Chapter: 17

I have frequently spoken of subordinate regulations, or of the laws of nature, and it will be well to give an example. In general our new philosophers make, use of that famous law that God always preserves the same amount of motion in the universe. In fact it is a very plausible law, and in times past I held it for indubitable. But since then I have learned in what its fault consists. Monsieur Descartes and many other clever mathematicians have thought that the quantity of motion, that is to say the velocity multiplied by the mass of the moving body, is exactly equivalent to the moving force, or to speak in mathematical terms, that the force varies as the velocity multiplied by the mass. Now it is reasonable to believe that the same force is always preserved in the universe. So also, by considering the phenomena it will be readily seen that a mechanical perpetual motion is impossible, because the force in such a machine, being always diminished a little by friction and being therefore doomed to ultimate exhaustion, would necessarily have to recoup its losses, and consequently would be constantly increasing of itself without any new impulsion from without; and we see furthermore that the force of a body is diminished only in proportion as it gives up force, either to a contiguous body or to its own parts, in so far as they have a separate movement. The mathematicians to whom I have referred think that what can be said of force can be said of the quantity of motion. In order, however, to show the difference I make two suppositions: in the first, place, that a body falling from a certain height acquires a force enabling it to remount to the same height provided that its direction is turned that way, or provided that there are no hindrances. For instance, a pendulum would rise exactly to the height from which it has fallen, provided the resistance of the air and certain other small particles did not diminish a little its acquired force. Leibniz Chapter 17

I suppose in the second place that it will take as much force to lift a body A weighing one pound to the height CD, four feet, as to raise a body B weighing four pounds to the height EF, one foot. These two suppositions are granted by our new philosophers. It is therefore manifest that the body A falling from the height CD acquires exactly as much fdrce as the body B falling from the height EF, for the body B at F, having by the first supposition sufficient force to return to E, has therefore the force to carry a body of four pounds to the distance of one foot, EF. And likewise the body A at D, having the force to return to C, has also the force required to carry a body weighing one pound, its own weight, back to C, a distance of four feet. Now by the second supposition the force of these two bodies is equal. Let us now see if the quantity of motion is the same in each case. It is here that we will be surprised to find a very great difference, for it has been proved by Galileo that the velocity acquired by the fall CD is double the velocity acquired by the fall EF, although the hight is four times as great. Multiplying, therefore, the body. A, whose mass is 1, by its velocity, which is 2, the product or the quantity of movement will be 2, and on the other hand, if we multiply the body B, whose mass is 4, by its velocity, which is 1, the product or quantity of motion will be 4. Hence the quantity of the motion of the body A at the point D is half the quantity of motion of the body B at the point F, yet their forces are equal, and there is therefore a great difference between the quantity of motion and the force. This is what we set out to show. We can see therefore how the force ought to be estimated by the quantity of the effect which it is able to produce, for example by the height to which a body of certain weight can be raised. This is a very different thing from the velocity which can be imparted to it, and in order to impart to it double the velocity we must have more than double the force. Nothing is simpler than this proof and Monsieur Descartes has fallen into error here, only because he trusted too much to his thoughts even when they had not been ripened by reflection. But it astonishes me that his disciples have not noticed this error, and I am afraid that they are gradually beginning to imitate certain Peripatetics whom they ridicule, and that they are accustoming themselves to consult the book’s of their master rather than reason or nature.