the
  other pair. Hence--

  _The lines joining any point on a conic to the two foci are equally
  inclined to the tangent and normal at that point._

  In case of the parabola this becomes--

  _The line joining any point on a parabola to the focus and the
  diameter through the point, are equally inclined to the tangent and
  normal at that point._

  From the definition of a focus it follows that--

  _The segment of a tangent between the directrix and the point of
  contact is seen from the focus belonging to the directrix under a
  right angle_, because the lines joining the focus to the ends of this
  segment are conjugate with regard to the conic, and therefore
  perpendicular.

  With equal ease the following theorem is proved:

  _The two lines which join the points of contact of two tangents each
  to one focus, but not both to the same, are seen from the intersection
  of the tangents under equal angles._

  § 86. Other focal properties of a conic are obtained by the following
  considerations:

  [Illustration: FIG. 35.]

  Let F (fig. 35) be a focus to a conic, f the corresponding directrix,
  A and B the points of contact of two tangents meeting at T, and P the
  point where the line AB cuts the directrix. Then TF will be the polar
  of P (because polars of F and T meet at P). Hence TF and PF are
  conjugate lines through a focus, and therefore perpendicular. They are
  further harmonic conjugates with regard to FA and FB (§§ 64 and 13),
  so that they bisect the angles formed by these lines. This by the way
  proves--

  _The segments between the point of intersection of two tangents to a
  conic and their points of contact are seen from a focus under equal
  angles._

  If we next draw through A and B lines parallel to TF, then the points
  A1, B1 where these cut the directrix will be harmonic conjugates with
  regard to P and the point where FT cuts the directrix. The lines FT
  and FP bisect therefore also the angles between FA1 and FB1. From this
  it follows easily that the triangles FAA1 and FBB1 are equiangular,
  and therefore similar, so that FA : AA1 = FB : BB1.

  The triangles AA1A2 and BB1B2 formed by drawing perpendiculars from A
  and B to the directrix are also similar, so that AA1 : AA2 = = BB1 :
  BB2. This, combined with the above proportion, gives FA : AA2 = FB :
  BB2. Hence the theorem:

  _The ratio of the distances of any point on a conic from a focus and
  the corresponding directrix is constant._

  To determine this ratio we consider its value for a vertex on the
  principal axis. In an ellipse the focus lies between the two vertices
  on this axis, hence the focus is nearer to a vertex than to the
  corresponding directrix. Similarly, in an hyperbola a vertex is nearer
  to the directrix than to the focus. In a parabola the vertex lies
  halfway between directrix and focus.

  It follows in an ellipse the ratio between the distance of a point
  from the focus to that from the directrix is less than unity, in the
  parabola it equals unity, and in the hyperbola it is greater than
  unity.

  It is here the same which focus we take, because the two foci lie
  symmetrical to the axis of the conic. If now P is any point on the
  conic having the distances r1 and r2 from the foci and the distances
  d1 and d2 from the corresponding directrices, then r1/d1 = r2/d2 =
  e, where e is constant. Hence also r1 ± r2 / d1 ± d2 = e.

  In the ellipse, which lies between the directrices, d1 + d2 is
  constant, therefore also r1 +r2. In the hyperbola on the other hand d1
  - d2 is constant, equal to the distance between the directrices,
  therefore in this case r1 - r2 is constant.

  If we call the distances of a point on a conic from the focus its
  focal distances we have the theorem:

  _In an ellipse the sum of the focal distances is constant; and in an
  hyperbola the difference of the focal distances is constant._

  _This constant sum or difference equals in both cases the length of
  the principal axis._


  PENCIL OF CONICS

  § 87. Through four points A, B, C, D in a plane, of which no three lie
  in a line, an infinite number of conics may be drawn, viz. through
  these four points and any fifth one single conic. This system of
  conics is called a pencil of conics. Similarly, all conics touching
  four fixed lines form a system such that any fifth tangent determines
  one and only one conic. We have here the theorems:

  The pairs of points in which        The pairs of tangents which
  any line is cut by a system of      can be drawn from a point to
  conics through four fixed points    a system of conics touching four
  are in involution.                  fixed lines are in involution.

  [Illustration: FIG. 36.]

  We prove the first theorem only. Let ABCD (fig. 36) be the four-point,
  then any line t will cut two opposite sides AC, BD in the points E,
  E', the pair AD, BC in points F, F', and any conic of the system in M,
  N, and we have A(CD, MN) = B(CD, MN).

  If we cut these pencils by t we get

     (EF, MN) = (F'E', MN)

  or (EF, MN) = (E'F', NM).

  But this is, according to § 77 (7), the condition that M, N are
  corresponding points in the involution determined by the point pairs
  E, E', F, F' in which the line t cuts pairs of opposite sides of the
  four-point ABCD. This involution is independent of the particular
  conic chosen.

  § 88. There follow several important theorems:

  _Through four points two, one, or no conics may be drawn which touch
  any given line, according as the involution determined by the given
  four-point on the line has real, coincident or imaginary foci._

  _Two, one, or no conics may be drawn which touch four given lines and
  pass through a given point, according as the involution determined by
  the given four-side at the point has real, coincident or imaginary
  focal rays._

  For the conic through four points which touches a given line has its
  point of contact at a focus of the involution determined by the
  four-point on the line.

  As a special case we get, by taking the line at infinity:

  _Through four points of which none is at infinity either two or no
  parabolas may be drawn._

  The problem of drawing a conic through four points and touching a
  given line is solved by determining the points of contact on the line,
  that is, by determining the foci of the involution in which the line
  cuts the sides of the four-point. The corresponding remark holds for
  the problem of drawing the conics which touch four lines and pass
  through a given point.


  RULED QUADRIC SURFACES

  § 89. We have considered hitherto projective rows which lie in the
  same plane, in which case lines joining corresponding points envelop a
  conic. We shall now consider projective rows whose bases do not meet.
  In this case, corresponding points will be joined by lines which do
  not lie in a plane, but on some surface, which like every surface
  generated by lines is called a _ruled_ surface. This surface clearly
  contains the bases of the two rows.

  If the points in either row be joined to the base of the other, we
  obtain two axial pencils which are also projective, those planes being
  corresponding which pass through corresponding points in the given
  rows. If A', A be two corresponding points, [alpha], [alpha]' the
  planes in the axial pencils passing through them, then AA' will be the
  line of intersection of the corresponding planes [alpha], [alpha]' and
  also the line joining corresponding points in the rows.

  If we cut the whole figure by a plane this will cut the axial pencils
  in two projective flat pencils, and the curve of the second order
  generated by these will be the curve in which the plane cuts the
  surface. Hence

  _The locus of lines joining corresponding points in two projective
  rows which do not lie in the same plane is a surface which contains
  the bases of the rows, and which can also be generated by the lines of
  intersection of corresponding planes in two projective axial pencils.
  This surface is cut by every plane in a curve of the second order,
  hence either in a conic or in a line-pair. No line which does not lie
  altogether on the surface can have more than two points in common with
  the surface, which is therefore said to be of the second order or is
  called a ruled quadric surface._

  That no line which does not lie on the surface can cut the surface in
  more than two points is seen at once if a plane be drawn through the
  line, for this will cut the surface in a conic. It follows also that a
  line which contains more than two points of the surface lies
  altogether on the surface.

  § 90. Through any point in space one line can always be drawn cutting
  two given lines which do not themselves meet.

  If therefore three lines in space be given of which no two meet, then
  through every point in either one line may be drawn cutting the other
  two.

  _If a line moves so that it always cuts three given lines of which no
  two meet, then it generates a ruled quadric surface._

  Let a, b, c be the given lines, and p, q, r ... lines cutting them in
  the points A, A', A" ...; B, B', B" ...; C, C', C" ... respectively;
  then the planes through a containing p, q, r, and the planes through b
  containing the same lines, may be taken as corresponding planes in two
  axial pencils which are projective, because both pencils cut the line
  c in the same row, C, C', C" ...; the surface can therefore be
  generated by projective axial pencils.

  Of the lines p, q, r ... no two can meet, for otherwise the lines a,
  b, c which cut them would also lie in their plane. There is a single
  infinite number of them, for one passes through each point of a. These
  lines are said to form a set of lines on the surface.

  If now three of the lines p, q, r be taken, then every line d cutting
  them will have three points in common with the surface, and will
  therefore lie altogether on it. This gives rise to a second set of
  lines on the surface. From what has been said the theorem follows:

  _A ruled quadric surface contains two sets of straight lines. Every
  line of one set cuts every line of the other, but no two lines of the
  same set meet._

  _Any two lines of the same set may be taken as bases of two projective
  rows, or of two projective pencils which generate the surface. They
  are cut by the lines of the other set in two projective rows._

  The plane at infinity like every other plane cuts the surface either
  in a conic proper or in a line-pair. In the first case the surface is
  called an _Hyperboloid of one sheet_, in the second an _Hyperbolic
  Paraboloid_.

  The latter may be generated by a line cutting three lines of which one
  lies at infinity, that is, cutting two lines and remaining parallel to
  a given plane.


  QUADRIC SURFACES

  § 91. The conics, the cones of the second order, and the ruled quadric
  surfaces complete the figures which can be generated by projective
  rows or flat and axial pencils, that is, by those aggregates of
  elements which are of one dimension (§§ 5, 6). We shall now consider
  the simpler figures which are generated by aggregates of two
  dimensions. The space at our disposal will not, however, allow us to
  do more than indicate a few of the results.

  § 92. We establish a correspondence between the lines and planes in
  pencils in space, or reciprocally between the points and lines in two
  or more planes, but consider principally pencils.

  In two pencils we may either make planes correspond to planes and
  lines to lines, or else planes to lines and lines to planes. If hereby
  the condition be satisfied that to a flat, or axial, pencil
  corresponds in the first case a projective flat, or axial, pencil, and
  in the second a projective axial, or flat, pencil, the pencils are
  said to be _projective_ in the first case and _reciprocal_ in the
  second.

  For instance, two pencils which join two points S1 and S2 to the
  different points and lines in a given plane [pi] are projective (and
  in perspective position), if those lines and planes be taken as
  corresponding which meet the plane [pi] in the same point or in the
  same line. In this case every plane through both centres S1 and S2 of
  the two pencils will correspond to itself. If these pencils are
  brought into any other position they will be projective (but not
  perspective).

  _The correspondence between two projective pencils is uniquely
  determined, if to four rays (or planes) in the one the corresponding
  rays (or planes) in the other are given, provided that no three rays
  of either set lie in a plane._

  Let a, b, c, d be four rays in the one, a', b', c', d' the
  corresponding rays in the other pencil. We shall show that we can find
  for every ray e in the first a single corresponding ray e' in the
  second. To the axial pencil a (b, c, d ...) formed by the planes which
  join a to b, c, d ..., respectively corresponds the axial pencil a'
  (b', c', d' ... ), and this correspondence is determined. Hence, the
  plane a'e' which corresponds to the plane ae is determined. Similarly
  the plane b'e' may be found and both together determine the ray e'.

  Similarly the correspondence between two reciprocal pencils is
  determined if for four rays in the one the corresponding planes in the
  other are given.

  § 93. We may now combine--

  1. Two reciprocal pencils.

    Each ray cuts its corresponding plane in a point, the locus of these
    points is a quadric surface.

  2. Two projective pencils.

    Each plane cuts its corresponding plane in a line, but a ray as a
    rule does not cut its corresponding ray. The locus of points where a
    ray cuts its corresponding ray is a twisted cubic. The lines where a
    plane cuts its corresponding plane are secants.

  3. Three projective pencils.

    The locus of intersection of corresponding planes is a cubic
    surface.

  Of these we consider only the first two cases.

  § 94. If two pencils are reciprocal, then to a plane in either
  corresponds a line in the other, to a flat pencil an axial pencil, and
  so on. Every line cuts its corresponding plane in a point. If S1 and
  S2 be the centres of the two pencils, and P be a point where a line a1
  in the first cuts its corresponding plane [alpha]2, _then the line b2
  in the pencil S2 which passes through P will meet its corresponding
  plane ß1 in P_. For b2 is a line in the plane [alpha]2. The
  corresponding plane ß1 must therefore pass through the line a1, hence
  through P.

  The points in which the lines in S1 cut the planes corresponding to
  them in S2 are therefore the same as the points in which the lines in
  S2 cut the planes corresponding to them in S1.

  _The locus of these points is a surface which is cut by a plane in a
  conic or in a line-pair and by a line in not more than two points
  unless it lies altogether on the surface. The surface itself is
  therefore called a quadric surface, or a surface of the second order._

  To prove this we consider any line p in space.

  The flat pencil in S1 which lies in the plane drawn through p and the
  corresponding axial pencil in S2 determine on p two projective rows,
  and those points in these which coincide with their corresponding
  points lie on the surface. But there exist only two, or one, or no
  such points, unless every point coincides with its corresponding
  point. In the latter case the line lies altogether on the surface.

  This proves also that a plane cuts the surface in a curve of the
  second order, as no line can have more than two points in common with
  it. To show that this is a curve of the same kind as those considered
  before, we have to show that it can be generated by projective flat
  pencils. We prove first that this is true for any plane through the
  centre of one of the pencils, and afterwards that every point on the
  surface may be taken as the centre of such pencil. Let then [alpha]1
  be a plane through S1. To the flat pencil in S1 which it contains
  corresponds in S2 a projective axial pencil with axis a2 and this cuts
  [alpha]1 in a second flat pencil. These two flat pencils in [alpha]1
  are projective, and, in general, neither concentric nor perspective.
  They generate therefore a conic. But if the line a2 passes through S1
  the pencils will have S1 as common centre, and may therefore have two,
  or one, or no lines united with their corresponding lines. The section
  of the surface by the plane [alpha]1 will be accordingly a line-pair
  or a single line, or else the plane [alpha]1 will have only the point
  S1 in common with the surface.

  Every line l1 through S1 cuts the surface in two points, viz. first in
  S1 and then at the point where it cuts its corresponding plane. If now
  the corresponding plane passes through S1, as in the case just
  considered, then the two points where l1 cuts the surface coincide at
  S1, and the line is called a tangent to the surface with S1 as point
  of contact. Hence if l1 be a tangent, it lies in that plane [tau]1
  which corresponds to the line S2S1 as a line in the pencil S2. The
  section of this plane has just been considered. It follows that--

  _All tangents to quadric surface at the centre of one of the
  reciprocal pencils lie in a plane which is called the tangent plane to
  the surface at that point as point of contact._

  _To the line joining the centres of the two pencils as a line in one
  corresponds in the other the tangent plane at its centre._

  _The tangent plane to a quadric surface either cuts the surface in two
  lines, or it has only a single line, or else only a single point in
  common with the surface._

  _In the first case the point of contact is said to be hyperbolic, in
  the second parabolic, in the third elliptic._

  § 95. It remains to be proved that every point S on the surface may be
  taken as centre of one of the pencils which generate the surface. Let
  S be any point on the surface [Phi]' generated by the reciprocal
  pencils S1 and S2. We have to establish a reciprocal correspondence
  between the pencils S and S1, so that the surface generated by them is
  identical with [Phi]. To do this we draw two planes [alpha]1 and ß1
  through S1, cutting the surface [Phi] in two conics which we also
  denote by [alpha]1 and ß1. These conics meet at S1, and at some other
  point T where the line of intersection of [alpha]1 and ß1 cuts the
  surface.

  In the pencil S we draw some plane [sigma] which passes through T, but
  not through S1 or S2. It will cut the two conics first at T, and
  therefore each at some other point which we call A and B respectively.
  These we join to S by lines a and b, and now establish the required
  correspondence between the pencils S1 and S as follows:--To S1T shall
  correspond the plane [sigma], to the plane [alpha]1 the line a, and to
  ß1 the line b, hence to the flat pencil in [alpha]1 the axial pencil
  a. These pencils are made projective by aid of the conic in [alpha]1.

  In the same manner the flat pencil in ß1 is made projective to the
  axial pencil b by aid of the conic in ß1, corresponding elements being
  those which meet on the conic. This determines the correspondence, for
  we know for more than four rays in S1 the corresponding planes in S.
  The two pencils S and S1 thus made reciprocal generate a quadric
  surface [Phi]', which passes through the point S and through the two
  conics [alpha]1 and ß1.

  The two surfaces [Phi] and [Phi]' have therefore the points S and S1
  and the conics [alpha]1 and ß1 in common. To show that they are
  identical, we draw a plane through S and S2, cutting each of the
  conics [alpha]1 and ß1 in two points, which will always be possible.
  This plane cuts [Phi] and [Phi]' in two conics which have the point S
  and the points where it cuts [alpha]1 and ß1 in common, that is five
  points in all. The conics therefore coincide.

  This proves that all those points P on [Phi]' lie on [Phi] which have
  the property that the plane SS2P cuts the conics [alpha]1, ß1 in two
  points each. If the plane SS2P has not this property, then we draw a
  plane SS1P. This cuts each surface in a conic, and these conics have
  in common the points S, S1, one point on each of the conics [alpha]1,
  ß1, and one point on one of the conics through S and S2 which lie on
  both surfaces, hence five points. They are therefore coincident, and
  our theorem is proved.

  § 96. The following propositions follow:--

  _A quadric surface has at every point a tangent plane._

  _Every plane section of a quadric surface is a conic or a line-pair._

  _Every line which has three points in common with a quadric surface
  lies on the surface._

  _Every conic which has five points in common with a quadric surface
  lies on the surface._

  _Through two conics which lie in different planes, but have two points
  in common, and through one external point always one quadric surface
  may be drawn._

  § 97. _Every plane which cuts a quadric surface in a line-pair is a
  tangent plane._ For every line in this plane through the centre of the
  line-pair (the point of intersection of the two lines) cuts the
  surface in two coincident points and is therefore a tangent to the
  surface, _the centre of the line-pair being the point of contact_.

  _If a quadric surface contains a line, then every plane through this
  line cuts the surface in a line-pair (or in two coincident lines)._
  For this plane cannot cut the surface in a conic. Hence:--

  _If a quadric surface contains one line p then it contains an infinite
  number of lines, and through every point Q on the surface, one line q
  can be drawn which cuts p._ For the plane through the point Q and the
  line p cuts the surface in a line-pair which must pass through Q and
  of which p is one line.

  _No two such lines q on the surface can meet_. For as both meet p
  their plane would contain p and therefore cut the surface in a
  triangle.

  _Every line which cuts three lines q will be on the surface_; for it
  has three points in common with it.

  _Hence the quadric surfaces which contain lines are the same as the
  ruled quadric surfaces considered in_ §§ 89-93, but with one important
  exception. In the last investigation we have left out of consideration
  the possibility of a plane having only one line (two coincident lines)
  in common with a quadric surface.

  § 98. To investigate this case we suppose first that there is one
  point A on the surface through which two different lines a, b can be
  drawn, which lie altogether on the surface.

  If P is any other point on the surface which lies neither on a nor b,
  then the plane through P and a will cut the surface in a second line
  a' which passes through P and which cuts a. Similarly there is a line
  b' through P which cuts b. These two lines a' and b' _may_ coincide,
  but then they must coincide with PA.

  If this happens for one point P, it happens for every other point Q.
  For if two different lines could be drawn through Q, then by the same
  reasoning the line PQ would be altogether on the surface, hence two
  lines would be drawn through P against the assumption. From this
  follows:--

  _If there is one point on a quadric surface through which one, but
  only one, line can be drawn on the surface, then through every point
  one line can be drawn, and all these lines meet in a point. The
  surface is a cone of the second order_.

  _If through one point on a quadric surface, two, and only two, lines
  can be drawn on the surface, then through every point two lines may be
  drawn, and the surface is ruled quadric surface._

  _If through one point on a quadric surface no line on the surface can
  be drawn, then the surface contains no lines._

  Using the definitions at the end of § 95, we may also say:--

  _On a quadric surface the points are all hyperbolic, or all parabolic,
  or all elliptic._

  As an example of a quadric surface with elliptical points, we mention
  the sphere which may be generated by two reciprocal pencils, where to
  each line in one corresponds the plane perpendicular to it in the
  other.

  § 99. _Poles and Polar Planes._--The theory of poles and polars with
  regard to a conic is easily extended to quadric surfaces.

  Let P be a point in space not on the surface, which we suppose not to
  be a cone. On every line through P which cuts the surface in two
  points we determine the harmonic conjugate Q of P with regard to the
  points of intersection. Through one of these lines we draw two planes
  [alpha] and ß. The locus of the points Q in [alpha] is a line a, the
  polar of P with regard to the conic in which [alpha] cuts the surface.
  Similarly the locus of points Q in ß is a line b. This cuts a, because
  the line of intersection of [alpha] and ß contains but one point Q.
  The locus of all points Q therefore is a plane. _This plane is called
  the polar plane of the point P, with regard to the quadric surface. If
  P lies on the surface we take the tangent plane of P as its polar._

  The following propositions hold:--

  1. _Every point has a polar plane_, which is constructed by drawing
  the polars of the point with regard to the conics in which two planes
  through the point cut the surface.

  2. _If Q is a point in the polar of P, then P is a point in the polar
  of Q_, because this is true with regard to the conic in which a plane
  through PQ cuts the surface.

  3. _Every plane is the polar plane of one point, which is called the
  Pole of the plane._

  The pole to a plane is found by constructing the polar planes of three
  points in the plane. Their intersection will be the pole.

  4. _The points in which the polar plane of P cuts the surface are
  points of contact of tangents drawn from P to the surface_, as is
  easily seen. Hence:--

  5. _The tangents drawn from a point P to a quadric surface form a cone
  of the second order_, for the polar plane of P cuts it in a conic.

  6. _If the pole describes a line a, its polar plane will turn about
  another line a'_, as follows from 2. _These lines a and a' are said to
  be conjugate with regard to the surface._

  § 100. The pole of the line at infinity is called the _centre_ of the
  surface. If it lies at the infinity, the plane at infinity is a
  tangent plane, and the surface is called a _paraboloid_.

  _The polar plane to any point at infinity passes through the centre,
  and is called a diametrical plane._

  _A line through the centre is called a diameter. It is bisected at the
  centre. The line conjugate to it lies at infinity._

  _If a point moves along a diameter its polar plane turns about the
  conjugate line at infinity_; that is, _it moves parallel to itself,
  its centre moving on the first line._

  _The middle points of parallel chords lie in a plane_, viz. in the
  polar plane of the point at infinity through which the chords are
  drawn.

  _The centres of parallel sections lie in a diameter which is a line
  conjugate to the line at infinity in which the planes meet._


  TWISTED CUBICS

  § 101. If two pencils with centres S1 and S2 are made projective, then
  to a ray in one corresponds a ray in the other, to a plane a plane, to
  a flat or axial pencil a projective flat or axial pencil, and so on.

  There is a double infinite number of lines in a pencil. We shall see
  that a single infinite number of lines in one pencil meets its
  corresponding ray, and that the points of intersection form a curve in
  space.

  Of the double infinite number of planes in the pencils each will meet
  its corresponding plane. This gives a system of a double infinite
  number of lines in space. We know (§ 5) that there is a quadruple
  infinite number of lines in space. From among these we may select
  those which satisfy one or more given conditions. The systems of lines
  thus obtained were first systematically investigated and classified by
  Plücker, in his _Geometrie des Raumes_. He uses the following names:--

  A _treble infinite_ number of lines, that is, all lines which satisfy
  one condition, are said to form a _complex of lines_; e.g. all lines
  cutting a given line, or all lines touching a surface.

  A _double infinite_ number of lines, that is, all lines which satisfy
  two conditions, or which are common to two complexes, are said to form
  a _congruence of lines_; e.g. all lines in a plane, or all lines
  cutting two curves, or all lines cutting a given curve twice.

  A _single infinite_ number of lines, that is, all lines which satisfy
  three conditions, or which belong to three complexes, form a _ruled
  surface_; e.g. one set of lines on a ruled quadric surface, or
  developable surfaces which are formed by the tangents to a curve.

  It follows that all lines in which corresponding planes in two
  projective pencils meet form a congruence. We shall see this
  congruence consists of all lines which cut a twisted cubic twice, or
  of all _secants_ to a twisted cubic.

  § 102. Let l1 be the line S1S2 as a line in the pencil S1. To it
  corresponds a line l2 in S2. _At each of the centres two corresponding
  lines meet._ The two axial pencils with l1 and l2 as axes are
  projective, and, as, their axes meet at S2, the intersections of
  corresponding planes form a cone of the second order (§ 58), with S2
  as centre. If [pi]1 and [pi]2 be corresponding planes, then their
  intersection will be a line p2 which passes through S2. Corresponding
  to it in S1 will be a line p1 which lies in the plane [pi]1, and which
  therefore meets p2 at some point P. Conversely, if p2 be any line in
  S2 which meets its corresponding line p1 at a point P, then to the
  plane l2p2 will correspond the plane l1p1, that is, the plane S1S2P.
  These planes intersect in p2, so that p2 is a line on the quadric cone
  generated by the axial pencils l1 and l2. Hence:--

  _All lines in one pencil which meet their corresponding lines in the
  other form a cone of the second order which has its centre at the
  centre of the first pencil, and passes through the centre of the
  second._

  From this follows that the points in which corresponding rays meet lie
  on two cones of the second order which have the ray joining their
  centres in common, and form therefore, together with the line S1S2 or
  l1, the intersection of these cones. Any plane cuts each of the cones
  in a conic. These two conics have necessarily that point in common in
  which it cuts the line l1, and therefore besides either one or three
  other points. It follows that the curve is of the third order as a
  plane may cut it in three, but not in more than three, points.
  Hence:--

  _The locus of points in which corresponding lines on two projective
  pencils meet is a curve of the third order or a "twisted cubic" k,
  which passes through the centres of the pencils, and which appears as
  the intersection of two cones of the second order, which have one line
  in common._

  _A line belonging to the congruence determined by the pencils is a
  secant of the cubic; it has two, or one, or no points in common with
  this cubic, and is called accordingly a secant proper, a tangent, or a
  secant improper of the cubic._ A secant improper may be considered, to
  use the language of coordinate geometry, as a secant with imaginary
  points of intersection.

  § 103. If a1 and a2 be any two corresponding lines in the two pencils,
  then corresponding planes in the axial pencils having a1 and a2 as
  axes generate a ruled quadric surface. If P be any point on the cubic
  k, and if p1, p2 be the corresponding rays in S1 and S2 which meet at
  P, then to the plane a1p1 in S1 corresponds a2p2 in S2. These
  therefore meet in a line through P.

  This may be stated thus:--

  _Those secants of the cubic which cut a ray a1, drawn through the
  centre S1 of one pencil, form a ruled quadric surface which passes
  through both centres, and which contains the twisted cubic k. Of such
  surfaces an infinite number exists. Every ray through S1 or S2 which
  is not a secant determines one of them._

  If, however, the rays a1 and a2 are secants meeting at A, then the
  ruled quadric surface becomes a cone of the second order, having A as
  centre. Or _all lines of the congruence which pass through a point on
  the twisted cubic k form a cone of the second order_. In other words,
  the projection of a twisted cubic from any point in the curve on to
  any plane is a conic.

  If a1 is not a secant, but made to pass through any point Q in space,
  the ruled quadric surface determined by a1 will pass through Q. _There
  will therefore be one line of the congruence passing through Q, and
  only one._ For if two such lines pass through Q, then the lines S1Q
  and S2Q will be corresponding lines; hence Q will be a point on the
  cubic k, and an infinite number of secants will pass through it.
  Hence:--

  _Through every point in space not on the twisted cubic one and only
  one secant to the cubic can be drawn._

  § 104. The fact that all the secants through a point on the cubic form
  a quadric cone shows that the centres of the projective pencils
  generating the cubic are not distinguished from any other points on
  the cubic. If we take any two points S, S' on the cubic, and draw the
  secants through each of them, we obtain two quadric cones, which have
  the line SS' in common, and which intersect besides along the cubic.
  If we make these two pencils having S and S' as centres projective by
  taking four rays on the one cone as corresponding to the four rays on
  the other which meet the first on the cubic, the correspondence is
  determined. These two pencils will generate a cubic, and the two cones
  of secants having S and S' as centres will be identical with the above
  cones, for each has five rays in common with one of the first, viz.
  the line SS' and the four lines determined for the correspondence;
  therefore these two cones intersect in the original cubic. This gives
  the theorem:--

  _On a twisted cubic any two points may be taken as centres of
  projective pencils which generate the cubic, corresponding planes
  being those which meet on the same secant._

  Of the two projective pencils at S and S' we may keep the first fixed,
  and move the centre of the other along the curve. The pencils will
  hereby remain projective, and a plane [alpha] in S will be cut by its
  corresponding plane [alpha]' always in the same secant a. Whilst S'
  moves along the curve the plane [alpha]' will turn about a, describing
  an axial pencil.

  AUTHORITIES.--In this article we have given a purely geometrical
  theory of conics, cones of the second order, quadric surfaces, &c. In
  doing so we have followed, to a great extent, Reye's _Geometrie der
  Lage_, and to this excellent work those readers are referred who wish
  for a more exhaustive treatment of the subject. Other works especially
  valuable as showing the development of the subject are: Monge,
  _Géométrie descriptive_: Carnot, _Géométrie de position_ (1803),
  containing a theory of transversals; Poncelet's great work _Traité des
  propriétés projectives des figures_ (1822); Möbins, _Barycentrischer
  Calcul_ (1826); Steiner, _Abhängigkeit geometrischer Gestalten_
  (1832), containing the first full discussion of the projective
  relations between rows, pencils, &c.; Von Staudt, _Geometrie der Lage_
  (1847) and _Beiträge zur Geometrie der Lage_ (1856-1860), in which a
  system of geometry is built up from the beginning without any
  reference to number, so that ultimately a number itself gets a
  geometrical definition, and in which imaginary elements are
  systematically introduced into pure geometry; Chasles, _Aperçu
  historique_ (1837), in which the author gives a brilliant account of
  the progress of modern geometrical methods, pointing out the
  advantages of the different purely geometrical methods as compared
  with the analytical ones, but without taking as much account of the
  German as of the French authors; Id., _Rapport sur les progrès de la
  géométrie_ (1870), a continuation of the _Aperçu_; Id., _Traité de
  géométrie supérieure_ (1852); Cremona, _Introduzione ad una teoria
  geometrica delle curve piane_ (1862) and its continuation _Preliminari
  di una teoria geometrica delle superficie_ (German translations by
  Curtze). As more elementary books, we mention: Cremona, _Elements of
  Projective Geometry_, translated from the Italian by C. Leudesdorf
  (2nd ed., 1894); J.W. Russell, _Pure Geometry_ (2nd ed., 1905).
       (O. H.)


III. DESCRIPTIVE GEOMETRY

This branch of geometry is concerned with the methods for representing
solids and other figures in three dimensions by drawings in one plane.
The most important method is that which was invented by Monge towards
the end of the 18th century. It is based on parallel projections to a
plane by rays perpendicular to the plane. Such a projection is called
orthographic (see PROJECTION, § 18). If the plane is horizontal the
projection is called the plan of the figure, and if the plane is
vertical the elevation. In Monge's method a figure is represented by its
plan and elevation. It is therefore often called drawing in plan and
elevation, and sometimes simply orthographic projection.

  § 1. We suppose then that we have two planes, one horizontal, the
  other vertical, and these we call the planes of plan and of elevation
  respectively, or the horizontal and the vertical plane, and denote
  them by the letters [pi]1 and [pi]2. Their line of intersection is
  called the axis, and will be denoted by xy.

  If the surface of the drawing paper is taken as the plane of the plan,
  then the vertical plane will be the plane perpendicular to it through
  the axis xy. To bring this also into the plane of the drawing paper we
  turn it about the axis till it coincides with the horizontal plane.
  This process of turning one plane down till it coincides with another
  is called _rabatting_ one to the other. Of course there is no
  necessity to have one of the two planes horizontal, but even when this
  is not the case it is convenient to retain the above names.

  [Illustration: FIG. 37.]

  [Illustration: FIG. 38.]

  The whole arrangement will be better understood by referring to fig.
  37. A point A in space is there projected by the perpendicular AA1 and
  AA2 to the planes [pi]1 and [pi]2 so that A1 and A2 are the horizontal
  and vertical projections of A.

  If we remember that a line is perpendicular to a plane that is
  perpendicular to every line in the plane if only it is perpendicular
  to any two intersecting lines in the plane, we see that the axis which
  is perpendicular both to AA1 and to AA2 is also perpendicular to A1A0
  and to A2A0 because these four lines are all in the same plane. Hence,
  if the plane [pi]2 be turned about the axis till it coincides with the
  plane [pi]1, then A2A0 will be the continuation of A1A0. This position
  of the planes is represented in fig. 38, in which the line A1A2 is
  perpendicular to the axis x.

  Conversely any two points A1, A2 in a line perpendicular to the axis
  will be the projections of some point in space when the plane [pi]2 is
  turned about the axis till it is perpendicular to the plane [pi]1,
  because in this position the two perpendiculars to the planes [pi]1
  and [pi]2 through the points A1 and A2 will be in a plane and
  therefore meet at some point A.

  _Representation of Points._--We have thus the following method of
  representing in a single plane the position of points in space:--_we
  take in the plane a line xy as the axis, and then any pair of points
  A1, A2 in the plane on a line perpendicular to the axis represent a
  point A in space_. If the line A1A2 cuts the axis at A0, and if at A1
  a perpendicular be erected to the plane, then the point A will be in
  it at a height A1A = A0A2 above the plane. This gives the position of
  the point A relative to the plane [pi]1. In the same way, if in a
  perpendicular to [pi]2 through A2 a point A be taken such that A2A =
  A0A1, then this will give the point A relative to the plane [pi]2.

  [Illustration: FIG. 39.]

  § 2. The two planes [pi]1, [pi]2 in their original position divide
  space into four parts. These are called the four quadrants. We suppose
  that the plane [pi]2 is turned as indicated in fig. 37, so that the
  point P comes to Q and R to S, then the quadrant in which the point A
  lies is called the first, and we say that in the first quadrant a
  point lies above the horizontal and in front of the vertical plane.
  Now we go round the axis in the sense in which the plane [pi]2 is
  turned and come in succession to the second, third and fourth
  quadrant. In the second a point lies above the plane of the plan and
  behind the plane of elevation, and so on. In fig. 39, which represents
  a side view of the planes in fig. 37 the quadrants are marked, and in
  each a point with its projection is taken. Fig. 38 shows how these are
  represented when the plane [pi]2 is turned down. We see that

  _A point lies in the first quadrant if the plan lies below, the
  elevation above the axis; in the second if plan and elevation both lie
  above; in the third if the plan lies above, the elevation below; in
  the fourth if plan and elevation both lie below the axis._

  _If a point lies in the horizontal plane_, its elevation lies in the
  axis and the plan coincides with the point itself. _If a point lies in
  the vertical plane_, its plan lies in the axis and the elevation
  coincides with the point itself. _If a point lies in the axis_, both
  its plan and elevation lie in the axis and coincide with it.

  Of each of these propositions, which will easily be seen to be true,
  the converse holds also.

  § 3. _Representation of a Plane._--As we are thus enabled to represent
  points in a plane, we can represent any finite figure by representing
  its separate points. It is, however, not possible to represent a plane
  in this way, for the projections of its points completely cover the
  planes [pi]1 and [pi]2, and no plane would appear different from any
  other. But any plane [alpha] cuts each of the planes [pi]1, [pi]2 in a
  line. These are called the traces of the plane. They cut each other in
  the axis at the point where the latter cuts the plane [alpha].

  _A plane is determined by its two traces, which are two lines that
  meet on the axis_, and, conversely, _any two lines which meet on the
  axis determine a plane_.

  _If the plane is parallel to the axis its traces are parallel to the
  axis._ Of these one may be at infinity; then the plane will cut one of
  the planes of projection at infinity and will be parallel to it. Thus
  a plane parallel to the horizontal plane of the plan has only one
  finite trace, viz. that with the plane of elevation.

