of four dimensions, so that it will not be quite so simple a geometry as that to which we are accustomed.[1]
But the point to which I would especially direct attention is this. Evidently the proposition which I have given you about time-triangles cannot be dissociated from the corresponding proposition about space-triangles. When we give up the mediaeval geocentric standpoint, we must recognize that they belong to one geometry, of which our ordinary space-geometry is only a part or projection. But if you examine the proposition about time-triangles, you will see that it is a statement about the behaviour of clocks when they move about, a subject which obviously comes under the heading of mechanics. When we deal with the four-dimensional world we can no longer distinguish between geometry and mechanics. They become the same subject. When we have completely mastered the geometry of the world of events, we shall have inevitably learnt the mechanics of it. That is why Einstein, studying the geometry of the world and discovering that it was strictly non-Euclidean, found that he was at the same time studying the mechanical force of gravitation. And when he had made up his mind which of the possible varieties of non-Euclidean geometry was obeyed, and so settled the laws of the new geometry, the same decision settled the law of gravitation—a law approximating to, but not identical with, the law which Newton had given.
Here a wide vista opens before us. We see that two great divisions of mathematical physics, viz. geometry and mechanics, have met in the four-dimensional world. It is not merely that mechanical problems can be treated by formulae originally belonging to pure
- ↑ This involves only a comparatively trifling generalization of Euclidean geometry, not to be confused with the 'non-Euclidean' geometry introduced later in the lecture.
