At a great distance from matter, the special theory will still be true, and therefore space will be Euclidean, since, if we put , the special theory gives the Euclidean formula for distance. The neighbourhood of gravitating matter is shown by a non-Euclidean character of the region concerned. This, however, requires some preliminary explanations, more especially an explanation of the method of tensors, which will form the subject of the next chapter. Everything in the general theory of relativity is dependent upon the existence of the above formula for . The formula itself is of the nature of an empirical generalization; no a priori justification for it is suggested. It is a generalization of the theorem of Pythagoras, which could formerly be proved. But the proof rested upon Euclid's axioms, which there is no reason to regard as exactly true. More than that, there is difficulty in assigning a meaning to his fundamental concepts, such as the "straight" line. The old geometry assumed a static space, which it could do because space and time were supposed to be separable. It is natural to think of motion as following a path in space which is there before and after the motion: a tram moves along pre-existing tram-lines. This view of motion, however, is no longer tenable. A moving point is a series of positions in space-time; a later moving point cannot pursue the "same" course, since its time co-ordinate is different, which means that, in another equally legitimate system of co-ordinates, its space co-ordinates also will be different. We think of a tram as performing the same journey every day, because we think of the earth as fixed; but from the sun's point of view, the tram never repeats a former journey. "We cannot step twice into the same rivers," as[Pg 62] Heraclitus says. It is thus obvious that, in place of Euclid's static straight line, we shall have to substitute a movement having some special property defined in terms of space-time, not of space. The movement required is a "geodesic," concerning which we shall have more to say later.