  [Illustration: FIG. 40.]

  _If the plane passes through the axis both its traces coincide with
  the axis._ This is the only case in which the representation of the
  plane by its two traces fails. A third plane of projection is
  therefore introduced, which is best taken perpendicular to the other
  two. We call it simply the third plane and denote it by [pi]3. As it
  is perpendicular to [pi]1, it may be taken as the plane of elevation,
  its line of intersection [gamma] with [pi]1 being the axis, and be
  turned down to coincide with [pi]1. This is represented in fig. 40. OC
  is the axis xy whilst OA and OB are the traces of the third plane.
  They lie in one line [gamma]. The plane is rabatted about [gamma] to
  the horizontal plane. A plane [alpha] through the axis xy will then
  show in it a trace [alpha]3. In fig. 40 the lines OC and OP will thus
  be the traces of a plane through the axis xy, which makes an angle POQ
  with the horizontal plane.

  We can also find the trace which any other plane makes with [pi]3. In
  rabatting the plane [pi]3 its trace OB with the plane [pi]2 will come
  to the position OD. Hence a plane ß having the traces CA and CB will
  have with the third plane the trace ß3, or AD if OD = OB.

  It also follows immediately that--

  _If a plane [alpha] is perpendicular to the horizontal plane, then
  every point in it has its horizontal projection in the horizontal
  trace of the plane_, as all the rays projecting these points lie in
  the plane itself.

  _Any plane which is perpendicular to the horizontal plane has its
  vertical trace perpendicular to the axis._

  _Any plane which is perpendicular to the vertical plane has its
  horizontal trace perpendicular to the axis and the vertical
  projections of all points in the plane lie in this trace._

  § 4. _Representation of a Line._--A line is determined either by two
  points in it or by two planes through it. We get accordingly two
  representations of it either by projections or by traces.

  First.--_A line a is represented by its projections a1 and a2 on the
  two planes [pi]1 and [pi]2._ These may be any two lines, for, bringing
  the planes [pi]1, [pi]2 into their original position, the planes
  through these lines perpendicular to [pi]1 and [pi]2 respectively will
  intersect in some line a which has a1, a2 as its projections.

  Secondly.--_A line a is represented by its traces--that is, by the
  points in which it cuts the two planes [pi]1, [pi]2._ Any two points
  may be taken as the traces of a line in space, for it is determined
  when the planes are in their original position as the line joining the
  two traces. This representation becomes undetermined if the two traces
  coincide in the axis. In this case we again use a third plane, or else
  the projections of the line.

  The fact that there are different methods of representing points and
  planes, and hence two methods of representing lines, suggests the
  principle of duality (section ii., _Projective Geometry_, § 41). It is
  worth while to keep this in mind. It is also worth remembering that
  traces of planes or lines always lie in the planes or lines which they
  represent. Projections do not as a rule do this excepting when the
  point or line projected lies in one of the planes of projection.

  Having now shown how to represent points, planes and lines, we have to
  state the conditions which must hold in order that these elements may
  lie one in the other, or else that the figure formed by them may
  possess certain metrical properties. It will be found that the former
  are very much simpler than the latter.

  Before we do this, however, we shall explain the notation used; for it
  is of great importance to have a systematic notation. We shall denote
  points in space by capitals A, B, C; planes in space by Greek letters
  [alpha], ß, [gamma]; lines in space by small letters a, b, c;
  horizontal projections by suffixes 1, like A1, a1; vertical
  projections by suffixes 2, like A2, a2; traces by single and double
  dashes [alpha]' [alpha]", a', a". Hence P1 will be the horizontal
  projection of a point P in space; a line a will have the projections
  a1, a2 and the traces a' and a"; a plane [alpha] has the traces
  [alpha]' and [alpha]".

  § 5. _If a point lies in a line, the projections of the point lie in
  the projections of the line._

  _If a line lies in a plane, the traces of the line lie in the traces
  of the plane._

  These propositions follow at once from the definitions of the
  projections and of the traces.

  If a point lies in two lines its projections must lie in the
  projections of both. Hence

  _If two lines, given by their projections, intersect, the intersection
  of their planes and the intersection of their elevations must lie in a
  line perpendicular to the axis_, because they must be the projections
  of the point common to the two lines.

  Similarly--_If two lines given by their traces lie in the same plane
  or intersect, then the lines joining their horizontal and vertical
  traces respectively must meet on the axis_, because they must be the
  traces of the plane through them.

  § 6. _To find the projections of a line which joins two points A, B
  given by their projections A1, A2 and B1, B2_, we join A1, B1 and A2,
  B2; these will be the projections required. For example, the traces of
  a line are two points in the line whose projections are known or at
  all events easily found. They are the traces themselves and the feet
  of the perpendiculars from them to the axis.

  Hence _if a' a" (fig. 41) are the traces of a line a, and if the
  perpendiculars from them cut the axis in P and Q respectively, then
  the line a'Q will be the horizontal and a"P the vertical projection of
  the line_.

  [Illustration: FIG. 41.]

  Conversely, if the projections a1, a2 of a line are given, and if
  these cut the axis in Q and P respectively, then _the perpendiculars
  Pa' and Qa" to the axis drawn through these points cut the projections
  a1 and a2 in the traces a' and a"_.

  _To find the line of intersection of two planes_, we observe that this
  line lies in both planes; its traces must therefore lie in the traces
  of both. Hence the points where the horizontal traces of the given
  planes meet will be the horizontal, and the point where the vertical
  traces meet the vertical trace of the line required.

  § 7. _To decide whether a point A, given by its projections, lies in a
  plane [alpha], given by its traces_, we draw a line p by joining A to
  some point in the plane [alpha] and determine its traces. If these lie
  in the traces of the plane, then the line, and therefore the point A,
  lies in the plane; otherwise not. This is conveniently done by joining
  A1 to some point p' in the trace [alpha]'; this gives p1; and the
  point where the perpendicular from p' to the axis cuts the latter we
  join to A2; this gives p2. If the vertical trace of this line lies in
  the vertical trace of the plane, then, and then only, does the line p,
  and with it the point A, lie in the plane [alpha].

  § 8. _Parallel planes have parallel traces_, because parallel planes
  are cut by any plane, hence also by [pi]1 and by [pi]2, in parallel
  lines.

  _Parallel lines have parallel projections_, because points at infinity
  are projected to infinity.

  _If a line is parallel to a plane, then lines through the traces of
  the line and parallel to the traces of the plane must meet on the
  axis_, because these lines are the traces of a plane parallel to the
  given plane.

  § 9. _To draw a plane through two intersecting lines or through two
  parallel lines_, we determine the traces of the lines; the lines
  joining their horizontal and vertical traces respectively will be the
  horizontal and vertical traces of the plane. They will meet, at a
  finite point or at infinity, on the axis if the lines do intersect.

  _To draw a plane through a line and a point without the line_, we join
  the given point to any point in the line and determine the plane
  through this and the given line.

  _To draw a plane through three points which are not in a line_, we
  draw two of the lines which each join two of the given points and draw
  the plane through them. If the traces of all three lines AB, BC, CA be
  found, these must lie in two lines which meet on the axis.

  § 10. We have in the last example got more points, or can easily get
  more points, than are necessary for the determination of the figure
  required--in this case the traces of the plane. This will happen in a
  great many constructions and is of considerable importance. It may
  happen that some of the points or lines obtained are not convenient in
  the actual construction. The horizontal traces of the lines AB and AC
  may, for instance, fall very near together, in which case the line
  joining them is not well defined. Or, one or both of them may fall
  beyond the drawing paper, so that they are practically non-existent
  for the construction. In this case the traces of the line BC may be
  used. Or, if the vertical traces of AB and AC are both in convenient
  position, so that the vertical trace of the required plane is found
  and one of the horizontal traces is got, then we may join the latter
  to the point where the vertical trace cuts the axis.

  The draughtsman must remember that the lines which he draws are not
  mathematical lines without thickness, and therefore every drawing is
  affected by some errors. It is therefore very desirable to be able
  constantly to check the latter. Such checks always present themselves
  when the same result can be obtained by different constructions, or
  when, as in the above case, some lines must meet on the axis, or if
  three points must lie in a line. A careful draughtsman will always
  avail himself of these checks.

  § 11. _To draw a plane through a given point parallel to a given plane
  [alpha]_, we draw through the point two lines which are parallel to
  the plane [alpha], and determine the plane through them; or, as we
  know that the traces of the required plane are parallel to those of
  the given one (§ 8), we need only draw one line l through the point
  parallel to the plane and find one of its traces, say the vertical
  trace l"; a line through this parallel to the vertical trace of
  [alpha] will be the vertical trace ß" of the required plane ß, and a
  line parallel to the horizontal trace of [alpha] meeting ß" on the
  axis will be the horizontal trace ß'.

  [Illustration: FIG. 42.]

  Let A1 A2 (fig. 42) be the given point, [alpha]' [alpha]" the given
  plane, a line l1 through A1, parallel to [alpha]' and a horizontal
  line l2 through A2 will be the projections of a line l through A
  parallel to the plane, because the horizontal plane through this line
  will cut the plane [alpha] in a line c which has its horizontal
  projection c1 parallel to [alpha]'.

  § 12. We now come to the metrical properties of figures.

  _A line is perpendicular to a plane if the projections of the line are
  perpendicular to the traces of the plane._ We prove it for the
  horizontal projection. If a line p is perpendicular to a plane
  [alpha], every plane through p is perpendicular to [alpha]; hence also
  the vertical plane which projects the line p to p1. As this plane is
  perpendicular both to the horizontal plane and to the plane [alpha],
  it is also perpendicular to their intersection--that is, to the
  horizontal trace of [alpha]. It follows that every line in this
  projecting plane, therefore also p1, the plan of p, is perpendicular
  to the horizontal trace of [alpha].

  _To draw a plane through a given point A perpendicular to a given line
  p_, we first draw through some point O in the axis lines [gamma]',
  [gamma]" perpendicular respectively to the projections p1 and p2 of
  the given line. These will be the traces of a plane [gamma] which is
  perpendicular to the given line. We next draw through the given point
  A a plane parallel to the plane [gamma]; this will be the plane
  required.

  Other metrical properties depend on the determination of the real size
  or shape of a figure.

  In general the projection of a figure differs both in size and shape
  from the figure itself. But figures in a plane parallel to a plane of
  projection will be identical with their projections, and will thus be
  given in their true dimensions. In other cases there is the problem,
  constantly recurring, either to find the true shape and size of a
  plane figure when plan and elevation are given, or, conversely, to
  find the latter from the known true shape of the figure itself. To do
  this, the plane is turned about one of its traces till it is laid down
  into that plane of projection to which the trace belongs. This is
  technically called rabatting the plane respectively into the plane of
  the plan or the elevation. As there is no difference in the treatment
  of the two cases, we shall consider only the case of rabatting a plane
  [alpha] into the plane of the plan. The plan of the figure is a
  parallel (orthographic) projection of the figure itself. The results
  of parallel projection (see PROJECTION, §§ 17 and 18) may therefore
  now be used. The trace [alpha]' will hereby take the place of what
  formerly was called the axis of projection. Hence we see that
  corresponding points in the plan and in the rabatted plane are joined
  by lines which are perpendicular to the trace [alpha]' and that
  corresponding lines meet on this trace. We also see that the
  correspondence is completely determined if we know for one point or
  one line in the plan the corresponding point or line in the rabatted
  plane.

  Before, however, we treat of this we consider some special cases.

  § 13. _To determine the distance between two points A, B given by
  their projections A1, B1 and A2, B2, or, in other words, to determine
  the true length of a line the plan and elevation of which are given._

  [Illustration: FIG. 43.]

  _Solution._--The two points A, B in space lie vertically above their
  plans A1, B1 (fig. 43) and A1A = A0A2, B1B = B0B2. The four points A,
  B, A1, B1 therefore form a plane quadrilateral on the base A1B1 and
  having right angles at the base. This plane we rabatt about A1B1 by
  drawing A1A and B1B perpendicular to A1B1 and making A1A = A0A2, B1B =
  B0B2. Then AB will give the length required.

  The construction might have been performed in the elevation by making
  A2A = A0A1 and B2B = B0B1 on lines perpendicular to A2B2. Of course AB
  must have the same length in both cases.

  This figure may be turned into a model. Cut the paper along A1A, AB
  and BB1, and fold the piece A1ABB1 over along A1B1 till it stands
  upright at right angles to the horizontal plane. The points A, B will
  then be in their true position in space relative to [pi]1. Similarly
  if B2BAA2 be cut out and turned along A2B2 through a right angle we
  shall get AB in its true position relative to the plane [pi]2. Lastly
  we fold the whole plane of the paper along the axis x till the plane
  [pi]2 is at right angles to [pi]1. In this position the two sets of
  points AB will coincide if the drawing has been accurate.

  Models of this kind can be made in many cases and their construction
  cannot be too highly recommended in order to realize orthographic
  projection.

  § 14. _To find the angle between two given lines a, b of which the
  projections a1, b1 and a2, b2 are given._

  [Illustration: FIG. 44.]

  _Solution._--Let a1, b1 (fig. 44) meet in P1, a2, b2 in T, then if the
  line P1T is not perpendicular to the axis the two lines will not meet.
  In this case we draw a line parallel to b to meet the line a. This is
  easiest done by drawing first the line P1P2 perpendicular to the axis
  to meet a2 in P2, and then drawing through P2 a line c2 parallel to
  b2; then b1, c2 will be the projections of a line c which is parallel
  to b and meets a in P. The plane [alpha] which these two lines
  determine we rabatt to the plan. We determine the traces a' and c' of
  the lines a and c; then a'c' is the trace [alpha]' of their plane. On
  rabatting the point P comes to a point S on the line P1Q perpendicular
  to a'c', so that QS = QP. But QP is the hypotenuse of a triangle PP1Q
  with a right angle P1. This we construct by making QR = P0P2; then P1R
  = PQ. The lines a'S and c'S will therefore include angles equal to
  those made by the given lines. It is to be remembered that two lines
  include two angles which are supplementary. Which of these is to be
  taken in any special case depends upon the circumstances.

  _To determine the angle between a line and a plane_, we draw through
  any point in the line a perpendicular to the plane (§ 12) and
  determine the angle between it and the given line. The complement of
  this angle is the required one.

  _To determine the angle between two planes_, we draw through any point
  two lines perpendicular to the two planes and determine the angle
  between the latter as above.

  In special cases it is simpler to determine at once the angle between
  the two planes by taking a plane section perpendicular to the
  intersection of the two planes and rabatt this. This is especially the
  case if one of the planes is the horizontal or vertical plane of
  projection.

  Thus in fig. 45 the angle P1QR is the angle which the plane [alpha]
  makes with the horizontal plane.

  § 15. We return to the general case of rabatting a plane [alpha] of
  which the traces [alpha]' [alpha]" are given.

  [Illustration: FIG. 45.]

  Here it will be convenient to determine first the position which the
  trace [alpha]"--which is a line in [alpha]--assumes when rabatted.
  Points in this line coincide with their elevations. Hence it is given
  in its true dimension, and we can measure off along it the true
  distance between two points in it. If therefore (fig. 45) P is any
  point in [alpha]" originally coincident with its elevation P2, and if
  O is the point where [alpha]" cuts the axis xy, so that O is also in
  [alpha]', then the point P will after rabatting the plane assume such
  a position that OP = OP2. At the same time the plan is an orthographic
  projection of the plane [alpha]. Hence the line joining P to the plan
  P1 will after rabatting be perpendicular to [alpha]'. But P1 is known;
  it is the foot of the perpendicular from P2 to the axis xy. We draw
  therefore, to find P, from P1 a perpendicular P1Q to [alpha]' and find
  on it a point P such that OP = OP2. Then the line OP will be the
  position of [alpha]" when rabatted. This line corresponds therefore to
  the plan of [alpha]"--that is, to the axis xy, corresponding points on
  these lines being those which lie on a perpendicular to [alpha]'.

  We have thus one pair of corresponding lines and can now find for any
  point B1 in the plan the corresponding point B in the rabatted plane.
  We draw a line through B1, say B1P1, cutting [alpha]' in C. To it
  corresponds the line CP, and the point where this is cut by the
  projecting ray through B1, perpendicular to [alpha]', is the required
  point B.

  Similarly any figure in the rabatted plane can be found when the plan
  is known; but this is usually found in a different manner without any
  reference to the general theory of parallel projection. As this method
  and the reasoning employed for it have their peculiar advantages, we
  give it also.

  Supposing the planes [pi]1 and [pi]2 to be in their positions in space
  perpendicular to each other, we take a section of the whole figure by
  a plane perpendicular to the trace [alpha]' about which we are going
  to rabatt the plane [alpha]. Let this section pass through the point Q
  in [alpha]'. Its traces will then be the lines QP1 and P1P2 (fig. 9).
  These will be at right angles, and will therefore, together with the
  section QP2 of the plane [alpha], form a right-angled triangle QP1P2
  with the right angle at P1, and having the sides P1Q and P1P2 which
  both are given in their true lengths. This triangle we rabatt about
  its base P1Q, making P1R = P1P2. The line QR will then give the true
  length of the line QP in space. If now the plane [alpha] be turned
  about [alpha]' the point P will describe a circle about Q as centre
  with radius QP = QR, in a plane perpendicular to the trace [alpha]'.
  Hence when the plane [alpha] has been rabatted into the horizontal
  plane the point P will lie in the perpendicular P1Q to [alpha]', so
  that QP = QR.

  If A1 is the plan of a point A in the plane [alpha], and if A1 lies in
  QP1, then the point A will lie vertically above A1 in the line QP. On
  turning down the triangle QP1P2, the point A will come to A0, the line
  A1A0 being perpendicular to QP1. Hence A will be a point in QP such
  that QA = QA0.

  If B1 is the plan of another point, but such that A1B1 is parallel to
  [alpha]', then the corresponding line AB will also be parallel to
  [alpha]'. Hence, if through A a line AB be drawn parallel to [alpha]',
  and B1B perpendicular to [alpha]', then their intersection gives the
  point B. Thus of any point given in plan the real position in the
  plane [alpha], when rabatted, can be found by this second method. This
  is the one most generally given in books on geometrical drawing. The
  first method explained is, however, in most cases preferable as it
  gives the draughtsman a greater variety of constructions. It requires
  a somewhat greater amount of theoretical knowledge.

  If instead of our knowing the plan of a figure the latter is itself
  given, then the process of finding the plan is the reverse of the
  above and needs little explanation. We give an example.

  § 16. _It is required to draw the plan and elevation of a polygon of
  which the real shape and position in a given plane [alpha] are known._

  We first rabatt the plane [alpha] (fig. 46) as before so that P1 comes
  to P, hence OP1 to OP. Let the given polygon in [alpha] be the figure
  ABCDE. We project, not the vertices, but the sides. To project the
  line AB, we produce it to cut [alpha]' in F and OP in G, and draw GG1
  perpendicular to [alpha]'; then G1 corresponds to G, therefore FG1 to
  FG. In the same manner we might project all the other sides, at least
  those which cut OF and OP in convenient points. It will be best,
  however, first to produce all the sides to cut OP and [alpha]' and
  then to draw all the projecting rays through A, B, C ... perpendicular
  to [alpha]', and in the same direction the lines G, G1, &c. By drawing
  FG we get the points A1, B1 on the projecting ray through A and B. We
  then join B to the point M where BC produced meets the trace [alpha]'.
  This gives C1. So we go on till we have found E1. The line A1 E1 must
  then meet AE in [alpha]', and this gives a check. If one of the sides
  cuts [alpha]' or OP beyond the drawing paper this method fails, but
  then we may easily find the projection of some other line, say of a
  diagonal, or directly the projection of a point, by the former
  methods. The diagonals may also serve to check the drawing, for two
  corresponding diagonals must meet in the trace [alpha]'.

  [Illustration: FIG. 46.]

  Having got the plan we easily find the elevation. The elevation of G
  is above G1 in [alpha]", and that of F is at F2 in the axis. This
  gives the elevation F2G2 of FG and in it we get A2B2 in the verticals
  through A1 and B1. As a check we have OG = OG2. Similarly the
  elevation of the other sides and vertices are found.

  § 17. We proceed to give some applications of the above principles to
  the representation of solids and of the solution of problems connected
  with them.

  _Of a pyramid are given its base, the length of the perpendicular from
  the vertex to the base, and the point where this perpendicular cuts
  the base; it is required first to develop the whole surface of the
  pyramid into one plane, and second to determine its section by a plane
  which cuts the plane of the base in a given line and makes a given
  angle with it._

  1. As the planes of projection are not given we can take them as we
  like, and we select them in such a manner that the solution becomes as
  simple as possible. We take the plane of the base as the horizontal
  plane and the vertical plane perpendicular to the plane of the
  section. Let then (fig. 47) ABCD be the base of the pyramid, V1 the
  plan of the vertex, then the elevations of A, B, C, D will be in the
  axis at A2, B2, C2, D2, and the vertex at some point V2 above V1 at a
  known distance from the axis. The lines V1A, V1B, &c., will be the
  plans and the lines V2A2, V2B2, &c., the elevations of the edges of
  the pyramid, of which thus plan and elevation are known.

  We develop the surface into the plane of the base by turning each
  lateral face about its lower edge into the horizontal plane by the
  method used in § 14. If one face has been turned down, say ABV to ABP,
  then the point Q to which the vertex of the next face BCV comes can be
  got more simply by finding on the line V1Q perpendicular to BC the
  point Q such that BQ = BP, for these lines represent the same edge BV
  of the pyramid. Next R is found by making CR = CQ, and so on till we
  have got the last vertex--in this case S. The fact that AS must equal
  AP gives a convenient check.

  2. The plane [alpha] whose section we have to determine has its
  horizontal trace given perpendicular to the axis, and its vertical
  trace makes the given angle with the axis. This determines it. To find
  the section of the pyramid by this plane there are two methods
  applicable: we find the sections of the plane either with the faces or
  with the edges of the pyramid. We use the latter.

  As the plane [alpha] is perpendicular to the vertical plane, the trace
  [alpha]" contains the projection of every figure in it; the points
  E2, F2, G2, H2 where this trace cuts the elevations of the edges will
  therefore be the elevations of the points where the edges cut [alpha].
  From these we find the plans E1, F1, G1, H1, and by joining them the
  plan of the section. If from E1, F1 lines be drawn perpendicular to
  AB, these will determine the points E, F on the developed face in
  which the plane [alpha] cuts it; hence also the line EF. Similarly on
  the other faces. Of course BF must be the same length on BP and on BQ.
  If the plane [alpha] be rabatted to the plan, we get the real shape of
  the section as shown in the figure in EFGH. This is done easily by
  making F0F = OF2, &c. If the figure representing the development of
  the pyramid, or better a copy of it, is cut out, and if the lateral
  faces be bent along the lines AB, BC, &c., we get a model of the
  pyramid with the section marked on its faces. This may be placed on
  its plan ABCD and the plane of elevation bent about the axis x. The
  pyramid stands then in front of its elevations. If next the plane
  [alpha] with a hole cut out representing the true section be bent
  along the trace [alpha]' till its edge coincides with [alpha]", the
  edges of the hole ought to coincide with the lines EF, FG, &c., on the
  faces.

  § 18. Polyhedra like the pyramid in § 17 are represented by the
  projections of their edges and vertices. But solids bounded by curved
  surfaces, or surfaces themselves, cannot be thus represented.

  For a surface we may use, as in case of the plane, its traces--that
  is, the curves in which it cuts the planes of projection. We may also
  project points and curves on the surface. A ray cuts the surface
  generally in more than one point; hence it will happen that some of
  the rays touch the surface, if two of these points coincide. The
  points of contact of these rays will form some curve on the surface,
  and this will appear from the centre of projection as the boundary of
  the surface or of part of the surface. The outlines of all surfaces of
  solids which we see about us are formed by the points at which rays
  through our eye touch the surface. The projections of these contours
  are therefore best adapted to give an idea of the shape of a surface.

  [Illustration: FIG. 47.]

  Thus the tangents drawn from any finite centre to a sphere form a
  right circular cone, and this will be cut by any plane in a conic. It
  is often called the projection of a sphere, but it is better called
  the contour-line of the sphere, as it is the boundary of the
  projections of all points on the sphere.

  If the centre is at infinity the tangent cone becomes a right circular
  cylinder touching the sphere along a great circle, and if the
  projection is, as in our case, orthographic, then the section of this
  cone by a plane of projection will be a circle equal to the great
  circle of the sphere. We get such a circle in the plan and another in
  the elevation, their centres being plan and elevation of the centre of
  the sphere.

  Similarly the rays touching a cone of the second order will lie in two
  planes which pass through the vertex of the cone, the contour-line of
  the projection of the cone consists therefore of two lines meeting in
  the projection of the vertex. These may, however, be invisible if no
  real tangent rays can be drawn from the centre of projection; and this
  happens when the ray projecting the centre of the vertex lies within
  the cone. In this case the traces of the cone are of importance. Thus
  in representing a cone of revolution with a vertical axis we get in
  the plan a circular trace of the surface whose centre is the plan of
  the vertex of the cone, and in the elevation the contour, consisting
  of a pair of lines intersecting in the elevation of the vertex of the
  cone. The circle in the plan and the pair of lines in the elevation do
  not determine the surface, for an infinite number of surfaces might be
  conceived which pass through the circular trace and touch two planes
  through the contour lines in the vertical plane. The surface becomes
  only completely defined if we write down to the figure that it shall
  represent a cone. The same holds for all surfaces. Even a plane is
  fully represented by its traces only under the silent understanding
  that the traces are those of a plane.

  § 19. Some of the simpler problems connected with the representation
  of surfaces are the determination of plane sections and of the curves
  of intersection of two such surfaces. The former is constantly used in
  nearly all problems concerning surfaces. Its solution depends of
  course on the nature of the surface.

  To determine the curve of intersection of two surfaces, we take a
  plane and determine its section with each of the two surfaces,
  rabatting this plane if necessary. This gives two curves which lie in
  the same plane and whose intersections will give us points on both
  surfaces. It must here be remembered that two curves in space do not
  necessarily intersect, hence that the points in which their
  projections intersect are not necessarily the projections of points
  common to the two curves. This will, however, be the case if the two
  curves lie in a common plane. By taking then a number of plane
  sections of the surfaces we can get as many points on their curve of
  intersection as we like. These planes have, of course, to be selected
  in such a way that the sections are curves as simple as the case
  permits of, and such that they can be easily and accurately drawn.
  Thus when possible the sections should be straight lines or circles.
  This not only saves time in drawing but determines all points on the
  sections, and therefore also the points where the two curves meet,
  with equal accuracy.

  § 20. We give a few examples how these sections have to be selected. A
  cone is cut by every plane through the vertex in lines, and if it is a
  cone of revolution by planes perpendicular to the axis in circles.

  A cylinder is cut by every plane parallel to the axis in lines, and if
  it is a cylinder of revolution by planes perpendicular to the axis in
  circles.

  A sphere is cut by every plane in a circle.

  Hence in case of two cones situated anywhere in space we take sections
  through both vertices. These will cut both cones in lines. Similarly
  in case of two cylinders we may take sections parallel to the axis of
  both. In case of a sphere and a cone of revolution with vertical axis,
  horizontal sections will cut both surfaces in circles whose plans are
  circles and whose elevations are lines, whilst vertical sections
  through the vertex of the cone cut the latter in lines and the sphere
  in circles. To avoid drawing the projections of these circles, which
  would in general be ellipses, we rabatt the plane and then draw the
  circles in their real shape. And so on in other cases.

  Special attention should in all cases be paid to those points in which
  the tangents to the projection of the curve of intersection are
  parallel or perpendicular to the axis x, or where these projections
  touch the contour of one of the surfaces.     (O. H.)


IV. ANALYTICAL GEOMETRY

1. In the name _geometry_ there is a lasting record that the science had
its origin in the knowledge that two distances may be compared by
measurement, and in the idea that measurement must be effectual in the
dissociation of different directions as well as in the comparison of
distances in the same direction. The distance from an observer's eye of
an object seen would be specified as soon as it was ascertained that a
rod, straight to the eye and of length taken as known, could be given
the direction of the line of vision, and had to be moved along it a
certain number of times through lengths equal to its own in order to
reach the object from the eye. Moreover, if a field had for two of its
boundaries lines straight to the eye, one running from south to north
and the other from west to east, the position of a point in the field
would be specified if the rod, when directed west, had to be shifted
from the point one observed number of times westward to meet the former
boundary, and also, when directed south, had to be shifted another
observed number of times southward to meet the latter. Comparison by
measurement, the beginning of geometry, involved counting, the basis of
arithmetic; and the science of number was marked out from the first as
of geometrical importance.

But the arithmetic of the ancients was inadequate as a science of
number. Though a length might be recognized as known when measurement
certified that it was so many times a standard length, it was not every
length which could be thus specified in terms of the same standard
length, even by an arithmetic enriched with the notion of fractional
number. The idea of possible incommensurability of lengths was
introduced into Europe by Pythagoras; and the corresponding idea of
irrationality of number was absent from a crude arithmetic, while there
were great practical difficulties in the way of its introduction. Hence
perhaps it arose that, till comparatively modern times, appeal to
arithmetical aid in geometrical reasoning was in all possible ways
restrained. Geometry figured rather as the helper of the more difficult
science of arithmetic.

2. It was reserved for algebra to remove the disabilities of arithmetic,
and to restore the earliest ideas of the land-measurer to the position
of controlling ideas in geometrical investigation. This unified science
of pure number made comparatively little headway in the hands of the
ancients, but began to receive due attention shortly after the revival
of learning. It expresses whole classes of arithmetical facts in single
statements, gives to arithmetical laws the form of equations involving
symbols which may mean any known or sought numbers, and provides
processes which enable us to analyse the information given by an
equation and derive from that equation other equations, which express
laws that are in effect consequences or causes of a law started from,
but differ greatly from it in form. Above all, for present purposes, it
deals not only with integral and fractional number, but with number
regarded as capable of continuous growth, just as distance is capable of
continuous growth. The difficulty of the arithmetical expression of
irrational number, a difficulty considered by the modern school of
analysts to have been at length surmounted (see FUNCTION), is not vital
to it. It can call the ratio of the diagonal of a square to a side, for
instance, or that of the circumference of a circle to a diameter, a
number, and let a or x denote that number, just as properly as it may
allow either letter to denote any rational number which may be greater
or less than the ratio in question by a difference less than any minute
one we choose to assign.

Counting only, and not the counting of objects, is of the essence of
arithmetic, and of algebra. But it is lawful to count objects, and in
particular to count equal lengths by measure. The widened idea is that
even when a or x is an irrational number we may speak of a or x unit
lengths by measure. We may give concrete interpretation to an
algebraical equation by allowing its terms all to mean numbers of times
the same unit length, or the same unit area, or &c. and in any equation
lawfully derived from the first by algebraical processes we may do the
same. Descartes in his _Géométrie_ (1637) was the first to systematize
the application of this principle to the inherent first notions of
geometry; and the methods which he instituted have become the most
potent methods of all in geometrical research. It is hardly too much to
say that, when known facts as to a geometrical figure have once been
expressed in algebraical terms, all strictly consequential facts as to
the figure can be deduced by almost mechanical processes. Some may well
be unexpected consequences; and in obtaining those of which there has
been suggestion beforehand the often bewildering labour of constant
attention to the figure is obviated. These are the methods of what is
now called _analytical_, or sometimes _algebraical_, _geometry_.

3. The modern use of the term "analytical" in geometry has obscured, but
not made obsolete, an earlier use, one as old as Plato. There is nothing
algebraical in this analysis, as distinguished from synthesis, of the
Greeks, and of the expositors of pure geometry. It has reference to an
order of ideas in demonstration, or, more frequently, in discovering
means to effect the geometrical construction of a figure with an
assigned special property. We have to suppose hypothetically that the
construction has been performed, drawing a rough figure which exhibits
it as nearly as is practicable. We then analyse or critically examine
the figure, treated as correct, and ascertain other properties which it
can only possess in association with the one in question. Presently one
of these properties will often be found which is of such a character
that the construction of a figure possessing it is simple. The means of
effecting synthetically a construction such as was desired is thus
brought to light by what Plato called _analysis_. Or again, being asked
to prove a theorem A, we ascertain that it must be true if another
theorem B is, that B must be if C is, and so on, thus eventually finding
that the theorem A is the consequence, through a chain of
intermediaries, of a theorem Z of which the establishment is easy. This
geometrical analysis is not the subject of the present article; but in
the reasoning from form to form of an equation or system of equations,
with the object of basing the algebraical proof of a geometrical fact on
other facts of a more obvious character, the same logic is utilized, and
the name "analytical geometry" is thus in part explained.

4. In algebra real positive number was alone at first dealt with, and in
geometry actual signless distance. But in algebra it became of
importance to say that every equation of the first degree has a root,
and the notion of negative number was introduced. The negative unit had
to be defined as what can be added to the positive unit and produce the
sum zero. The corresponding notion was readily at hand in geometry,
where it was clear that a unit distance can be measured to the left or
down from the farther end of a unit distance already measured to the
right or up from a point O, with the result of reaching O again. Thus,
to give full interpretation in geometry to the algebraically negative,
it was only necessary to associate distinctness of sign with
oppositeness of direction. Later it was discovered that algebraical
reasoning would be much facilitated, and that conclusions as to the real
would retain all their soundness, if a pair of imaginary units ±[root]-1
of what might be called number were allowed to be contemplated, the pair
being defined, though not separately, by the two properties of having
the real sum 0 and the real product 1. Only in these two real
combinations do they enter in conclusions as to the real. An advantage
gained was that every quadratic equation, and not some quadratics only,
could be spoken of as having two roots. These admissions of new units
into algebra were final, as it admitted of proof that all equations of
degrees higher than two have the full numbers of roots possible for
their respective degrees in any case, and that every root has a value
included in the form a + b [root]-1, with a, b, real. The corresponding
enrichment could be given to geometry, with corresponding advantages and
the same absence of danger, and this was done. On a line of measurement
of distance we contemplate as existing, not only an infinite continuum
of points at real distances from an origin of measurement O, but a
doubly infinite continuum of points, all but the singly infinite
continuum of real ones imaginary, and imaginary in conjugate pairs, a
conjugate pair being at imaginary distances from O, which have a real
arithmetic and a real geometric mean. To geometry enriched with this
conception all algebra has its application.

5. Actual geometry is one, two or three-dimensional, i.e. lineal, plane
or solid. In one-dimensional geometry positions and measurements in a
single line only are admitted. Now descriptive constructions for points
in a line are impossible without going out of the line. It has therefore
been held that there is a sense in which no science of geometry strictly
confined to one dimension exists. But an algebra of one variable can be
applied to the study of distances along a line measured from a chosen
point on it, so that the idea of construction as distinct from
measurement is not essential to a one-dimensional geometry aided by
algebra. In geometry of two dimensions, the flat of the land-measurer,
the passage from one point O to any other point, can be effected by two
successive marches, one east or west and one north or south, and, as
will be seen, an algebra of two variables suffices for geometrical
exploitation. In geometry of three dimensions, that of space, any point
can be reached from a chosen one by three marches, one east or west, one
north or south, and one up or down; and we shall see that an algebra of
three variables is all that is necessary. With three dimensions actual
geometry stops; but algebra can supply any number of variables. Four or
more variables have been used in ways analogous to those in which one,
two and three variables are used for the purposes of one, two and
three-dimensional geometry, and the results have been expressed in
quasi-geometrical language on the supposition that a higher space can be
conceived of, though not realized, in which four independent directions
exist, such that no succession of marches along three of them can effect
the same displacement of a point as a march along the fourth; and
similarly for higher numbers than four. Thus analytical, though not
actual, geometries exist for four and more dimensions. They are in fact
algebras furnished with nomenclature of a geometrical cast, suggested by
convenient forms of expression which actual geometry has, in return for
benefits received, conferred on algebras of one, two and three
variables.

