where , , are in general functions of and , not constants. Gauss showed that there are certain functions of , , which have the same value however the co-ordinates and may be defined; these functions express properties of the surface, which can theoretically be discovered by measurements carried out on the surface, without reference to external space. Riemann extended this method to space. He supposed that the theorem of Pythagoras may be not exact, and that the correct formula for the distance between two points may be such as results from Gauss's formula by adding another variable. He showed that this supposition could be made the basis of non-Euclidean geometry. The whole subject of[Pg 60] non-Euclidean geometry remained, however, without visible relevance to physics until it was utilized in Einstein's theory of gravitation, which results from the combination of Riemann's ideas with the substitution of space-time "interval" for distance in space and time, which had already been made in the special theory of relativity. In the special theory of relativity, as we saw, the interval between two space-time points, one of which is the origin, is , where: if the interval is space-like, and: if the interval is time-like. In practice, the latter form is always taken. Any system of co-ordinates allowed by the special theory gives the same value for the interval between two given space-time points. But we are now allowing much greater latitude in the choice of co-ordinates, and we are assuming that the special theory represents only an approximation, being not strictly true except in the absence of a gravitational field. We still assume that, for small distances, there is a quadratic function of the co-ordinate differences which has a physical significance, and has the same value however the co-ordinates may be assigned, subject to the condition of continuity already explained. That is, if , , , are the co-ordinates of a point, and , , , are the co-ordinates of a neighbouring point, we assume that there is a quadratic function: which has the same value however the co-ordinates may be assigned; we then define as the "interval" between the two neighbouring points. The 's will be functions of the co-ordinates (in general not constants), and for convenience we take . Just as Gauss was able to deduce the[Pg 61] geometry of a surface from his formula, so we can deduce the geometry of space-time from our formula. But as we include time, our geometry is not merely geometry, but physics; in other words, it combines history with geography.