Therefore when a body is being acted on by an accelerating force it tends to go faster and faster as it proceeds, and therefore its velocity increases with the time. But the space passed through increases faster still, for as the time runs on not only does the space passed through increase, but the rate of passing also gets bigger. It goes on increasing at an increasing rate. It is like a man who has an increasing income and always goes on saving it. His total mounts up not merely in proportion to the time, but the very rate of increase also increases with the time, so that the total increase is in proportion to the time multiplied into the time, in other words to the square of the time. So then, if I let a body drop from rest under the action of any force capable of producing an acceleration, the space passed through will be as the square of the time. Now let us see what the speed will be if the force is gravity, that is the attraction of the earth. Turning back to what was said about Galileo, it will be remembered that he showed that all bodies, big and small, light and heavy, fell to the earth at the same speeds. What is that speed? Let us denominate by G the number of feet per second of increase of motion produced in a body by the earth’s action during one second. Then the velocity at the end of that second will be V = GT. The space fallen through will be S = 1/2 GT². What I want to know then is this: how far will a body under the action of gravity fall in a second of time? This, of course, is a matter for measurement. If we can get a machine to measure seconds, we shall be able to do it; but inasmuch as falling bodies begin by falling sixteen feet in the first second and afterwards go on falling quicker and quicker, the measurements are difficult. Galileo wanted to see if he could make it easier to observe. He said to himself, “If I can only water down the force of gravity and make it weaker, so that the body will move very slowly under its action, then the time of falling will be easier to observe.” But how to do it? This is one of those things the discovery of which at once marks the inventor. [Illustration: FIG. 25.] The idea of Galileo was, instead of letting the body drop vertically, to make it roll slowly down an incline, for a body put upon an incline is not urged down the incline with the same force which tends to make it fall vertically. Can any law be discovered tending to show what the force is with which gravity tends to drag a mass down an incline?