We will confine ourselves to the dimensions of actual geometry, and will
devote no space to the one-dimensional, except incidentally as existing
within the two-dimensional. The analytical method will now be explained
for the cases of two and three dimensions in succession. The form of it
originated by Descartes, and thence known as Cartesian, will alone be
considered in much detail.


  I. _Plane Analytical Geometry._

  [Illustration: FIG. 48.]

  [Illustration: FIG. 49.]

  6. _Coordinates._--It is assumed that the points, lines and figures
  considered lie in one and the same plane, which plane therefore need
  not be in any way referred to. In the plane a point O, and two lines
  x'Ox, y'Oy, intersecting in O, are taken once for all, and regarded as
  fixed. O is called the origin, and x'Ox, y'Oy the axes of x and y
  respectively. Other positions in the plane are specified in relation
  to this fixed origin and these fixed axes. From any point P we suppose
  PM drawn parallel to the axis of y to meet the axis of x in M, and may
  also suppose PN drawn parallel to the axis of x to meet the axis of y
  in N, so that OMPN is a parallelogram. The position of P is determined
  when we know OM ( = NP) and MP ( = ON). If OM is x times the unit of a
  scale of measurement chosen at pleasure, and MP is y times the unit,
  so that x and y have numerical values, we call x and y the (Cartesian)
  coordinates of P. To distinguish them we often speak of y as the
  ordinate, and of x as the abscissa.

  It is necessary to attend to signs; x has one sign or the other
  according as the point P is on one side or the other of the axis of y,
  and y one sign or the other according as P is on one side or the other
  of the axis of x. Using the letters N, E, S, W, as in a map, and
  considering the plane as divided into four quadrants by the axes, the
  signs are usually taken to be:

    x   y   For quadrant

    +   +      N  E
    +   -      S  E
    -   +      N  W
    -   -      S  W

  A point is referred to as the point (a, b), when its coordinates are x
  = a, y = b. A point may be fixed, or it may be variable, i.e. be
  regarded for the time being as free to move in the plane. The
  coordinates (x, y) of a variable point are algebraic variables, and
  are said to be "current coordinates."

  The axes of x and y are usually (as in fig. 48) taken at right angles
  to one another, and we then speak of them as rectangular axes, and of
  x and y as "rectangular coordinates" of a point P; OMPN is then a
  rectangle. Sometimes, however, it is convenient to use axes which are
  oblique to one another, so that (as in fig. 49) the angle xOy between
  their positive directions is some known angle [omega] distinct from a
  right angle, and OMPN is always an oblique parallelogram with given
  angles; and we then speak of x and y as "oblique coordinates." The
  coordinates are as a rule taken to be rectangular in what follows.

  7. _Equations and loci._ If (x, y) is the point P, and if we are given
  that x = 0, we are told that, in fig. 48 or fig. 49, the point M lies
  at O, whatever value y may have, i.e. we are told the one fact that P
  lies on the axis of y. Conversely, if P lies anywhere on the axis of
  y, we have always OM = 0, i.e. x = 0. Thus the equation x = 0 is one
  satisfied by the coordinates (x, y) of every point in the axis of y,
  and not by those of any other point. We say that x = 0 is the equation
  of the axis of y, and that the axis of y is the locus represented by
  the equation x = 0. Similarly y = 0 is the equation of the axis of x.
  An equation x = a, where a is a constant, expresses that P lies on a
  parallel to the axis of y through a point M on the axis of x such that
  OM = a. Every line parallel to the axis of y has an equation of this
  form. Similarly, every line parallel to the axis of x has an equation
  of the form y = b, where b is some definite constant.

  These are simple cases of the fact that a single equation in the
  current coordinates of a variable point (x, y) imposes one limitation
  on the freedom of that point to vary. The coordinates of a point taken
  at random in the plane will, as a rule, not satisfy the equation, but
  infinitely many points, and in most cases infinitely many real ones,
  have coordinates which do satisfy it, and these points are exactly
  those which lie upon some locus of one dimension, a straight line or
  more frequently a curve, which is said to be represented by the
  equation. Take, for instance, the equation y = mx, where m is a given
  constant. It is satisfied by the coordinates of every point P, which
  is such that, in fig. 48, the distance MP, with its proper sign, is m
  times the distance OM, with its proper sign, i.e. by the coordinates
  of every point in the straight line through O which we arrive at by
  making a line, originally coincident with x'Ox, revolve about O in the
  direction opposite to that of the hands of a watch through an angle of
  which m is the tangent, and by those of no other points. That line is
  the locus which it represents. Take, more generally, the equation y =
  [phi](x), where [phi](x) is any given non-ambiguous function of x.
  Choosing any point M on x'Ox in fig. 1, and giving to x the value of
  the numerical measure of OM, the equation determines a single
  corresponding y, and so determines a single point P on the line
  through M parallel to y'Oy. This is one point whose coordinates
  satisfy the equation. Now let M move from the extreme left to the
  extreme right of the line x'Ox, regarded as extended both ways as far
  as we like, i.e. let x take all real values from -[oo] to [oo]. With
  every value goes a point P, as above, on the parallel to y'Oy through
  the corresponding M; and we thus find that there is a path from the
  extreme left to the extreme right of the figure, all points P along
  which are distinguished from other points by the exceptional property
  of satisfying the equation by their coordinates. This path is a locus;
  and the equation y = [phi](x) represents it. More generally still,
  take an equation f(x, y) = 0 which involves both x and y under a
  functional form. Any particular value given to x in it produces from
  it an equation for the determination of a value or values of y, which
  go with that value of x in specifying a point or points (x, y), of
  which the coordinates satisfy the equation f(x, y) = 0. Here again, as
  x takes all values, the point or points describe a path or paths,
  which constitute a locus represented by the equation. Except when y
  enters to the first degree only in f(x, y), it is not to be expected
  that all the values of y, determined as going with a chosen value of
  x, will be necessarily real; indeed it is not uncommon for all to be
  imaginary for some ranges of values of x. The locus may largely
  consist of continua of imaginary points; but the real parts of it
  constitute a real curve or real curves. Note that we have to allow x
  to admit of all imaginary, as well as of all real, values, in order to
  obtain all imaginary parts of the locus.

  A locus or curve may be algebraically specified in another way; viz.
  we may be given two equations x = f([theta]), y = F([theta]), which
  express the coordinates of any point of it as two functions of the
  same variable parameter [theta] to which all values are open. As
  [theta] takes all values in turn, the point (x, y) traverses the
  curve.

  It is a good exercise to trace a number of curves, taken as defined by
  the equations which represent them. This, in simple cases, can be done
  approximately by plotting the values of y given by the equation of a
  curve as going with a considerable number of values of x, and
  connecting the various points (x, y) thus obtained. But methods exist
  for diminishing the labour of this tentative process.

  Another problem, which will be more attended to here, is that of
  determining the equations of curves of known interest, taken as
  defined by geometrical properties. It is not a matter for surprise
  that the curves which have been most and longest studied geometrically
  are among those represented by equations of the simplest character.

  8. _The Straight Line._--This is the simplest type of locus. Also the
  simplest type of equation in x and y is Ax + By + C = 0, one of the
  first degree. Here the coefficients A, B, C are constants. They are,
  like the current coordinates, x, y, numerical. But, in giving
  interpretation to such an equation, we must of course refer to numbers
  Ax, By, C of unit magnitudes of the same kind, of units of counting
  for instance, or unit lengths or unit squares. It will now be seen
  that every straight line has an equation of the first degree, and that
  every equation of the first degree represents a straight line.

  [Illustration: FIG. 50.]

  It has been seen (§ 7) that lines parallel to the axes have equations
  of the first degree, free from one of the variables. Take now a
  straight line ABC inclined to both axes. Let it make a given angle
  [alpha] with the positive direction of the axis of x, i.e. in fig. 50
  let this be the angle through which Ax must be revolved
  counter-clockwise about A in order to be made coincident with the
  line. Let C, of coordinates (h, k), be a fixed point on the line, and
  P(x, y) any other point upon it. Draw the ordinates CD, PM of C and P,
  and let the parallel to the axis of x through C meet PM, produced if
  necessary, in R. The right-angled triangle CRP tells us that, with the
  signs appropriate to their directions attached to CR and RP,

    RP = CR tan [alpha], i.e. MP - DC = (OM - OD) tan [alpha],

  and this gives that

    y - k = tan [alpha] (x - h),

  an equation of the first degree satisfied by x and y. No point not on
  the line satisfies the same equation; for the line from C to any point
  off the line would make with CR some angle ß different from [alpha],
  and the point in question would satisfy an equation y - k = tan ß(x -
  h), which is inconsistent with the above equation.

  The equation of the line may also be written y = mx + b, where m = tan
  [alpha], and b = k - h tan [alpha]. Here b is the value obtained for y
  from the equation when 0 is put for x, i.e. it is the numerical
  measure, with proper sign, of OB, the intercept made by the line on
  the axis of y, measured from the origin. For different straight lines,
  m and b may have any constant values we like.

  Now the general equation of the first degree Ax + By + C = 0 may be
  written y = -(A/B)x - C/B, unless B = 0, in which case it represents a
  line parallel to the axis of y; and -A/B, -C/B are values which can be
  given to m and b, so that every equation of the first degree
  represents a straight line. It is important to notice that the general
  equation, which in appearance contains three constants A, B, C, in
  effect depends on two only, the ratios of two of them to the third. In
  virtue of this last remark, we see that two distinct conditions
  suffice to determine a straight line. For instance, it is easy from
  the above to see that

    x    y
    -- + -- = 1
    a    b

  is the equation of a straight line determined by the two conditions
  that it makes intercepts OA, OB on the two axes, of which a and b are
  the numerical measures with proper signs: note that in fig. 50 a is
  negative. Again,

             y2 - y1
    y - y1 = ------- (x - x1),
             x2 - x1

  i.e.

    (y1 - y2)x - (x1 - x2)y + x1y2 - x2y1 = 0,

  represents the line determined by the data that it passes through two
  given points (x1, y1) and (x2, y2). To prove this find m in the
  equation y - y1 = m(x - x1) of a line through (x1, y1), from the
  condition that (x2, y2) lies on the line.

  In this paragraph the coordinates have been assumed rectangular. Had
  they been oblique, the doctrine of similar triangles would have given
  the same results, except that in the forms of equation y - k = m(x -
  h), y = mx + b, we should not have had m = tan [alpha].

  9. _The Circle._--It is easy to write down the equation of a given
  circle. Let (h, k) be its given centre C, and [rho] the numerical
  measure of its given radius. Take P (x, y) any point on its
  circumference, and construct the triangle CRP, in fig. 50 as above.
  The fact that this is right-angled tells us that

    CR² + RP² = CP²,

  and this at once gives the equation

    (x - h)² + (y - k)² = [rho]².

  A point not upon the circumference of the particular circle is at some
  distance from (h, k) different from [rho], and satisfies an equation
  inconsistent with this one; which accordingly represents the
  circumference, or, as we say, the circle.

  The equation is of the form

    x² + y² + 2Ax + 2By + C = 0.

  Conversely every equation of this form represents a circle: we have
  only to take -A, -B, A² + B² - C for h, k, [rho]² respectively, to
  obtain its centre and radius. But this statement must appear too
  unrestricted. Ought we not to require A² + B² - C to be positive?
  Certainly, if by circle we are only to mean the visible round
  circumference of the geometrical definition. Yet, analytically, we
  contemplate altogether imaginary circles, for which [rho]² is
  negative, and circles, for which [rho] = 0, with all their reality
  condensed into their centres. Even when [rho]² is positive, so that a
  visible round circumference exists, we do not regard this as
  constituting the whole of the circle. Giving to x any value whatever
  in (x - h)² + (y - k)² = [rho]², we obtain two values of y, real,
  coincident or imaginary, each of which goes with the abscissa x as the
  ordinate of a point, real or imaginary, on what is represented by the
  equation of the circle.

  The doctrine of the imaginary on a circle, and in geometry generally,
  is of purely algebraical inception; but it has been in its entirety
  accepted by modern pure geometers, and signal success has attended the
  efforts of those who, like K.G.C. von Staudt, have striven to base its
  conclusions on principles not at all algebraical in form, though of
  course cognate to those adopted in introducing the imaginary into
  algebra.

  A circle with its centre at the origin has an equation x² + y² =
  [rho]².

  In oblique coordinates the general equation of a circle is x² + 2xy
  cos [omega] + y² + 2Ax + 2By + C = 0.

  10. The conic sections are the next simplest loci; and it will be seen
  later that they are the loci represented by equations of the second
  degree. Circles are particular cases of conic sections; and they have
  just been seen to have for their equations a particular class of
  equations of the second degree. Another particular class of such
  equations is that included in the form (Ax + By + C)(A'x + B'y + C') =
  0, which represents two straight lines, because the product on the
  left vanishes if, and only if, one of the two factors does, i.e. if,
  and only if, (x, y) lies on one or other of two straight lines. The
  condition that ax² + 2hxy + by² + 2gx + 2fy + c = 0, which is often
  written (a, b, c, f, g, h)(x, y, I)² = 0, takes this form is abc +
  2fgh-af²-bg² - ch² = 0. Note that the two lines may, in particular
  cases, be parallel or coincident.

  Any equation like F1(x, y) F2(x, y) ... F_n(x, y) = 0, of which the
  left-hand side breaks up into factors, represents all the loci
  separately represented by F1(x, y) = 0, F2(x, y) = 0, ... F_n(x, y) =
  0. In particular an equation of degree n which is free from x
  represents n straight lines parallel to the axis of x, and one of
  degree n which is homogeneous in x and y, i.e. one which upon division
  by x^n, becomes an equation in the ratio y/x, represents n straight
  lines through the origin.

  Curves represented by equations of the third degree are called cubic
  curves. The general equation of this degree will be written (*)(x, y,
  I)³ = 0.

  11. _Descriptive Geometry._--A geometrical proposition is either
  descriptive or metrical: in the former case the statement of it is
  independent of the idea of magnitude (length, inclination, &c.), and
  in the latter it has reference to this idea. The method of coordinates
  seems to be by its inception essentially metrical. Yet in dealing by
  this method with descriptive propositions we are eminently free from
  metrical considerations, because of our power to use general
  equations, and to avoid all assumption that measurements implied are
  any particular measurements.

  [Illustration: FIG. 51.]

  12. It is worth while to illustrate this by the instance of the
  well-known theorem of the radical centre of three circles. The theorem
  is that, given any three circles A, B, C (fig. 51), the common chords
  [alpha][alpha]', ß[beta]', [gamma][gamma]' of the three pairs of
  circles meet in a point.

  The geometrical proof is metrical throughout:--

  Take O the point of intersection of [alpha][alpha]', ß[beta]', and
  joining this with [gamma]', suppose that [gamma]'O does not pass
  through [gamma], but that it meets the circles A, B in two distinct
  points [gamma]2, [gamma]1 respectively. We have then the known
  metrical property of intersecting chords of a circle; viz. in circle
  C, where [alpha][alpha]', ß[beta]', are chords meeting at a point O,

    O[alpha]·O[alpha]' = Oß·Oß',

  where, as well as in what immediately follows, O[alpha], &c. denote,
  of course, _lengths_ or _distances_.

  Similarly in circle A,

    Oß·Oß' = O[gamma]2·O[gamma]',

  and in circle B,

    O[alpha]·O[alpha]' = O[gamma]1·O[gamma]'.

  Consequently O[gamma]1·O[gamma]' = O[gamma]2·O[gamma]', that is,
  O[gamma]1 = O[gamma]2, or the points [gamma]1 and [gamma]2 coincide;
  that is, they each coincide with [gamma].

  We contrast this with the analytical method:--

  Here it only requires to be known that an equation Ax + By + C = 0
  represents a line, and an equation x² + y² + Ax + By + C = 0
  represents a circle. A, B, C have, in the two cases respectively,
  metrical significations; but these we are not concerned with. Using S
  to denote the function x² + y² + Ax + By + C, the equation of a circle
  is S = o. Let the equation of any other circle be S', = x² + y² + A'x
  + B'y + C' = 0; the equation S - S' = 0 is a linear equation (S - S'
  is in fact = (A - A')x + (B - B')y + C - C), and it thus represents a
  line; this equation is satisfied by the coordinates of each of the
  points of intersection of the two circles (for at each of these points
  S = 0 and S' = 0, therefore also S - S' = 0); hence the equation S -
  S' = 0 is that of the line joining the two points of intersection of
  the two circles, or say it is the equation of the common chord of the
  two circles. Considering then a third circle S", = x² + y² + A"x + B"y
  + C" = 0, the equations of the common chords are S-S' = 0, S - S" = 0,
  S' - S" = 0 (each of these a linear equation); at the intersection of
  the first and second of these lines S = S' and S = S", therefore also
  S' = S", or the equation of the third line is satisfied by the
  coordinates of the point in question; that is, the three chords
  intersect in a point O, the coordinates of which are determined by the
  equations S = S' = S".

  It further appears that if the two circles S = 0, S' = 0 do not
  intersect in any real points, they must be regarded as intersecting in
  two imaginary points, such that the line joining them is the real line
  represented by the equation S - S' = 0; or that two circles, whether
  their intersections be real or imaginary, have always a real common
  chord (or radical axis), and that for _any_ three circles the common
  chords intersect in a point (of course real) which is the radical
  centre. And by this very theorem, given two circles with imaginary
  intersections, we can, by drawing circles which meet each of them in
  real points, construct the radical axis of the first-mentioned two
  circles.

  13. The principle employed in showing that the equation of the common
  chord of two circles is S - S' = 0 is one of very extensive
  application, and some more illustrations of it may be given.

  Suppose S = 0, S' = 0 are lines (that is, let S, S' now denote linear
  functions Ax + By + C, A'x + B'y + C'), then S - kS' = 0 (k an
  arbitrary constant) is the equation of any line passing through the
  point of intersection of the two given lines. Such a line may be made
  to pass through any given point, say the point (x0, y0); if S0, S'0
  are what S, S' respectively become on writing for (x, y) the values
  (x0, y0), then the value of k is k = S0 ÷ S'0. The equation in fact is
  SS'0 - S0S' = 0; and starting from this equation we at once verify it
  _a posteriori_; the equation is a linear equation satisfied by the
  values of (x, y) which make S = 0, S' = 0; and satisfied also by the
  values (x0, y0); and it is thus the equation of the line in question.

  If, as before, S = 0, S' = 0 represent circles, then (k being
  arbitrary) S - kS' = 0 is the equation of any circle passing through
  the two points of intersection of the two circles; and to make this
  pass through a given point (x0, y0) we have again k = S0 ÷ S'0. In the
  particular case k = 1, the circle becomes the common chord (more
  accurately it becomes the common chord together with the line
  infinity; see § 23 below).

  If S denote the general quadric function,

    S = ax² +2hxy + by² + 2fy + 2gx + c,

  then the equation S = 0 represents a conic; assuming this, then, if S'
  = 0 represents another conic, the equation S - kS' = 0 represents
  _any_ conic through the four points of intersection of the two conics.

  [Illustration: FIG. 52.]

  14. The object still being to illustrate the mode of working with
  coordinates for descriptive purposes, we consider the theorem of the
  polar of a point in regard to a circle. Given a circle and a point O
  (fig. 52), we draw through O any two lines meeting the circle in the
  points A, A' and B, B' respectively, and then taking Q as the
  intersection of the lines AB' and A'B, the theorem is that the locus
  of the point Q is a right line depending only upon O and the circle,
  but independent of the particular lines OAA' and OBB'.

  Taking O as the origin, and for the axes any two lines through O at
  right angles to each other, the equation of the circle will be

    x² + y² + 2Ax + 2By + C = 0;

  and if the equation of the line OAA' is taken to be y = mx, then the
  points A, A' are found as the intersections of the straight line with
  the circle; or to determine x we have

    x²(1 + m²) + 2x(A + Bm) + C = 0.

  If(x1, y1) are the coordinates of A, and (x2, y2) of A', then the
  roots of this equation are x1, x2, whence easily

    1    1        A + Bm
    -- + --  = -2 ------.
    x1   x2         C

  And similarly, if the equation of the line OBB' is taken to be y =
  m'x1 and the coordinates of B, B' to be (x3, y3) and (x4, y4)
  respectively, then

    1    1         A + Bm'
    -- + --  =  -2 -------.
    x3   x4           C'

  We have then by § 8

    x(y1 - y4) - y(x1 - x4) + x1y4 - x4y1 = 0,

    x(y2 - y3) - y(x2 - x3) + x2y3 - x3y2 = 0,

  as the equations of the lines AB' and A'B respectively. Reducing by
  means of the relations y1 - mx1 = 0, y2 - mx2 = 0, y3 - m'x3 = 0, y4 -
  m'x4 = 0, the two equations become

    x(mx1 - m'x4) - y(x1 - x4) + (m'- m)x1x4 = 0,

    x(mx2 - m'x3) - y(x2 - x3) + (m'- m)x2x3 = 0,

  and if we divide the first of these equations by x1x4, and the second
  by x2x3 and then add, we obtain
      _                               _      _                       _
     |   / 1    1  \       / 1    1 \  |    |  1    1     / 1    1 \  |
    x| m(  -- + --  ) - m'(  -- + -- ) | - y|  -- + -- - (  -- + -- ) |
     |_  \ x3   x4 /       \ x1   x2/ _|    |_ x3   x4    \ x1   x2/ _|

      + 2m' - 2m = 0,

  or, what is the same thing,

     / 1    1  \              / 1    1  \
    (  -- + --  )(y - m'x) - (  -- + --  )(y - mx) + 2m' - 2m = 0,
     \ x1   x2 /              \ x3   x4 /

  which by what precedes is the equation of a line through the point Q.
  Substituting herein for 1/x1 + 1/x2, 1/x3 + 1/x4 their foregoing
  values, the equation becomes

    -(A + Bm)(y - m'x) + (A + Bm')(y - mx) + C(m' - m) = 0;

  that is,

    (m - m')(Ax + By + C) = 0;

  or finally it is Ax + By + C = 0, showing that the point Q lies in a
  line the position of which is independent of the particular lines
  OAA', OBB' used in the construction. It is proper to notice that there
  is no correspondence to each other of the points A, A' and B, B'; the
  grouping might as well have been A, A' and B', B; and it thence
  appears that the line Ax + By + C = 0 just obtained is in fact the
  line joining the point Q with the point R which is the intersection of
  AB and A'B'.

  15. In § 8 it has been seen that two conditions determine the equation
  of a straight line, because in Ax + By + C = 0 one of the coefficients
  may be divided out, leaving only two parameters to be determined.
  Similarly five conditions instead of six determine an equation of the
  second degree (a, b, c, f, g, h)(x, y, 1)² = 0, and nine instead of
  ten determine a cubic (*)(x, y, 1)³ = 0. It thus appears that a cubic
  can be made to pass through 9 given points, and that the cubic so
  passing through 9 given points is completely determined. There is,
  however, a remarkable exception. Considering two given cubic curves S
  = 0, S' = 0, these intersect in 9 points, and through these 9 points
  we have the whole series of cubics S - kS' = 0, where k is an
  arbitrary constant: k may be determined so that the cubic shall pass
  through a given tenth point (k = S0 ÷ S'0, if the coordinates are (x0,
  y0), and S0, S'0 denote the corresponding values of S, S'). The
  resulting curve SS'0 - S'S0 = 0 may be regarded as the cubic
  determined by the conditions of passing through 8 of the 9 points and
  through the given point (x0, y0); and from the equation it thence
  appears that the curve passes through the remaining one of the 9
  points. In other words, we thus have the theorem, any cubic curve
  which passes through 8 of the 9 intersections of two given cubic
  curves passes through the 9th intersection.

  The applications of this theorem are very numerous; for instance, we
  derive from it Pascal's theorem of the inscribed hexagon. Consider a
  hexagon inscribed in a conic. The three alternate sides constitute a
  cubic, and the other three alternate sides another cubic. The cubics
  intersect in 9 points, being the 6 vertices of the hexagon, and the 3
  Pascalian points, or intersections of the pairs of opposite sides of
  the hexagon. Drawing a line through two of the Pascalian points, the
  conic and this line constitute a cubic passing through 8 of the 9
  points of intersection, and it therefore passes through the remaining
  point of intersection--that is, the third Pascalian point; and since
  obviously this does not lie on the conic, it must lie on the
  line--that is, we have the theorem that the three Pascalian points (or
  points of intersection of the pairs of opposite sides) lie on a line.

  16. _Metrical Theory resumed. Projections and Perpendiculars._--It
  is a metrical fact of fundamental importance, already used in § 8,
  that, if a finite line PQ be projected on any other line OO' by
  perpendiculars PP', QQ' to OO', the length of the projection P'Q' is
  equal to that of PQ multiplied by the cosine of the acute angle
  between the two lines. Also the algebraical sum of the projections of
  the sides of any closed polygon upon any line is zero, because as a
  point goes round the polygon, from any vertex A to A again, the point
  which is its projection on the line passes from A' the projection of A
  to A' again, i.e. traverses equal distances along the line in positive
  and negative senses. If we consider the polygon as consisting of two
  broken lines, each extending from the same initial to the same
  terminal point, the sum of the projections of the lines which compose
  the one is equal, in sign and magnitude, to the sum of the projections
  of the lines composing the other. Observe that the projection on a
  line of a length perpendicular to the line is zero.

  Let us hence find the equation of a straight line such that the
  perpendicular OD on it from the origin is of length [rho] taken as
  positive, and is inclined to the axis of x at an angle xOD = [alpha],
  measured counter-clockwise from Ox. Take any point P (x, y) on the
  line, and construct OM and MP as in fig. 48. The sum of the
  projections of OM and MP on OD is OD itself; and this gives the
  equation of the line

    x cos [alpha] + y sin [alpha] = [rho].

  Observe that cos [alpha] and sin [alpha] here are the sin [alpha] and
  -cos [alpha], or the -sin [alpha] and cos [alpha] of § 8 according to
  circumstances.

  We can write down an expression for the perpendicular distance from
  this line of any point (x', y') which does not lie upon it. If the
  parallel through (x', y') to the line meet OD in E, we have x' cos
  [alpha] + y' sin [alpha] = OE, and the perpendicular distance required
  is OD - OE, i.e. [rho] - x' cos [alpha] - y' sin [alpha]; it is the
  perpendicular distance taken positively or negatively according as
  (x', y') lies on the same side of the line as the origin or not.

  The general equation Ax + By + C = 0 may be given the form x cos
  [alpha] + y sin [alpha] - [rho] = 0 by dividing it by [root](A² + B³).
  Thus (Ax' + By' + C) ÷ [root](A² + B²) is in absolute value the
  perpendicular distance of (x', y') from the line Ax + By + C = 0.
  Remember, however, that there is an essential ambiguity of sign
  attached to a square root. The expression found gives the distance
  taken positively when (x', y') is on the origin side of the line, if
  the sign of C is given to [root](A² + B²).

  17. _Transformation of Coordinates._--We often need to adopt new axes
  of reference in place of old ones; and the above principle of
  projections readily expresses the old coordinates of any point in
  terms of the new.

  [Illustration: FIG. 53.]

  Suppose, for instance, that we want to take for new origin the point
  O' of old coordinates OA = h, AO' = k, and for new axes of X and Y
  lines through O' obtained by rotating parallels to the old axes of x
  and y through an angle [theta] counter-clockwise. Construct (fig. 53)
  the old and new coordinates of any point P. Expressing that the
  projections, first on the old axis of x and secondly on the old axis
  of y, of OP are equal to the sums of the projections, on those axes
  respectively, of the parts of the broken line OO'M'P, we obtain:

    x = h + X cos [theta] + Y cos ([theta] + ½[pi]) = h + X cos [theta] -
      Y sin [theta],

  and

    y = k + X cos (½[pi] - [theta]) + Y cos [theta] = k + X sin [theta] +
      Y cos [theta].

  Be careful to observe that these formulae do not apply to every
  conceivable change of reference from one set of rectangular axes to
  another. It might have been required to take O'X, O'Y' for the
  positive directions of the new axes, so that the change of directions
  of the axes could not be effected by rotation. We must then write -Y
  for Y in the above.

  Were the new axes oblique, making angles [alpha], ß respectively with
  the old axis of x, and so inclined at the angle ß - [alpha], the same
  method would give the formulae

    x = h + X cos [alpha] + Y cos ß, y = k + X sin [alpha] + Y sin ß.

  18. _The Conic Sections._--The conics, as they are now called, were at
  first defined as curves of intersection of planes and a cone; but
  Apollonius substituted a definition free from reference to space of
  three dimensions. This, in effect, is that a conic is the locus of a
  point the distance of which from a given point, called the focus, has
  a given ratio to its distance from a given line, called the directrix
  (see CONIC SECTION). If e : 1 is the ratio, e is called the
  eccentricity. The distances are considered signless.

  Take (h, k) for the focus, and x cos [alpha] + y sin [alpha] - p = 0
  for the directrix. The absolute values of [root] {(x - h)² + (y - k)²}
  and p - x cos [alpha] -y sin [alpha] are to have the ratio e : 1; and
  this gives

    (x - h)² + (y - k)² = e²(p - x cos [alpha] - y sin [alpha])²

  as the general equation, in rectangular coordinates, of a conic.

  It is of the second degree, and is the general equation of that
  degree. If, in fact, we multiply it by an unknown [lambda], we can, by
  solving six simultaneous equations in the six unknowns [lambda], h, k,
  e, p, [alpha], so choose values for these as to make the coefficients
  in the equation equal to those in any equation of the second degree
  which may be given. There is no failure of this statement in the
  special case when the given equation represents two straight lines, as
  in § 10, but there is speciality: if the two lines intersect, the
  intersection and either bisector of the angle between them are a focus
  and directrix; if they are united in one line, any point on the line
  and a perpendicular to it through the point are: if they are parallel,
  the case is a limiting one in which e and h² + k² have become infinite
  while e^(-2)(h² + k²) remains finite. In the case (§ 9) of an equation
  such as represents a circle there is another instance of proceeding to
  a limit: e has to become 0, while ep remains finite: moreover [alpha]
  is indeterminate. The centre of a circle is its focus, and its
  directrix has gone to infinity, having no special direction. This last
  fact illustrates the necessity, which is also forced on plane geometry
  by three-dimensional considerations, of treating all points at
  infinity in a plane as lying on a single straight line.

  Sometimes, in reducing an equation to the above focus and directrix
  form, we find for h, k, e, p, tan [alpha], or some of them, only
  imaginary values, as quadratic equations have to be solved; and we
  have in fact to contemplate the existence of entirely imaginary
  conics. For instance, no real values of x and y satisfy x² + 2y² + 3 =
  0. Even when the locus represented is real, we obtain, as a rule, four
  sets of values of h, k, e, p, of which two sets are imaginary; a real
  conic has, besides two real foci and corresponding directrices, two
  others that are imaginary.

  In oblique as well as rectangular coordinates equations of the second
  degree represent conics.

  19. _The three Species of Conics._--A real conic, which does not
  degenerate into straight lines, is called an ellipse, parabola or
  hyperbola according as e <, = , or > 1. To trace the three forms it is best so to
  choose the axes of reference as to simplify their equations.

  In the case of a parabola, let 2c be the distance between the given
  focus and directrix, and take axes referred to which these are the
  point (c, 0) and the line x = - c. The equation becomes (x - c)² + y²
  = (x + c)², i.e. y² = 4cx.

  In the other cases, take a such that a(e ~ e^(-1)) is the distance of
  focus from directrix, and so choose axes that these are (ae, 0) and x
  = ae^(-1), thus getting the equation(x - ae)² + y² = e²(x - ae^(-1))²,
  i.e. (1 - e²)x² + y² = a²(1 - e²). When e < 1, i.e. in the case of an
  ellipse, this may be written x²/a² + y²/b² = 1, where b² = a²(1 - e²);
  and when e > 1, i.e. in the case of an hyperbola, x²/a² - y²/b² = 1,
  where b² = a²(e² - 1). The axes thus chosen for the ellipse and
  hyperbola are called the principal axes.

  In figs. 54, 55, 56 in order, conics of the three species, thus
  referred, are depicted.

  [Illustration: FIG. 54]

  [Illustration: FIG. 55]

  [Illustration: FIG. 56.]

  The oblique straight lines in fig. 56 are the _asymptotes_ x/a = ±y/b
  of the hyperbola, lines to which the curve tends with unlimited
  closeness as it goes to infinity. The hyperbola would have an equation
  of the form xy = c if referred to its asymptotes as axes, the
  coordinates being then oblique, unless a = b, in which case the
  hyperbola is called rectangular. An ellipse has two imaginary
  asymptotes. In particular a circle x² + y² = a², a particular ellipse,
  has for asymptotes the imaginary lines x = ±y [root]-1. These run from
  the centre to the so-called circular points at infinity.

  20. _Tangents and Curvature._--Let (x', y') and (x' + h, y' + k) be
  two neighbouring points P, P' on a curve. The equation of the line on
  which both lie is h(y - y') = k(x - x'). Now keep P fixed, and let P'
  move towards coincidence with it along the curve. The connecting line
  will tend towards a limiting position, to which it can never attain as
  long as P and P' are distinct. The line which occupies this limiting
  position is the tangent at P. Now if we subtract the equation of the
  curve, with (x', y') for the coordinates in it, from the like equation
  in (x' + h, y' + k), we obtain a relation in h and k, which will, as a
  rule, be of the form 0 = Ah + Bk + terms of higher degrees in h and k,
  where A, B and the other coefficients involve x' and y'. This gives
  k/h = -A/B + terms which tend to vanish as h and k do, so that -A : B
  is the limiting value tended to by k : h. Hence the equation of the
  tangent is B(y - y') + A(x - x') = 0.

  The _normal_ at (x', y') is the line through it at right angles to the
  tangent, and its equation is A(y - y') - B(x - x') = 0.

  In the case of the conic (a, b, c, f, g, h) (x, y, 1)² = 0 we find
  that A/B = (ax' + hy' + g)/(hx' + by' + f).

  We can obtain the coordinates of Q, the intersection of the normals
  QP, QP' at (x', y') and (x' + h, y' + k), and then, using the limiting
  value of k : h, deduce those of its limiting position as P' moves up
  to P. This is the _centre of curvature_ of the curve at P (x', y'),
  and is so called because it is the centre of the circle of closest
  contact with the curve at that point. That it is so follows from the
  facts that the closest circle is the limit tended to by the circle
  which touches the curve at P and passes through P', and that the arc
  from P to P' of this circle lies between the circles of centre Q and
  radii QP, QP', which circles tend, not to different limits as P' moves
  up to P, but to one. The distance from P to the centre of curvature is
  the _radius of curvature_.

  21. _Differential Plane Geometry._--The language and notation of the
  differential calculus are very useful in the study of tangents and
  curvature. Denoting by ([xi], [eta]) the current coordinates, we find,
  as above, that the tangent at a point (x, y) of a curve is [eta] - y =
  ([xi] - x)dy/dx, where dy/dx is found from the equation of the curve.
  If this be f(x, y) = 0 the tangent is ([xi] - x) (dPf/dPx) + ([eta] - y)
  (dPf/dPy) = 0. If [rho] and ([alpha], ß) are the radius and centre of
  curvature at (x, y), we find that q([alpha] - x) = -p(1 + p²), q(ß -
  y) = 1 + p², q²[rho]² = (1 + p²)³, where p, q denote dy/dx, d²y/dx²
  respectively. (See INFINITESIMAL CALCULUS.)

