SEC. II.] OF NATURAL PHILOSOPHY. 103 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 ve locity 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 pro portions 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 de monstrating any other thing that is likewise geometrical. It may also be objected, that if the ultimate ratios of evanescent quan tities 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 incommensurables, in the 10th Book of his Elements. But this objection is founded on a false supposition. For those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing without limit do always converge ; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in wfinitum. This thing will appear more evident in quantities infinitely great. If two quantities, whose dif ference is given, be augmented in infin&um, the ultimate ratio of these quantities will be given, to wit, the ratio of equality ; but it does not from thence follow, that the ultimate or greatest quantities themselves, whose ratio that is, will be given. Therefore if in what follows, for the sake of being more easily understood, I should happen to mention quantities as least, or evanescent, or ultimate, you are not to suppose that quantities of any determinate magnitude are meant, but such as are conceived to be al ways diminished without end. SECTION II. Of the Invention of Centripetal Forces. PROPOSITION I. THEOREM 1. The areas, which revolving bodies describe by radii drawn to an ^mmovable centra of force do lie in tJ:e same immovable planes, and are proportional to the times in which they are described. For suppose the time to be divided into equal parts, and in the first part of that time let the body by its innate force describe the right line AB In the second part of that time, the same would (by Law I.), if not hindered, proceel directly to c, alo ILJ; the line Be equal to AB ; so that by the radii AS, BS, cS, draw. i to the centre, the equal areas ASB, BSc, would be de