It is obvious from experiences with which we are all familiar that an accelerated motion is much more difficult to deal with than a uniform one. When you are in a train which is traveling steadily, the motion is not noticeable so long as you do not look out of the window; but when the brakes are applied suddenly you are precipitated forwards, and you become aware that something is happening without having to notice anything outside the train. Similarly in a lift everything seems ordinary while it is moving steadily, but at starting and stopping, when its motion is accelerated, you have odd sensations in the pit of the stomach. (We call a motion “accelerated” when it is getting slower as well as when it is getting quicker; when it is getting slower the acceleration is negative.) The same thing applies to dropping a weight in the cabin of a ship. So long as the ship is moving uniformly, the weight will behave, relatively to the cabin, just as if the ship were at rest: if it starts from the middle of the ceiling, it will hit the middle of the floor. But if there is an acceleration everything is changed. If the boat is increasing its speed very rapidly, the weight will seem to an observer in the cabin to fall in a curve directed towards the stern; if the speed is being rapidly diminished, the curve will be directed towards the bow. All these facts are familiar, and they led Galileo and Newton to regard an accelerated motion as something radically different, in its own nature, from a uniform motion. But this distinction could only be maintained by regarding motion as absolute, not relative. If all motion is relative, the earth is accelerated relatively to the lift just as truly as the lift relatively to the earth. Yet the people on the ground have no sensations in the pits of their stomachs when the lift starts to go up. This illustrates the difficulty of our problem. In fact, though few physicists in modern times have believed in absolute motion, the technique of mathematical physics still embodied Newton’s belief in it, and a revolution in method was required to obtain a technique free from this assumption. This revolution was accomplished in Einstein’s general theory of relativity. It is somewhat optional where we begin in explaining the new ideas which Einstein introduced, but perhaps we shall do best by taking the conception of “interval.” This conception, as it appears in the special theory of relativity, is already a generalization of the traditional notion of distance in space and time; but it is necessary to generalize it still further. However, it is necessary first to explain a certain amount of history, and for this purpose we must go back as far as Pythagoras.