  In any given case we can, at all events in theory, eliminate x, y
  between the above equations for [alpha] - x and ß - y, and the
  equation of the curve. The resulting equation in ([alpha], ß)
  represents the locus of the centre of curvature. This is the _evolute_
  of the curve.

  22. _Polar Coordinates._--In plane geometry the distance of any point
  P from a fixed origin (or pole) O, and the inclination xOP of OP to a
  fixed line Ox, determine the point: r, the numerical measure of OP,
  the _radius vector_, and [theta], the circular measure of xOP, the
  _inclination_, are called polar coordinates of P. The formulae x = r
  cos [theta], y = r sin [theta] connect Cartesian and polar
  coordinates, and make transition from either system to the other easy.
  In polar coordinates the equations of a circle through O, and of a
  conic with O as focus, take the simple forms r = 2a cos
  ([theta]-[alpha]), r {1 - e cos ([theta]-[alpha])} = l. The use of
  polar coordinates is very convenient in discussing curves which have
  properties of symmetry akin to that of a regular polygon, such curves
  for instance as r = a cos m [theta], with m integral, and also the
  curves called spirals, which have equations giving r as functions of
  [theta] itself, and not merely of sin [theta] and cos [theta]. In the
  geometry of motion under central forces the advantage of working with
  polar coordinates is great.

  23. _Trilinear and Areal Coordinates._--Consider a fixed triangle ABC,
  and regard its sides as produced without limit. Denote, as in
  trigonometry, by a, b, c the positive numbers of units of a chosen
  scale contained in the lengths BC, CA, AB, by A, B, C the angles, and
  by [Delta] the area, of the triangle. We might, as in § 6, take CA, CB
  as axes of x and y, inclined at an angle C. Any point P (x, y) in the
  plane is at perpendicular distances y sin C and x sin C from CA and
  CB. Call these ß and [alpha] respectively. The signs of ß and [alpha]
  are those of y and x, i.e. ß is positive or negative according as P
  lies on the same side of CA as B does or the opposite, and similarly
  for [alpha]. An equation in (x, y) of any degree may, upon replacing
  in it x and y by [alpha] cosec C and ß cosec C, be written as one of
  the same degree in ([alpha], ß). Now let [gamma] be the perpendicular
  distance of P from the third side AB, taken as positive or negative as
  P is on the C side of AB or not. The geometry of the figure tells us
  that a[alpha] + bß + c[gamma] = 2[Delta]. By means of this relation in
  [alpha], ß, [gamma] we can give an equation considered countless other
  forms, involving two or all of [alpha], ß, [gamma]. In particular we
  may make it _homogeneous_ in [alpha], ß, [gamma]: to do this we have
  only to multiply the terms of every degree less than the highest
  present in the equation by a power of (a[alpha] + bß +
  c[gamma])/2[Delta] just sufficient to raise them, in each case, to the
  highest degree.

  We call ([alpha], ß, [gamma]) _trilinear coordinates_, and an equation
  in them the trilinear equation of the locus represented. Trilinear
  equations are, as a rule, dealt with in their homogeneous forms. An
  advantage thus gained is that we need not mean by ([alpha], ß,
  [gamma]) the actual measures of the perpendicular distances, but any
  properly signed numbers which have the same ratio two and two as these
  distances.

  In place of [alpha], ß, [gamma] it is lawful to use, as coordinates
  specifying the position of a point in the plane of a triangle of
  reference ABC, any given multiples of these. For instance, we may use
  x = a[alpha]/2[Delta], y = bß/2[Delta], z = c[gamma]/2[Delta], the
  properly signed ratios of the triangular areas PBC, PCA, PAB to the
  triangular area ABC. These are called the _areal_ coordinates of P. In
  areal coordinates the relation which enables us to make any equation
  homogeneous takes the simple form x + y + z = 1; and, as before, we
  need mean by x, y, z, in a homogeneous equation, only signed numbers
  in the right ratios.

  Straight lines and conics are represented in trilinear and in areal,
  because in Cartesian, coordinates by equations of the first and second
  degrees respectively, and these degrees are preserved when the
  equations are made homogeneous. What must be said about points
  infinitely far off in order to make universal the statement, to which
  there is no exception as long as finite distances alone are
  considered, that _every_ homogeneous equation of the first degree
  represents a straight line? Let the point of areal coordinates (x',
  y', z') move infinitely far off, and mean by x, y, z finite quantities
  in the ratios which x', y', z' tend to assume as they become infinite.
  The relation x' + y' + z' = 1 gives that the limiting state of things
  tended to is expressed by x + y + z = 0. This particular equation of
  the first degree is satisfied by no point at a finite distance; but we
  see the propriety of saying that it has to be taken as satisfied by
  all the points conceived of as actually at infinity. Accordingly the
  special property of these points is expressed by saying that they lie
  on a special straight line, of which the areal equation is x + y + z =
  0. In trilinear coordinates this _line at infinity_ has for equation
  a[alpha] + bß + c[gamma] = 0.

  On the one special line at infinity parallel lines are treated as
  meeting. There are on it two special (imaginary) points, the circular
  points at infinity of § 19, through which all circles pass in the same
  sense. In fact if S = O be one circle, in areal coordinates, S + (x +
  y + z)(lx + my + nz) = 0 may, by proper choice of l, m, n, be made any
  other; since the added terms are once lx + my + nz, and have the
  generality of any expression like a'x + b'y + c' in Cartesian
  coordinates. Now these two circles intersect in the two points where
  either meets x + y + z = 0 as well as in two points on the radical
  axis lx + my + nz = 0.

  24. Let us consider the perpendicular distance of a point ([alpha]',
  ß', [gamma]') from a line l[alpha] + mß + n[gamma]. We can take
  rectangular axes of Cartesian coordinates (for clearness as to
  equalities of angle it is best to choose an origin inside ABC), and
  refer to them, by putting expressions p - x cos[theta] - y sin[theta],
  &c., for [alpha] &c.; we can then apply § 16 to get the perpendicular
  distance; and finally revert to the trilinear notation. The result is
  to find that the required distance is

    (l[alpha]' + mß' + n[gamma]')/{l, m, n},

  where {l, m, n}² = l² + m² + n² - 2mn cos A - 2nl cos B - 2lm cos C.

  In areal coordinates the perpendicular distance from (x', y', z') to
  lx + my + nz = 0 is 2[Delta](lx' + my' + nz')/{al, bm, cn}. In both
  cases the coordinates are of course actual values.

  Now let [xi], [eta], [zeta] be the perpendiculars on the line from the
  vertices A, B, C, i.e. the points (1, 0, 0), (0, 1, 0), (0, 0, 1),
  with signs in accord with a convention that oppositeness of sign
  implies distinction between one side of the line and the other. Three
  applications of the result above give

    [xi]/l = 2[Delta]/{al, bm, cn} = [eta]/m = [zeta]/n;

  and we thus have the important fact that [xi]x' + [eta]y' + [zeta]z'
  is the perpendicular distance between a point of areal coordinates
  (x'y'z') and a line on which the perpendiculars from A, B, C are [xi],
  [eta], [zeta] respectively. We have also that [xi]x + [eta]y + [zeta]z
  = 0 is the areal equation of the line on which the perpendiculars are
  [xi], [eta], [zeta]; and, by equating the two expressions for the
  perpendiculars from (x', y', z') on the line, that in all cases
  {a[xi], b[eta], c[zeta]}² = 4[Delta]².

  25. _Line-coordinates. Duality._--A quite different order of ideas
  may be followed in applying analysis to geometry. The notion of a
  straight line specified may precede that of a point, and points may be
  dealt with as the intersections of lines. The specification of a line
  may be by means of coordinates, and that of a point by an equation,
  satisfied by the coordinates of lines which pass through it. Systems
  of _line-coordinates_ will here be only briefly considered. Every such
  system is allied to some system of point-coordinates; and space will
  be saved by giving prominence to this fact, and not recommencing _ab
  initio_.

  Suppose that any particular system of point-coordinates, in which lx +
  my + nz = 0 may represent any straight line, is before us: notice that
  not only are trilinear and areal coordinates such systems, but
  Cartesian coordinates also, since we may write x/z, y/z for the
  Cartesian x, y, and multiply through by z. The line is exactly
  assigned if l, m, n, or their mutual ratios, are known. Call (l, m, n)
  the _coordinates_ of the line. Now keep x, y, z constant, and let the
  coordinates of the line vary, but always so as to satisfy the
  equation. This equation, which we now write xl + ym + zn = 0, is
  satisfied by the coordinates of every line through a certain fixed
  point, and by those of no other line; it is the equation of that point
  in the line-coordinates l, m, n.

  Line-coordinates are also called _tangential_ coordinates. A curve is
  the envelope of lines which touch it, as well as the locus of points
  which lie on it. A homogeneous equation of degree above the first in
  l, m, n is a relation connecting the coordinates of every line which
  touches some curve, and represents that curve, regarded as an
  envelope. For instance, the condition that the line of coordinates (l,
  m, n), i.e. the line of which the allied point-coordinate equation is
  lx + my + nz = 0, may touch a conic (a, b, c, f, g, h) (x, y, z)² = 0,
  is readily found to be of the form (A, B, C, F, G, H) (l, m, n)² = 0,
  i.e. to be of the second degree in the line-coordinates. It is not
  hard to show that the _general_ equation of the second degree in l, m,
  n thus represents a conic; but the degenerate conics of
  line-coordinates are not line-pairs, as in point-coordinates, but
  point-pairs.

  The degree of the point-coordinate equation of a curve is the _order_
  of the curve, the number of points in which it cuts a straight line.
  That of the line-coordinate equation is its _class_, the number of
  tangents to it from a point. The order and class of a curve are
  generally different when either exceeds two.

  26. The system of line-coordinates allied to the areal system of
  point-coordinates has special interest.

  The l, m, n of this system are the perpendiculars [xi], [eta], [zeta]
  of § 24; and x'[xi] + y'[eta] + z'[zeta] = 0 is the equation of the
  point of areal coordinates (x', y', z'), i.e. is a relation which the
  perpendiculars from the vertices of the triangle of reference on every
  line through the point, but no other line, satisfy. Notice that a
  non-homogeneous equation of the first degree in [xi], [eta], [zeta]
  does not, as a homogeneous one does, represent a point, but a circle.
  In fact x'[xi] + y'[eta] + z'[zeta] = R expresses the constancy of the
  perpendicular distance of the fixed point x'[xi] + y'[eta] + z'[zeta]
  = 0 from the variable line ([xi], [eta], [zeta]), i.e. the fact that
  ([xi], [eta], [zeta]) touches a circle with the fixed point for
  centre. The relation in any [xi], [eta], [zeta] which enables us to
  make an equation homogeneous is not linear, as in point-coordinates,
  but quadratic, viz. it is the relation {a[xi], b[eta], c[zeta]}² =
  4[Delta]² of § 24. Accordingly the homogeneous equation of the above
  circle is

    4[Delta]²(x'[xi] + y'[eta] + z'[zeta])² = R²{a[xi], b[eta], c[zeta]}².

  Every circle has an equation of this form in the present system of
  line-coordinates. Notice that the equation of any circle is satisfied
  by those coordinates of lines which satisfy both x'[xi] + y'[eta] +
  z'[zeta] = 0, the equation of its centre, and {a[xi], b[eta],
  c[zeta]}² = 0. This last equation, of which the left-hand side
  satisfies the condition for breaking up into two factors, represents
  the two imaginary circular points at infinity, through which all
  circles and their asymptotes pass.

  There is strict duality in descriptive geometry between
  point-line-locus and line-point-envelope theorems. But in metrical
  geometry duality is encumbered by the fact that there is in a plane
  one special line only, associated with distance, while of special
  points, associated with direction, there are two: moreover the line is
  real, and the points both imaginary.


  II. _Solid Analytical Geometry._

  27. Any point in space may be specified by three coordinates. We
  consider three fixed planes of reference, and generally, as in all
  that follows, three which are at right angles two and two. They
  intersect, two and two, in lines x'Ox, y'Oy, z'Oz, called the axes of
  x, y, z respectively, and divide all space into eight parts called
  octants. If from any point P in space we draw PN parallel to zOz' to
  meet the plane xOy in N, and then from N draw NM parallel to yOy' to
  meet x'Ox in M, the coordinates (x, y, z) of P are the numerical
  measures of OM, MN, NP; in the case of rectangular coordinates these
  are the perpendicular distances of P from the three planes of
  reference. The sign of each coordinate is positive or negative as P
  lies on one side or the other of the corresponding plane. In the
  octant delineated the signs are taken all positive.

  [Illustration: FIG. 57.]

  [Illustration: FIG. 58.]

  In fig. 57 the delineation is on a plane of the paper taken parallel
  to the plane zOx, the points of a solid figure being projected on that
  plane by parallels to some chosen line through O in the positive
  octant. Sometimes it is clearer to delineate, as in fig. 58, by
  projection parallel to that line in the octant which is equally
  inclined to Ox, Oy, Oz upon a plane of the paper perpendicular to it.
  It is possible by parallel projection to delineate equal scales along
  Ox, Oy, Oz by scales having any ratios we like along lines in a plane
  having any mutual inclinations we like.

  [Illustration: FIG. 59.]

  For the delineation of a surface of simple form it frequently suffices
  to delineate the sections by the coordinate planes; and, in
  particular, when the surface has symmetry about each coordinate plane,
  to delineate the quarter-sections belonging to a single octant. Thus
  fig. 59 conveniently represents an octant of the wave surface, which
  cuts each coordinate plane in a circle and an ellipse. Or we may
  delineate a series of contour lines, i.e. sections by planes parallel
  to xOy, or some other chosen plane; of course other sections may be
  indicated too for greater clearness. For the delineation of a curve a
  good method is to represent, as above, a series of points P thereof,
  each accompanied by its ordinate PN, which serves to refer it to the
  plane of xy. The employment of stereographic projection is also
  interesting.

  28. In plane geometry, reckoning the line as a curve of the first
  order, we have only the point and the curve. In solid geometry,
  reckoning a line as a curve of the first order, and the plane as a
  surface of the first order, we have the point, the curve and the
  surface; but the increase of complexity is far greater than would
  hence at first sight appear. In plane geometry a curve is considered
  in connexion with lines (its tangents); but in solid geometry the
  curve is considered in connexion with lines and planes (its tangents
  and osculating planes), and the surface also in connexion with lines
  and planes (its tangent lines and tangent planes); there are surfaces
  arising out of the line--cones, skew surfaces, developables, doubly
  and triply infinite systems of lines, and whole classes of theories
  which have nothing analogous to them in plane geometry: it is thus a
  very small part indeed of the subject which can be even referred to in
  the present article.

  In the case of a surface we have between the coordinates (x, y, z) a
  single, or say a onefold relation, which can be represented by a
  single relation [f](x, y, z) = 0; or we may consider the coordinates
  expressed each of them as a given function of two variable parameters
  p, q; the form z = [f](x, y) is a particular case of each of these
  modes of representation; in other words, we have in the first mode
  [f](x, y, z) = z - [f](x, y), and in the second mode x = p, y = q for
  the expression of two of the coordinates in terms of the parameters.

  In the case of a curve we have between the coordinates (x, y, z) a
  twofold relation: two equations [f](x, y, z) = 0, [phi](x, y, z) = 0
  give such a relation; i.e. the curve is here considered as the
  intersection of two surfaces (but the curve is not always the complete
  intersection of two surfaces, and there are hence difficulties); or,
  again, the coordinates may be given each of them as a function of a
  single variable parameter. The form y = [phi](x), z = [psi](x), where
  two of the coordinates are given in terms of the third, is a
  particular case of each of these modes of representation.

  29. The remarks under plane geometry as to descriptive and metrical
  propositions, and as to the non-metrical character of the method of
  coordinates when used for the proof of a descriptive proposition,
  apply also to solid geometry; and they might be illustrated in like
  manner by the instance of the theorem of the radical centre of four
  spheres. The proof is obtained from the consideration that S and S'
  being each of them a function of the form x² + y² + z² + ax + by + cz
  + d, the difference S-S' is a mere linear function of the coordinates,
  and consequently that S-S' = 0 is the equation of the plane containing
  the circle of intersection of the two spheres S = 0 and S' = 0.

  [Illustration: FIG. 60.]

  30. _Metrical Theory._--The foundation in solid geometry of the
  metrical theory is in fact the before-mentioned theorem that if a
  finite right line PQ be projected upon any other line OO' by lines
  perpendicular to OO', then the length of the projection P'Q' is equal
  to the length of PQ into the cosine of its inclination to P'Q'--or (in
  the form in which it is now convenient to state the theorem) the
  perpendicular distance P'Q' of two parallel planes is equal to the
  inclined distance PQ into the cosine of the inclination. The principle
  of § 16, that the algebraical sum of the projections of the sides of
  any closed polygon on any line is zero, or that the two sets of sides
  of the polygon which connect a vertex A and a vertex B have the same
  sum of projections on the line, in sign and magnitude, as we pass from
  A to B, is applicable when the sides do not all lie in one plane.

  31. Consider the skew quadrilateral QMNP, the sides QM, MN, NP being
  respectively parallel to the three rectangular axes Ox, Oy, Oz; let
  the lengths of these sides be [xi], [eta], [zeta], and that of the
  side QP be = [rho]; and let the cosines of the inclinations (or say
  the cosine-inclinations) of [rho] to the three axes be [alpha], ß,
  [gamma]; then projecting successively on the three sides and on QP we
  have

    [xi], [eta], [zeta] = [rho][alpha], [rho]ß, [rho][gamma],

  and

    [rho] = [alpha][xi] + ß[eta] + [gamma][zeta],

  whence [rho]² = [xi]² + [eta]² + [zeta]², which is the relation
  between a distance [rho] and its projections [xi], [eta], [zeta] upon
  three rectangular axes. And from the same equations we obtain [alpha]²
  + ß² + [gamma]² = 1, which is a relation connecting the
  cosine-inclinations of a line to three rectangular axes.

  Suppose we have through Q any other line QT, and let the
  cosine-inclinations of this to the axes be [alpha]', ß', [gamma]', and
  [delta] be its cosine-inclination to QP; also let [rho] be the length
  of the projection of QP upon QT; then projecting on QT we have

    [rho] = [alpha]'[xi] + ß'[eta] + [gamma]'[zeta] = [rho][delta].

  And in the last equation substituting for [xi], [eta], [zeta] their
  values [rho][alpha], [rho]ß, [rho][gamma] we find

    [delta] = [alpha][alpha]' + ß[beta]' + [gamma][gamma]',

  which is an expression for the mutual cosine-inclination of two lines,
  the cosine-inclinations of which to the axes are [alpha], ß, [gamma]
  and [alpha]', ß', [gamma]' respectively. We have of course [alpha]² +
  ß² + [gamma]² = 1 and [alpha]'² + ß'² + [gamma]'² = 1; and hence also

    1 - [delta]² = ([alpha]² + ß² + [gamma]²)([alpha]'² + ß'² + [gamma]'²)
      - ([alpha][alpha]' + ß[beta]' + [gamma][gamma]')²,

    = (ß[gamma]' - ß'[gamma])² + ([gamma][alpha]' - [gamma]'[alpha])² +
      ([alpha]ß' - [alpha]'ß)²;

  so that the sine of the inclination can only be expressed as a square
  root. These formulae are the foundation of spherical trigonometry.

  32. _Straight Lines, Planes and Spheres._--The foregoing formulae give
  at once the equations of these loci.

  For first, taking Q to be a fixed point, coordinates (a, b, c), and
  the cosine-inclinations ([alpha], ß, [gamma]) to be constant, then P
  will be a point in the line through Q in the direction thus
  determined; or, taking (x, y, z) for its coordinates, these will be
  the current coordinates of a point in the line. The values of [xi],
  [eta], [zeta] then are x - a, y - b, z - c, and we thus have

     x - a    y - b    z - c
    ------- = ----- = ------- (= [rho]),
    [alpha]     ß     [gamma]

  which (omitting the last equation, = [rho]) are the equations of the
  line through the point (a, b, c), the cosine-inclinations to the axes
  being [alpha], ß, [gamma], and these quantities being connected by the
  relation [alpha]² + ß² + [gamma]² = 1. This equation may be omitted,
  and then [alpha], ß, [gamma], instead of being equal, will only be
  proportional, to the cosine-inclinations.

  Using the last equation, and writing

    x, y, z = a + [alpha][rho], b + ß[rho], c + [gamma][rho],

  these are expressions for the current coordinates in terms of a
  parameter [rho], which is in fact the distance from the fixed point
  (a, b, c).

  It is easy to see that, if the coordinates (x, y, z) are connected by
  any two linear equations, these equations can always be brought into
  the foregoing form, and hence that the two linear equations represent
  a line.

  Secondly, taking for greater simplicity the point Q to be coincident
  with the origin, and [alpha]', ß', [gamma]', p to be constant, then p
  is the perpendicular distance of a plane from the origin, and
  [alpha]', ß', [gamma]' are the cosine-inclinations of this distance to
  the axes ([alpha]'² + ß'² + [gamma]'² = 1). P is any point in this
  plane, and taking its coordinates to be (x, y, z) then ([xi], [eta],
  [zeta]) are = (x, y, z), and the foregoing equation p = [alpha]'[xi] +
  ß'[eta] + [gamma]'[zeta] becomes

    [alpha]'x + ß'y + [gamma]'z = p,

  which is the equation of the plane in question.

  If, more generally, Q is not coincident with the origin, then, taking
  its coordinates to be (a, b, c), and writing p1 instead of p, the
  equation is

    [alpha]'(x - a) + ß'(y - b) + [gamma]'(z - c) = p1;

  and we thence have p1 = p - (a[alpha]' + bß' + c[gamma]'), which is an
  expression for the perpendicular distance of the point (a, b, c) from
  the plane in question.

  It is obvious that any linear equation Ax + By + Cz + D = O between
  the coordinates can always be brought into the foregoing form, and
  hence that such an equation represents a plane.

  Thirdly, supposing Q to be a fixed point, coordinates (a, b, c), and
  the distance QP = [rho], to be constant, say this is = d, then, as
  before, the values of [xi], [eta], [zeta] are x - a, y - b, z - c, and
  the equation [xi]² + [eta]² + [zeta]² = [rho]² becomes

    (x - a)² + (y - b)² + (z - c)² = d²,

  which is the equation of the sphere, coordinates of the centre = (a,
  b, c), and radius = d.

  A quadric equation wherein the terms of the second order are x² + y² +
  z², viz. an equation

    x² + y² + z² + Ax + By + Cz + D = 0,

  can always, it is clear, be brought into the foregoing form; and it
  thus appears that this is the equation of a sphere, coordinates of the
  centre -½A, -½B, -½C, and squared radius = ¼(A² + B² + C²) - D.

  33. _Cylinders, Cones, ruled Surfaces._--If the two equations of a
  straight line involve a parameter to which any value may be given, we
  have a singly infinite system of lines. They cover a surface, and the
  equation of the surface is obtained by eliminating the parameter
  between the two equations.

  If the lines all pass through a given point, then the surface is a
  cone; and, in particular, if the lines are all parallel to a given
  line, then the surface is a cylinder.

  Beginning with this last case, suppose the lines are parallel to the
  line x = mz, y = nz, the equations of a line of the system are x = mz
  + a, y = nz + b,--where a, b are supposed to be functions of the
  variable parameter, or, what is the same thing, there is between them
  a relation f(a, b) = 0: we have a = x - mz, b = y - nz, and the result
  of the elimination of the parameter therefore is [f](x - mz, y - nz) =
  0, which is thus the general equation of the cylinder the generating
  lines whereof are parallel to the line x = mz, y = nz. The equation of
  the section by the plane z = 0 is [f](x, y) = 0, and conversely if the
  cylinder be determined by means of its curve of intersection with the
  plane z = 0, then, taking the equation of this curve to be f(x, y) =
  0, the equation of the cylinder is [f](x - mz, y - nz) = 0. Thus, if
  the curve of intersection be the circle (x - [alpha])² + (y - ß)² =
  [gamma]², we have (x - mz - [alpha])² + (y - nz - ß)² = [gamma]² as
  the equation of an oblique cylinder on this base, and thus also (x -
  [alpha])² + (y - ß)² = [gamma]² as the equation of the right cylinder.

  If the lines all pass through a given point (a, b, c), then the
  equations of a line are x - a = [alpha](z - c), y - b = ß(z - c),
  where [alpha], ß are functions of the variable parameter, or, what is
  the same thing, there exists between them an equation f([alpha], ß) =
  0; the elimination of the parameter gives, therefore, f[(x - a)/(x -
  c'), (y - b)/(z - c)] = 0; and this equation, or, what is the same
  thing, any homogeneous equation f(x - a, y - b, z - c) = 0, or, taking
  f to be a rational and integral function of the order n, say (*)(x -
  a, y - b, z - c)^n = 0, is the general equation of the cone having the
  point (a, b, c) for its vertex. Taking the vertex to be at the origin,
  the equation is (*)(x, y, z)^n = 0; and, in particular, (*)(x, y, z)²
  = 0 is the equation of a cone of the second order, or quadricone,
  having the origin for its vertex.

  34. In the general case of a singly infinite system of lines, the
  locus is a ruled surface (or _regulus_). Now, when a line is changing
  its position in space, it may be looked upon as in a state of turning
  about some point in itself, while that point is, as a rule, in a state
  of moving out of the plane in which the turning takes place. If
  instantaneously it is only in a state of turning, it is usual, though
  not strictly accurate, to say that it intersects its consecutive
  position. A regulus such that consecutive lines on it do not
  intersect, in this sense, is called a skew surface, or _scroll_; one
  on which they do is called a developable surface or _torse_.

  Suppose, for instance, that the equations of a line (depending on the
  variable parameter [theta]) are x/a + y/c = [theta] (1 + y/b), x/a -
  z/c = 1/[theta] (1 - y/b); then, eliminating [theta] we have x²/a² -
  z²/c² = 1 - y²/b², or say, x²/a² + z²/b² - z²/c² = 1, the equation of
  a quadric surface, afterwards called the hyperboloid of one sheet;
  this surface is consequently a scroll. It is to be remarked that we
  have upon the surface a second singly infinite series of lines; the
  equations of a line of this second system (depending on the variable
  parameter [phi]) are

    x    z          /    y \   x    z      1    /    y \
    -- + -- = [phi]( 1 - -- ), -- - -- = ----- ( 1 + -- ).
    a    c          \    b /   a    c    [phi]  \    b /

  It is easily shown that any line of the one system intersects every
  line of the other system.

  Considering any curve (of double curvature) whatever, the tangent
  lines of the curve form a singly infinite system of lines, each line
  intersecting the consecutive line of the system,--that is, they form a
  developable, or torse; the curve and torse are thus inseparably
  connected together, forming a single geometrical figure. An osculating
  plane of the curve (see § 38 below) is a tangent plane of the torse
  all along a generating line.

  35. _Transformation of Coordinates._--There is no difficulty in
  changing the origin, and it is for brevity assumed that the origin
  remains unaltered. We have, then, two sets of rectangular axes, Ox,
  Oy, Oz, and Ox1, Oy1, Ozx1, the mutual cosine-inclinations being shown
  by the diagram--

        |    x    |    y  |    z    |
    ----+---------+-------+---------+
    x1  | [alpha] |   ß   | [gamma] |
    ----+---------+-------+---------+
    y1  | [alpha] |   ß'  | [gamma]'|
    ----+---------+-------+---------+
    z1  | [alpha]"|   ß"  | [gamma]"|
    ----+---------+-------+---------+

  that is, [alpha], ß, [gamma] are the cosine-inclinations of Ox1 to Ox,
  Oy, Oz; [alpha]', ß', [gamma]' those of Oy1, &c.

  And this diagram gives also the linear expressions of the coordinates
  (x1, y1, z1) or (x, y, z) of either set in terms of those of the other
  set; we thus have

    x1 = [alpha] x + ß y + [gamma] z,
     x = [alpha]x1 + [alpha]'y1 + [alpha]"z1,

    y1 = [alpha]'x + ß'y + [gamma]'z,
     y = ßx1 + ß'y1 + ß"z1,

    z1 = [alpha]"x + ß"y + [gamma]"z,
     z = [gamma]x1 + [gamma]'y1 + [gamma]"z1,

  which are obtained by projection, as above explained. Each of these
  equations is, in fact, nothing else than the before-mentioned equation
  p = [alpha]'[xi] + ß'[eta] + [gamma]'[zeta], adapted to the problem in
  hand.

  But we have to consider the relations between the nine coefficients.
  By what precedes, or by the consideration that we must have
  identically x² + y² + z² = x1² + y1² + z1², it appears that these
  satisfy the relations--

    a² + ß² + [gamma]² = 1,
    [alpha]² + [alpha]'² + [alpha]"² = 1,

    [alpha]'² + ß'² + [gamma]'² = 1,
    ß² + ß'² + ß"² = 1,

    [alpha]"² + ß"² + [gamma]"² = 1,
    [gamma]² + [gamma]'² + [gamma]"² = 1,

    a'a" + ß'ß" + [gamma]'[gamma]" = 0,
    ß[gamma] +ß'[gamma]' + ß"[gamma]" = 0,

    [alpha]"[alpha] + ß"ß + [gamma]"[gamma] = 0,
    [gamma][alpha] + [gamma]'[alpha]' + [gamma]"[alpha]" = 0,

    [alpha][alpha]' + ß[beta]' + [gamma][gamma]' = 0,
    [alpha]ß +[alpha]'ß' + [alpha]"ß" = 0,

  either set of six equations being implied in the other set.

  It follows that the square of the determinant

    |[alpha],  ß,  [gamma] |
    |                      |
    |[alpha]', ß', [gamma]'|
    |                      |
    |[alpha]", ß", [gamma]"|

  is = 1; and hence that the determinant itself is = ± 1. The
  distinction of the two cases is an important one: if the determinant
  is = + 1, then the axes Ox1, Oy1, Oz1 are such that they can by a
  rotation about O be brought to coincide with Ox, Oy, Oz respectively;
  if it is = -1, then they cannot. But in the latter case, by measuring
  x1, y1, z1 in the opposite directions we change the signs of all the
  coefficients and so make the determinant to be = + 1; hence the former
  case need alone be considered, and it is accordingly assumed that the
  determinant is = + 1. This being so, it is found that we have the
  equality [alpha] = ß'[gamma]" - ß"[gamma]', and eight like ones,
  obtained from this by cyclical interchanges of the letters [alpha], ß,
  [gamma], and of unaccented, singly and doubly accented letters.

  36. The nine cosine-inclinations above are, as has been seen,
  connected by six equations. It ought then to be possible to express
  them all in terms of three parameters. An elegant means of doing this
  has been given by Rodrigues, who has shown that the tabular expression
  of the formulae of transformation may be written

        |            x             |            y             |            z             |
    ----+--------------------------+--------------------------+--------------------------+
    x1  |1 + [lambda]² - µ² - [nu]²|    2([lambda]µ - [nu])   |   2([nu][lambda] + µ)    |
    ----+--------------------------+--+-----------------------+-----+--------------------+
    y1  |    2([lambda]µ + [nu])   |1 - [lambda]² + µ² - [nu]²|   2(µ[nu] + [lambda])    |
    ----+--------------------------+--+-----------------------+-----+--------------------+
    z1  |    2([nu][lambda] - µ)   |    2(µ[nu] + [lambda])   |1 - [lambda]² - µ² + [nu]²|
    ----+--------------------------+--------------------------+--------------------------+
       ÷(1 + [lambda]² + µ² + [nu]²),

  the meaning being that the coefficients in the transformation are
  fractions, with numerators expressed as in the table, and the common
  denominator.

  37. _The Species of Quadric Surfaces_.--Surfaces represented by
  equations of the second degree are called _quadric_ surfaces. Quadric
  surfaces are either _proper_ or _special_. The special ones arise when
  the coefficients in the general equation are limited to satisfy
  certain special equations; they comprise (1) plane-pairs, including in
  particular one plane twice repeated, and (2) cones, including in
  particular cylinders; there is but one form of cone, but cylinders may
  be elliptic, parabolic or hyperbolic.

  A discussion of the general equation of the second degree shows that
  the _proper_ quadric surfaces are of five kinds, represented
  respectively, when referred to the most convenient axes of reference,
  by equations of the five types (a and b positive):

             x²   y²
    (1)  z = -- + --, elliptic paraboloid.
             2a   2b

             x²   y²
    (2)  z = -- - --, hyperbolic paraboloid.
             2a   2b

         x²   y²   z²
    (3)  -- + -- + -- = 1, ellipsoid.
         a²   b²   c²

         x²   y²   z²
    (4)  -- + -- - -- = 1, hyperboloid of one sheet.
         a²   b²   c²

         x²   y²   z²
    (5)  -- + -- - -- = -1, hyperboloid of two sheets.
         a²   b²   c²

  It is at once seen that these are distinct surfaces; and the equations
  also show very readily the general form and mode of generation of the
  several surfaces.

  [Illustration: FIG. 61.]

  In the elliptic paraboloid (fig. 61) the sections by the planes of zx
  and zy are the parabolas

        x²      y²
    z = --, z = --
        2a      2b

  having the common axes Oz; and the section by any plane z = [gamma]
  parallel to that of xy is the ellipse

              x²   y²
    [gamma] = -- + --;
              2a   2b

  so that the surface is generated by a variable ellipse moving parallel
  to itself along the parabolas as directrices.

  [Illustration: FIG. 62.]

  [Illustration: FIG. 63.]

  In the hyperbolic paraboloid (figs. 62 and 63) the sections by the
  planes of zx, zy are the parabolas z = x²/2a, z = - y²/2b, having the
  opposite axes Oz, Oz', and the section by a plane z = [gamma] parallel
  to that of xy is the hyperbola [gamma] = x²/2a - y²/2b, which has its
  transverse axis parallel to Ox or Oy according as [gamma] is positive
  or negative. The surface is thus generated by a variable hyperbola
  moving parallel to itself along the parabolas as directrices. The form
  is best seen from fig. 63, which represents the sections by planes
  parallel to the plane of xy, or say the contour lines; the continuous
  lines are the sections above the plane of xy, and the dotted lines the
  sections below this plane. The form is, in fact, that of a saddle.

  [Illustration: FIG. 64.]

  In the ellipsoid (fig. 64) the sections by the planes of zx, zy, and
  xy are each of them an ellipse, and the section by any parallel plane
  is also an ellipse. The surface may be considered as generated by an
  ellipse moving parallel to itself along two ellipses as directrices.

  In the hyperboloid of one sheet (fig. 65), the sections by the planes
  of zx, zy are the hyperbolas

    x²   z²      y²   z²
    -- - -- = 1, -- - -- = 1,
    c²   c²      b²   c²

  having a common conjugate axis zOz'; the section by the plane of x, y,
  and that by any parallel plane, is an ellipse; and the surface may be
  considered as generated by a variable ellipse moving parallel to
  itself along the two hyperbolas as directrices. If we imagine two
  equal and parallel circular disks, their points connected by strings
  of equal lengths, so that these are the generators of a right circular
  cylinder, and if we turn one of the disks about its centre through an
  angle in its plane, the strings in their new positions will be one
  system of generators of a hyperboloid of one sheet, for which a = b;
  and if we turn it through the same angle in the opposite direction, we
  get in like manner the generators of the other system; there will be
  the same general configuration when a = | b. The hyperbolic paraboloid
  is also covered by two systems of rectilinear generators as a method
  like that used in § 34 establishes without difficulty. The figures
  should be studied to see how they can lie.

  [Illustration: FIG. 65.]

  [Illustration: FIG. 66.]

  In the hyperboloid of two sheets (fig. 66) the sections by the planes
  of zx and zy are the hyperbolas

    z²   x²      z²   y²
    -- - -- = 1, -- - -- = 1,
    c²   a²      c²   b²

  having a common transverse axis along z'Oz; the section by any plane z
  = ±[gamma] parallel to that of xy is the ellipse

    x²   y²   [gamma]²
    -- + -- = -------- - 1,
    a²   b²      c²

  provided [gamma]² > c², and the surface, consisting of two distinct
  portions or sheets, may be considered as generated by a variable
  ellipse moving parallel to itself along the hyperbolas as directrices.

