An Essay on the Foundations of Geometry

half the circumference of the universe. In our former discussion, we assumed that any relation to outside space must be a relation of position—and a relation of position must be altered by motion. But with a finite space, in which we have absolute magnitude, another relation becomes possible, namely, a relation of magnitude. Antipodal points, accordingly, like coincident points, no longer determine a unique straight line. And it is instructive to observe that there is, in consequence, an ambiguity in the expression for distance, like the ordinary ambiguity in angular measurement. If 1/k2 be the space constant, and d be one value for the distance between two points, 2πkn ± d, where n is any integer, is an equally good value. Distance is, in short, a periodic function like angle. Thus such a state of things rather confirms than destroys my contention, that distance depends on a curve uniquely determined by two points. For as soon as we drop this unique determination, we see ambiguities creeping into our expression for distance. Distance still has a set of discrete values, corresponding to the fact that, given one point, the straight line is uniquely determined for all other points but one, the antipodal point. It is tempting to go on, and say: If through every pair of points there were an infinite number of the curves used in measuring distance, distance would be able, for the same pair of points, to take, not only a discrete series, but an infinite continuous series of values. 171. This, however, is mere speculation. I come now to the pièce de résistance of my argument. The ambiguity in spherical space arose, as we saw, from a relation of magnitude to the rest of space—such a relation being unaltered by a motion of the two points, and therefore falling outside our introductory reasoning. But what is this relation of magnitude? Simply a relation of the distance between the two points to a distance given in the nature of the space in question. It follows that such a relation presupposes a measure of distance, and need not, therefore, be contemplated in any argument which deals with the à priori requisites for the possibility of definite distances[174]. 172. I have now shown, I hope conclusively, that spherical space affords no objection to the apriority of my axiom. Any two points have one relation, their distance, which is independent of the rest of space, and this relation requires, as its measure, a curve uniquely determined by those two points. I might have taken the bull by the horns, and said: Two points can have no relation but what is given by lines which join them, and therefore, if they have a relation independent of the rest of space, there must be one line joining them which they completely determine. Thus James says[175]: "Just as, in the field of quantity, the relation between two numbers is another number, so in the field of space the relations are facts of the same order with the facts they relate.... When we speak of the relation of direction of two points towards each other, we mean simply the sensation of the line that joins the two points together. The line is the relation.... The relation of position between the top and bottom points of a vertical line is that line, and nothing else." If I had been willing to use this doctrine at the beginning, I might have avoided all discussion. A unique relation between two points must in this case, involve a unique line between them. But it seemed better to avoid a doctrine not universally accepted, the more so as I was approaching the question from the logical, not the psychological, side. After disposing of the objections, however, it is interesting to find this confirmation of the above theory from so different a standpoint. Indeed, I believe James's doctrine could be proved to be a logical necessity, as well as a psychological fact. For what sort of thing can a spatial relation between two distinct points be? It must be something spatial, and it must, since points are wholly constituted by their relations, be something at least as real and tangible as the points it relates. There seems nothing which can satisfy these requirements, except a line joining them. Hence, once more, a unique relation must involve a unique line. That is, linear magnitude is logically impossible, unless space allows of curves uniquely determined by any two of their points. 173. (3) But farther, the existence of curves uniquely determined by two points can be deduced from the nature of any form of externality[176]. For we saw, in discussing Free Mobility, that this axiom, together with homogeneity and the relativity of position, can be so deduced, and we saw in the beginning of our discussion on distance, that the existence of a unique relation between two points could be deduced from the homogeneity of space. Since position is relative, we may say, any two points must have some relation to each other: since our form of externality is homogeneous, this relation can be kept unchanged while the two points move in the form, i.e., change their relations to other points; hence their relation to each other is an intrinsic relation, independent of their relations to other points. But since our form is merely a complex of relations, a relation of externality must appear in the form, with the same evidence as anything else in the form; thus if the form be intuitive or sensational, the relation must be immediately presented, and not a mere inference. Hence the intrinsic relation between two points must be a unique figure in our form, i.e. in spatial terms, the straight line joining the two points. 