An Essay on the Foundations of Geometry

that two points must determine a unique curve, the straight line. In metrical Geometry, the corresponding axiom is, that two points must determine a unique spatial quantity, distance. I propose to prove, in what follows, (1) that if distance, as a quantity completely determined by two points, did not exist, spatial magnitude would not be measurable; (2) that distance can only be determined by two points, if there is an actual curve in space determined by those two points; (3) that the existence of such a curve can be deduced from the conception of a form of externality, and (4) that the application of quantity to such a curve necessarily leads to a certain magnitude, namely distance, uniquely determined by any two points which determine the curve. The conclusion will be, if these propositions can be successfully maintained, that the axiom of distance is à priori in the same double sense as the axiom of Free Mobility, i.e. it is presupposed in the possibility of measurement, and it is necessarily true of any possible form of externality. 163. (1) The possibility of spatial measurement allows us to infer the existence of a magnitude uniquely determined by any two points. The proof of this depends on the axiom of Free Mobility, or its equivalent, the homogeneity of space. We have seen that these are involved in the possibility of spatial measurement; we may employ them, therefore, in any argument as to the conditions of this possibility. Now to begin with, two points must, if Geometry is to be possible, have some relation to each other, for we have seen that such relations alone constitute position or localization. But if two points have a relation to each other, this must be an intrinsic relation. For it follows, from the axiom of Free Mobility, that two points, forming a figure congruent with the given pair, can be constructed in any part of space. If this were not possible, we have seen that metrical Geometry could not exist. But both the figures may be regarded as composed of two points and their relation; if the two figures are congruent, therefore, it follows that the relation is quantitatively the same for both figures, since congruence is the test of spatial equality. Hence the two points have a quantitative relation, which is such that they can traverse all space in a combined motion without in any way altering that relation. But in such a general motion, any external relation of the two points, any relation involving other points or figures in space, must be altered[170]. Hence the relation between the two points, being unaltered, must be an intrinsic relation, a relation involving no other point or figure in space; and this intrinsic relation we call distance[171]. 164. It might be objected, to the above argument, that it involves a petitio principii. For it has been assumed that the two points and their relation form a figure, to which other figures can be congruent. Now if two points have no intrinsic relation, it would seem that they cannot form such a figure. The argument, therefore, apparently assumes what it had to prove. Why, it may be asked, should not three points be required, before we obtain any relation, which Free Mobility allows us to construct afresh in other parts of space? The answer to this, as to the corresponding question in the first section of this chapter, lies, I think, in the passivity of space, or the mutual independence of its parts. For it follows, from this independence, that any figure, or any assemblage of points, may be discussed without reference to other figures or points. This principle is the basis of infinite divisibility, of the use of quantity in Geometry, and of all possibility of isolating particular figures for discussion. It follows that two points cannot be dependent, as to their relation, on any other points or figures, for if they were so dependent, we should have to suppose some action of such points or figures on the two points considered, which would contradict the mutual independence of different positions. To illustrate by an example: the relation of two given points does not depend on the other points of the straight line on which the given points lie. For only through their relation, i.e. through the straight line which they determine, can the other points of the straight line be known to have any peculiar connection with the given pair. 165. But why, it may be asked, should there be only one such relation between two points? Why not several? The answer to this lies in the fact that points are wholly constituted by relations, and have no intrinsic nature of their own[172]. A point is defined by its relations to other points, and when once the relations necessary for definition have been given, no fresh relations to the points used in definition are possible, since the point defined has no qualities from which such relations could flow. Now one relation to any one other point is as good for definition as more would be, since however many we had, they would all remain unaltered in a combined motion of both points. Hence there can only be one relation determined by any two points. 166. (2) We have thus established our first proposition—two points have one and only one relation uniquely determined by those two points. This relation we call their distance apart. It remains to consider the conditions of the measurement of distance, i.e., how far a unique value for distance involves a curve uniquely determined by the two points. In the first place, some curve joining the two points is involved in the above notion of a combined motion of the two points, or of two other points forming a figure congruent with the first two. For without some such curve, the two point-pairs cannot be known as congruent, nor can we have any test by which to discover when a point-pair is moving as a single figure[173]. Distance must be measured, therefore, by some line which joins the two points. But need this be a line which the two points completely determine? 