  38. _Differential Geometry of Curves._--For convenience consider the
  coordinates (x, y, z) of a point on a curve in space to be given as
  functions of a variable parameter [theta], which may in particular be
  one of themselves. Use the notation x', x" for dx/d[theta],
  d²x/d[theta]², and similarly as to y and z. Only a few formulae will
  be given. Call the current coordinates ([xi], [eta], [zeta]).

  The _tangent_ at (x, y, z) is the line tended to as a limit by the
  connector of (x, y, z) and a neighbouring point of the curve when the
  latter moves up to the former: its equations are

    ([xi] - x)/x' = ([eta] - y)/y' = ([zeta] - z)/z'.

  The _osculating plane_ at (x, y, z) is the plane tended to as a limit
  by that through (x, y, z) and two neighbouring points of the curve as
  these, remaining distinct, both move up to (x, y, z): its one equation
  is

    ([xi] - x)(y'z" - y"z') + ([eta] - y)(z'x" - z"x') + ([zeta] - z)
      (x'y" - x"y') = 0.

  The _normal plane_ is the plane through (x, y, z) at right angles to
  the tangent line, i.e. the plane

    x'([xi] - x) + y'([eta] - y) + z'([zeta] - z) = 0.

  It cuts the osculating plane in a line called the _principal normal_.
  Every line through (x, y, z) in the normal plane is a normal. The
  normal perpendicular to the osculating plane is called the _binormal_.
  A tangent, principal normal, and binormal are a convenient set of
  rectangular axes to use as those of reference, when the nature of a
  curve near a point on it is to be discussed.

  Through (x, y, z) and three neighbouring points, all on the curve,
  passes a single sphere; and as the three points all move up to (x, y,
  z) continuing distinct, the sphere tends to a limiting size and
  position. The limit tended to is the sphere of closest contact with
  the curve at (x, y, z); its centre and radius are called the centre
  and radius of _spherical curvature_. It cuts the osculating plane in a
  circle, called the _circle of absolute curvature_; and the centre and
  radius of this circle are the centre and radius of absolute curvature.
  The centre of absolute curvature is the limiting position of the point
  where the principal normal at (x, y, z) is cut by the normal plane at
  a neighbouring point, as that point moves up to (x, y, z).

  39. _Differential Geometry of Surfaces._--Let (x, y, z) be any chosen
  point on a surface [f](x, y, z) = 0. As a second point of the surface
  moves up to (x, y, z), its connector with (x, y, z) tends to a
  limiting position, a tangent line to the surface at (x, y, z). All
  these tangent lines at (x, y, z), obtained by approaching (x, y, z)
  from different directions on a surface, lie in one plane

    dP[f]              dP[f]               dP[f]
    ----- ([xi] - x) + ----- ([eta] - y) + ----- ([zeta] - z) = 0.
     dPx                dPy                 dPz

  This plane is called the _tangent plane_ at (x, y, z). One line
  through (x, y, z) is at right angles to the tangent plane. This is the
  normal

                /dP[f]                /dP[f]                   /dP[f]
    ([xi] - x) / ----- = ([eta] - y) / ----- = ([zeta] - z) = / -----.
              /   dPx               /   dPy                  /   dPz

  The tangent plane is cut by the surface in a curve, real or imaginary,
  with a node or double point at (x, y, z). Two of the tangent lines
  touch this curve at the node. They are called the "chief tangents"
  (_Haupt-tangenten_) at (x, y, z); they have closer contact with the
  surface than any other tangents.

  In the case of a quadric surface the curve of intersection of a
  tangent and the surface is of the second order and has a node, it must
  therefore consist of two straight lines. Consequently a quadric
  surface is covered by two sets of straight lines, a pair through every
  point on it; these are imaginary for the ellipsoid, hyperboloid of two
  sheets, and elliptic paraboloid.

  A surface of any order is covered by two singly infinite systems of
  curves, a pair through every point, the tangents to which are all
  chief tangents at their respective points of contact. These are called
  _chief-tangent curves_; on a quadric surface they are the above
  straight lines.

  40. The tangents at a point of a surface which bisect the angles
  between the chief tangents are called the _principal tangents_ at the
  point. They are at right angles, and together with the normal
  constitute a convenient set of rectangular axes to which to refer the
  surface when its properties near the point are under discussion. At a
  special point which is such that the chief tangents there run to the
  circular points at infinity in the tangent plane, the principal
  tangents are indeterminate; such a special point is called an umbilic
  of the surface.

  There are two singly infinite systems of curves on a surface, a pair
  cutting one another at right angles through every point upon it, all
  tangents to which are principal tangents of the surface at their
  respective points of contact. These are called _lines of curvature_,
  because of a property next to be mentioned.

  As a point Q moves in an arbitrary direction on a surface from
  coincidence with a chosen point P, the normal at it, as a rule, at
  once fails to meet the normal at P; but, if it takes the direction of
  a line of curvature through P, this is instantaneously not the case.
  We have thus on the normal two centres of curvature, and the distances
  of these from the point on the surface are the two _principal radii of
  curvature_ of the surface at that point; these are also the radii of
  curvature of the sections of the surface by planes through the normal
  and the two principal tangents respectively; or say they are the radii
  of curvature of the normal sections through the two principal tangents
  respectively. Take at the point the axis of z in the direction of the
  normal, and those of x and y in the directions of the principal
  tangents respectively, then, if the radii of curvature be a, b (the
  signs being such that the coordinates of the two centres of curvature
  are z = a and z = b respectively), the surface has in the
  neighbourhood of the point the form of the paraboloid

        x²   y²
    z = -- + --,
        2a   2b

  and the chief-tangents are determined by the equation 0 = x²/2a +
  y²/2b. The two centres of curvature may be on the same side of the
  point or on opposite sides; in the former case a and b have the same
  sign, the paraboloid is elliptic, and the chief-tangents are
  imaginary; in the latter case a and b have opposite signs, the
  paraboloid is hyperbolic, and the chief-tangents are real.

  The normal sections of the surface and the paraboloid by the same
  plane have the same radius of curvature; and it thence readily follows
  that the radius of curvature of a normal section of the surface by a
  plane inclined at an angle [theta] to that of zx is given by the
  equation

      1     cos² [theta]   sin² [theta]
    ----- = ------------ + ------------.
    [rho]        a               b

  The section in question is that by a plane through the normal and a
  line in the tangent plane inclined at an angle [theta] to the
  principal tangent along the axis of x. To complete the theory,
  consider the section by a plane having the same trace upon the tangent
  plane, but inclined to the normal at an angle [phi]; then it is shown
  without difficulty (Meunier's theorem) that the radius of curvature of
  this inclined section of the surface is = [rho] cos [phi].

  AUTHORITIES.--The above article is largely based on that by Arthur
  Cayley in the 9th edition of this work. Of early and important recent
  publications on analytical geometry, special mention is to be made of
  R. Descartes, _Géométrie_ (Leyden, 1637); John Wallis, _Tractatus de
  sectionibus conicis nova methodo expositis_ (1655, _Opera
  mathematica_, i., Oxford, 1695); de l'Hospital, _Traité analytique des
  sections coniques_ (Paris, 1720); Leonhard Euler, _Introductio in
  analysin infinitorum_, ii. (Lausanne, 1748); Gaspard Monge,
  "Application d'algèbre à la géométrie" (_Journ. École Polytech._,
  1801); Julius Plücker, _Analytisch-geometrische Entwickelungen_, 3
  Bde. (Essen, 1828-1831); _System der analytischen Geometrie_ (Berlin,
  1835); G. Salmon, _A Treatise on Conic Sections_ (Dublin, 1848; 6th
  ed., London, 1879); Ch. Briot and J. Bouquet, _Leçons de géométrie
  analytique_ (Paris, 1851; 16th ed., 1897); M. Chasles, _Traité de
  géométrie supérieure_ (Paris, 1852); Wilhelm Fiedler, _Analytische
  Geometrie der Kegelschnitte_ nach G. Salmon frei bearbeitet (Leipzig,
  5te Aufl., 1887-1888); N.M. Ferrers, _An Elementary Treatise on
  Trilinear Coordinates_ (London, 1861); Otto Hesse, _Vorlesungen aus
  der analytischen Geometrie_ (Leipzig, 1865, 1881); W.A. Whitworth,
  _Trilinear Coordinates and other Methods of Modern Analytical
  Geometry_ (Cambridge, 1866); J. Booth, _A Treatise on Some New
  Geometrical Methods_ (London, i., 1873; ii., 1877); A. Clebsch-F.
  Lindemann, _Vorlesungen über Geometrie_, Bd. i. (Leipzig, 1876, 2te
  Aufl., 1891); R. Baltser, _Analytische Geometrie_ (Leipzig, 1882);
  Charlotte A. Scott, _Modern Methods of Analytical Geometry_ (London,
  1894); G. Salmon, _A Treatise on the Analytical Geometry of three
  Dimensions_ (Dublin, 1862; 4th ed., 1882); Salmon-Fiedler,
  _Analytische Geometrie des Raumes_ (Leipzig, 1863; 4te Aufl., 1898);
  P. Frost, _Solid Geometry_ (London, 3rd ed., 1886; 1st ed., Frost and
  J. Wolstenholme). See also E. Pascal, _Repertorio di matematiche
  superiori, II. Geometria_ (Milan, 1900), and articles now appearing in
  the _Encyklopädie der mathematischen Wissenschaften_, Bd. iii. 1, 2.
       (E. B. El.)


V. LINE GEOMETRY

Line geometry is the name applied to those geometrical investigations in
which the straight line replaces the point as element. Just as ordinary
geometry deals primarily with points and systems of points, this theory
deals in the first instance with straight lines and systems of straight
lines. In two dimensions there is no necessity for a special line
geometry, inasmuch as the straight line and the point are
interchangeable by the principle of duality; but in three dimensions the
straight line is its own reciprocal, and for the better discussion of
systems of lines we require some new apparatus, e.g., a system of
coordinates applicable to straight lines rather than to points. The
essential features of the subject are most easily elucidated by
analytical methods: we shall therefore begin with the notion of line
coordinates, and in order to emphasize the merits of the system of
coordinates ultimately adopted, we first notice a system without these
advantages, but often useful in special investigations.

  In ordinary Cartesian coordinates the two equations of a straight line
  may be reduced to the form y = rx + s, z = tx + u, and r, s, t, u may
  be regarded as the four coordinates of the line. These coordinates
  lack symmetry: moreover, in changing from one base of reference to
  another the transformation is not linear, so that the degree of an
  equation is deprived of real significance. For purposes of the general
  theory we employ homogeneous coordinates; if x1y1z1w1 and x2y2z2w2 are
  two points on the line, it is easily verified that the six
  determinants of the array

    |x1y1z1w1|
    |x2y2z2w2|

  are in the same ratios for all point-pairs on the line, and further,
  that when the point coordinates undergo a linear transformation so
  also do these six determinants. We therefore adopt these six
  determinants for the coordinates of the line, and express them by the
  symbols l, [lambda], m, µ, n, [nu] where l = x1w2 - x2w1, [lambda] =
  y1z2 - y2z1, &c. There is the further advantage that if a1b1c1d1 and
  a2b2c2d2 be two planes through the line, the six determinants

    |a1b1c1d1|
    |a2b2c2d2|

  are in the same ratios as the foregoing, so that except as regards a
  factor of proportionality we have [lambda] = b1c2 - b2c1, l = c1d2 -
  c2d1, &c. The identical relation l[lambda] + mµ + n[nu] = o reduces
  the number of independent constants in the six coordinates to four,
  for we are only concerned with their mutual ratios; and the quadratic
  character of this relation marks an essential difference between point
  geometry and line geometry. The condition of intersection of two lines
  is

    l[lambda]' + l'[lambda] + mµ' + m'µ + n[nu]' + n'[nu] = 0

  where the accented letters refer to the second line. If the
  coordinates are Cartesian and l, m, n are direction cosines, the
  quantity on the left is the mutual moment of the two lines.

  Since a line depends on four constants, there are three distinct types
  of configurations arising in line geometry--those containing a
  triply-infinite, a doubly-infinite and a singly-infinite number of
  lines; they are called Complexes, Congruences, and Ruled Surfaces or
  Skews respectively. A _Complex_ is thus a system of lines satisfying
  one condition--that is, the coordinates are connected by a single
  relation; and the degree of the complex is the degree of this equation
  supposing it to be algebraic. The lines of a complex of the nth degree
  which pass through any point lie on a cone of the nth degree, those
  which lie in any plane envelop a curve of the nth class and there are
  n lines of the complex in any plane pencil; the last statement
  combines the former two, for it shows that the cone is of the nth
  degree and the curve is of the nth class. To find the lines common to
  four complexes of degrees n1, n2, n3, n4, we have to solve five
  equations, viz. the four complex equations together with the quadratic
  equation connecting the line coordinates, therefore the number of
  common lines is 2n1n2n3n4. As an example of complexes we have the
  lines meeting a twisted curve of the nth degree, which form a complex
  of the nth degree.

  A _Congruence_ is the set of lines satisfying two conditions: thus a
  finite number m of the lines pass through any point, and a finite
  number n lie in any plane; these numbers are called the degree and
  class respectively, and the congruence is symbolically written (m, n).

  The simplest example of a congruence is the system of lines
  constituted by all those that pass through m points and those that lie
  in n planes; through any other point there pass m of these lines, and
  in any other plane there lie n, therefore the congruence is of degree
  m and class n. It has been shown by G.H. Halphen that the number of
  lines common to two congruences is mm' + nn', which may be verified by
  taking one of them to be of this simple type. The lines meeting two
  fixed lines form the general (1, 1) congruence; and the chords of a
  twisted cubic form the general type of a (1, 3) congruence; Halphen's
  result shows that two twisted cubics have in general ten common
  chords. As regards the analytical treatment, the difficulty is of the
  same nature as that arising in the theory of curves in space, for a
  congruence is not in general the complete intersection of two
  complexes.

  A _Ruled Surface_, _Regulus_ or _Skew_ is a configuration of lines
  which satisfy three conditions, and therefore depend on only one
  parameter. Such lines all lie on a surface, for we cannot draw one
  through an arbitrary point; only one line passes through a point of
  the surface; the simplest example, that of a quadric surface, is
  really two skews on the same surface.

  The degree of a ruled surface _qua_ line geometry is the number of its
  generating lines contained in a linear complex. Now the number which
  meets a given line is the degree of the surface _qua_ point geometry,
  and as the lines meeting a given line form a particular case of linear
  complex, it follows that the degree is the same from whichever point
  of view we regard it. The lines common to three complexes of degrees,
  n1n2n3, form a ruled surface of degree 2n1n2n3; but not every ruled
  surface is the complete intersection of three complexes.


    Linear complex.

  In the case of a complex of the first degree (or linear complex) the
  lines through a fixed point lie in a plane called the polar plane or
  nul-plane of that point, and those lying in a fixed plane pass through
  a point called the nul-point or pole of the plane. If the nul-plane of
  A pass through B, then the nul-plane of B will pass through A; the
  nul-planes of all points on one line l1 pass through another line l2.
  The relation between l1 and l2 is reciprocal; any line of the complex
  that meets one will also meet the other, and every line meeting both
  belongs to the complex. They are called conjugate or polar lines with
  respect to the complex. On these principles can be founded a theory of
  reciprocation with respect to a linear complex.

  This may be aptly illustrated by an elegant example due to A. Voss.
  Since a twisted cubic can be made to satisfy twelve conditions, it
  might be supposed that a finite number could be drawn to touch four
  given lines, but this is not the case. For, suppose one such can be
  drawn, then its reciprocal with respect to any linear complex
  containing the four lines is a curve of the third class, i.e. another
  twisted cubic, touching the same four lines, which are unaltered in
  the process of reciprocation; as there is an infinite number of
  complexes containing the four lines, there is an infinite number of
  cubics touching the four lines, and the problem is poristic.

  The following are some geometrical constructions relating to the
  unique linear complex that can be drawn to contain five arbitrary
  lines:

  To construct the nul-plane of any point O, we observe that the two
  lines which meet any four of the given five are conjugate lines of the
  complex, and the line drawn through O to meet them is therefore a ray
  of the complex; similarly, by choosing another four we can find
  another ray through O: these rays lie in the nul-plane, and there is
  clearly a result involved that the five lines so obtained all lie in
  one plane. A reciprocal construction will enable us to find the
  nul-point of any plane. Proceeding now to the metrical properties and
  the statical and dynamical applications, we remark that there is just
  one line such that the nul-plane of any point on it is perpendicular
  to it. This is called the central axis; if d be the shortest distance,
  [theta] the angle between it and a ray of the complex, then d tan
  [theta] = p, where p is a constant called the pitch or parameter. Any
  system of forces can be reduced to a force R along a certain line, and
  a couple G perpendicular to that line; the lines of nul-moment for the
  system form a linear complex of which the given line is the central
  axis and the quotient G/R is the pitch. Any motion of a rigid body can
  be reduced to a screw motion about a certain line, i.e. to an angular
  velocity [omega] about that line combined with a linear velocity u
  along the line. The plane drawn through any point perpendicular to the
  direction of its motion is its nul-plane with respect to a linear
  complex having this line for central axis, and the quotient u/[omega]
  for pitch (cf. Sir R.S. Ball, _Theory of Screws_).

  The following are some properties of a configuration of two linear
  complexes:

  The lines common to the two-complexes also belong to an infinite
  number of linear complexes, of which two reduce to single straight
  lines. These two lines are conjugate lines with respect to each of the
  complexes, but they may coincide, and then some simple modifications
  are required. The locus of the central axis of this system of
  complexes is a surface of the third degree called the cylindroid,
  which plays a leading part in the theory of screws as developed
  synthetically by Ball. Since a linear complex has an invariant of the
  second degree in its coefficients, it follows that two linear
  complexes have a lineo-linear invariant. This invariant is
  fundamental: if the complexes be both straight lines, its vanishing is
  the condition of their intersection as given above; if only one of
  them be a straight line, its vanishing is the condition that this line
  should belong to the other complex. When it vanishes for any two
  complexes they are said to be in _involution_ or _apolar_; the
  nul-points P, Q of any plane then divide harmonically the points in
  which the plane meets the common conjugate lines, and each complex is
  its own reciprocal with respect to the other. As regards a
  configuration of these linear complexes, the common lines from one
  system of generators of a quadric, and the doubly infinite system of
  complexes containing the common lines, include an infinite number of
  straight lines which form the other system of generators of the same
  quadric.


    General line coordinates.

  If the equation of a linear complex is Al + Bm + Cn + D[lambda] + Eµ +
  F[nu] = 0, then for a line not belonging to the complex we may regard
  the expression on the left-hand side as a multiple of the moment of
  the line with respect to the complex, the word moment being used in
  the statical sense; and we infer that when the coordinates are
  replaced by linear functions of themselves the new coordinates are
  multiples of the moments of the line with respect to six fixed
  complexes. The essential features of this coordinate system are the
  same as those of the original one, viz. there are six coordinates
  connected by a quadratic equation, but this relation has in general a
  different form. By suitable choice of the six fundamental complexes,
  as they may be called, this connecting relation may be brought into
  other simple forms of which we mention two: (i.) When the six are
  mutually in involution it can be reduced to x1² + x2² + x3² + x4² +
  x5² + x6² = 0; (ii.) When the first four are in involution and the
  other two are the lines common to the first four it is x1² + x2² + x3²
  + x4² - 2x5x6 = 0. These generalized coordinates might be explained
  without reference to actual magnitude, just as homogeneous point
  coordinates can be; the essential remark is that the equation of any
  coordinate to zero represents a linear complex, a point of view which
  includes our original system, for the equation of a coordinate to zero
  represents all the lines meeting an edge of the fundamental
  tetrahedron.

  The system of coordinates referred to six complexes mutually in
  involution was introduced by Felix Klein, and in many cases is more
  useful than that derived directly from point coordinates; e.g. in the
  discussion of quadratic complexes: by means of it Klein has developed
  an analogy between line geometry and the geometry of spheres as
  treated by G. Darboux and others. In fact, in that geometry a point is
  represented by _five_ coordinates, connected by a relation of the same
  type as the one just mentioned when the five fundamental spheres are
  mutually at right angles and the equation of a sphere is of the first
  degree. Extending this to four dimensions of space, we obtain an exact
  analogue of line geometry, in which (i.) a point corresponds to a
  line; (ii.) a linear complex to a hypersphere; (iii.) two linear
  complexes in involution to two orthogonal hyperspheres; (iv.) a linear
  complex and two conjugate lines to a hypersphere and two inverse
  points. Many results may be obtained by this principle, and more still
  are suggested by trying to extend the properties of circles to spheres
  in three and four dimensions. Thus the elementary theorem, that, given
  four lines, the circles circumscribed to the four triangles formed by
  them are concurrent, may be extended to six hyperplanes in four
  dimensions; and then we can derive a result in line geometry by
  translating the inverse of this theorem. Again, just as there is an
  infinite number of spheres touching a surface at a given point, two of
  them having contact of a closer nature, so there is an infinite number
  of linear complexes touching a non-linear complex at a given line, and
  _three_ of these have contact of a closer nature (cf. Klein, _Math.
  Ann._ v.).

  Sophus Lie has pointed out a different analogy with sphere geometry.
  Suppose, in fact, that the equation of a sphere of radius r is

    x² + y² + z² + 2ax + 2by + 2cz + d = 0,

  so that r² = a² + b² + c² - d; then introducing the quantity e to make
  this equation homogeneous, we may regard the sphere as given by the
  six coordinates a, b, c, d, e, r connected by the equation a² + b² +
  c² - r² - de = 0, and it is easy to see that two spheres touch, if the
  polar form 2aa1 + 2bb1 + 2cc1 - 2rr1 - de1 - d1e vanishes. Comparing
  this with the equation x1² + x2² + x3² + x4² - 2x5x6 = 0 given above,
  it appears that this sphere geometry and line geometry are identical,
  for we may write a = x1, b = x2, c = x3, r = x4(/[delta] - 1), d = x5,
  e = ½x6; but it is to be noticed that a sphere is really replaced by
  two lines whose coordinates only differ in the sign of x4, so that
  they are polar lines with respect to the complex x4 = 0. Two spheres
  which touch correspond to two lines which intersect, or more
  accurately to two pairs of lines (p, p') and (q, q'), of which the
  pairs (p, q) and (p', q') both intersect. By this means the problem of
  describing a sphere to touch four given spheres is reduced to that of
  drawing a pair of lines (t, t') (of which t intersects one line of the
  four pairs (pp'), (qq'), (rr'), (ss'), and t' intersects the remaining
  four). We may, however, ignore the accented letters in translating
  theorems, for a configuration of lines and its polar with respect to a
  linear complex have the same projective properties. In Lie's
  transformation a linear complex corresponds to the totality of spheres
  cutting a given sphere at a given angle. A most remarkable result is
  that lines of curvature in the sphere geometry become asymptotic lines
  in the line geometry.

  Some of the principles of line geometry may be brought into clearer
  light by admitting the ideas of space of four and five dimensions.

  Thus, regarding the coordinates of a line as homogeneous coordinates
  in five dimensions, we may say that line geometry is equivalent to
  geometry on a quadric surface in five dimensions. A linear complex is
  represented by a hyperplane section; and if two such complexes are in
  involution, the corresponding hyperplanes are conjugate with respect
  to the fundamental quadric. By projecting this quadric
  stereographically into space of four dimensions we obtain Klein's
  analogy. In the same way geometry in a linear complex is equivalent to
  geometry on a quadric in four dimensions; when two lines intersect the
  representative points are on the same generator of this quadric.
  Stereographic projection, therefore, converts a curve in a linear
  complex, i.e. one whose tangents all belong to the complex, into one
  whose tangents intersect a fixed conic: when this conic is the
  imaginary circle at infinity the curve is what Lie calls a minimal
  curve. Curves in a linear complex have been extensively studied. The
  osculating plane at any point of such a curve is the nul-plane of the
  point with respect to the complex, and points of superosculation
  always coincide in pairs at the points of contact of stationary
  tangents. When a point of such a curve is given, the osculating plane
  is determined, hence all the curves through a given point with the
  same tangent have the same torsion.


    Non-linear complexes.

  The lines through a given point that belong to a complex of the nth
  degree lie on a cone of the nth degree: if this cone has a double line
  the point is said to be a singular point. Similarly, a plane is said
  to be singular when the envelope of the lines in it has a double
  tangent. It is very remarkable that the same surface is the locus of
  the singular points and the envelope of the singular planes: this
  surface is called the singular surface, and both its degree and class
  are in general 2n(n - 1)², which is equal to four for the quadratic
  complex.

  The singular lines of a complex F = 0 are the lines common to F and
  the complex

    [delta]F    [delta]F       [delta]F  [delta]F   [delta]F   [delta]F
    -------- --------------- + --------  -------- + --------  ----------- = 0.
    [delta]l [delta][lambda]   [delta]m  [delta]µ   [delta]n  [delta][nu]

  As already mentioned, at each line l of a complex there is an infinite
  number of tangent linear complexes, and they all contain the lines
  adjacent to l. If now l be a singular line, these complexes all reduce
  to straight lines which form a plane pencil containing the line l.
  Suppose the vertex of the pencil is A, its plane a, and one of its
  lines [xi], then l' being a complex line near l, meets [xi], or more
  accurately the mutual moment of l', and is of the second order of
  small quantities. If P be a point on l, a line through P quite near l
  in the plane a will meet [xi] and is therefore a line of the complex;
  hence the complex-cones of all points on l touch a and the
  complex-curves of all planes through l touch l at A. It follows that l
  is a double line of the complex-cone of A, and a double tangent of the
  complex-curve of a. Conversely, a double line of a cone or curve is a
  singular line, and a singular line clearly touches the curves of all
  planes through it in the same point. Suppose now that the consecutive
  line l' is also a singular line, A' being the allied singular point,
  a' the singular plane and [xi]' any line of the pencil (A', a') so
  that [xi]' is a tangent line at l' to the complex: the mutual moments
  of the pairs l', [xi] and l, [xi] are each of the second order; hence
  the plane a' meets the lines l and [xi]' in two points very near A.
  This being true for all singular planes, near a the point of contact
  of a with its envelope is in A, i.e. the locus of singular points is
  the same as the envelope of singular planes. Further, when a line
  touches a complex it touches the singular surface, for it belongs to a
  plane pencil like (Aa), and thus in Klein's analogy the analogue of a
  focus of a hyper-surface being a bitangent line of the complex is also
  a bitangent line of the singular surface. The theory of cosingular
  complexes is thus brought into line with that of confocal surfaces in
  four dimensions, and guided by these principles the existence of
  cosingular quadratic complexes can easily be established, the analysis
  required being almost the same as that invented for confocal cyclides
  by Darboux and others. Of cosingular complexes of higher degree
  nothing is known.

  Following J. Plücker, we give an account of the lines of a quadratic
  complex that meet a given line.

  The cones whose vertices are on the given line all pass through eight
  fixed points and envelop a surface of the fourth degree; the conics
  whose planes contain the given line all lie on a surface of the fourth
  class and touch eight fixed planes. It is easy to see by elementary
  geometry that these two surfaces are identical. Further, the given
  line contains four singular points A1, A2, A3, A4, and the planes into
  which their cones degenerate are the eight common tangent planes
  mentioned above; similarly, there are four singular planes, a1, a2,
  a3, a4, through the line, and the eight points into which their conics
  degenerate are the eight common points above. The locus of the pole of
  the line with respect to all the conics in planes through it is a
  straight line called the _polar line_ of the given one; and through
  this line passes the polar plane of the given line with respect to
  each of the cones. The name polar is applied in the ordinary
  analytical sense; any line has an infinite number of polar complexes
  with respect to the given complex, for the equation of the latter can
  be written in an infinite number of ways; one of these polars is a
  straight line, and is the polar line already introduced. The surface
  on which lie all the conics through a line l is called the Plücker
  surface of that line: from the known properties of (2, 2)
  correspondences it can be shown that the Plücker surface of l cuts l1
  in a range of the same cross ratio as that of the range in which the
  Plücker surface of l1 cuts l. Applying this to the case in which l1 is
  the polar of l, we find that the cross ratios of (A1, A2, A3, A4) and
  (a1, a2, a3, a4) are equal. The identity of the locus of the A's with
  the envelope of the a's follows at once; moreover, a line meets the
  singular surface in four points having the same cross ratio as that of
  the four tangent planes drawn through the line to touch the surface.
  The Plücker surface has eight nodes, eight singular tangent planes,
  and is a double line. The relation between a line and its polar line
  is not a reciprocal one with respect to the complex; but W. Stahl has
  pointed out that the relation is reciprocal as far as the singular
  surface is concerned.


    Quadratic complexes.

  To facilitate the discussion of the general quadratic complex we
  introduce Klein's canonical form. We have, in fact, to deal with two
  quadratic equations in six variables; and by suitable linear
  transformations these can be reduced to the form

    a1x1² + a2x2² + a3x3² + a4x4² + a5x5² +  a6x6² = 0
     x1²  +  x2²  +  x3²  +  x4²  +  x5²  +   x6²  = 0

  subject to certain exceptions, which will be mentioned later.

  Taking the first equation to be that of the complex, we remark that
  both equations are unaltered by changing the sign of any coordinate;
  the geometrical meaning of this is, that the quadratic complex is its
  own reciprocal with respect to each of the six fundamental complexes,
  for changing the sign of a coordinate is equivalent to taking the
  polar of a line with respect to the corresponding fundamental complex.
  It is easy to establish the existence of six systems of bitangent
  linear complexes, for the complex l1x1 + l2x2 + l3x3 + l4x4 + l5x5 +
  l6x6 = 0 is a bitangent when

                  l2²       l3²       l4²       l5²       l6²
    l1 = 0, and ------- + ------- + ------- + ------- + ------- = 0
                a2 - a1   a3 - a1   a4 - a1   a5 - a1   a6 - a1

  and its lines of contact are conjugate lines with respect to the first
  fundamental complex. We therefore infer the existence of six systems
  of bitangent lines of the complex, of which the first is given by

              x2²       x3²       x4²       x5²       x6²
    x1 = 0, ------- + ------- + ------- + ------- + ------- = 0.
            a2 - a1   a3 - a1   a4 - a1   a5 - a1   a6 - a1

  Each of these lines is a bitangent of the singular surface, which is
  therefore completely determined as being the focal surface of the (2,
  2) congruence above. It is thence easy to verify that the two
  complexes [Sigma]ax² = 0 and [Sigma]bx² = 0 are cosingular if b_r =
  a_r[lambda] + µ/a_r[nu] + [rho].

  The singular surface of the general quadratic complex is the famous
  quartic, with sixteen nodes and sixteen singular tangent planes, first
  discovered by E.E. Kümmer.

  We cannot give a full account of its properties here, but we deduce at
  once from the above that its bitangents break up into six (2, 2)
  congruences, and the six linear complexes containing these are
  mutually in involution. The nodes of the singular surface are points
  whose complex cones are coincident planes, and the complex conic in a
  singular tangent plane consists of two coincident points. This
  configuration of sixteen points and planes has many interesting
  properties; thus each plane contains six points which lie on a conic,
  while through each point there pass six planes which touch a quadric
  cone. In many respects the Kümmer quartic plays a part in three
  dimensions analogous to the general quartic curve in two; it further
  gives a natural representation of certain relations between
  hyperelliptic functions (cf. R.W.H.T. Hudson, _Kümmer's Quartic_,
  1905).


    Classification of quadratic complexes.

  As might be expected from the magnitude of a form in six variables,
  the number of projectivally distinct varieties of quadratic complexes
  is very great; and in fact Adolf Weiler, by whom the question was
  first systematically studied on lines indicated by Klein, enumerated
  no fewer than forty-nine different types. But the principle of the
  classification is so important, and withal so simple, that we give a
  brief sketch which indicates its essential features.

  We have practically to study the intersection of two quadrics F and F'
  in six variables, and to classify the different cases arising we make
  use of the results of Karl Weierstrass on the equivalence conditions
  of two pairs of quadratics. As far as at present required, they are as
  follows: Suppose that the factorized form of the determinantal
  equation Disct (F + [lambda]F') = 0 is

    ([lambda] - [alpha])^(s1 + s2 + s3 ...)
      ([lambda] - ß)^(t1 + t2 + t3 + ...) ...

  where the root [alpha] occurs s1 + s2 + s3 ... times in the
  determinant, s2 + s3 ... times in every first minor, s3 + ... times in
  every second minor, and so on; the meaning of each exponent is then
  perfectly definite. Every factor of the type ([lambda] - [alpha])^s is
  called an _elementartheil_ (elementary divisor) of the determinant,
  and the condition of equivalence of two pairs of quadratics is simply
  that their determinants have the same elementary divisors. We write
  the pair of forms symbolically thus [(s1s2 ...), (t1t2 ...), ...],
  letters in the inner brackets referring to the same factor. Returning
  now to the two quadratics representing the complex, the sum of the
  exponents will be six, and two complexes are put in the same class if
  they have the same symbolical expression; i.e. the actual values of
  the roots of the determinantal equation need not be the same for both,
  but their manner of occurrence, as far as here indicated, must be
  identical in the two. The enumeration of all possible cases is thus
  reduced to a simple question in combinatorial analysis, and the actual
  study of any particular case is much facilitated by a useful rule of
  Klein's for writing down in a simple form two quadratics belonging to
  a given class--one of which, of course, represents the equation
  connecting line coordinates, and the other the equation of the
  complex. The general complex is naturally [111111]; the complex of
  tangents to a quadric is [(111), (111)] and that of lines meeting a
  conic is [(222)]. Full information will be found in Weiler's memoir,
  _Math. Ann._ vol. vii.

  The detailed study of each variety of complex opens up a vast subject;
  we only mention two special cases, the harmonic complex and the
  tetrahedral complex.

  The harmonic complex, first studied by Battaglini, is generated in an
  infinite number of ways by the lines cutting two quadrics
  harmonically. Taking the most general case, and referring the quadrics
  to their common self-conjugate tetrahedron, we can find its equation
  in a simple form, and verify that this complex really depends only on
  seventeen constants, so that it is not the most general quadratic
  complex. It belongs to the general type in so far as it is discussed
  above, but the roots of the determinant are in involution. The
  singular surface is the "tetrahedroid" discussed by Cayley. As a
  particular case, from a metrical point of view, we have L.F. Painvin's
  complex generated by the lines of intersection of perpendicular
  tangent planes of a quadric, the singular surface now being Fresnel's
  wave surface. The tetrahedral or Reye complex is the simplest and best
  known of proper quadratic complexes. It is generated by the lines
  which cut the faces of a tetrahedron in a constant cross ratio, and
  therefore by those subtending the same cross ratio at the four
  vertices. The singular surface is made up of the faces or the vertices
  of the fundamental tetrahedron, and each edge of this tetrahedron is a
  double line of the complex. The complex was first discussed by K.T.
  Reye as the assemblage of lines joining corresponding points in a
  homographic transformation of space, and this point of view leads to
  many important and elegant properties. A (metrically) particular case
  of great interest is the complex generated by the normals to a family
  of confocal quadrics, and for many investigations it is convenient to
  deal with this complex referred to the principal axes. For example,
  Lie has developed the theory of curves in a Reye complex (i.e. curves
  whose tangents belong to the complex) as solutions of a differential
  equation of the form (b - c)xdydz + (c - a)ydzdx + (a - b)zdxdy = 0,
  and we can simplify this equation by a logarithmic transformation.
  Many theorems connecting complexes with differential equations have
  been given by Lie and his school. A line complex, in fact, corresponds
  to a Mongian equation having [oo]^3 line integrals.


