The problem can be discussed equally well in two dimensions. In Gauss's theory of surfaces, a sphere and an ellipsoid, e.g. are distinguishable by the fact that there is an irreducible difference between the formulæ for which hold for the two surfaces when expressed in terms of two co-ordinates; this expresses the fact that the measure of curvature is constant in the case of the sphere, but not in the case of the ellipsoid. Yet from a purely ordinal point of view, such as that of analysis situs, the two figures are indistinguishable. What, exactly, is added to make the difference? This problem is essentially the same as that which arises in the general theory of relativity. In part, the answer in this case is simple. What is added is the comparability of distances in different directions. So long as our apparatus is purely ordinal, we can say of three points which have the order that is nearer to than is, but we cannot say anything analogous of three points which are not in a row—I do not say "in a straight line," because the concept involved is more general, as will appear later. But although this is part of the answer, it does not seem to be the whole, since our relation also enabled us to compare distances not having a common origin. [Pg 114] It seems that what distinguishes distance as required in geometry from such a relation as "subtending a given angle at a given point" is the absence of reference to anything external. When the distance between two points is equal to the distance between two others, we are supposed to have a fact which does not demand reference to some other point or points. In fact, this is the reason why the "interval" has been substituted for distance: the latter, as hitherto conceived, was found to depend upon the motion of the co-ordinate frame, and thus to be not an intrinsic geometrical relation. The distance, if it is to serve its purpose, must be a function of the two points exclusively, and must not involve any other geometrical data. Here, for relativity purposes, "geometry" includes "kinematics." The angle which two points subtend at a given point becomes a function of three points as soon as the given point is thought of as variable. There must be no such way of turning the distance between two points into a function involving other variables also.