An Essay on the Foundations of Geometry

as numbers arbitrarily assigned, on some system not further investigated, to different points. The difficulty has been treated at length by Sir R. Ball (Theory of the Content, Trans. R. I. A. 1889), who urges that if the values of our coordinates already involve the usual measure of distance, then to give a new definition, while retaining the usual coordinates, is to incur a contradiction. He says (op. cit. p. 1): "In the study of non-Euclidean Geometry I have often felt a difficulty which has, I know, been shared by others. In that theory it seems as if we try to replace our ordinary notion of distance between two points by the logarithm of a certain anharmonic ratio[46]. But this ratio itself involves the notion of distance measured in the ordinary way. How, then, can we supersede our old notion of distance by the non-Euclidean notion, inasmuch as the very definition of the latter involves the former?" 36. This objection is valid, we must admit, so long as anharmonic ratio is defined in the ordinary metrical manner. It would be valid, for example, against any attempt to found a new definition of distance on Cremona's account of anharmonic ratio[47], in which it appears as a metrical property unaltered by projective transformation. If a logical error is to be avoided, in fact, all reference to spatial magnitude of any kind must be avoided; for all spatial magnitude, as will be shown hereafter[48], is logically dependent on the fundamental magnitude of distance. Anharmonic ratio and coordinates must alike be defined by purely descriptive properties, if the use afterwards made of them is to be free from metrical presuppositions, and therefore from the objections of Sir R. Ball. Such a definition has been satisfactorily given by Klein[49], who appeals, for the purpose, to v. Staudt's quadrilateral construction[50]. By this construction, which I have reproduced in outline in Chapter III. Section A, § 112 ff., we obtain a purely descriptive definition of harmonic and anharmonic ratio, and, given a pair of points, we can obtain the harmonic conjugate to any third point on the same straight line. On this construction, the introduction of projective coordinates is based. Starting with any three points on a straight line, we assign to them arbitrarily the numbers 0, 1, ∞. We then find the harmonic conjugate to the first with respect to 1, ∞, and assign to it the number 2. The object of assigning this number rather than any other, is to obtain the value –1 for the anharmonic ratio of the four numbers corresponding to the four points[51]. We then find the harmonic conjugate to the point 1, with respect to 2, ∞, and assign to it the number 3; and so on. Klein has shown that by this construction, we can obtain any number of points, and can construct a point corresponding to any given number, fractional or negative. Moreover, when two sets of four points have the same anharmonic ratio, descriptively defined[52], the corresponding numbers also have the same anharmonic ratio. By introducing such a numerical system on two straight lines, or on three, we obtain the coordinates of any point in a plane, or in space. By this construction, which is of fundamental importance to projective Geometry, the logical error, upon which Sir R. Ball bases his criticism, is satisfactorily avoided. Our coordinates are introduced by a purely descriptive method, and involve no presupposition whatever as to the measurement of distance. 37. With this coordinate system, then, to define distance as a certain function of the coordinates is not to be guilty of a vicious circle. But it by no means follows that the definition of distance is arbitrary. All reference to distance has been hitherto excluded, to avoid metrical ideas; but when distance is introduced, metrical ideas inevitably reappear, and we have to remember that our coordinates give no information, primâ facie, as to any of these metrical ideas. It is open to us, of course, if we choose, to continue to exclude distance in the ordinary sense, as the quantity of a finite straight line, and to define the word distance in any way we please. But the conception, for which the word has hitherto stood, will then require a new name, and the only result will be a confusion between the apparent meaning of our propositions, to those who retain the associations belonging to the old sense of the word, and the real meaning, resulting from the new sense in which the word is used. This confusion, I believe, has actually occurred, in the case of those who regard the question between Euclid and Metageometry as one of the definition of distance. Distance is a quantitative relation, and as such presupposes identity of quality. But projective Geometry deals only with quality—for which reason it is called descriptive—and cannot distinguish between two figures which are qualitatively alike. Now the meaning of qualitative likeness, in Geometry, is the possibility of mutual transformation by a collineation[53]. Any two pairs of points on the same straight line, therefore, are qualitatively alike; their only qualitative relation is the straight line, which both pairs have in common; and it is exactly the qualitative identity of the relations of the two pairs, which enables the difference of their relations to be exhaustively dealt with by quantity, as a difference of distance. But where quantity is excluded, any two pairs of points on the same straight line appear as alike, and even any two sets of three: for any three points on a straight line can be projectively transformed into any other three. It is only with four points in a line that we acquire a projective property distinguishing them from other sets of four, and this property is anharmonic ratio, descriptively defined. The projective Geometer, therefore, sees no reason to give a name to the relation between two points, in so far as this relation is anything over and above the unlimited straight line on which they lie; and when he introduces the notion of distance, he defines it, in the only way in which projective principles allow him to define it, as a relation