    Congruences.

  As the coordinates of a line belonging to a congruence are functions
  of two independent parameters, the theory of congruences is analogous
  to that of surfaces, and we may regard it as a fundamental inquiry to
  find the simplest form of surface into which a given congruence can be
  transformed. Most of those whose properties have been extensively
  discussed can be represented on a plane by a birational
  transformation. But in addition to the difficulties of the theory of
  algebraic surfaces, a subject still in its infancy, the theory of
  congruences has other difficulties in that a congruence is seldom
  completely represented, even by two equations.

  A fundamental theorem is that the lines of a congruence are in general
  bitangents of a surface; in fact, since the condition of intersection
  of two consecutive straight lines is ld[lambda] + dmdµ + dnd[nu] = 0,
  a line l of the congruence meets two adjacent lines, say l1 and l2.
  Suppose l, l1 lie in the plane pencil (A1a1) and l, l2 in the plane
  pencil (A2a2), then the locus of the A's is the same as the envelope
  of the a's, but a2 is the tangent plane at A1 and a1 at A2. This
  surface is called the focal surface of the congruence, and to it all
  the lines l are bitangent. The distinctive property of the points A is
  that two of the congruence lines through them coincide, and in like
  manner the planes a each contain two coincident lines. The focal
  surface consists of two sheets, but one or both may degenerate into
  curves; thus, for example, the normals to a surface are bitangents of
  the surface of centres, and in the case of Dupin's cyclide this
  surface degenerates into two conics.

  In the discussion of congruences it soon becomes necessary to
  introduce another number r, called the rank, which expresses the
  number of plane pencils each of which contains an arbitrary line and
  two lines of the congruence. The order of the focal surface is 2m(n -
  1) - 2r, and its class is m(m - 1) - 2r. Our knowledge of congruences
  is almost exclusively confined to those in which either m or n does
  not exceed two. We give a brief account of those of the second order
  without singular lines, those of order unity not being especially
  interesting. A congruence generally has singular points through which
  an infinite number of lines pass; a singular point is said to be of
  order r when the lines through it lie on a cone of the rth degree. By
  means of formulae connecting the number of singular points and their
  orders with the class m of quadratic congruence Kümmer proved that the
  class cannot exceed seven. The focal surface is of degree four and
  class 2m; this kind of quartic surface has been extensively studied by
  Kümmer, Cayley, Rohn and others. The varieties (2, 2), (2, 3), (2, 4),
  (2, 5) all belong to at least one Reye complex; and so also does the
  most important class of (2, 6) congruences which includes all the
  above as special cases. The congruence (2, 2) belongs to a linear
  complex and forty different Reye complexes; as above remarked, the
  singular surface is Kümmer's sixteen-nodal quartic, and the same
  surface is focal for six different congruences of this variety. The
  theory of (2, 2) congruences is completely analogous to that of the
  surfaces called cyclides in three dimensions. Further particulars
  regarding quadratic congruences will be found in Kümmer's memoir of
  1866, and the second volume of Sturm's treatise. The properties of
  quadratic congruences having singular lines, i.e. degenerate focal
  surfaces, are not so interesting as those of the above class; they
  have been discussed by Kümmer, Sturm and others.


    Ruled surfaces.

  Since a ruled surface contains only [infinity]¹ elements, this theory
  is practically the same as that of curves. If a linear complex
  contains more than n generators of a ruled surface of the nth degree,
  it contains all the generators, hence for n = 2 there are three
  linearly independent complexes, containing all the generators, and
  this is a well-known property of quadric surfaces. In ruled cubics the
  generators all meet two lines which may or may not coincide; these two
  cases correspond to the two main classes of cubics discussed by Cayley
  and Cremona. As regards ruled quartics, the generators must lie in one
  and may lie in two linear complexes. The first class is equivalent to
  a quartic in four dimensions and is always rational, but the latter
  class has to be subdivided into the elliptic and the rational, just
  like twisted quartic curves. A quintic skew may not lie in a linear
  complex, and then it is unicursal, while of sextics we have two
  classes not in a linear complex, viz. the elliptic variety, having
  thirty-six places where a linear complex contains six consecutive
  generators, and the rational, having six such places.

  The general theory of skews in two linear complexes is identical with
  that of curves on a quadric in three dimensions and is known. But for
  skews lying in only one linear complex there are difficulties; the
  curve now lies in four dimensions, and we represent it in three by
  stereographic projection as a curve meeting a given plane in n points
  on a conic. To find the maximum deficiency for a given degree would
  probably be difficult, but as far as degree eight the space-curve
  theory of Halphen and Nöther can be translated into line geometry at
  once. When the skew does not lie in a linear complex at all the theory
  is more difficult still, and the general theory clearly cannot advance
  until further progress is made in the study of twisted curves.

  REFERENCES.--The earliest works of a general nature are Plücker, _Neue
  Geometrie des Raumes_ (Leipzig, 1868); and Kümmer, "Über die
  algebraischen Strahlensysteme," _Berlin Academy_ (1866). Systematic
  development on purely synthetic lines will be found in the three
  volumes of Sturm, _Liniengeometrie_ (Leipzig, 1892, 1893, 1896); vol.
  i. deals with the linear and Reye complexes, vols. ii. and iii. with
  quadratic congruences and complexes respectively. For a highly
  suggestive review by Gino Loria see _Bulletin des sciences
  mathématiques_ (1893, 1897). A shorter treatise, giving a very
  interesting account of Klein's coordinates, is the work of Koenigs,
  _La Géométrie réglée et ses applications_ (Paris, 1898). English
  treatises are C.M. Jessop, _Treatise on the Line Complex_ (1903);
  R.W.H.T. Hudson, _Kümmer's Quartic_ (1905). Many references to memoirs
  on line geometry will be found in Hagen, _Synopsis der höheren
  Mathematik_, ii. (Berlin, 1894); Loria, _Il passato ed il presente
  delle principali teorie geometriche_ (Milan, 1897); a clear résumé of
  the principal results is contained in the very elegant volume of
  Pascal, _Repertorio di mathematiche superiori_, ii. (Milan, 1900).
  Another treatise dealing extensively with line geometry is Lie,
  _Geometrie der Berührungstransformationen_ (Leipzig, 1896). Many
  memoirs on the subject have appeared in the _Mathematische Annalen_; a
  full list of these will be found in the index to the first fifty
  volumes, p. 115. Perhaps the two memoirs which have left most
  impression on the subsequent development of the subject are Klein,
  "Zur Theorie der Liniencomplexe des ersten und zweiten Grades," _Math.
  Ann._ ii.; and Lie, "Über Complexe, insbesondere Linien- und
  Kugelcomplexe," _Math. Ann._ v.     (J. H. Gr.)


VI. NON-EUCLIDEAN GEOMETRY

The various metrical geometries are concerned with the properties of the
various types of congruence-groups, which are defined in the study of
the _axioms_ of _geometry_ and of their immediate consequences. But this
point of view of the subject is the outcome of recent research, and
historically the subject has a different origin. Non-Euclidean geometry
arose from the discussion, extending from the Greek period to the
present day, of the various assumptions which are implicit in the
traditional Euclidean system of geometry. In the course of these
investigations it became evident that metrical geometries, each
internally consistent but inconsistent in many respects with each other
and with the Euclidean system, could be developed. A short historical
sketch will explain this origin of the subject, and describe the famous
and interesting progress of thought on the subject. But previously a
description of the chief characteristic properties of elliptic and of
hyperbolic geometries will be given, assuming the standpoint arrived at
below under VII. _Axioms of Geometry_.

First assume the equation to the absolute (cf. _loc. cit._) to be w² -
x² - y² - z² = 0. The absolute is then real, and the geometry is
hyberbolic.

  The distance (d12) between the two points (x1, y1, z1, w1) and (x2,
  y2, z2, w2) is given by

    cosh (d12/[gamma]) = (w1w2 - x1x2 - y1y2 - z1z2)/[(w1² - x1² - y1² - Z1²)
      (w2² - x2² - y2² - z2²)]½     (1)

  The only points to which the metrical geometry applies are those
  within the region enclosed by the quadric; the other points are
  "improper ideal points." The angle ([theta]12) between two planes, l1x
  + m1y + n1z + r1w = 0 and l2x + m2y + n2z + r2w = 0, is given by

  cos [theta]12 = (l1l2 + m1m2 + n1n2 - r1r2)/{(l1² + m1² + n1² - r1²)
    (l2² + m2² + n2² - r2²)}^½       (2)

  These planes only have a real angle of inclination if they possess a
  line of intersection within the actual space, i.e. if they intersect.
  Planes which do not intersect possess a shortest distance along a line
  which is perpendicular to both of them. If this shortest distance is
  [delta]12, we have

    cosh ([delta]12/[gamma]) = (l1l2 + m1m2 + n1n2 - r1r2)/(l1² + m1² + n1² - r1²)
      (l2² + m2² + n2² - r2²)½     (3)

  [Illustration: FIG. 67.]

  Thus in the case of the two planes one and only one of the two,
  [theta]12 and [delta]12, is real. The same considerations hold for
  coplanar straight lines (see VII. _Axioms of Geometry_). Let O (fig.
  67) be the point (0, 0, 0, 1), OX the line y = 0, z = 0, OY the line z
  = 0, x = 0, and OZ the line x = 0, y = 0. These are the coordinate
  axes and are at right angles to each other. Let P be any point, and
  let [rho] be the distance OP, [theta] the angle POZ, and [phi] the
  angle between the planes ZOX and ZOP. Then the coordinates of P can be
  taken to be

    sinh ([rho]/[gamma]) sin [theta] cos [phi], sinh ([rho]/[gamma]) sin [theta]
      sin [phi], sinh ([rho]/[gamma]) cos[theta], cosh ([rho]/[gamma]).

  [Illustration: FIG. 68.]

  If ABC is a triangle, and the sides and angles are named according to
  the usual convention, we have

    sinh (a/[gamma])/sin A = sinh (b/[gamma])/sin B = sinh (c/[gamma])/sin C, (4)

  and also

    cosh (a/[gamma]) = cosh (b/[gamma]) cosh (c/[gamma]) -
      sinh (b/[gamma]) sinh (c/[gamma]) cos A,   (5)

  with two similar equations. The sum of the three angles of a triangle
  is always less than two right angles. The area of the triangle ABC is
  [lambda]²([pi] - A - B - C). If the base BC of a triangle is kept
  fixed and the vertex A moves in the fixed plane ABC so that the area
  ABC is constant, then the locus of A is a line of equal distance from
  BC. This locus is not a straight line. The whole theory of similarity
  is inapplicable; two triangles are either congruent, or their angles
  are not equal two by two. Thus the elements of a triangle are
  determined when its three angles are given. By keeping A and B and the
  line BC fixed, but by making C move off to infinity along BC, the
  lines BC and AC become parallel, and the sides a and b become
  infinite. Hence from equation (5) above, it follows that two parallel
  lines (cf. Section VII. _Axioms of Geometry_) must be considered as
  making a zero angle with each other. Also if B be a right angle, from
  the equation (5), remembering that, in the limit,

    cosh (a/[gamma])/cosh (b/[gamma]) = cosh (a/[gamma])/sinh (b/[gamma]) = 1,

  we have cos A = tanh (c/2[gamma]) .... (6).

  The angle A is called by N.I. Lobatchewsky the "angle of parallelism."

  The whole theory of lines and planes at right angles to each other is
  simply the theory of conjugate elements with respect to the absolute,
  where ideal lines and planes are introduced.

  Thus if l and l' be any two conjugate lines with respect to the
  absolute (of which one of the two must be improper, say l'), then any
  plane through l' and containing proper points is perpendicular to l.
  Also if p is any plane containing proper points, and P is its pole,
  which is necessarily improper, then the lines through P are the
  normals to P. The equation of the sphere, centre (x1, y1, z1, w1) and
  radius [rho], is

    (w1²- x1²- y1²- z1²)(w² - x² - y² - z²) cosh²([rho]/[gamma]) = (w1w -
      x1x - y1y -z1z)²   (7).

  The equation of the surface of equal distance ([sigma]) from the plane
  lx + my + nz + rw = 0 is

    (l² + m² + n² - r²)(w² - x² - y² - z²) sinh²([sigma]/[gamma]) = (rw +
      lx + my + nz)²    (8).

  A surface of equal distance is a sphere whose centre is improper; and
  both types of surface are included in the family

    k²(w² - x² - y² - z²) = (ax + by + cz + dw)²   (9).

  But this family also includes a third type of surfaces, which can be
  looked on either as the limits of spheres whose centres have
  approached the absolute, or as the limits of surfaces of equal
  distance whose central planes have approached a position tangential to
  the absolute. These surfaces are called limit-surfaces. Thus (9)
  denotes a limit-surface, if d² - a² - b² - c² = 0. Two limit-surfaces
  only differ in position. Thus the two limit-surfaces which touch the
  plane YOZ at O, but have their concavities turned in opposite
  directions, have as their equations

    w² - x² - y² - z² = (w ± x)².

  The geodesic geometry of a sphere is elliptic, that of a surface of
  equal distance is hyperbolic, and that of a limit-surface is parabolic
  (i.e. _Euclidean_). The equation of the surface (cylinder) of equal
  distance ([delta]) from the line OX is

    (w² - x²) tanh²([delta]/[gamma]) - y² - z² = 0.

  This is not a ruled surface. Hence in this geometry it is not possible
  for two straight lines to be at a constant distance from each other.

  Secondly, let the equation of the absolute be x² + y² + z² + w² = 0.
  The absolute is now imaginary and the geometry is elliptic.

  The distance (d12) between the two points (x1, y1, z1, w1) and (x2,
  y2, z2, w2) is given by

    cos (d12/[gamma]) = ±(x1x2 + y1y2 + z1z2 + w1w2)
      / {(x1² + y1² + z1² + w1²) {(x2² + y2² + z2² + w2²)}^½   (10).

  Thus there are two distances between the points, and if one is d12,
  the other is [pi][gamma]-d12. Every straight line returns into itself,
  forming a closed series. Thus there are two segments between any two
  points, together forming the whole line which contains them; one
  distance is associated with one segment, and the other distance with
  the other segment. The complete length of every straight line is
  [pi][gamma].

  The angle between the two planes l1x + m1y + n1z + r + 1w = 0 and l2x
  + m2y + n2z + r2w = 0 is

    cos [theta]12 = (l1l2 + m1m2 + n1n2 + r1r2)/ {(l1² + m1² + n1² +r1²)
      (l2² + m2² + n2² + r2²)}^½ (11).

  The polar plane with respect to the absolute of the point (x1, y1, z1,
  w1) is the real plane x1x + y1y + z1z + w1w = 0, and the pole of the
  plane l1x + m1y + n1z + r1w = 0 is the point (l1, m1, n1, r1). Thus
  (from equations 10 and 11) it follows that the angle between the polar
  planes of the points (x1, ...) and (x2, ...) is d12/[gamma], and that
  the distance between the poles of the planes (l1, ...) and (l2, ...)
  is [gamma][theta]12. Thus there is complete reciprocity between points
  and planes in respect to all properties. This complete reign of the
  principle of duality is one of the great beauties of this geometry.
  The theory of lines and planes at right angles is simply the theory of
  conjugate elements with respect to the absolute. A tetrahedron
  self-conjugate with respect to the absolute has all its intersecting
  elements (edges and planes) at right angles. If l and l' are two
  conjugate lines, the planes through one are the planes perpendicular
  to the other. If P is the pole of the plane p, the lines through P are
  the normals to the plane p. The distance from P to p is ½[pi][gamma].
  Thus every sphere is also a surface of equal distance from the polar
  of its centre, and conversely. A plane does not divide space; for the
  line joining any two points P and Q only cuts the plane once, in L
  say, then it is always possible to go from P to Q by the segment of
  the line PQ which does not contain L. But P and Q may be said to be
  separated by a plane p, if the point in which PQ cuts p lies on the
  shortest segment between P and Q. With this sense of "separation," it
  is possible[2] to find three points P, Q, R such that P and Q are
  separated by the plane p, but P and R are not separated by p, nor are
  Q and R.

  Let A, B, C be any three non-collinear points, then four triangles are
  defined by these points. Thus if a, b, c and A, B, C are the elements
  of any one triangle, then the four triangles have as their elements:

    (1)      a,                  b,                  c,             A,           B,           C.

    (2)      a,            [pi][gamma] - b,   [pi][gamma] - c,      A,       [pi] - B,    [pi] - C.

    (3) [pi][gamma] - a,          b,          [pi][gamma] - c,   [pi] - A,       B,       [pi] - C.

    (4) [pi][gamma] - a,   [pi][gamma] - b,          c,          [pi] - A,   [pi] - B,        C.

  The formulae connecting the elements are

    sin A/sin (a/[gamma]) = sin B/sin (b/[gamma]) = sin C/sin (c/[gamma]),
         (12)

  and

    cos (a/[gamma]) = cos (b/[gamma]) cos (c/[gamma]) + sin (b/[gamma])
      sin (c/[gamma]) cos A,   (13)

  with two similar equations.

  Two cases arise, namely (I.) according as one of the four triangles
  has as its sides the shortest segments between the angular points, or
  (II.) according as this is not the case. When case I. holds there is
  said to be a "principal triangle."[3] If all the figures considered
  lie within a sphere of radius ¼[pi][gamma] only case I. can hold, and
  the principal triangle is the triangle wholly within this sphere, also
  the peculiarities in respect to the separation of points by a plane
  cannot then arise. The sum of the three angles of a triangle ABC is
  always greater than two right angles, and the area of the triangle is
  [gamma]²(A + B + C--[pi]). Thus as in hyperbolic geometry the theory
  of similarity does not hold, and the elements of a triangle are
  determined when its three angles are given. The coordinates of a point
  can be written in the form

    sin ([rho]/[gamma]) sin [Phi] cos [phi], sin ([rho]/[gamma]) sin [Phi]
      sin [phi], sin ([rho]/[gamma]) cos [Phi], cos ([rho]/[gamma]),

  where [rho], [Phi] and [phi] have the same meanings as in the
  corresponding formulae in hyperbolic geometry. Again, suppose a watch
  is laid on the plane OXY, face upwards with its centre at O, and the
  line 12 to 6 (as marked on dial) along the line YOY. Let the watch be
  continually pushed along the plane along the line OX, that is, in the
  direction 9 to 3. Then the line XOX being of finite length, the watch
  will return to O, but at its first return it will be found to be face
  downwards on the other side of the plane, with the line 12 to 6
  reversed in direction along the line YOY. This peculiarity was first
  pointed out by Felix Klein. The theory of parallels as it exists in
  hyperbolic space has no application in elliptic geometry. But another
  property of Euclidean parallel lines holds in elliptic geometry, and
  by the use of it parallel lines are defined. For the equation of the
  surface (cylinder) of equal distance ([delta]) from the line XOX is

    (x² + w²) tan²([delta]/[gamma]) - (y² + z²) = 0.

  This is also the surface of equal distance, ½[pi][gamma]-[delta], from
  the line conjugate to XOX. Now from the form of the above equation
  this is a ruled surface, and through every point of it two generators
  pass. But these generators are lines of equal distance from XOX. Thus
  throughout every point of space two lines can be drawn which are lines
  of equal distance from a given line l. This property was discovered by
  W.K. Clifford. The two lines are called Clifford's right and left
  parallels to l through the point. This property of parallelism is
  reciprocal, so that if m is a left parallel to l, then l is a left
  parallel to m. Note also that two parallel lines l and m are not
  coplanar. Many of those properties of Euclidean parallels, which do
  not hold for Lobatchewsky's parallels in hyperbolic geometry, do hold
  for Clifford's parallels in elliptic geometry. The geodesic geometry
  of spheres is elliptic, the geodesic geometry of surfaces of equal
  distance from lines (cylinders) is Euclidean, and surfaces of
  revolution can be found[4] of which the geodesic geometry is
  hyperbolic. But it is to be noticed that the connectivity of these
  surfaces is different to that of a Euclidean plane. For instance there
  are only [&infin]² congruence transformations of the cylindrical
  surfaces of equal distance into themselves, instead of the [&infin]³
  for the ordinary plane. It would obviously be possible to state
  "axioms" which these geodesics satisfy, and thus to define
  independently, and not as loci, quasi-spaces of these peculiar types.
  The existence of such Euclidean quasi-geometries was first pointed out
  by Clifford.[5]

In both elliptic and hyperbolic geometry the spherical geometry, i.e.
the relations between the angles formed by lines and planes passing
through the same point, is the same as the "spherical trigonometry" in
Euclidean geometry. The constant [gamma], which appears in the formulae
both of hyperbolic and elliptic geometry, does not by its variation
produce different types of geometry. There is only one type of elliptic
geometry and one type of hyperbolic geometry; and the magnitude of the
constant [gamma] in each case simply depends upon the magnitude of the
arbitrary unit of length in comparison with the natural unit of length
which each particular instance of either geometry presents. The
existence of a natural unit of length is a peculiarity common both to
hyperbolic and elliptic geometries, and differentiates them from
Euclidean geometry. It is the reason for the failure of the theory of
similarity in them. If [gamma] is very large, that is, if the natural
unit is very large compared to the arbitrary unit, and if the lengths
involved in the figures considered are not large compared to the
arbitrary unit, then both the elliptic and hyperbolic geometries
approximate to the Euclidean. For from formulae (4) and (5) and also
from (12) and (13) we find, after retaining only the lowest powers of
small quantities, as the formulae for any triangle ABC,

  a/ sin A = b/ sin B = c/ sin C,

and

  a² = b² + c² - 2bc cos A,

with two similar equations. Thus the geometries of small figures are in
both types Euclidean.


  Theory of parallels before Gauss.

_History._--"In pulcherrimo Geometriae corpore," wrote Sir Henry Savile
in 1621, "duo sunt naevi, duae labes nec quod sciam plures, in quibus
eluendis et emaculendis cum veterum tum recentiorum ... vigilavit
industria." These two blemishes are the theory of parallels and the
theory of proportion. The "industry of the moderns," in both respects,
has given rise to important branches of mathematics, while at the same
time showing that Euclid is in these respects more free from blemish
than had been previously credible. It was from endeavours to improve the
theory of parallels that non-Euclidean geometry arose; and though it has
now acquired a far wider scope, its historical origin remains
instructive and interesting. Euclid's "axiom of parallels" appears as
Postulate V. to the first book of his _Elements_, and is stated thus,
"And that, if a straight line falling on two straight lines make the
angles, internal and on the same side, less than two right angles, the
two straight lines, being produced indefinitely, meet on the side on
which are the angles less than two right angles." The original Greek is
[Greek: kai ean eis duo eutheias eutheia empiptousa tas entos kai epi ta
auta merê gônias duo orthôn elassonas poiê, ekballomenas tas duo
eutheias ep' apeiron sympiptein, eph' ha merê eisin hai tôn duo orthôn
elassones].

To Euclid's successors this axiom had signally failed to appear
self-evident, and had failed equally to appear indemonstrable. Without
the use of the postulate its converse is proved in Euclid's 28th
proposition, and it was hoped that by further efforts the postulate
itself could be also proved. The first step consisted in the discovery
of equivalent axioms. Christoph Clavius in 1574 deduced the axiom from
the assumption that a line whose points are all equidistant from a
straight line is itself straight. John Wallis in 1663 showed that the
postulate follows from the possibility of similar triangles on different
scales. Girolamo Saccheri (1733) showed that it is sufficient to have a
single triangle, the sum of whose angles is two right angles. Other
equivalent forms may be obtained, but none shows any essential
superiority to Euclid's. Indeed plausibility, which is chiefly aimed at,
becomes a positive demerit where it conceals a real assumption.


  Saccheri.

A new method, which, though it failed to lead to the desired goal,
proved in the end immensely fruitful, was invented by Saccheri, in a
work entitled _Euclides ab omni naevo vindicatus_ (Milan, 1733). If the
postulate of parallels is involved in Euclid's other assumptions,
contradictions must emerge when it is denied while the others are
maintained. This led Saccheri to attempt a _reductio ad absurdum_, in
which he mistakenly believed himself to have succeeded. What is
interesting, however, is not his fallacious conclusion, but the
non-Euclidean results which he obtains in the process. Saccheri
distinguishes three hypotheses (corresponding to what are now known as
Euclidean or parabolic, elliptic and hyperbolic geometry), and proves
that some one of the three must be universally true. His three
hypotheses are thus obtained: equal perpendiculars AC, BD are drawn from
a straight line AB, and CD are joined. It is shown that the angles ACD,
BDC are equal. The first hypothesis is that these are both right angles;
the second, that they are both obtuse; and the third, that they are both
acute. Many of the results afterwards obtained by Lobatchewsky and
Bolyai are here developed. Saccheri fails to be the founder of
non-Euclidean geometry only because he does not perceive the possible
truth of his non-Euclidean hypotheses.


  Lambert.

Some advance is made by Johann Heinrich Lambert in his _Theorie der
Parallellinien_ (written 1766; posthumously published 1786). Though he
still believed in the necessary truth of Euclidean geometry, he
confessed that, in all his attempted proofs, something remained
undemonstrated. He deals with the same three hypotheses as Saccheri,
showing that the second holds on a sphere, while the third would hold on
a sphere of purely imaginary radius. The second hypothesis he succeeds
in condemning, since, like all who preceded Bernhard Riemann, he is
unable to conceive of the straight line as finite and closed. But the
third hypothesis, which is the same as Lobatchewsky's, is not even
professedly refuted.[6]


  Three periods of non-Euclidean geometry.

Non-Euclidean geometry proper begins with Karl Friedrich Gauss. The
advance which he made was rather philosophical than mathematical: it was
he (probably) who first recognized that the postulate of parallels is
possibly false, and should be empirically tested by measuring the angles
of large triangles. The history of non-Euclidean geometry has been aptly
divided by Felix Klein into three very distinct periods. The
first--which contains only Gauss, Lobatchewsky and Bolyai--is
characterized by its synthetic method and by its close relation to
Euclid. The attempt at indirect proof of the disputed postulate would
seem to have been the source of these three men's discoveries; but when
the postulate had been denied, they found that the results, so far from
showing contradictions, were just as self-consistent as Euclid. They
inferred that the postulate, if true at all, can only be proved by
observations and measurements. Only one kind of non-Euclidean space is
known to them, namely, that which is now called hyperbolic. The second
period is analytical, and is characterized by a close relation to the
theory of surfaces. It begins with Riemann's inaugural dissertation,
which regards space as a particular case of a _manifold_; but the
characteristic standpoint of the period is chiefly emphasized by Eugenio
Beltrami. The conception of measure of curvature is extended by Riemann
from surfaces to spaces, and a new kind of space, finite but unbounded
(corresponding to the second hypothesis of Saccheri and Lambert), is
shown to be possible. As opposed to the second period, which is purely
metrical, the third period is essentially projective in its method. It
begins with Arthur Cayley, who showed that metrical properties are
projective properties relative to a certain fundamental quadric, and
that different geometries arise according as this quadric is real,
imaginary or degenerate. Klein, to whom the development of Cayley's work
is due, showed further that there are two forms of Riemann's space,
called by him the elliptic and the spherical. Finally, it has been shown
by Sophus Lie, that if figures are to be freely movable throughout all
space in [oo]^6 ways, no other three-dimensional spaces than the above
four are possible.


  Gauss.

Gauss published nothing on the theory of parallels, and it was not
generally known until after his death that he had interested himself in
that theory from a very early date. In 1799 he announces that Euclidean
geometry would follow from the assumption that a triangle can be drawn
greater than any given triangle. Though unwilling to assume this, we
find him in 1804 still hoping to prove the postulate of parallels. In
1830 he announces his conviction that geometry is not an a priori
science; in the following year he explains that non-Euclidean geometry
is free from contradictions, and that, in this system, the angles of a
triangle diminish without limit when all the sides are increased. He
also gives for the circumference of a circle of radius r the formula
[pi]k(e^(r/k) - e^(r-/k)), where k is a constant depending upon the
nature of the space. In 1832, in reply to the receipt of Bolyai's
_Appendix_, he gives an elegant proof that the amount by which the sum
of the angles of a triangle falls short of two right angles is
proportional to the area of the triangle. From these and a few other
remarks it appears that Gauss possessed the foundations of hyperbolic
geometry, which he was probably the first to regard as perhaps true. It
is not known with certainty whether he influenced Lobatchewsky and
Bolyai, but the evidence we possess is against such a view.[7]


  Lobatchewsky.

The first to publish a non-Euclidean geometry was Nicholas Lobatchewsky,
professor of mathematics in the new university of Kazañ.[8] In the place
of the disputed postulate he puts the following: "All straight lines
which, in a plane, radiate from a given point, can, with respect to any
other straight line in the same plane, be divided into two classes, the
_intersecting_ and the _non-intersecting_. The _boundary line_ of the
one and the other class is called _parallel to the given line_." It
follows that there are two parallels to the given line through any
point, each meeting the line at infinity, like a Euclidean parallel.
(Hence a line has two distinct points at infinity, and not one only as
in ordinary geometry.) The two parallels to a line through a point make
equal acute angles with the perpendicular to the line through the point.
If p be the length of the perpendicular, either of these angles is
denoted by [Pi](p). The determination of [Pi](p) is the chief problem
(cf. equation (6) above); it appears finally that, with a suitable
choice of the unit of length,

  tan ½ [Pi](p) = e^(-p).

Before obtaining this result it is shown that spherical trigonometry is
unchanged, and that the normals to a circle or a sphere still pass
through its centre. When the radius of the circle or sphere becomes
infinite all these normals become parallel, but the circle or sphere
does not become a straight line or plane. It becomes what Lobatchewsky
calls a limit-line or limit-surface. The geometry on such a surface is
shown to be Euclidean, limit-lines replacing Euclidean straight lines.
(It is, in fact, a surface of zero measure of curvature.) By the help of
these propositions Lobatchewsky obtains the above value of [Pi](p), and
thence the solution of triangles. He points out that his formulae result
from those of spherical trigonometry by substituting ia, ib, ic, for the
sides a, b, c.


  Bolyai.

John Bolyai, a Hungarian, obtained results closely corresponding to
those of Lobatchewsky. These he published in an appendix to a work by
his father, 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_.[9] This work was published in 1831, but its
conception dates from 1823. It reveals a profounder appreciation of the
importance of the new ideas, but otherwise differs little from
Lobatchewsky's. Both men point out that Euclidean geometry as a limiting
case of their own more general system, that the geometry of very small
spaces is always approximately Euclidean, that no a priori grounds exist
for a decision, and that observation can only give an approximate
answer. Bolyai gives also, as his title indicates, a geometrical
construction, in hyperbolic space, for the quadrature of the circle, and
shows that the area of the greatest possible triangle, which has all its
sides parallel and all its angles zero, is [pi][iota]², where i is what
we should now call the space-constant.


  Riemann.

The works of Lobatchewsky and Bolyai, though known and valued by Gauss,
remained obscure and ineffective until, in 1866, they were translated
into French by J. Hoüel. But at this time Riemann's dissertation, _Über
die Hypothesen, welche der Geometrie zu Grunde liegen_,[10] was already
about to be published. In this work Riemann, without any knowledge of
his predecessors in the same field, inaugurated a far more profound
discussion, based on a far more general standpoint; and by its
publication in 1867 the attention of mathematicians and philosophers was
at last secured. (The dissertation dates from 1854, but owing to changes
which Riemann wished to make in it, it remained unpublished until after
his death.)


  Definition of a manifold.

Riemann's work contains two fundamental conceptions, that of a manifold
and that of the _measure of curvature_ of a continuous manifold
possessed of what he calls flatness in the smallest parts. By means of
these conceptions space is made to appear at the end of a gradual series
of more and more specialized conceptions. Conceptions of magnitude, he
explains, are only possible where we have a general conception capable
of determination in various ways. The manifold consists of all these
various determinations, each of which is an element of the manifold. The
passage from one element to another may be discrete or continuous; the
manifold is called discrete or continuous accordingly. Where it is
discrete two portions of it can be compared, as to magnitude, by
counting; where continuous, by measurement. But measurement demands
superposition, and consequently some magnitude independent of its place
in the manifold. In passing, in a continuous manifold, from one element
to another in a determinate way, we pass through a series of
intermediate terms, which form a one-dimensional manifold. If this whole
manifold be similarly caused to pass over into another, each of its
elements passes through a one-dimensional manifold, and thus on the
whole a two-dimensional manifold is generated. In this way we can
proceed to n dimensions. Conversely, a manifold of n dimensions can be
analysed into one of one dimension and one of (n - 1) dimensions. By
repetitions of this process the position of an element may be at last
determined by n magnitudes. We may here stop to observe that the above
conception of a manifold is akin to that due to Hermann Grassmann in the
first edition (1847) of his _Ausdehnungslehre_.[11]


  Measure of curvature.

Both concepts have been elaborated and superseded by the modern
procedure in respect to the axioms of geometry, and by the conception of
abstract geometry involved therein. Riemann proceeds to specialize the
manifold by considerations as to measurement. If measurement is to be
possible, some magnitude, we saw, must be independent of position; let
us consider manifolds in which lengths of lines are such magnitudes, so
that every line is measurable by every other. The coordinates of a point
being x1, x2, ... x_n, let us confine ourselves to lines along which the
ratios dx1 : dx2 : ... : dx_n alter continuously. Let us also assume that
the element of length, ds, is unchanged (to the first order) when all
its points undergo the same infinitesimal motion. Then if all the
increments dx be altered in the same ratio, ds is also altered in this
ratio. Hence ds is a homogeneous function of the first degree of the
increments dx. Moreover, ds must be unchanged when all the dx change
sign. The simplest possible case is, therefore, that in which ds is the
square root of a quadratic function of the dx. This case includes space,
and is alone considered in what follows. It is called the case of
flatness in the smallest parts. Its further discussion depends upon the
measure of curvature, the second of Riemann's fundamental conceptions.
This conception, derived from the theory of surfaces, is applied as
follows. Any one of the shortest lines which issue from a given point
(say the origin) is completely determined by the initial ratios of the
dx. Two such lines, defined by dx and [delta]x say, determine a pencil,
or one-dimensional series, of shortest lines, any one of which is
defined by [lambda]dx + µ[delta]x, where the parameter [lambda] : µ may
have any value. This pencil generates a two-dimensional series of
points, which may be regarded as a surface, and for which we may apply
Gauss's formula for the measure of curvature at any point. Thus at every
point of our manifold there is a measure of curvature corresponding to
every such pencil; but all these can be found when n.[/(n-1)]/2 of them
are known. If figures are to be freely movable, it is necessary and
sufficient that the measure of curvature should be the same for all
points and all directions at each point. Where this is the case, if
[alpha] be the measure of curvature, the linear element can be put into
the form

  ds = [root]([Sigma]dx²)/(1 + ¼[alpha][Sigma]x²).

If [alpha] be positive, space is finite, though still unbounded, and
every straight line is closed--a possibility first recognized by
Riemann. It is pointed out that, since the possible values of a form a
continuous series, observations cannot prove that our space is strictly
Euclidean. It is also regarded as possible that, in the infinitesimal,
the measure of curvature of our space should be variable.

There are four points in which this profound and epoch-making work is
open to criticism or development--(1) the idea of a manifold requires
more precise determination; (2) the introduction of coordinates is
entirely unexplained and the requisite presuppositions are unanalysed;
(3) the assumption that ds is the square root of a quadratic function of
dx1, dx2, ... is arbitrary; (4) the idea of superposition, or
congruence, is not adequately analysed. The modern solution of these
difficulties is properly considered in connexion with the general
subject of the axioms of geometry.


  Helmholtz.

