Again, the events which are parts of a single light-ray have a definite time-order, in spite of the fact that the interval between any two of them is zero. This appears as follows. Suppose a light-ray proceeds from the sun to the moon and is thence reflected to the earth: it reaches the earth later than a direct ray which left the sun at the same time. There is therefore a definite sense in saying that the ray reached the moon later than it left the sun—i.e. we can say that the ray went from the sun to the moon, not from the moon to the sun. Generalizing, we may say: If and are part of one light-ray, and light-rays from and , distinct from the previous light-ray, contain events , whose interval is time-like, then the time-order of , is the same whatever these new light-rays may be—i.e. we shall have always before , or always before . In the first case, we say that the "sense" of the ray is from to in the second, from to . This illustrates the difficulties which would arise if we were to attempt to found our geometry on interval alone. We must also take account of the purely ordinal properties of the space-time manifold. These properties give a wide separation between the departure of a light-ray from the sun and its[Pg 71] arrival on the earth, although the "interval" between these two events is zero. Reverting now to the method of tensors and its possible eventual simplification, it seems probable that we have an example of a general tendency to over-emphasize numbers, which has existed in mathematics ever since the time of Pythagoras, though it was temporarily less prominent in later Greek geometry as exemplified in Euclid. Euclid's theory of proportion does not, of course, dispense with numbers, since it uses "equimultiples"; but at any rate it requires only integers, not irrationals. Owing to the fact that arithmetic is easy, Greek methods in geometry have been in the background since Descartes, and co-ordinates have come to seem indispensable. But mathematical logic has shown that number is logically irrelevant in many problems where it formerly seemed essential, notably mathematical induction, limits, and continuity. A new technique, which seems difficult because it is unfamiliar, is required when numbers are not used; but there is a compensating gain in logical purity. It should be possible to apply a similar process of purification to physics. The method of tensors first assigns co-ordinates, and then shows how to obtain results which, though expressed in terms of co-ordinates, do not really depend upon them. There must be a less indirect technique possible, in which we use no more apparatus than is logically necessary, and have a language which will only express such facts as are now expressed in the language of tensors, not such as depend upon the choice of co-ordinates. I do not say that such a method, if discovered, would be preferable in practice, but I do say that it would give a better expression