between four points. As he nevertheless wishes the word to give him the power of distinguishing between different pairs of points, he agrees to take two out of the four points as fixed. In this way, the only variables in distance are the two remaining points, and distance appears, therefore, as a function of two variables, namely the coordinates of the two variable points. When we have further defined our function so that distance may be additive, we have a function with many of the properties of distance in the ordinary sense. This function, therefore, the projective Geometer regards as the only proper definition of distance. We can see, in fact, from the manner in which our projective coordinates were introduced, that some function of these coordinates must express distance in the ordinary sense. For they were introduced serially, so that, as we proceeded from the zero-point towards the infinity-point, our coordinates continually grew. To every point, a definite coordinate corresponded: to the distance between two variable points, therefore, as a function dependent on no other variables, must correspond some definite function of the coordinates, since these are themselves functions of their points. The function discussed above, therefore, must certainly include distance in the ordinary sense. But the arbitrary and conventional nature of distance, as maintained by Poincaré and Klein, arises from the fact that the two fixed points, required to determine our distance in the projective sense, may be arbitrarily chosen, and although, when our choice is once made, any two points have a definite distance, yet, according as we make that choice, distance will become a different function of the two variable points. The ambiguity thus introduced is unavoidable on projective principles; but are we to conclude, from this, that it is really unavoidable? Must we not rather conclude that projective Geometry cannot adequately deal with distance? If A, B, C, be three different points on a line, there must be some difference between the relation of A to B and of A to C, for otherwise, owing to the qualitative identity of all points, B and C could not be distinguished. But such a difference involves a relation, between A and B, which is independent of other points on the line; for unless we have such a relation, the other points cannot be distinguished as different. Before we can distinguish the two fixed points, therefore, from which the projective definition starts, we must already suppose some relation, between any two points on our line, in which they are independent of other points; and this relation is distance in the ordinary sense[54]. When we have measured this quantitative relation by the ordinary methods of metrical Geometry, we can proceed to decide what base-points must be chosen, on our line, in order that the projective function discussed above may have the same value as ordinary distance. But the choice of these base-points, when we are discussing distance in the ordinary sense, is not arbitrary, and their introduction is only a technical device. Distance, in the ordinary sense, remains a relation between two points, not between four; and it is the failure to perceive that the projective sense differs from, and cannot supersede, the ordinary sense, which has given rise to the views of Klein and Poincaré. The question is not one of convention, but of the irreducible metrical properties of space. To sum up: Quantities, as used in projective Geometry, do not stand for spatial magnitudes, but are conventional symbols for purely qualitative spatial relations. But distance, quâ quantity, presupposes identity of quality, as the condition of quantitative comparison. Distance in the ordinary sense is, in short, that quantitative relation, between two points on a line, by which their difference from other points can be defined. The projective definition, however, being unable to distinguish a collection of less than four points from any other on the same straight line, makes distance depend on two other points besides those whose relation it defines. No name remains, therefore, for distance in the ordinary sense, and many projective Geometers, having abolished the name, believe the thing to be abolished also, and are inclined to deny that two points have a unique relation at all. This confusion, in projective Geometry, shows the importance of a name, and should make us chary of allowing new meanings to obscure one of the fundamental properties of space. 38. It remains to discuss the manner in which non-Euclidean Geometries result from the projective definition of distance, as also the true interpretation to be given to this view of Metageometry. It is to be observed that the projective methods which follow Cayley deal throughout with a Euclidean plane, on which they introduce different measures of distance. Hence arises, in any interpretation of these methods, an apparent subordination of the non-Euclidean spaces, as though these were less self-subsistent than Euclid's. This subordination is not intended in what follows; on the contrary, the correlation with Euclidean space is regarded as valuable, first, because Euclidean space has been longer studied and is more familiar, but secondly, because this correlation proves, when truly interpreted, that the other spaces are self-subsistent. We may confine ourselves chiefly, in discussing this interpretation, to distances measured along a single straight line. But we must be careful to remember that the metrical definition of distance—which, according to the view here advocated, is the only adequate definition—is the same in Euclidean and in non-Euclidean spaces; to argue in its favour is not, therefore, to argue in favour of Euclid. The projective scheme of coordinates consists of a series of numbers, of which each represents a certain anharmonic ratio and denotes one and only one point, and which increase uniformly with the distance from a fixed origin, until they become infinite on reaching a certain point. Now Cayley showed that, in Euclidean Geometry, distance may be expressed as the limit of the logarithm of the