It is assumed that every point of space-time can have four real numbers assigned to it, and conversely that any four real numbers (at any rate within certain limits) are the co-ordinates of a point. This amounts to [Pg 58]the assumption that the number of points is , where is the number of finite integers; that is to say, the number of points is the number of the Cantorian continuum. Every class of terms is the field of various multiple relations which arrange the class in a four-dimensional continuum—or an -dimensional continuum, for that matter. But we require a little more than this. Of all the ways of arranging the points of space-time in a four-dimensional continuum, there is only one that has physical significance; the others exist only for mathematical logic. That means that there must be among points relations derivable from an empirical basis, which generate a four-dimensional continuum. These will be the ordinal relations spoken of in the last paragraph but one. We assume, therefore, that these ordinal relations generate a continuum, and that co-ordinates are so assigned that neighbouring points have neighbouring co-ordinates. More exactly the co-ordinates of the limit of a set of points are the limits of the co-ordinates of the set. This is not a law of nature, but a prescription as to the manner in which co-ordinates are assigned. It leaves great latitude, but not complete latitude. It allows any system of co-ordinates to be replaced by another system in which the new co-ordinates are any continuous functions of the old co-ordinates, but it excludes discontinuous functions. We now assume that any two neighbouring points have a metrical relation, called their "interval," whose square is a quadratic function of the differences of their co-ordinates. This is a generalization of the theorem of Pythagoras, which has come by way of Gauss and Riemann. It will be worth while to consider the historical development for a moment. By the theorem of Pythagoras, if two points in a plane have co-ordinates (), () and is their distance apart: By an immediately obvious extension, if two points in space[Pg 59] have co-ordinates (), (), their distance apart is , where: If the distance apart is small, we write , , for , , and for ; thus: Gauss considered a problem concerned with surfaces, which arises naturally out of the above. On a surface, the position of a point can be fixed by two co-ordinates, which need not involve reference to anything outside the surface. Thus on the earth position is fixed by latitude and longitude. Suppose and are two such co-ordinates which fix position on a surface. Then in general we shall not have: for the distance between neighbouring points; in general, we cannot get a formula of this kind however we may define and . We can get a formula of this kind on a cylinder or a cone, and generally on what are called "developable" surfaces, but not, e.g., on a sphere. The general formula takes the shape: