proportion, before the quantities have vaniſhed, is not the ultimate, and when they are vaniſhed, is none. But by the ſame argument it may be alledged, that a body arriving at a certain place, and there ſlopping, has no ultimate velocity: becauſe the velocity, before the body comes to the place, is not its ultimate velocity; when it has arrived, is none. But the anſwer is eaſy; for by the ultimate velocity is meant that with which the body is moved, neither beſore it arrives at its laſt place and the motion ceaſes, nor after, but at the very inſtant it arrives; that is, that velocity with which the body arrives at its laſt place, and with which the motion ceaſes. And in like manner, by the ultimate ratio of evaneſcent quantities is to be underſtood the ratio of the quantities, not before they vaniſh, nor afterwards, but with which they vaniſh. In like manner the firſt ratio of naſcent quantities is that with which they begin to be. And the firſt or laſt ſum is that with which they begin and ceaſe to be (or to be augmented or diminiſhed.) 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 ceaſe to be. And ſince ſuch limits are certain and definite, to determine the ſame is a problem ſtrictly geometrical. But whatever is geometrical we may be allowed to uſe in determining and demonſtrating any other thing that is likewiſe geometrical.
It may alſo be objected, that if the ultimate ratio's of evaneſcent quantities are given, their ultimate magnitudes will be alſo given: and ſo all quantities will conſiſt of indiviſibles, which is contrary to what Euclid has demonſtrated concerning incommenſurable, in the 10th book of his Elements. But this objection is
