An Essay on the Foundations of Geometry

anharmonic ratio of the two points and the (coincident) points at infinity on their straight line; while, if we assumed that the points at infinity were distinct, we obtained the formula for distance in hyperbolic or spherical Geometry, according as these points were real or imaginary. Hence it follows that, with the projective definition of distance, we shall obtain precisely the formulae of hyperbolic, parabolic or spherical Geometry, according as we choose the point, to which the value +∞ is assigned, at a finite, infinite or imaginary distance (in the ordinary sense) from the point to which we assign the value 0. Our straight line remains, all the while, an ordinary Euclidean straight line. But we have seen that the projective definition of distance fits with the true definition only when the two fixed points to which it refers are suitably chosen. Now the ordinary meaning of distance is required in non-Euclidean as in Euclidean Geometries—indeed, it is only in metrical properties that these Geometries differ. Hence our Euclidean straight line, though it may serve to illustrate other Geometries than Euclid's, can only be dealt with correctly by Euclid. Where we give a different definition of distance from Euclid's, we are still in the domain of purely projective properties, and derive no information as to the metrical properties of our straight line. But the importance, to Metageometry, of this new interpretation, lies in the fact that, having independently established the metrical formulae of non-Euclidean spaces, we find, as in Beltrami's Saggio, that these spaces can be related, by a homographic correspondence, with the points of Euclidean space; and that this can be effected in such a manner as to give, for the distance between two points of our non-Euclidean space, the hyperbolic or spherical measure of distance for the corresponding points of Euclidean space. 39. On the whole, then, a modification of Sir R. Ball's view, which is practically a generalized statement of Beltrami's method, seems the most tenable. He imagines what, with Grassmann, he calls a Content, i.e. a perfectly general three-dimensional manifold, and then correlates its elements, one by one, with points in Euclidean space. Thus every element of the Content acquires, as its coordinates, the ordinary Euclidean coordinates of the corresponding point in Euclidean space. By means of this correlation, our calculations, though they refer to the Content, are carried on, as in Beltrami's Saggio, in ordinary Euclidean space. Thus the confusion disappears, but with it, the supposed Euclidean interpretation also disappears. Sir R. Ball's Content, if it is to be a space at all, must be a space radically different from Euclid's[55]; to speak, as Klein does, of ordinary planes with hyperbolic or elliptic measures of distance, is either to incur a contradiction, or to forego any metrical meaning of distance. Instead of ordinary planes, we have surfaces like Beltrami's, of constant measure of curvature; instead of Euclid's space, we have hyperbolic or spherical space. At the same time, it remains true that we can, by Klein's method, give a Euclidean meaning to every symbolic proposition in non-Euclidean Geometry. For by substituting, for distance, the logarithm above alluded to, we obtain, from the non-Euclidean result, a result which follows from the ordinary Euclidean axioms. This correspondence removes, once for all, the possibility of a lurking contradiction in Metageometry, since, to a proposition in the one, corresponds one and only one proposition in the other, and contradictory results in one system, therefore, would correspond to contradictory results in the other. Hence Metageometry cannot lead to contradictions, unless Euclidean Geometry, at the same moment, leads to corresponding contradictions. Thus the Euclidean plane with hyperbolic or elliptic measure of distance, though either contradictory or not metrical as an independent notion, has, as a help in the interpretation of non-Euclidean results, a very high degree of utility. 40. We have still to discuss Klein's third kind of non-Euclidean Geometry, which he calls elliptic. The difference between this and spherical Geometry is difficult to grasp, but it may be illustrated by a simpler example. A plane, as every one knows, can be wrapped, without stretching, on a cylinder, and straight lines in the plane become, by this operation, geodesics on the cylinder. The Geometries of the plane and the cylinder, therefore, have much in common. But since the generating circle of the cylinder, which is one of its geodesics, is finite, only a portion of the plane is used up in wrapping it once round the cylinder. Hence, if we endeavour to establish a point-to-point correspondence between the plane and the cylinder, we shall find an infinite series of points on the plane for a single point on the cylinder. Thus it happens that geodesics, though on the plane they have only one point in common, may on the cylinder have an infinite number of intersections. Somewhat similar to this is the relation between the spherical and elliptic Geometries. To any one point in elliptic space, two points correspond in spherical space. Thus geodesics, which in spherical space may have two points in common, can never, in elliptic space, have more than one intersection. But Klein's method can only prove that elliptic Geometry holds of the ordinary Euclidean plane with elliptic measure of distance. Klein has made great endeavours to enforce the distinction between the spherical and elliptic Geometries[56], but it is not immediately evident that the latter, as distinct from the former, is valid. In the first place, Klein's elliptic Geometry, which arises as one of the alternative metrical systems on a Euclidean plane or in a Euclidean space, does not by itself suffice, if the above discussion has been correct, to prove the possibility of an elliptic space, i.e. of a space having a point-to-point correspondence with the Euclidean space, and having as the ordinary distance between two of its points the elliptic definition of the distance between corresponding points of the