167. We are accustomed to the definition of the straight line as the shortest distance between two points, which implies that distance might equally well be measured by curved lines. This implication I believe to be false, for the following reasons. When we speak of the length of a curve, we can give a meaning to our words only by supposing the curve divided into infinitesimal rectilinear arcs, whose sum gives the length of an equivalent straight line; thus unless we presuppose the straight line, we have no means of comparing the lengths of different curves, and can therefore never discover the applicability of our definition. It might be thought, perhaps, that some other line, say a circle, might be used as the basis of measurement. But in order to estimate in this way the length of any curve other than a circle, we should have to divide the curve into infinitesimal circular arcs. Now two successive points do not determine a circle, so that an arc of two points would have an indeterminate length. It is true that, if we exclude infinitesimal radii for the measuring circles, the lengths of the infinitesimal arcs would be determinate, even if the circles varied, but that is only because all the small circular arcs through two consecutive points coincide with the straight line through those two points. Thus, even with the help of the arbitrary restriction to a finite radius, all that happens is that we are brought back to the straight line. If, to mend matters, we take three consecutive points of our curve, and reckon distance by the arc of the circle of curvature, the notion of distance loses its fundamental property of being a relation between two points. For two consecutive points of the arc could not then be said to have any corresponding distance apart—three points would be necessary before the notion of distance became applicable. Thus the circle is not a possible basis for measurement, and similar objections apply, of course, with increased force, to any other curve. All this argument is designed to show, in detail, the logical impossibility of measuring distance by any curve not completely defined by the two points whose distance apart is required. If in the above we had taken distance as measured by circles of given radius, we should have introduced into its definition a relation to other points besides the two whose distance was to be measured, which we saw to be a logical fallacy. Moreover, how are we to know that all the circles have equal radii, until we have an independent measure of distance? 168. A straight line, then, is not the shortest distance, but is simply the distance between two points—so far, this conclusion has stood firm. But suppose we had two or more curves through two points, and that all these curves were congruent inter se. We should then say, in accordance with the definition of spatial equality, that the lengths of all these curves were equal. Now it might happen that, although no one of the curves was uniquely determined by the two end-points, yet the common length of all the curves was so determined. In this case, what would hinder us from calling this common length the distance apart, although no unique figure in space corresponded to it? This is the case contemplated by spherical Geometry, where, as on a sphere, antipodes can be joined by an infinite number of geodesics, all of which are of equal length. The difficulty supposed is, therefore, not a purely imaginary one, but one which modern Geometry forces us to face. I shall consequently discuss it at some length. 169. To begin with, I must point out that my axiom is not quite equivalent to Euclid's. Euclid's axiom states that two straight lines cannot enclose a space, i.e., cannot have more than one common point. Now if every two points, without exception, determine a unique straight line, it follows, of course, that two different straight lines can have only one point in common—so far, the two axioms are equivalent. But it may happen, as in spherical space, that two points in general determine a unique straight line, but fail to do so when they have to each other the special relation of being antipodes. In such a system every pair of straight lines in the same plane meet in two points, which are each other's antipodes; but two points, in general, still determine a unique straight line. We are still able, therefore, to obtain distances from unique straight lines, except in limiting cases; and in such cases, we can take any point intermediate between the two antipodes, join it by the same straight line to both antipodes, and measure its distance from those antipodes in the usual way. The sum of these distances then gives a unique value for the distance between the antipodes. Thus even in spherical space, we are greatly assisted by the axiom of the straight line; all linear measurement is effected by it, and exceptional cases can be treated, through its help, by the usual methods for limits. Spherical space, therefore, is not so adverse as it at first appeared to be to the à priori necessity of the axiom. Nevertheless we have, so far, not attacked the kernel of the objection which spherical space suggested. To this attack it is now our duty to proceed. 170. It will be remembered that, in our à priori proof that two points must have one definite relation, we held it impossible for those two points to have, to the rest of space, any relation which would be unaltered by motion. Now in spherical space, in the particular case where the two points are antipodes, they have a relation, unaltered by motion, to the rest of space—the relation, namely, that their distance is