In ordinary Euclidean geometry, there is exactly one point on a plane which is equidistant from three given points on the plane; it is the centre of the circumscribed circle. In three dimensions, there is one point equidistant from four given points; in four, from five. This last holds also in the special theory of relativity, and even in the general theory so long as the distances concerned are small. If we take a point () near the origin, another point () is equidistant from this point and the origin if (where the have their values at the origin), which is a simple equation in . Four such equations give a unique set of values for (). Thus there is just one point equidistant from five given points close together. Moreover, a simple equation, which we may take to be that of the part of a plane near the origin, gives the locus of points near the origin and equidistant from it and a neighbouring point. In fact, as we should expect, for small distances everything proceeds as in elementary geometry, given the formula for . But the mere assumption that there is such a relation as between point-pairs does not yield these results, since it does not imply the interrelation of distances which is given by the formula for . Nevertheless, it does suffice theoretically as a basis of measurement, since, as we have seen, it enables us to halve distances and double them, and therefore to assign numbers to them. This shows that the geometry of relativity, even in its most general and abstract form, assumes a good deal more than the mere possibility of measurement, which, in itself, is of very little value. In itself, it does not lead to a geometry; this only results when there is some interconnection between different measures. It may be asked whether, when the geometry of relativity is generalized to the utmost, any genuinely quantitative element remains in its formulæ. We start with an ordered four-dimensional manifold, and we assign co-ordinates subject to[Pg 113] the sole restriction that their order-relations are to reproduce those of the given manifold. We then proceed to find formulæ (tensor-equations) which hold equally in all systems of co-ordinates satisfying the above condition. It might seem a possibility that such formulæ really express only ordinal relations, and that the sole advantage of co-ordinates lies in the fact that they provide names for the terms of a manifold of the required sort. (They do not provide names for all of them; the number of names is , and therefore only a vanishing proportion of real numbers can be named—i.e. expressed by means of a formula of finite complexity which employs integers.) This possibility requires investigation.