The publication of Riemann's dissertation was closely followed by two
works of Hermann von Helmholtz,[12] again undertaken in ignorance of the
work of predecessors. In these a proof is attempted that ds must be a
rational integral quadratic function of the increments of the
coordinates. This proof has since been shown by Lie to stand in need of
correction (see VII. _Axioms of Geometry_). Helmholtz's remaining works
on the subject[13] are of almost exclusively philosophical interest. We
shall return to them later.


  Beltrami.

The only other writer of importance in the second period is Eugenio
Beltrami, by whom Riemann's work was brought into connexion with that of
Lobatchewsky and Bolyai. As he gave, by an elegant method, a convenient
Euclidean interpretation of hyperbolic plane geometry, his results will
be stated at some length[14]. The _Saggio_ shows that Lobatchewsky's
plane geometry holds in Euclidean geometry on surfaces of constant
negative curvature, straight lines being replaced by geodesics. Such
surfaces are capable of a conformal representation on a plane, by which
geodesics are represented by straight lines. Hence if we take, as
coordinates on the surface, the Cartesian coordinates of corresponding
points on the plane, the geodesics must have linear equations.

  Hence it follows that

    ds² = R²w^(-4){([alpha]² - v²)du² + 2uvdudv + ([alpha]² - u²)dv²}

  where w² = [alpha]² - u² - v², and (-1)/R² is the measure of curvature
  of our surface (note that k = [gamma] as used above). The angle
  between two geodesics u = const., v = const. is [theta], where

    cos [theta] = uv/[root]{([alpha]² - u²)([alpha]² - v²)}, sin [theta] =
      aw/[root]{(a² - u²)(a² - v²)}.

  Thus u = 0 is orthogonal to all geodesies v = const., and vice versa.
  In order that sin [theta] may be real, w² must be positive; thus
  geodesics have no real intersection when the corresponding straight
  lines intersect outside the circle u² + v² = [alpha]². When they
  intersect on this circle, [theta] = 0. Thus Lobatchewsky's parallels
  are represented by straight lines intersecting on the circle. Again,
  transforming to polar coordinates u = r cos µ, v = r sin µ, and
  calling [rho] the geodesic distance of u, v from the origin, we have,
  for a geodesic through the origin,

    d[rho] = Radr/(a² - r²), [rho] = ½R log(a + r)/(a - r), r = a tan h
      ([rho]/R).

  Thus points on the surface corresponding to points in the plane on the
  limiting circle r = a, are all at an infinite distance from the
  origin. Again, considering r constant, the arc of a geodesic circle
  subtending an angle µ at the origin is

    [sigma] = Rrµ/[root](a² - r²) = µR sin h ([rho]/R),

  whence the circumference of a circle of radius [rho] is 2[pi]R sin h
  ([rho]/R). Again, if [alpha] be the angle between any two geodesics

    V - v = m(U - u), V - v = n(U - u),

  then tan [alpha] = a(n - m)w/{(1 + mn)a² - (v - mu) (v - nu)}.

  Thus [alpha] is imaginary when u, v is outside the limiting circle,
  and is zero when, and only when, u, v is on the limiting circle. All
  these results agree with those of Lobatchewsky and Bolyai. The maximum
  triangle, whose angles are all zero, is represented in the auxiliary
  plane by a triangle inscribed in the limiting circle. The angle of
  parallelism is also easily obtained. The perpendicular to v = 0 at a
  distance [delta] from the origin is u = a tan h ([delta]/R), and the
  parallel to this through the origin is u = v sin h ([delta]/R). Hence
  [Pi] ([delta]), the angle which this parallel makes with v = 0, is
  given by

    tan [Pi]([delta]) . sin h ([delta]/R) = 1, or tan ½[Pi]([delta]) =
      e^(-[delta]/R)

  which is Lobatchewsky's formula. We also obtain easily for the area of
  a triangle the formula R²([pi] - A - B - C).

  Beltrami's treatment connects two curves which, in the earlier
  treatment, had no connexion. These are limit-lines and curves of
  constant distance from a straight line. Both may be regarded as
  circles, the first having an infinite, the second an imaginary radius.
  The equation to a circle of radius [rho] and centre u0v0 is

    (a² - uu0 - vv0)² = cos h² ([rho]/R)w0²w² = C²w²     (say).

  This equation remains real when [rho] is a pure imaginary, and remains
  finite when w0 = 0, provided [rho] becomes infinite in such a way that
  w0 cos h ([rho]/R) remains finite. In the latter case the equation
  represents a limit-line. In the former case, by giving different
  values to C, we obtain concentric circles with the imaginary centre
  u0v0. One of these, obtained by putting C = 0, is the straight line a²
  - uu0 - vv0 = 0. Hence the others are each throughout at a constant
  distance from this line. (It may be shown that all motions in a
  hyperbolic plane consist, in a general sense, of rotations; but three
  types must be distinguished according as the centre is real, imaginary
  or at infinity. All points describe, accordingly, one of the three
  types of circles.)

  The above Euclidean interpretation fails for three or more dimensions.
  In the _Teoria fondamentale_, accordingly, where n dimensions are
  considered, Beltrami treats hyperbolic space in a purely analytical
  spirit. The paper shows that Lobatchewsky's space of any number of
  dimensions has, in Riemann's sense, a constant negative measure of
  curvature. Beltrami starts with the formula (analogous to that of the
  _Saggio_)

    ds² = R²x^(-2)(dx² + dx1² + dx2² + ... + dx_n²)

  where x² + x1² + x2² + ... + x_n² = a².

  He shows that geodesics are represented by linear equations between
  x1, x2, ..., x_n, and that the geodesic distance [rho] between two
  points x and x' is given by

          [rho]                a² - x1x'1 - x2x'2 - ... - x_n x'_n
     cosh ----- = -----------------------------------------------------------------
            R     {(a² - x1² - x2² - ... - x_n²)(a² - x'1² - x'2² - ... - x'_n²)}^½

  (a formula practically identical with Cayley's, though obtained by a
  very different method). In order to show that the measure of curvature
  is constant, we make the substitutions

    x1 = r[lambda]1, x2 = r[lambda]2 ... x_n = r[lambda]_n, where
      [Sigma][lambda]² = 1.

  Hence
                 _______
    ds² = (Radr/(a² - r²)])² + R²r²d[Delta]²/(a² - r²).

  where

    d[Delta]² = [Sigma]d[lambda]².

  Also calling [rho] the geodesic distance from the origin, we have

         [rho]         a               [rho]          r
    cosh ----- = ---------------, sinh ----- = ---------------.
           R     [root](a² - r²)         R     [root](a² - r²)

  Hence

    ds² = d[rho]² + (R sin h ([rho]/R))²d[Delta]².

  Putting

    z1 = [rho][lambda]1, z2 = [rho][lambda]2, ... z_n = [rho][lambda]_n,

  we obtain
                               _                       _
                          1   |  /  R        [rho]\²    |
    ds² = [Sigma]dz² + ------ | ( ----- sinh ----- ) - 1| [Sigma](z_i dz_k - z_k dz_i)².
                       [rho]² |_ \[rho]        R  /    _|

  Hence when [rho] is small, we have approximately

                        1
    ds² = [Sigma]dz² + ---[Sigma](z_i dz_k - z_k dz_i)²    (1).
                       3R²

  Considering a surface element through the origin, we may choose our
  axes so that, for this element,

    z3 = Z4 = ... = z_n = 0.

  Thus

                         1
    ds² = dz1² + dz2² + ---(z1dz2 - z2dz1)²    (2).
                        3R²

  Now the area of the triangle whose vertices are (0, 0), (z1, z2),
  (dz1, dz2) is ½(z1, dz2 - z2dz1). Hence the quotient when the terms of
  the fourth order in (2) are divided by the square of this triangle is
  4/3R²; hence, returning to general axes, the same is the quotient when
  the terms of the fourth order in (1) are divided by the square of the
  triangle whose vertices are (0, 0, ... 0), (z1, z2, z3, ... z_n),
  (dz1, dz2, dz3 ... dz_n). But -¾ of this quotient is defined by
  Riemann as the measure of curvature.[15] Hence the measure of
  curvature is -1/R², i.e. is constant and negative. The properties of
  parallels, triangles, &c., are as in the _Saggio_. It is also shown
  that the analogues of limit surfaces have zero curvature; and that
  spheres of radius [rho] have constant positive curvature 1/R² sinh²
  ([rho]/R), so that spherical geometry may be regarded as contained in
  the pseudo-spherical (as Beltrami calls Lobatchewsky's system).


  Transition to the projective method.

The _Saggio_, as we saw, gives a Euclidean interpretation confined to
two dimensions. But a consideration of the auxiliary plane suggests a
different interpretation, which may be extended to any number of
dimensions. If, instead of referring to the pseudosphere, we merely
_define_ distance and angle, in the Euclidean plane, as those functions
of the coordinates which gave us distance and angle on the pseudosphere,
we find that the geometry of our plane has become Lobatchewsky's. All
the points of the limiting circle are now at infinity, and points beyond
it are imaginary. If we give our circle an imaginary radius the geometry
on the plane becomes elliptic. Replacing the circle by a sphere, we
obtain an analogous representation for three dimensions. Instead of a
circle or sphere we may take any conic or quadric. With this definition,
if the fundamental quadric be [Sigma]_(xx) = 0, and if [Sigma]_(xx)' be
the polar form of [Sigma]_(xx), the distance [rho] between x and x' is
given by the projective formula

  cos([rho]/k) = [Sigma]_xx'/{[Sigma]_(xx)·[Sigma]_x'x'}^½.

That this formula is projective is rendered evident by observing that
e^(-2i[rho]/k) is the anharmonic ratio of the range consisting of the
two points and the intersections of the line joining them with the
fundamental quadric. With this we are brought to the third or projective
period. The method of this period is due to Cayley; its application to
previous non-Euclidean geometry is due to Klein. The projective method
contains a generalization of discoveries already made by Laguerre[16] in
1853 as regards Euclidean geometry. The arbitrariness of this procedure
of deriving metrical geometry from the properties of conics is removed
by Lie's theory of congruence. We then arrive at the stage of thought
which finds its expression in the modern treatment of the axioms of
geometry.


  The two kinds of elliptic space.

The projective method leads to a discrimination, first made by
Klein,[17] of two varieties of Riemann's space; Klein calls these
elliptic and spherical. They are also called the polar and antipodal
forms of elliptic space. The latter names will here be used. The
difference is strictly analogous to that between the diameters and the
points of a sphere. In the polar form two straight lines in a plane
always intersect in one and only one point; in the antipodal form they
intersect always in two points, which are antipodes. According to the
definition of geometry adopted in section VII. (_Axioms of Geometry_),
the antipodal form is not to be termed "geometry," since any pair of
coplanar straight lines intersect each other in two points. It may be
called a "quasi-geometry." Similarly in the antipodal form two diameters
always determine a plane, but two points on a sphere do not determine a
great circle when they are antipodes, and two great circles always
intersect in two points. Again, a plane does not form a boundary among
lines through a point: we can pass from any one such line to any other
without passing through the plane. But a great circle does divide the
surface of a sphere. So, in the polar form, a complete straight line
does not divide a plane, and a plane does not divide space, and does
not, like a Euclidean plane, have two sides.[18] But, in the antipodal
form, a plane is, in these respects, like a Euclidean plane.

It is explained in section VII. in what sense the metrical geometry of
the material world can be considered to be determinate and not a matter
of arbitrary choice. The scientific question as to the best available
evidence concerning the nature of this geometry is one beset with
difficulties of a peculiar kind. We are obstructed by the fact that all
existing physical science assumes the Euclidean hypothesis. This
hypothesis has been involved in all actual measurements of large
distances, and in all the laws of astronomy and physics. The principle
of simplicity would therefore lead us, in general, where an observation
conflicted with one or more of those laws, to ascribe this anomaly, not
to the falsity of Euclidean geometry, but to the falsity of the laws in
question. This applies especially to astronomy. On the earth our means
of measurement are many and direct, and so long as no great accuracy is
sought they involve few scientific laws. Thus we acquire, from such
direct measurements, a very high degree of probability that the
space-constant, if not infinite, is yet large as compared with
terrestrial distances. But astronomical distances and triangles can only
be measured by means of the received laws of astronomy and optics, all
of which have been established by assuming the truth of the Euclidean
hypothesis. It therefore remains possible (until a detailed proof of the
contrary is forthcoming) that a large but finite space-constant, with
different laws of astronomy and optics, would have equally explained the
phenomena. We cannot, therefore, accept the measurements of stellar
parallaxes, &c., as conclusive evidence that the space-constant is large
as compared with stellar distances. For the present, on grounds of
simplicity, we may rightly adopt this view; but it must remain possible
that, in view of some hitherto undiscovered discrepancy, a slight
correction of the sort suggested might prove the simplest alternative.
But conversely, a finite parallax for very distant stars, or a negative
parallax for any star, could not be accepted as conclusive evidence that
our geometry is non-Euclidean, unless it were shown--and this seems
scarcely possible--that no modification of astronomy or optics could
account for the phenomenon. Thus although we may admit a probability
that the space-constant is large in comparison with stellar distances, a
conclusive proof or disproof seems scarcely possible.

Finally, it is of interest to note that, though it is theoretically
possible to prove, by scientific methods, that our geometry is
non-Euclidean, it is wholly impossible to prove by such methods that it
is accurately Euclidean. For the unavoidable errors of observation must
always leave a slight margin in our measurements. A triangle might be
found whose angles were certainly greater, or certainly less, than two
right angles; but to prove them _exactly_ equal to two right angles must
always be beyond our powers. If, therefore, any man cherishes a hope of
proving the exact truth of Euclid, such a hope must be based, not upon
scientific, but upon philosophical considerations.

  BIBLIOGRAPHY.--The bibliography appended to section VII. should be
  consulted in this connexion. Also, in addition to the citations
  already made, the following works may be mentioned.

  For Lobatchewsky's writings, cf. _Urkunden zur Geschichte der
  nichteuklidischen Geometrie_, i., _Nikolaj Iwanowitsch Lobatschefsky_,
  by F. Engel and P. Stäckel (Leipzig, 1898). For John Bolyai's
  _Appendix_, cf. _Absolute Geometrie nach Johann Bolyai_, by J.
  Frischauf (Leipzig, 1872), and also the new edition of his father's
  large work, _Tentamen_ ..., published by the Mathematical Society of
  Budapest; the second volume contains the appendix. Cf. also J.
  Frischauf, _Elemente der absoluten Geometrie_ (Leipzig, 1876); M.L.
  Gérard, _Sur la géométrie non-Euclidienne_ (thesis for doctorate)
  (Paris, 1892); de Tilly, _Essai sur les principes fondamentales de la
  géométrie et de la mécanique_ (Bordeaux, 1879); Sir R.S. Ball, "On the
  Theory of Content," _Trans. Roy. Irish Acad._ vol. xxix. (1889); F.
  Lindemann, "Mechanik bei projectiver Maasbestimmung," _Math. Annal._
  vol. vii.; W.K. Clifford, "Preliminary Sketch of Biquaternions,"
  _Proc. of Lond. Math. Soc._ (1873), and _Coll. Works_; A. Buchheim,
  "On the Theory of Screws in Elliptic Space," _Proc. Lond. Math. Soc._
  vols. xv., xvi., xvii.; H. Cox, "On the Application of Quaternions and
  Grassmann's Algebra to different Kinds of Uniform Space," _Trans.
  Camb. Phil. Soc._ (1882); M. Dehn, "Die Legendarischen Sätze über die
  Winkelsumme im Dreieck," Math. Ann. vol. 53 (1900), and "Über den
  Rauminhalt," _Math. Annal._ vol. 55 (1902).

  For expositions of the whole subject, cf. F. Klein, _Nicht-Euklidische
  Geometrie_ (Göttingen, 1893); R. Bonola, _La Geometria non-Euclidea_
  (Bologna, 1906); P. Barbarin, _La Géométrie non-Euclidienne_ (Paris,
  1902); W. Killing, _Die nicht-Euklidischen Raumformen in analytischer
  Behandlung_ (Leipzig, 1885). The last-named work also deals with
  geometry of more than three dimensions; in this connexion cf. also G.
  Veronese, _Fondamenti di geometria a più dimensioni ed a più specie_
  _di unità rettilinee_ ... (Padua, 1891, German translation, Leipzig,
  1894); G. Fontené, _L'Hyperespace à (n-1) dimensions_ (Paris, 1892);
  and A.N. Whitehead, _loc. cit._ Cf. also E. Study, "Über
  nicht-Euklidische und Liniengeometrie," _Jahr. d. Deutsch. Math. Ver._
  vol. xv. (1906); W. Burnside, "On the Kinematics of non-Euclidean
  Space," _Proc. Lond. Math. Soc._ vol. xxvi. (1894). A bibliography on
  the subject up to 1878 has been published by G.B. Halsted, _Amer.
  Journ. of Math._ vols. i. and ii.; and one up to 1900 by R. Bonola,
  _Index operum ad geometriam absolutam spectantium_ ... (1902, and
  Leipzig, 1903).     (B. A. W. R.; A. N. W.)


VII. AXIOMS OF GEOMETRY

  Theories of space.

Until the discovery of the non-Euclidean geometries (Lobatchewsky, 1826
and 1829; J. Bolyai, 1832; B. Riemann, 1854), geometry was universally
considered as being exclusively the science of existent space. (See
section VI. _Non-Euclidean Geometry_.) In respect to the science, as
thus conceived, two controversies may be noticed. First, there is the
controversy respecting the absolute and relational theories of space.
According to the absolute theory, which is the traditional view (held
explicitly by Newton), space has an existence, in some sense whatever it
may be, independent of the bodies which it contains. The bodies occupy
space, and it is not intrinsically unmeaning to say that any definite
body occupies _this_ part of space, and not _that_ part of space,
without reference to other bodies occupying space. According to the
relational theory of space, of which the chief exponent was
Leibnitz,[19] space is nothing but a certain assemblage of the relations
between the various particular bodies in space. The idea of space with
no bodies in it is absurd. Accordingly there can be no meaning in saying
that a body is _here_ and not _there_, apart from a reference to the
other bodies in the universe. Thus, on this theory, absolute motion is
intrinsically unmeaning. It is admitted on all hands that in practice
only relative motion is directly measurable. Newton, however, maintains
in the _Principia_ (scholium to the 8th definition) that it is
indirectly measurable by means of the effects of "centrifugal force" as
it occurs in the phenomena of rotation. This irrelevance of absolute
motion (if there be such a thing) to science has led to the general
adoption of the relational theory by modern men of science. But no
decisive argument for either view has at present been elaborated.[20]
Kant's view of space as being a form of perception at first sight
appears to cut across this controversy. But he, saturated as he was with
the spirit of the Newtonian physics, must (at least in both editions of
the _Critique_) be classed with the upholders of the absolute theory.
The form of perception has a type of existence proper to itself
independently of the particular bodies which it contains. For example he
writes:[21] "Space does not represent any quality of objects by
themselves, or objects in their relation to one another, i.e. space does
not represent any determination which is inherent in the objects
themselves, and would remain, even if all subjective conditions of
intuition were removed."


  Axioms.

The second controversy is that between the view that the axioms
applicable to space are known only from experience, and the view that in
some sense these axioms are given _a priori_. Both these views, thus
broadly stated, are capable of various subtle modifications, and a
discussion of them would merge into a general treatise on epistemology.
The cruder forms of the _a priori_ view have been made quite untenable
by the modern mathematical discoveries. Geometers now profess ignorance
in many respects of the exact axioms which apply to existent space, and
it seems unlikely that a profound study of the question should thus
obliterate _a priori_ intuitions.

Another question irrelevant to this article, but with some relevance to
the above controversy, is that of the derivation of our perception of
existent space from our various types of sensation. This is a question
for psychology.[22]

_Definition of Abstract Geometry._--Existent space is the subject matter
of only one of the applications of the modern science of abstract
geometry, viewed as a branch of pure mathematics. Geometry has been
defined[23] as "the study of series of two or more dimensions." It has
also been defined[24] as "the science of cross classification." These
definitions are founded upon the actual practice of mathematicians in
respect to their use of the term "Geometry." Either of them brings out
the fact that geometry is not a science with a determinate subject
matter. It is concerned with any subject matter to which the formal
axioms may apply. Geometry is not peculiar in this respect. All branches
of pure mathematics deal merely with types of relations. Thus the
fundamental ideas of geometry (e.g. those of _points_ and of _straight
lines_) are not ideas of determinate entities, but of any entities for
which the axioms are true. And a set of formal geometrical axioms cannot
in themselves be true or false, since they are not determinate
propositions, in that they do not refer to a determinate subject matter.
The axioms are propositional functions.[25] When a set of axioms is
given, we can ask (1) whether they are consistent, (2) whether their
"existence theorem" is proved, (3) whether they are independent. Axioms
are consistent when the contradictory of any axiom cannot be deduced
from the remaining axioms. Their existence theorem is the proof that
they are true when the fundamental ideas are considered as denoting some
determinate subject matter, so that the axioms are developed into
determinate propositions. It follows from the logical law of
contradiction that the proof of the existence theorem proves also the
consistency of the axioms. This is the only method of proof of
consistency. The axioms of a set are independent of each other when no
axiom can be deduced from the remaining axioms of the set. The
independence of a given axiom is proved by establishing the consistency
of the remaining axioms of the set, together with the contradictory of
the given axiom. The enumeration of the axioms is simply the enumeration
of the hypotheses[26] (with respect to the undetermined subject matter)
of which some at least occur in each of the subsequent propositions.

Any science is called a "geometry" if it investigates the theory of the
classification of a set of entities (the points) into classes (the
straight lines), such that (1) there is one and only one class which
contains any given pair of the entities, and (2) every such class
contains more than two members. In the two geometries, important from
their relevance to existent space, axioms which secure an order of the
points on any line also occur. These geometries will be called
"Projective Geometry" and "Descriptive Geometry." In projective geometry
any two straight lines in a plane intersect, and the straight lines are
closed series which return into themselves, like the circumference of a
circle. In descriptive geometry two straight lines in a plane do not
necessarily intersect, and a straight line is an open series without
beginning or end. Ordinary Euclidean geometry is a descriptive geometry;
it becomes a projective geometry when the so-called "points at infinity"
are added.


_Projective Geometry._

Projective geometry may be developed from two undefined fundamental
ideas, namely, that of a "point" and that of a "straight line." These
undetermined ideas take different specific meanings for the various
specific subject matters to which projective geometry can be applied.
The number of the axioms is always to some extent arbitrary, being
dependent upon the verbal forms of statement which are adopted. They
will be presented[27] here as twelve in number, eight being "axioms of
classification," and four being "axioms of order."

_Axioms of Classification._--The eight axioms of classification are as
follows:

1. Points form a class of entities with at least two members.

2. Any straight line is a class of points containing at least three
members.

3. Any two distinct points lie in one and only one straight line.

4. There is at least one straight line which does not contain all the
points.

5. If A, B, C are non-collinear points, and A' is on the straight line
BC, and B' is on the straight line CA, then the straight lines AA' and
BB' possess a point in common.

  _Definition._--If A, B, C are any three non-collinear points, the
  _plane_ ABC is the class of points lying on the straight lines joining
  A with the various points on the straight line BC.

6. There is at least one plane which does not contain all the points.

7. There exists a plane [alpha], and a point A not incident in [alpha],
such that any point lies in some straight line which contains both A and
a point in [alpha].

  _Definition._--Harm. (ABCD) symbolizes the following conjoint
  statements: (1) that the points A, B, C, D are collinear, and (2) that
  a quadrilateral can be found with one pair of opposite sides
  intersecting at A, with the other pair intersecting at C, and with its
  diagonals passing through B and D respectively. Then B and D are said
  to be "harmonic conjugates" with respect to A and C.

8. Harm. (ABCD) implies that B and D are distinct points.

In the above axioms 4 secures at least two dimensions, axiom 5 is the
fundamental axiom of the plane, axiom 6 secures at least three
dimensions, and axiom 7 secures at most three dimensions. From axioms
1-5 it can be proved that any two distinct points in a straight line
determine that line, that any three non-collinear points in a plane
determine that plane, that the straight line containing any two points
in a plane lies wholly in that plane, and that any two straight lines in
a plane intersect. From axioms 1-6 Desargue's well-known theorem on
triangles in perspective can be proved.

  The enunciation of this theorem is as follows: If ABC and A'B'C' are
  two coplanar triangles such that the lines AA', BB', CC' are
  concurrent, then the three points of intersection of BC and B'C' of CA
  and C'A', and of AB and A'B' are collinear; and conversely if the
  three points of intersection are collinear, the three lines are
  concurrent. The proof which can be applied is the usual projective
  proof by which a third triangle A"B"C" is constructed not coplanar
  with the other two, but in perspective with each of them.

  It has been proved[28] that Desargues's theorem cannot be deduced from
  axioms 1-5, that is, if the geometry be confined to two dimensions.
  All the proofs proceed by the method of producing a specification of
  "points" and "straight lines" which satisfies axioms 1-5, and such
  that Desargues's theorem does not hold.

  It follows from axioms 1-5 that Harm. (ABCD) implies Harm. (ADCB) and
  Harm. (CBAD), and that, if A, B, C be any three distinct collinear
  points, there exists at least one point D such that Harm. (ABCD). But
  it requires Desargues's theorem, and hence axiom 6, to prove that
  Harm. (ABCD) and Harm. (ABCD') imply the identity of D and D'.

The necessity for axiom 8 has been proved by G. Fano,[29] who has
produced a three dimensional geometry of fifteen points, i.e. a method
of cross classification of fifteen entities, in which each straight line
contains three points, and each plane contains seven straight lines. In
this geometry axiom 8 does not hold. Also from axioms 1-6 and 8 it
follows that Harm. (ABCD) implies Harm. (BCDA).

  _Definitions._--When two plane figures can be derived from one another
  by a single projection, they are said to be in _perspective_. When two
  plane figures can be derived one from the other by a finite series of
  perspective relations between intermediate figures, they are said to
  be _projectively_ related. Any property of a plane figure which
  necessarily also belongs to any projectively related figure, is called
  a _projective_ property.

  The following theorem, known from its importance as "the fundamental
  theorem of projective geometry," cannot be proved[30] from axioms 1-8.
  The enunciation is: "A projective correspondence between the points on
  two straight lines is completely determined when the correspondents of
  three distinct points on one line are determined on the other." This
  theorem is equivalent[31] (assuming axioms 1-8) to another theorem,
  known as Pappus's Theorem, namely: "If l and l' are two distinct
  coplanar lines, and A, B, C are three distinct points on l, and A',
  B', C' are three distinct points on l', then the three points of
  intersection of AA' and B'C, of A'B and CC', of BB' and C'A, are
  collinear." This theorem is obviously Pascal's well-known theorem
  respecting a hexagon inscribed in a conic, for the special case when
  the conic has degenerated into the two lines l and l'. Another theorem
  also equivalent (assuming axioms 1-8) to the fundamental theorem is
  the following:[32] If the three collinear pairs of points, A and A', B
  and B', C and C', are such that the three pairs of opposite sides of a
  complete quadrangle pass respectively through them, i.e. one pair
  through A and A' respectively, and so on, and if also the three sides
  of the quadrangle which pass through A, B, and C, are concurrent in
  one of the corners of the quadrangle, then another quadrangle can be
  found with the same relation to the three pairs of points, except that
  its three sides which pass through A, B, and C, are not concurrent.

  Thus, if we choose to take any one of these three theorems as an
  axiom, all the theorems of projective geometry which do not require
  ordinal or metrical ideas for their enunciation can be proved. Also a
  conic can be defined as the locus of the points found by the usual
  construction, based upon Pascal's theorem, for points on the conic
  through five given points. But it is unnecessary to assume here any
  one of the suggested axioms; for the fundamental theorem can be
  deduced from the axioms of order together with axioms 1-8.

_Axioms of Order._--It is possible to define (cf. Pieri, _loc. cit._)
the property upon which the order of points on a straight line depends.
But to secure that this property does in fact range the points in a
serial order, some axioms are required. A straight line is to be a
closed series; thus, when the points are in order, it requires two
points on the line to divide it into two distinct complementary
segments, which do not overlap, and together form the whole line.
Accordingly the problem of the definition of order reduces itself to the
definition of these two segments formed by any two points on the line;
and the axioms are stated relatively to these segments.

  _Definition._--If A, B, C are three collinear points, the points on
  the _segment_ ABC are defined to be those points such as X, for which
  there exist two points Y and Y' with the property that Harm. (AYCY')
  and Harm. (BYXY') both hold. The _supplementary segment_ ABC is
  defined to be the rest of the points on the line. This definition is
  elucidated by noticing that with our ordinary geometrical ideas, if B
  and X are any two points between A and C, then the two pairs of
  points, A and C, B and X, define an involution with real double
  points, namely, the Y and Y' of the above definition. The property of
  belonging to a segment ABC is projective, since the harmonic relation
  is projective.

The first three axioms of order (cf. Pieri, _loc. cit._) are:

9. If A, B, C are three distinct collinear points, the supplementary
segment ABC is contained within the segment BCA.

10. If A, B, C are three distinct collinear points, the common part of
the segments BCA and CAB is contained in the supplementary segment ABC.

11. If A, B, C are three distinct collinear points, and D lies In the
segment ABC, then the segment ADC is contained within the segment ABC.

From these axioms all the usual properties of a closed order follow. It
will be noticed that, if A, B, C are any three collinear points, C is
necessarily traversed in passing from A to B by one route along the
line, and is not traversed in passing from A to B along the other route.
Thus there is no meaning, as referred to closed straight lines, in the
simple statement that C lies between A and B. But there may be a
relation of separation between two pairs of collinear points, such as A
and C, and B and D. The couple B and D is said to separate A and C, if
the four points are collinear and D lies in the segment complementary to
the segment ABC. The property of the separation of pairs of points by
pairs of points is projective. Also it can be proved that Harm. (ABCD)
implies that B and D separate A and C.

  _Definitions._--A series of entities arranged in a serial order, open
  or closed, is said to be _compact_, if the series contains no
  immediately consecutive entities, so that in traversing the series
  from any one entity to any other entity it is necessary to pass
  through entities distinct from either. It was the merit of R. Dedekind
  and of G. Cantor explicitly to formulate another fundamental property
  of series. The Dedekind property[33] as applied to an open series can
  be defined thus: An open series possesses the Dedekind property, if,
  however, it be divided into two mutually exclusive classes u and v,
  which (1) contain between them the whole series, and (2) are such that
  every member of u precedes in the serial order every member of v,
  there is always a member of the series, belonging to one of the two, u
  or v, which precedes every member of v (other than itself if it belong
  to v), and also succeeds every member of u (other than itself if it
  belong to u). Accordingly in an open series with the Dedekind property
  there is always a member of the series marking the junction of two
  classes such as u and v. An open series is _continuous_ if it is
  compact and possesses the Dedekind property. A closed series can
  always be transformed into an open series by taking any arbitrary
  member as the first term and by taking one of the two ways round as
  the ascending order of the series. Thus the definitions of compactness
  and of the Dedekind property can be at once transferred to a closed
  series.

12. The last axiom of order is that there exists at least one straight
line for which the point order possesses the Dedekind property.

It follows from axioms 1-12 by projection that the Dedekind property is
true for all lines. Again the _harmonic system_ ABC, where A, B, C are
collinear points, is defined[34] thus: take the harmonic conjugates A',
B', C' of each point with respect to the other two, again take the
harmonic conjugates of each of the six points A, B, C, A', B', C' with
respect to each pair of the remaining five, and proceed in this way by
an unending series of steps. The set of points thus obtained is called
the harmonic system ABC. It can be proved that a harmonic system is
compact, and that every segment of the line containing it possesses
members of it. Furthermore, it is easy to prove that the fundamental
theorem holds for harmonic systems, in the sense that, if A, B, C are
three points on a line l, and A', B', C' are three points on a line l',
and if by any two distinct series of projections A, B, C are projected
into A', B', C', then any point of the harmonic system ABC corresponds
to the same point of the harmonic system A'B'C' according to both the
projective relations which are thus established between l and l'. It now
follows immediately that the fundamental theorem must hold for all the
points on the lines l and l', since (as has been pointed out) harmonic
systems are "everywhere dense" on their containing lines. Thus the
fundamental theorem follows from the axioms of order.

A system of numerical coordinates can now be introduced, possessing the
property that linear equations represent planes and straight lines. The
outline of the argument by which this remarkable problem (in that
"distance" is as yet undefined) is solved, will now be given. It is
first proved that the points on any line can in a certain way be
definitely associated with all the positive and negative real numbers,
so as to form with them a one-one correspondence. The arbitrary elements
in the establishment of this relation are the points on the line
associated with 0, 1 and [oo].

This association[35] is most easily effected by considering a class of
projective relations of the line with itself, called by F. Schur (_loc.
cit._) _prospectivities_.

  Let l (fig. 69) be the given line, m and n any two lines intersecting
  at U on l, S and S' two points on n. Then a projective relation
  between l and itself is formed by projecting l from S on to m, and
  then by projecting m from S' back on to l. All such projective
  relations, however m, n, S and S' be varied, are called
  "prospectivities," and U is the double point of the prospectivity. If
  a point O on l is related to A by a prospectivity, then all
  prospectivities, which (1) have the same double point U, and (2)
  relate O to A, give the same correspondent (Q, in figure) to any point
  P on the line l; in fact they are all the same prospectivity, however
  m, n, S, and S' may have been varied subject to these conditions. Such
  a prospectivity will be denoted by (OAU²).

  [Illustration: FIG. 69.]

  The sum of two prospectivities, written (OAU²) + (OBU²), is defined to
  be that transformation of the line l into itself which is obtained by
  first applying the prospectivity (OAU²) and then applying the
  prospectivity (OBU²). Such a transformation, when the two summands
  have the same double point, is itself a prospectivity with that double
  point.

  [Illustration: FIG. 70]

  With this definition of addition it can be proved that prospectivities
  with the same double point satisfy all the axioms of magnitude.
  Accordingly they can be associated in a one-one correspondence with
  the positive and negative real numbers. Let E (fig. 70) be any point
  on l, distinct from O and U. Then the prospectivity (OEU²) is
  associated with unity, the prospectivity (OOU²) is associated with
  zero, and (OUU²) with [infinity]. The prospectivities of the type
  (OPU²), where P is any point on the segment OEU, correspond to the
  positive numbers; also if P' is the harmonic conjugate of P with
  respect to O and U, the prospectivity (OP'U²) is associated with the
  corresponding negative number. (The subjoined figure explains this
  relation of the positive and negative prospectivities.) Then any point
  P on l is associated with the same number as is the prospectivity
  (OPU²).

  [Illustration: FIG. 71.]

  It can be proved that the order of the numbers in algebraic order of
  magnitude agrees with the order on the line of the associated points.
  Let the numbers, assigned according to the preceding specification, be
  said to be associated with the points according to the
  "numeration-system (OEU)." The introduction of a coordinate system for
  a plane is now managed as follows: Take any triangle OUV in the plane,
  and on the lines OU and OV establish the numeration systems (OE1U) and
  (OE2V), where E1 and E2 are arbitrarily chosen. Then (cf. fig. 71) if
  M and N are associated with the numbers x and y according to these
  systems, the coordinates of P are x and y. It then follows that the
  equation of a straight line is of the form ax + by + c = 0. Both
  coordinates of any point on the line UV are infinite. This can be
  avoided by introducing homogeneous coordinates X, Y, Z, where x = X/Z,
  and y = Y/Z, and Z = 0 is the equation of UV.

  [Illustration: FIG. 72.]

  The procedure for three dimensions is similar. Let OUVW (fig. 72) be
  any tetrahedron, and associate points on OU, OV, OW with numbers
  according to the numeration systems (OE1U), (OE2V), and (OE3W). Let
  the planes VWP, WUP, UVP cut OU, OV, OW in L, M, N respectively; and
  let x, y, z be the numbers associated with L, M, N respectively. Then
  P is the point (x, y, z). Also homogeneous coordinates can be
  introduced as before, thus avoiding the infinities on the plane UVW.

