Chapter XXXVI: Book XIII (8)

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

(w1^2- x1^2- y1^2- z1^2)(w^2 - x^2 - y^2 - z^2) cosh^2([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

(l^2 + m^2 + n^2 - r^2)(w^2 - x^2 - y^2 - z^2) sinh^2([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

k^2(w^2 - x^2 - y^2 - z^2) = (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 d^2 - a^2 - b^2 - c^2 = 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

w^2 - x^2 - y^2 - z^2 = (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

(w^2 - x^2) tanh^2([delta]/[gamma]) - y^2 - z^2 = 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 x^2 + y^2 + z^2 + w^2 =
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)
/ {(x1^2 + y1^2 + z1^2 + w1^2) {(x2^2 + y2^2 + z2^2 + w2^2)}^1/2 (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)/ {(l1^2 + m1^2 + n1^2 +r1^2)
(l2^2 + m2^2 + n2^2 + r2^2)}^1/2 (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 1/2[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 1/4[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

(x^2 + w^2) tan^2([delta]/[gamma]) - (y^2 + z^2) = 0.

This is also the surface of equal distance, 1/2[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

a^2 = b^2 + c^2 - 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 mere gonias duo orthon elassonas poie, ekballomenas tas duo eutheias ep' apeiron sympiptein, eph' ha mere eisin hai ton duo orthon 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 Kazan.[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 1/2 [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. Houel. But at this time Riemann's dissertation, _Uber 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 + [mu][delta]x, where the parameter [lambda] : [mu] 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]dx^2)/(1 + 1/4[alpha][Sigma]x^2).

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

ds^2 = R^2w^(-4){([alpha]^2 - v^2)du^2 + 2uvdudv + ([alpha]^2 - u^2)dv^2}

where w^2 = [alpha]^2 - u^2 - v^2, and (-1)/R^2 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 - u^2)([alpha]^2 - v^2)}, sin [theta] =
aw/[root]{(a^2 - u^2)(a^2 - v^2)}.

Thus u = 0 is orthogonal to all geodesies v = const., and vice versa.
In order that sin [theta] may be real, w^2 must be positive; thus
geodesics have no real intersection when the corresponding straight
lines intersect outside the circle u^2 + v^2 = [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 [mu], v = r sin [mu], and
calling [rho] the geodesic distance of u, v from the origin, we have,
for a geodesic through the origin,

d[rho] = Radr/(a^2 - r^2), [rho] = 1/2R 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 [mu] at the origin is

[sigma] = Rr[mu]/[root](a^2 - r^2) = [mu]R 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)a^2 - (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 1/2[Pi]([delta]) =
e^(-[delta]/R)

which is Lobatchewsky's formula. We also obtain easily for the area of
a triangle the formula R^2([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

(a^2 - uu0 - vv0)^2 = cos h^2 ([rho]/R)w0^2w^2 = C^2w^2 (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
a^2 - 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_)

ds^2 = R^2x^(-2)(dx^2 + dx1^2 + dx2^2 + ... + dx_n^2)

where x^2 + x1^2 + x2^2 + ... + x_n^2 = a^2.

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] a^2 - x1x'1 - x2x'2 - ... - x_n x'_n
cosh ----- = ---------------------------------------------------------------------------
R {(a^2 - x1^2 - x2^2 - ... - x_n^2)(a^2 - x'1^2 - x'2^2 - ... - x'_n^2)}^1/2

(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
_________
ds^2 = (Radr/(a^2 - r^2)])^2 + R^2r^2d[Delta]^2/(a^2 - r^2).

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](a^2 - r^2) R [root](a^2 - r^2)

Hence

ds^2 = 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 |
ds^2 = [Sigma]dz^2 + ------ | ( ----- 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
ds^2 = [Sigma]dz^2 + ----[Sigma](z_i dz_k - z_k dz_i)^2 (1).
3R^2

Considering a surface element through the origin, we may choose our
axes so that, for this element,

z3 = Z4 = ... = z_n = 0.

Thus

1
ds^2 = dz1^2 + dz2^2 + ----(z1dz2 - z2dz1)^2 (2).
3R^2

Now the area of the triangle whose vertices are (0, 0), (z1, z2),
(dz1, dz2) is 1/2(z1, dz2 - z2dz1). Hence the quotient when the terms
of the fourth order in (2) are divided by the square of this triangle
is 4/3R^2; 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 -3/4 of this quotient is defined by
Riemann as the measure of curvature.[15] Hence the measure of
curvature is -1/R^2, 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/R^2 sinh^2
([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'}^1/2.

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. Stackel (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.
Gerard, _Sur la geometrie non-Euclidienne_ (thesis for doctorate)
(Paris, 1892); de Tilly, _Essai sur les principes fondamentales de la
geometrie et de la mecanique_ (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 Satze uber die
Winkelsumme im Dreieck," Math. Ann. vol. 53 (1900), and "Uber den
Rauminhalt," _Math. Annal._ vol. 55 (1902).

For expositions of the whole subject, cf. F. Klein, _Nicht-Euklidische
Geometrie_ (Gottingen, 1893); R. Bonola, _La Geometria non-Euclidea_
(Bologna, 1906); P. Barbarin, _La Geometrie 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 piu dimensioni ed a piu specie_
_di unita rettilinee_ ... (Padua, 1891, German translation, Leipzig,
1894); G. Fontene, _L'Hyperespace a (n-1) dimensions_ (Paris, 1892);
and A.N. Whitehead, _loc. cit._ Cf. also E. Study, "Uber
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: