Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/12

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Therefore if the body, deſtitute of all gravity, move by its innate force only in free ſpaces, and there be given both its whole motion at the beginning, and alſo the motion remaining after ſome part of the way is gone over; there will be given alſo the whole ſpace which the body can deſcribe in an infinite time. For that ſpace will be to the ſpace now deſcribed, as the whole motion at the beginning is to the part loſt of that motion.

Quantities proportional to their differences are continually proportional.

Let A be to A—B as B to B—C and C to C—D, &c. and, by converſion, A will be to B as B to C and C to D, &c. Q.E.D.

If a body is reſiſted in the ratio of its veolocity, and moves, by its vis inſita only, through a ſimilar medium; and the times be taken equal; the velocities in the beginning of each of the times are in a geometrical progreſſion, and the ſpaces deſcribed in each of the times are as the velocities.

Let the time be divided into equal particles; and if at the very beginning of each particle we ſuppoſe the reſiſtance to act with one ſingle impulſe which is as the velocity; the decrement, of the velocity in each of the particles of time will be as the ſame velocity. Therefore the velocities are proportional to their differences, and therefore (by Lem. 1. Book. 2.)