  The cross ratio of a range of four collinear points can now be defined
  as a number characteristic of that range. Let the coordinates of any
  point P_r of the range P1 P2 P3 P4 be

    [lambda]_r a + µ_r + a'   [lambda]_r b + µ_r b'
    -----------------------,  ---------------------,
       [lambda]_r + µ_r         [lambda]_r + µ_r

      [lambda]_r c + µ_r c'
      ---------------------, (r = 1, 2, 3, 4)
        [lambda]_r + µ_r

  and let ([lambda]_r µ_s) be written for [lambda]_r µ_s -[lambda]_s
  µ_r. Then the cross ratio {P1 P2 P3 P4} is defined to be the number
  ([lambda]1µ2)([lambda]3µ4)/([lambda]1µ4)([lambda]3µ2). The equality of
  the cross ratios of the ranges (P1 P2 P3 P4) and (Q1 Q2 Q3 Q4) is
  proved to be the necessary and sufficient condition for their mutual
  projectivity. The cross ratios of all harmonic ranges are then easily
  seen to be all equal to -1, by comparing with the range (OE1UE'1) on
  the axis of x.

  Thus all the ordinary propositions of geometry in which distance and
  angular measure do not enter otherwise than in cross ratios can now be
  enunciated and proved. Accordingly the greater part of the analytical
  theory of conics and quadrics belongs to geometry at this stage The
  theory of distance will be considered after the principles of
  descriptive geometry have been developed.


_Descriptive Geometry._

Descriptive geometry is essentially the science of multiple order for
open series. The first satisfactory system of axioms was given by M.
Pasch.[36] An improved version is due to G. Peano.[37] Both these
authors treat the idea of the class of points constituting the segment
lying _between_ two points as an undefined fundamental idea. Thus in
fact there are in this system two fundamental ideas, namely, of points
and of segments. It is then easy enough to define the prolongations of
the segments, so as to form the complete straight lines. D.
Hilbert's[38] formulation of the axioms is in this respect practically
based on the same fundamental ideas. His work is justly famous for some
of the mathematical investigations contained in it, but his exposition
of the axioms is distinctly inferior to that of Peano. Descriptive
geometry can also be considered[39] as the science of a class of
relations, each relation being a two-termed serial relation, as
considered in the logic of relations, ranging the points between which
it holds into a linear open order. Thus the relations are the straight
lines, and the terms between which they hold are the points. But a
combination of these two points of view yields[40] the simplest
statement of all. Descriptive geometry is then conceived as the
investigation of an undefined fundamental relation between three terms
(points); and when the relation holds between three points A, B, C, the
points are said to be "in the [linear] order ABC."

O. Veblen's axioms and definitions, slightly modified, are as follows:--

1. If the points A, B, C are in the order ABC, they are in the order
CBA.

2. If the points A, B, C are in the order ABC, they are not in the order
BCA.

3. If the points A, B, C are in the order ABC, A is distinct from C.

4. If A and B are any two distinct points, there exists a point C such
that A, B, C are in the order ABC.

  _Definition._--The _line_ AB (A =| B) consists of A and B, and of all
  points X in one of the possible orders, ABX, AXB, XAB. The points X in
  the order AXB constitute the _segment_ AB.

5. If points C and D (C =| D) lie on the line AB, then A lies on the
line CD.

6. There exist three distinct points A, B, C not in any of the orders
ABC, BCA, CAB.

7. If three distinct points A, B, C (fig. 73) do not lie on the same
line, and D and E are two distinct points in the orders BCD and CEA,
then a point F exists in the order AFB, and such that D, E, F are
collinear.

[Illustration: FIG. 73.]

  _Definition._--If A, B, C are three non-collinear points, the _plane_
  ABC is the class of points which lie on any one of the lines joining
  any two of the points belonging to the _boundary_ of the triangle ABC,
  the boundary being formed by the segments BC, CA and AB. The
  _interior_ of the triangle ABC is formed by the points in segments
  such as PQ, where P and Q are points respectively on two of the
  segments BC, CA, AB.

8. There exists a plane ABC, which does not contain all the points.

  _Definition._--If A, B, C, D are four non-coplanar points, the space
  ABCD is the class of points which lie on any of the lines containing
  two points on the surface of the tetrahedron ABCD, the _surface_ being
  formed by the interiors of the triangles ABC, BCD, DCA, DAB.

9. There exists a space ABCD which contains all the points.

10. The Dedekind property holds for the order of the points on any
straight line.

It follows from axioms 1-9 that the points on any straight line are
arranged in an open serial order. Also all the ordinary theorems
respecting a point dividing a straight line into two parts, a straight
line dividing a plane into two parts, and a plane dividing space into
two parts, follow.

  Again, in any plane [alpha] consider a line l and a point A (fig. 74).

  [Illustration: FIG. 74.]

  Let any point B divide l into two half-lines l1 and l2. Then it can be
  proved that the set of half-lines, emanating from A and intersecting
  l1 (such as m), are bounded by two half-lines, of which ABC is one.
  Let r be the other. Then it can be proved that r does not intersect
  l1. Similarly for the half-line, such as n, intersecting l2. Let s be
  its bounding half-line. Then two cases are possible. (1) The
  half-lines r and s are collinear, and together form one complete line.
  In this case, there is one and only one line (viz. r + s) through A
  and lying in [alpha] which does not intersect l. This is the Euclidean
  case, and the assumption that this case holds is the _Euclidean
  parallel axiom_. But (2) the half-lines r and s may not be collinear.
  In this case there will be an infinite number of lines, such as k for
  instance, containing A and lying in [alpha], which do not intersect l.
  Then the lines through A in [alpha] are divided into two classes by
  reference to l, namely, the _secant_ lines which intersect l, and the
  _non-secant_ lines which do not intersect l. The two boundary
  non-secant lines, of which r and s are respectively halves, may be
  called the two parallels to l through A.

  The perception of the possibility of case 2 constituted the
  starting-point from which Lobatchewsky constructed the first explicit
  coherent theory of non-Euclidean geometry, and thus created a
  revolution in the philosophy of the subject. For many centuries the
  speculations of mathematicians on the foundations of geometry were
  almost confined to hopeless attempts to prove the "parallel axiom"
  without the introduction of some equivalent axiom.[41]

_Associated Projective and Descriptive Spaces._--A region of a
projective space, such that one, and only one, of the two supplementary
segments between any pair of points within it lies entirely within it,
satisfies the above axioms (1-10) of descriptive geometry, where the
points of the region are the descriptive points, and the portions of
straight lines within the region are the descriptive lines. If the
excluded part of the original projective space is a single plane, the
Euclidean parallel axiom also holds, otherwise it does not hold for the
descriptive space of the limited region. Again, conversely, starting
from an original descriptive space an associated projective space can be
constructed by means of the concept of _ideal points_.[42] These are
also called _projective points_, where it is understood that the simple
points are the points of the original descriptive space. An _ideal
point_ is the class of straight lines which is composed of two coplanar
lines a and b, together with the lines of intersection of all pairs of
intersecting planes which respectively contain a and b, together with
the lines of intersection with the plane ab of all planes containing any
one of the lines (other than a or b) already specified as belonging to
the ideal point. It is evident that, if the two original lines a and b
intersect, the corresponding ideal point is nothing else than the whole
class of lines which are concurrent at the point ab. But the essence of
the definition is that an ideal point has an existence when the lines a
and b do not intersect, so long as they are coplanar. An ideal point is
termed _proper_, if the lines composing it intersect; otherwise it is
_improper_.

A theorem essential to the whole theory is the following: if any two of
the three lines a, b, c are coplanar, but the three lines are not all
coplanar, and similarly for the lines a, b, d, then c and d are
coplanar. It follows that any two lines belonging to an ideal point can
be used as the pair of guiding lines in the definition. An ideal point
is said to be _coherent_ with a plane, if any of the lines composing it
lie in the plane. An _ideal line_ is the class of ideal points each of
which is coherent with two given planes. If the planes intersect, the
ideal line is termed _proper_, otherwise it is _improper_. It can be
proved that any two planes, with which any two of the ideal points are
both coherent, will serve as the guiding planes used in the definition.
The ideal planes are defined as in projective geometry, and all the
other definitions (for segments, order, &c.) of projective geometry are
applied to the ideal elements. If an ideal plane contains some proper
ideal points, it is called _proper_, otherwise it is _improper_. Every
ideal plane contains some improper ideal points.

It can now be proved that all the axioms of projective geometry hold of
the ideal elements as thus obtained; and also that the order of the
ideal points as obtained by the projective method agrees with the order
of the proper ideal points as obtained from that of the associated
points of the descriptive geometry. Thus a projective space has been
constructed out of the ideal elements, and the proper ideal elements
correspond element by element with the associated descriptive elements.
Thus the proper ideal elements form a region in the projective space
within which the descriptive axioms hold. Accordingly, by substituting
ideal elements, a descriptive space can always be considered as a region
within a projective space. This is the justification for the ordinary
use of the "points at infinity" in the ordinary Euclidean geometry; the
reasoning has been transferred from the original descriptive space to
the associated projective space of ideal elements; and with the
Euclidean parallel axiom the improper ideal elements reduce to the ideal
points on a single improper ideal plane, namely, the plane at
infinity.[43]

_Congruence and Measurement._--The property of physical space which is
expressed by the term "measurability" has now to be considered. This
property has often been considered as essential to the very idea of
space. For example, Kant writes,[44] "Space is represented as an
infinite given _quantity_." This quantitative aspect of space arises
from the measurability of distances, of angles, of surfaces and of
volumes. These four types of quantity depend upon the two first among
them as fundamental. The measurability of space is essentially connected
with the idea of _congruence_, of which the simplest examples are to be
found in the proofs of equality by the method of superposition, as used
in elementary plane geometry. The mere concepts of "part" and of "whole"
must of necessity be inadequate as the foundation of measurement, since
we require the comparison as to quantity of regions of space which have
no portions in common. The idea of congruence, as exemplified by the
method of superposition in geometrical reasoning, appears to be founded
upon that of the "rigid body," which moves from one position to another
with its internal spatial relations unchanged. But unless there is a
previous concept of the metrical relations between the parts of the
body, there can be no basis from which to deduce that they are
unchanged.

It would therefore appear as if the idea of the congruence, or metrical
equality, of two portions of space (as empirically suggested by the
motion of rigid bodies) must be considered as a fundamental idea
incapable of definition in terms of those geometrical concepts which
have already been enumerated. This was in effect the point of view of
Pasch.[45] It has, however, been proved by Sophus Lie[46] that
congruence is capable of definition without recourse to a new
fundamental idea. This he does by means of his theory of finite
continuous groups (see GROUPS, THEORY OF), of which the definition is
possible in terms of our established geometrical ideas, remembering that
coordinates have already been introduced. The displacement of a rigid
body is simply a mode of defining to the senses a one-one transformation
of all space into itself. For at any point of space a particle may be
conceived to be placed, and to be rigidly connected with the rigid body;
and thus there is a definite correspondence of any point of space with
the new point occupied by the associated particle after displacement.
Again two successive displacements of a rigid body from position A to
position B, and from position B to position C, are the same in effect as
one displacement from A to C. But this is the characteristic "group"
property. Thus the transformations of space into itself defined by
displacements of rigid bodies form a group.

Call this group of transformations a congruence-group. Now according to
Lie a congruence-group is defined by the following characteristics:--

1. A congruence-group is a finite continuous group of one-one
transformations, containing the identical transformation.

2. It is a sub-group of the general projective group, i.e. of the group
of which any transformation converts planes into planes, and straight
lines into straight lines.

3. An infinitesimal transformation can always be found satisfying the
condition that, at least throughout a certain enclosed region, any
definite line and any definite point on the line are latent, i.e.
correspond to themselves.

4. No infinitesimal transformation of the group exists, such that, at
least in the region for which (3) holds, a straight line, a point on it,
and a plane through it, shall all be latent.

The property enunciated by conditions (3) and (4), taken together, is
named by Lie "Free mobility in the infinitesimal." Lie proves the
following theorems for a projective space:--

  1. If the above four conditions are only satisfied by a group
  throughout part of projective space, this part either ([alpha]) must
  be the region enclosed by a real closed quadric, or (ß) must be the
  whole of the projective space with the exception of a single plane. In
  case ([alpha]) the corresponding congruence group is the continuous
  group for which the enclosing quadric is latent; and in case (ß) an
  imaginary conic (with a real equation) lying in the latent plane is
  also latent, and the congruence group is the continuous group for
  which the plane and conic are latent.

  2. If the above four conditions are satisfied by a group throughout
  the whole of projective space, the congruence group is the continuous
  group for which some imaginary quadric (with a real equation) is
  latent.

  By a proper choice of non-homogeneous co-ordinates the equation of any
  quadrics of the types considered, either in theorem 1 ([alpha]), or in
  theorem 2, can be written in the form 1 +c(x² + y² + z²) = 0, where c
  is negative for a real closed quadric, and positive for an imaginary
  quadric. Then the general infinitesimal transformation is defined by
  the three equations:

    dx/dt = u - [omega]3y + [omega]2z + cx(ux + vy + wz), \
    dy/dt = v - [omega]1z + [omega]3x + cy(ux + vy + wz),  > (A)
    dz/dt = w - [omega]2x + [omega]1y + cz(ux + vy + wz). /

  In the ease considered in theorem 1 (ß), with the proper choice of
  co-ordinates the three equations defining the general infinitesimal
  transformation are:

    dx/dt = u - [omega]3y + [omega]2z, \
    dy/dt = v - [omega]1z + [omega]3x,  > (B)
    dz/dt = w - [omega]2x + [omega]1y. /

  In this case the latent plane is the plane for which at least one of
  x, y, z are infinite, that is, the plane 0.x + 0.y + 0.z + a = 0; and
  the latent conic is the conic in which the cone x² + y² + z² = 0
  intersects the latent plane.

It follows from theorems 1 and 2 that there is not one unique
congruence-group, but an indefinite number of them. There is one
congruence-group corresponding to each closed real quadric, one to each
imaginary quadric with a real equation, and one to each imaginary conic
in a real plane and with a real equation. The quadric thus associated
with each congruence-group is called the _absolute_ for that group, and
in the degenerate case of 1 (ß) the absolute is the latent plane
together with the latent imaginary conic. If the absolute is real, the
congruence-group is _hyperbolic_; if imaginary, it is _elliptic_; if the
absolute is a plane and imaginary conic, the group is parabolic.
Metrical geometry is simply the theory of the properties of some
particular congruence-group selected for study.

  The definition of distance is connected with the corresponding
  congruence-group by two considerations in respect to a range of five
  points (A1, A2, P1, P2, P3), of which A1 and A2 are on the absolute.

  Let {A1P1A2P2} stand for the cross ratio (as defined above) of the
  range (A1P1A2P2), with a similar notation for the other ranges. Then

  (1)   log{A1P1A2P2} + log{A1P2A2P3} = log{A1P1A2P3},

  and

  (2), if the points A1, A2, P1, P2 are transformed into A'1, A'2, P'1,
  P'2 by any transformation of the congruence-group, ([alpha])
  {A1P(1}A2P2 = {A'1P'1A'2P'2}, since the transformation is projective,
  and (ß) A'1, A'2 are on the absolute since A1 and A2 are on it. Thus
  if we define the distance P1P2 to be ½k log {A1P1A2P2}, where A1 and
  A2 are the points in which the line P1P2 cuts the absolute, and k is
  some constant, the two characteristic properties of distance, namely,
  (1) the addition of consecutive lengths on a straight line, and (2)
  the invariability of distances during a transformation of the
  congruence-group, are satisfied. This is the well-known Cayley-Klein
  projective definition[47] of distance, which was elaborated in view of
  the addition property alone, previously to Lie's discovery of the
  theory of congruence-groups. For a hyperbolic group when P1 and P2 are
  in the region enclosed by the absolute, log {A1P1A2P2} is real, and
  therefore k must be real. For an elliptic group A1 and A2 are
  conjugate imaginaries, and log {A1P1A2P2} is a pure imaginary, and k
  is chosen to be [kappa]/[iota], where [kappa] is real and [iota] =
  [root]-.

  Similarly the angle between two planes, p1 and p2, is defined to be
  (1/2[iota]) log (t1p1t2p2), where t1 and t2 are tangent planes to the
  absolute through the line p1p2. The planes t1 and t2 are imaginary for
  an elliptic group, and also for an hyperbolic group when the planes p1
  and p2 intersect at points within the region enclosed by the absolute.
  The development of the consequences of these metrical definitions is
  the subject of non-Euclidean geometry.

  The definitions for the parabolic case can be arrived at as limits of
  those obtained in either of the other two cases by making k ultimately
  to vanish. It is also obvious that, if P1 and P2 be the points (x1,
  y1, z1) and (x2, y2, z2), it follows from equations (B) above that
  {(x1 - x2)² + (y1 - y2)² + (z1 - z2)²}^½ is unaltered by a congruence
  transformation and also satisfies the addition property for collinear
  distances. Also the previous definition of an angle can be adapted to
  this case, by making t1 and t2 to be the tangent planes through the
  line p1p2 to the imaginary conic. Similarly if p1 and p2 are
  intersecting lines, the same definition of an angle holds, where t1
  and t2 are now the lines from the point p1p2 to the two points where
  the plane p1p2 cuts the imaginary conic. These points are in fact the
  "circular points at infinity" on the plane. The development of the
  consequences of these definitions for the parabolic case gives the
  ordinary Euclidean metrical geometry.

Thus the only metrical geometry for the whole of projective space is of
the elliptic type. But the actual measure-relations (though not their
general properties) differ according to the elliptic congruence-group
selected for study. In a descriptive space a congruence-group should
possess the four characteristics of such a group throughout the whole of
the space. Then form the associated ideal projective space. The
associated congruence-group for this ideal space must satisfy the four
conditions throughout the region of the proper ideal points. Thus the
boundary of this region is the absolute. Accordingly there can be no
metrical geometry for the whole of a descriptive space unless its
boundary (in the associated ideal space) is a closed quadric or a plane.
If the boundary is a closed quadric, there is one possible
congruence-group of the hyperbolic type. If the boundary is a plane (the
plane at infinity), the possible congruence-groups are parabolic; and
there is a congruence-group corresponding to each imaginary conic in
this plane, together with a Euclidean metrical geometry corresponding to
each such group. Owing to these alternative possibilities, it would
appear to be more accurate to say that systems of quantities can be
found in a space, rather than that space is a quantity.

Lie has also deduced[48] the same results with respect to
congruence-groups from another set of defining properties, which
explicitly assume the existence of a quantitative relation (the
distance) between any two points, which is invariant for any
transformation of the congruence-group.[49]

The above results, in respect to congruence and metrical geometry,
considered in relation to existent space, have led to the doctrine[50]
that it is intrinsically unmeaning to ask which system of metrical
geometry is true of the physical world. Any one of these systems can be
applied, and in an indefinite number of ways. The only question before
us is one of convenience in respect to simplicity of statement of the
physical laws. This point of view seems to neglect the consideration
that science is to be relevant to the definite perceiving minds of men;
and that (neglecting the ambiguity introduced by the invariable slight
inexactness of observation which is not relevant to this special
doctrine) we have, in fact, presented to our senses a definite set of
transformations forming a congruence-group, resulting in a set of
measure relations which are in no respect arbitrary. Accordingly our
scientific laws are to be stated relevantly to that particular
congruence-group. Thus the investigation of the type (elliptic,
hyperbolic or parabolic) of this special congruence-group is a perfectly
definite problem, to be decided by experiment. The consideration of
experiments adapted to this object requires some development of
non-Euclidean geometry (see section VI., _Non-Euclidean Geometry_). But
if the doctrine means that, assuming some sort of objective reality for
the material universe, beings can be imagined, to whom _either_ all
congruence-groups are equally important, _or_ some other
congruence-group is specially important, the doctrine appears to be an
immediate deduction from the mathematical facts. Assuming a definite
congruence-group, the investigation of surfaces (or three-dimensional
loci in space of four dimensions) with geodesic geometries of the form
of metrical geometries of other types of congruence-groups forms an
important chapter of non-Euclidean geometry. Arising from this
investigation there is a widely-spread fallacy, which has found its way
into many philosophic writings, namely, that the possibility of the
geometry of existent three-dimensional space being other than Euclidean
depends on the physical existence of Euclidean space of four or more
dimensions. The foregoing exposition shows the baselessness of this
idea.

  BIBLIOGRAPHY.--For an account of the investigations on the axioms of
  geometry during the Greek period, see M. Cantor, _Vorlesungen über die
  Geschichte der Mathematik_, Bd. i. and iii.; T.L. Heath, _The Thirteen
  Books of Euclid's Elements, a New Translation from the Greek, with
  Introductory Essays and Commentary, Historical, Critical, and
  Explanatory_ (Cambridge, 1908)--this work is the standard source of
  information; W.B. Frankland, _Euclid, Book I., with a Commentary_
  (Cambridge, 1905)--the commentary contains copious extracts from the
  ancient commentators. The next period of really substantive importance
  is that of the 18th century. The leading authors are: G. Saccheri,
  S.J., _Euclides ab omni naevo vindicatus_ (Milan, 1733). Saccheri was
  an Italian Jesuit who unconsciously discovered non-Euclidean geometry
  in the course of his efforts to prove its impossibility. J.H. Lambert,
  _Theorie der Parallellinien_ (1766); A.M. Legendre, _Éléments de
  géométrie_ (1794). An adequate account of the above authors is given
  by P. Stäckel and F. Engel, _Die Theorie der Parallellinien von Euklid
  bis auf Gauss_ (Leipzig, 1895). The next period of time (roughly from
  1800 to 1870) contains two streams of thought, both of which are
  essential to the modern analysis of the subject. The first stream is
  that which produced the discovery and investigation of non-Euclidean
  geometries, the second stream is that which has produced the geometry
  of position, comprising both projective and descriptive geometry not
  very accurately discriminated. The leading authors on non-Euclidean
  geometry are K.F. Gauss, in private letters to Schumacher, cf. Stäckel
  and Engel, _loc. cit._; N. Lobatchewsky, rector of the university of
  Kazan, to whom the honour of the effective discovery of non-Euclidean
  geometry must be assigned. His first publication was at Kazan in 1826.
  His various memoirs have been re-edited by Engel; cf. _Urkunden zur
  Geschichte der nichteuklidischen Geometrie_ by Stäckel and Engel, vol.
  i. "Lobatchewsky." J. Bolyai discovered non-Euclidean geometry
  apparently in independence of Lobatchewsky. His memoir was published
  in 1831 as an appendix to a work by his father W. Bolyai, _Tentamen
  juventutem...._ This memoir has been separately edited by J.
  Frischauf, _Absolute Geometrie nach J. Bolyai_ (Leipzig, 1872); B.
  Riemann, _Über die Hypothesen, welche der Geometrie zu Grunde liegen_
  (1854); cf. _Gesamte Werke_, a translation in The Collected Papers of
  W.K. Clifford. This is a fundamental memoir on the subject and must
  rank with the work of Lobatchewsky. Riemann discovered elliptic
  metrical geometry, and Lobatchewsky hyperbolic geometry. A full
  account of Riemann's ideas, with the subsequent developments due to
  Clifford, F. Klein and W. Killing, will be found in _The Boston
  Colloquium for 1903_ (New York, 1905), article "Forms of Non-Euclidean
  Space," by F.S. Woods. A. Cayley, _loc. cit._ (1859), and F. Klein,
  "Über die sogenannte nichteuklidische Geometrie," _Math. Annal._ vols.
  iv. and vi. (1871 and 1872), between them elaborated the projective
  theory of distance; H. Helmholtz, "Über die thatsächlichen Grundlagen
  der Geometrie" (1866), and "Über die Thatsachen, die der Geometrie zu
  Grunde liegen" (1868), both in his _Wissenschaftliche Abhandlungen_,
  vol. ii., and S. Lie, _loc. cit._ (1890 and 1893), between them
  elaborated the group theory of congruence.

  The numberless works which have been written to suggest equivalent
  alternatives to Euclid's parallel axioms may be neglected as being of
  trivial importance, though many of them are marvels of geometric
  ingenuity.

  The second stream of thought confined itself within the circle of
  ideas of Euclidean geometry. Its origin was mainly due to a succession
  of great French mathematicians, for example, G. Monge, _Géométrie
  descriptive_ (1800); J.V. Poncelet, _Traité des proprietés projectives
  des figures_ (1822); M. Chasles, _Aperçu historique sur l'origine et
  le développement des méthodes en géométrie_ (Bruxelles, 1837), and
  _Traité de géométrie supérieure_ (Paris, 1852); and many others. But
  the works which have been, and are still, of decisive influence on
  thought as a store-house of ideas relevant to the foundations of
  geometry are K.G.C. von Staudt's two works, _Geometrie der Lage_
  (Nürnberg, 1847); and _Beiträge zur Geometrie der Lage_ (Nürnberg,
  1856, 3rd ed. 1860).

  The final period is characterized by the successful production of
  exact systems of axioms, and by the final solution of problems which
  have occupied mathematicians for two thousand years. The successful
  analysis of the ideas involved in serial continuity is due to R.
  Dedekind, _Stetigkeit und irrationale Zahlen_ (1872), and to G.
  Cantor, _Grundlagen einer allgemeinen Mannigfaltigkeitslehre_
  (Leipzig, 1883), and _Acta math._ vol. 2.

  Complete systems of axioms have been stated by M. Pasch, _loc. cit._;
  G. Peano, _loc. cit._; M. Pieri, _loc. cit._; B. Russell, _Principles
  of Mathematics_; O. Veblen, _loc. cit._; and by G. Veronese in his
  treatise, _Fondamenti di geometria_ (Padua, 1891; German transl. by A.
  Schepp, _Grundzüge der Geometrie_, Leipzig, 1894). Most of the leading
  memoirs on special questions involved have been cited in the text; in
  addition there may be mentioned M. Pieri, "Nuovi principii di
  geometria projettiva complessa," _Trans. Accad. R. d. Sci._ (Turin,
  1905); E.H. Moore, "On the Projective Axioms of Geometry," _Trans.
  Amer. Math. Soc._, 1902; O. Veblen and W.H. Bussey, "Finite Projective
  Geometries," _Trans. Amer. Math. Soc._, 1905; A.B. Kempe, "On the
  Relation between the Logical Theory of Classes and the Geometrical
  Theory of Points," _Proc. Lond. Math. Soc._, 1890; J. Royce, "The
  Relation of the Principles of Logic to the Foundations of Geometry,"
  _Trans. of Amer. Math. Soc._, 1905; A. Schoenflies, "Über die
  Möglichkeit einer projectiven Geometrie bei transfiniter
  (nichtarchimedischer) Massbestimmung," _Deutsch. M.-V. Jahresb._,
  1906.

  For general expositions of the bearings of the above investigations,
  cf. Hon. Bertrand Russell, _loc. cit._; L. Couturat, _Les Principes
  des mathématiques_ (Paris, 1905); H. Poincaré, _loc. cit._; Russell
  and Whitehead, _Principia mathematica_ (Cambridge, Univ. Press). The
  philosophers whose views on space and geometric truth deserve especial
  study are Descartes, Leibnitz, Hume, Kant and J.S. Mill.     (A. N. W.)


FOOTNOTES:

  [1] For Egyptian geometry see EGYPT, § _Science and Mathematics_.

  [2] Cf. A.N. Whitehead, _Universal Algebra_, Bk. vi. (Cambridge,
    1898).

  [3] Cf. A.N. Whitehead, _loc. cit._

  [4] Cf. A.N. Whitehead, "The Geodesic Geometry of Surfaces in
    non-Euclidean Space," _Proc. Lond. Math. Soc._ vol. xxix.

  [5] Cf. Klein, "Zur nicht-Euklidischen Geometrie," _Math. Annal._
    vol. xxxvii.

  [6] On the theory of parallels before Lobatchewsky, see Stäckel und
    Engel, _Theorie der Parallellinien von Euklid bis auf Gauss_
    (Leipzig, 1895). The foregoing remarks are based upon the materials
    collected in this work.

  [7] See Stäckel und Engel, _op. cit._, and "Gauss, die beiden Bolyai,
    und die nicht-Euklidische Geometrie," _Math. Annalen_, Bd. xlix.;
    also Engel's translation of Lobatchewsky (Leipzig, 1898), pp. 378 ff.

  [8] Lobatchewsky's works on the subject are the following:--"On the
    Foundations of Geometry," _Kazañ Messenger_, 1829-1830; "New
    Foundations of Geometry, with a complete Theory of Parallels,"
    _Proceedings of the University of Kazañ_, 1835 (both in Russian, but
    translated into German by Engel, Leipzig, 1898); "Géométrie
    imaginaire," Crelle's Journal, 1837; _Theorie der Parallellinien_
    (Berlin, 1840; 2nd ed., 1887; translated by Halsted, Austin, Texas,
    1891). His results appear to have been set forth in a paper (now
    lost) which he read at Kazañ in 1826.

  [9] Translated by Halsted (Austin, Texas, 4th ed., 1896.)

  [10] _Abhandlungen d. Königl. Ges. d. Wiss. zu Göttingen_, Bd. xiii.;
    _Ges. math. Werke_, pp. 254-269; translated by Clifford, _Collected
    Mathematical Papers_.

  [11] Cf. _Gesamm. math. und phys. Werke_, vol. i. (Leipzig, 1894).

  [12] _Wiss. Abh._ vol. ii. pp. 610, 618 (1866, 1868).

  [13] _Mind_, O.S., vols. i. and iii.; _Vorträge und Reden_, vol. ii.
    pp. 1, 256.

  [14] His papers are "Saggio di interpretazione della geometria
    non-Euclidea," _Giornale di matematiche_, vol. vi. (1868); "Teoria
    fondamentale degli spazii di curvatura costante," _Annali di
    matematica_, vol. ii. (1868-1869). Both were translated into French
    by J. Hoüel, _Annales scientifiques de l'École Normale supérieure_,
    vol. vi. (1869).

  [15] Beltrami shows also that this definition agrees with that of
    Gauss.

  [16] "Sur la théorie des foyers," _Nouv. Ann._ vol. xii.

  [17] _Math. Annalen_, iv. vi., 1871-1872.

  [18] For an investigation of these and similar properties, see
    Whitehead, _Universal Algebra_ (Cambridge, 1898), bk. vi. ch. ii. The
    polar form was independently discovered by Simon Newcomb in 1877.

  [19] For an analysis of Leibnitz's ideas on space, cf. B. Russell,
    _The Philosophy of Leibnitz_, chs. viii.-x.

  [20] Cf. Hon. Bertrand Russell, "Is Position in Time and Space
    Absolute or Relative?" _Mind_, n.s. vol. 10 (1901), and A.N.
    Whitehead, "Mathematical Concepts of the Material World," _Phil.
    Trans._ (1906), p. 205.

  [21] Cf. _Critique of Pure Reason_, 1st section: "Of Space,"
    conclusion A, Max Müller's translation.

  [22] Cf. Ernst Mach, _Erkenntniss und Irrtum_ (Leipzig); the relevant
    chapters are translated by T.J. McCormack, _Space and Geometry_
    (London, 1906); also A. Meinong, _Über die Stellung der
    Gegenstandstheorie im System der Wissenschaften_ (Leipzig, 1907).

  [23] Cf. Russell, _Principles of Mathematics_, § 352 (Cambridge,
    1903).

  [24] Cf. A.N. Whitehead, _The Axioms of Projective Geometry_, § 3
    (Cambridge, 1906).

  [25] Cf. Russell, _Princ. of Math._, ch. i.

  [26] Cf. Russell, _loc. cit._, and G. Frege, "Über die Grundlagen der
    Géométrie," _Jahresber. der Deutsch. Math. Ver._ (1906).

  [27] This formulation--though not in respect to number--is in all
    essentials that of M. Pieri, cf. "I principii della Geometria di
    Posizione," _Accad. R. di Torino_ (1898); also cf. Whitehead, _loc.
    cit._

  [28] Cf. G. Peano, "Sui fondamenti della Geometria," p. 73, _Rivista
    di matematica_, vol. iv. (1894), and D. Hilbert, _Grundlagen der
    Geometrie_ (Leipzig, 1899); and R.F. Moulton, "A Simple
    non-Desarguesian Plane Geometry," _Trans. Amer. Math. Soc._, vol.
    iii. (1902).

  [29] Cf. "Sui postulati fondamentali della geometria projettiva,"
    _Giorn. di matematica_, vol. xxx. (1891); also of Pieri, _loc. cit._,
    and Whitehead, _loc. cit._

  [30] Cf. Hilbert, _loc. cit._; for a fuller exposition of Hilbert's
    proof cf. K.T. Vahlen, _Abstrakte Geometrie_ (Leipzig, 1905), also
    Whitehead, _loc. cit._

  [31] Cf. H. Wiener, _Jahresber. der Deutsch. Math. Ver._ vol. i.
    (1890); and F. Schur, "Über den Fundamentalsatz der projectiven
    Geometrie," _Math. Ann._ vol. li. (1899).

  [32] Cf. Hilbert, _loc. cit._, and Whitehead, _loc. cit._

  [33] Cf. Dedekind, _Stetigkeit und irrationale Zahlen_ (1872).

  [34] Cf. v. Staudt, _Geometrie der Lage_ (1847).

  [35] Cf. Pasch, _Vorlesungen über neuere Geometrie_ (Leipzig, 1882),
    a classic work; also Fiedler, _Die darstellende Geometrie_ (1st ed.,
    1871, 3rd ed., 1888); Clebsch, _Vorlesungen über Geometrie_, vol.
    iii.; Hilbert, _loc. cit._; F. Schur, _Math. Ann. Bd._ lv. (1902);
    Vahlen, _loc. cit._; Whitehead, _loc. cit._

  [36] Cf. _loc. cit._

  [37] Cf. _I Principii di geometria_ (Turin, 1889) and "Sui fondamenti
    della geometria," _Rivista di mat._ vol. iv. (1894).

  [38] Cf. _loc. cit._

  [39] Cf. Vailati, _Rivista di mat._ vol. iv. and Russell, _loc. cit._
    § 376.

  [40] Cf. O. Veblen, "On the Projective Axioms of Geometry," _Trans.
    Amer. Math. Soc._ vol. iii. (1902).

  [41] Cf. P. Stäckel and F. Engel, _Die Theorie der Parallellinien von
    Euklid bis auf Gauss_ (Leipzig, 1895).

  [42] Cf. Pasch, _loc. cit._, and R. Bonola, "Sulla introduzione degli
    enti improprii in geometria projettive," _Giorn. di mat._ vol.
    xxxviii. (1900); and Whitehead, _Axioms of Descriptive Geometry_
    (Cambridge, 1907).

  [43] The original idea (confined to this particular case) of ideal
    points is due to von Staudt (_loc. cit._).

  [44] Cf. _Critique_, "Trans. Aesth." Sect. I.

  [45] Cf. _loc. cit._

  [46] Cf. _Über die Grundlagen der Geometrie_ (Leipzig, Ber., 1890);
    and _Theorie der Transformationsgruppen_ (Leipzig, 1893), vol. iii.

  [47] Cf. A. Cayley, "A Sixth Memoir on Quantics," _Trans. Roy. Soc._,
    1859, and _Coll. Papers_, vol. ii.; and F. Klein, _Math. Ann._ vol.
    iv., 1871.

  [48] Cf. _loc. cit._

  [49] For similar deductions from a third set of axioms, suggested in
    essence by Peano, Riv. mat. vol. iv. _loc. cit._ cf. Whitehead, _Desc.
    Geom. loc. cit._

  [50] Cf. H. Poincaré, _La Science et l'hypothèse_, ch. iii.