174. (4) Finally, we have to prove that the existence of such a curve necessarily leads, when quantity is applied to the relation between two points, to a unique magnitude, which those two points completely determine. With this, we shall be brought back to distance, from which we started, and shall complete the circle of our argument. We saw, in section A § 119, that the figure formed by two points is projectively indistinguishable from that formed by any two other points in the same straight line; the figure, in both cases, is, from the projective standpoint, simply the straight line on which the two points lie. The difference of relation, in the two cases, is not qualitative, since projective Geometry cannot deal with it; nevertheless, there is some difference of relation. For instance, if one point be kept fixed, while the other moves, there is obviously some change of relation. This change, since all parts of the straight line are qualitatively alike, must be a change of quantity. If two points, therefore, determine a unique figure, there must exist, for the distinction between the various other points of this figure, a unique quantitative relation between the two determining points, and therefore, since these points are arbitrary, between only two points. This relation is distance, with which our argument began, and to which it at least returns. 175. To sum up: If points are defined simply by relations to other points, i.e., if all position is relative, every point must have to every other point one, and only one, relation independent of the rest of space. This relation is the distance between the two points. Now a relation between two points can only be defined by a line joining them—nay further, it may be contended that a relation can only be a line joining them. Hence a unique relation involves a unique line, i.e., a line determined by any two of its points. Only in a space which admits of such a line is linear magnitude a logically possible conception. But when once we have established the possibility, in general, of drawing such lines, and therefore of measuring linear magnitudes, we may find that a certain magnitude has a peculiar relation to the constitution of space. The straight line may turn out to be of finite length, and in this case its length will give a certain peculiar magnitude, the space-constant. Two antipodal points, that is, points which bisect the entire straight line, will then have a relation of magnitude which, though unaltered by motion, is rendered peculiar by a certain constant relation to the rest of space. This peculiarity presupposes a measure of linear magnitude in general, and cannot, therefore, upset the apriority of the axiom of the straight line. But it destroys, for points having the peculiar antipodal relation to each other, the argument which proved that the relation between two points could not, since it was unchanged by motion, have reference to the rest of space. Thus it is intelligible that, for such special points, the axiom breaks down, and an infinite number of straight lines are possible between them; but unless we had started with assuming the general validity of the axiom, we could never have reached a position in which antipodal points could have been known to be peculiar, or, indeed, a position which would have enabled us to give any quantitative definition whatever of particular points. Distance and the straight line, as relations uniquely determined by two points, are thus à priori necessary to metrical Geometry. But further, they are properties which must belong to any form of externality. Since their necessity for Geometry was deduced from homogeneity and the relativity of position, and since these are necessary properties of any form of externality, the same argument proves both conclusions. We thus obtain, as in the case of Free Mobility, a double apriority: The axiom of Distance, and its implication, the axiom of the Straight Line, are, on the one hand, presupposed in the possibility of spatial magnitude, and cannot, therefore, be contradicted by any experience resulting from the measurement of space; while they are consequences, on the other hand, of the necessary properties of any form of externality which is to render possible experience of an external world. 176. In connection with the straight line, it will be convenient to discuss the conditions of a metrical coordinate system. The projective coordinate system, as we have seen, aims only at a convenient nomenclature for different points, and can be set up without introducing the notion of spatial quantity. But a metrical coordinate system does much more than this. It defines every point quantitatively, by its quantitative spatial relations to a certain coordinate figure. Only when the system of coordinates is thus metrical, i.e., when every coordinate represents some spatial magnitude, which is itself a relation of the point defined to some other point or figure—can operations with coordinates lead to a metrical result. When, as in projective Geometry, the coordinates are not spatial magnitudes, no amount of transformation can give a metrical result. I wish to prove, here, that a metrical coordinate system necessarily involves the straight line, and cannot, without a logical fallacy, be set up on any other basis. The projective system of coordinates, as we saw, is entirely based on the straight line; but the metrical system is more important, since its quantities embody actual information as to spatial magnitudes, which, in projective Geometry, is not the case. In the first place, a point's metrical coordinates constitute a complete quantitative definition of it; now a point can only be defined, as we have seen, by its relations to other points, and these relations can only be defined by means of the straight line. Consequently, any metrical system of coordinates must involve the straight line, as the basis of its definitions of points. This à priori argument, however, though I believe it to be quite sound, is not likely to carry conviction to any one persuaded of the opposite. Let us, therefore, examine metrical coordinate systems in detail, and show, in each case, their dependence on the straight line. We have already seen that the notion of distance is impossible without the straight line. We cannot, therefore,