This line of argument was developed by Riemann, in his dissertation “On the hypotheses which underlie geometry” (1854), which applied Gauss’s work on surfaces to different kinds of three-dimensional spaces. He showed that all the essential characteristics of a kind of space could be deduced from the formula for small distances. He assumed that, from the small distances in three given directions which would together carry you from one point to another not far from it, the distances between the two points could be calculated. For instance, if you know that you can get from one point to another by first moving a certain distance east, then a certain distance north, and finally a certain distance straight up in the air, you are to be able to calculate the distance from the one point to the other. And the rule for the calculation is to be an extension of the theorem of Pythagoras, in the sense that you arrive at the square of the required distance by adding together multiples of the squares of the component distances, together possibly with multiples of their products. From certain characteristics in the formula, you can tell what sort of space you have to deal with. These characteristics do not depend upon the particular method you have adopted for determining the positions of points. In order to arrive at what we want for the theory of relativity, we now have one more generalization to make: we have to substitute the “interval” between events for the distance between points. This takes us to space-time. We have already seen that, in the special theory of relativity, the square of the interval is found by subtracting the square of the distance between the events from the square of the distance that light would travel in the time between them. In the general theory, we do not assume this special form of interval, except at a great distance from matter. Elsewhere, we assume to begin with a general form, like that which Riemann used for distances. Moreover, like Riemann, Einstein only assumes his formula for _neighboring_ events, that is to say, events which have only a small interval between them. What goes beyond these initial assumptions depends upon observation of the actual motion of bodies, in ways which we shall explain in later chapters.