Euclidean space. To prove this possibility, we must adopt the direct method of Newcomb (Crelle's Journal, Vol. 83). Now in the first place Newcomb has not proved that his postulates are self-consistent; he has only failed to prove that they are contradictory[57]. This would leave elliptic space in the same position in which Lobatchewsky and Bolyai left hyperbolic space. But further there seems to be, at first sight, in two-dimensional elliptic space, a positive contradiction. To explain this, however, some account of the peculiarities of the elliptic plane will be necessary. The elliptic plane, regarded as a figure in three-dimensional elliptic space, is what is called a double surface[58], i.e. as Newcomb says (loc. cit. p. 298): "The two sides of a complete plane are not distinct, as in a Euclidean surface.... If ... a being should travel to distance 2D, he would, on his return, find himself on the opposite surface to that on which he started, and would have to repeat his journey in order to return to his original position without leaving the surface." Now if we imagine a two-dimensional elliptic space, the distinction between the sides of a plane becomes unmeaning, since it only acquires significance by reference to the third dimension. Nevertheless, some such distinction would be forced upon us. Suppose, for example, that we took a small circle provided with an arrow, as in the figure, and moved this circle once round the universe. Then the sense of the arrow would be reversed. We should thus be forced, either to regard the new position as distinct from the former, which transforms our plane into a spherical plane, or to attribute the reversal of the arrow to the action of a motion which restores our circle to its original place. It is to be observed that nothing short of moving round the universe would suffice to reverse the sense of the arrow. This reversal seems like an action of empty space, which would force us to regard the points which, from a three-dimensional point of view, are coincident though opposite, as really distinct, and so reduce the elliptic to the spherical plane. But motion, not space, really causes the change, and the elliptic plane is therefore not proved to be impossible. The question is not, however, of any great philosophic importance. 41. In connection with the reduction of metrical to projective Geometry, we have one more topic for discussion. This is the geometrical use of imaginaries, by means of which, except in the case of hyperbolic space, the reduction is effected. I have already contended, on other grounds, that this reduction, in spite of its immense technical importance, and in spite of the complete logical freedom of projective Geometry from metrical ideas, is purely technical, and is not philosophically valid. The same conclusion will appear, if we take up Cayley's challenge at the British Association, in his Presidential Address of 1883. In this address, Professor Cayley devoted most of his time to non-Euclidean systems. Non-Euclidean spaces, he declared, seemed to him mistaken à priori[59]; but non-Euclidean Geometries, here as in his mathematical works, were accepted as flowing from a change in the definition of distance. This view has been already discussed, and need not, therefore, be further criticised here. What I wish to speak about, is the question with which Cayley himself opened his address, namely, the geometrical use and meaning of imaginary quantities. From the manner in which he spoke of this question, it becomes imperative to treat it somewhat at length. For he said (pp. 8–9): "... The notion which is the really fundamental one (and I cannot too strongly emphasize the assertion) underlying and pervading the whole notion of modern analysis and Geometry, [is] that of imaginary magnitude in analysis, and of imaginary space (or space as the locus in quo of imaginary points and figures) in Geometry: I use in each case the word imaginary as including real.... Say even the conclusion were that the notion belongs to mere technical mathematics, or has reference to nonentities in regard to which no science is possible, still it seems to me that (as a subject of philosophical discussion) the notion ought not to be thus ignored; it should at least be shown that there is a right to ignore it." 42. This right it is now my purpose to demonstrate. But for fear non-mathematicians should miss the point of Cayley's remark (which has sometimes been erroneously supposed to refer to non-Euclidean spaces), I may as well explain, at the outset, that this question is radically distinct from, and only indirectly connected with, the validity or import of Metageometry. An imaginary quantity is one which involves √–1 : its most general form is a + √–1 b where a and b are real; Cayley uses the word imaginary so as to include real, in order to cover the special case where b = 0. It will be convenient, in what follows, to exclude this wider meaning, and assume that b is not zero. An imaginary point is one whose coordinates involve √–1, i.e. whose coordinates are imaginary quantities. An imaginary curve is one whose points are imaginary—or, in some special uses, one whose equation contains imaginary coefficients. The mathematical subtleties to which this notion leads need not be here discussed; the reader who is interested in them will find an excellent elementary account of their geometrical uses in Klein's Nicht-Euklid, II. pp. 38–46. But for our present purpose, we may confine ourselves to imaginary points. If these are found to have a merely technical import, and to be destitute of any philosophical meaning, then the same will hold of any collection of imaginary points, i.e. of any imaginary curve or surface. That the notion of imaginary points is of supreme importance in Geometry, will be seen by any one who reflects that the circular points are imaginary, and that the reduction of metrical to projective Geometry, which is one of Cayley's greatest achievements, depends on these points. But to discuss adequately their philosophical import is difficult to me, since I am unacquainted with any satisfactory philosophy of imaginaries in pure Algebra. I will therefore adopt