AN ESSAY ON QUANTITY
tremely unfortunate. For, first, Whereas all proof should be taken from principles that are common to both sides, in order to prove a thing we deny, he assumes a principle which we think further from the truth; namely, that the height to which the body rises is the whole effect of the impulse, and ought to be the whole measure of it. Secondly, His reasoning serves as well against him as for him. For may I not plead with as good reason, at least, thus? The velocity given by an impressed force, is the whole effect of that impressed force; and therefore the force must be as the velocity. Thirdly, Supposing the height to which the body is raised to be the measure of the force, this principle overturns the conclusion he would establish by it, as well as that which he opposes. For, supposing the first velocity of the body to be still the same, the height to which it rises will be increased, if the power of gravity is diminished; and diminished, if the power of gravity is increased. Bodies descend slower at the equator, and faster towards the poles, as is found by experiments made on pendulums. If, then, a body is driven upwards at the equator with a given velocity, and the same body is afterwards driven upwards at Leipsic with the same velocity, the height to which it rises in the former case will be greater than in the latter; and therefore, according to his reasoning, its force was greater in the former case; but the velocity in both was the same; consequently, the force is not as the square of the velocity, any more than as the velocity.
Sec. 8. Reflections on this controversy.
Upon the whole, I cannot but think the controvertists on both sides have had a very hard task; the one to prove, by mathematical reasoning and experiment, what ought to be taken for granted; the other, by the same means, to prove what might be granted, making some allowance for impropriety of expression, but can never be proved.
If some mathematician should take it in his head to affirm, that the velocity of a body is not as the space it passes over in a given time, but as the square of that space, you might bring mathematical arguments and experiments to confute him; but you would never by these force him to yield, if he was ingenuous in his way; because you have no common principles left you to argue from, and you differ from one another, not in a mathematical proposition, but in a mathematical definition.
Suppose a philosopher has considered only that measure of centripetal force which is proportional to the velocity generated by it in a given time, and from this measure deduces several propositions: another philosopher, in a distant country, who has the same general notion of centripetal force, takes the velocity generated by it, and the quantity of matter together, as the measure of it. From this he deduces several conclusions,