Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/101

 1. proportion, before the quantities have vanihed, is not the ultimate, and when they are vanihed, is none. But by the ame argument it may be alledged, that a body arriving at a certain place, and there lopping, has no ultimate velocity: becaue the velocity, before the body comes to the place, is not its ultimate velocity; when it has arrived, is none. But the anwer is eay; for by the ultimate velocity is meant that with which the body is moved, neither beore it arrives at its lat place and the motion ceaes, nor after, but at the very intant it arrives; that is, that velocity with which the body arrives at its lat place, and with which the motion ceaes. And in like manner, by the ultimate ratio of evanecent quantities is to be undertood the ratio of the quantities, not before they vanih, nor afterwards, but with which they vanih. In like manner the firt ratio of nacent quantities is that with which they begin to be. And the firt or lat um is that with which they begin and ceae to be (or to be augmented or diminihed.) There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and ceae to be. And ince uch limits are certain and definite, to determine the ame is a problem trictly geometrical. But whatever is geometrical we may be allowed to ue in determining and demontrating any other thing that is likewie geometrical.

It may alo be objected, that if the ultimate ratio's of evanecent quantities are given, their ultimate magnitudes will be alo given: and o all quantities will conit of indiviibles, which is contrary to what Euclid has demontrated concerning incommenurable, in the 10th book of his Elements. But this objection is Rh