Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/15

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Sect. I
of Natural Philoſophy.
5

(by Law 2) as the increments. of the velocities, that is, as the rectangles Ak, Kl, Lm, Mn, &c. and therefore (by Lem. 1. Book 1.) in a geometrical progreſſion; Therefore if the right lines Kk, Ll, Mm, Nn, &c. are produced ſo as to meet the Hyperbola in q, r, s, t, &c. the areas ABqK, KqrL, KqrL, LrsM, MstN, &c. will be equal, and therefore analogous to the equal times and equal gravitating forces. But the area ABqK (by Corol. 3. Lem.7 & 8. Book 1.) is to the area Bkq as Kq to , or AC to , that is as the force of gravity to the reſiſtance in the middle of the firſt time. And by the like reaſoning the areas qKLr, rLMs, sMnt, &c. are to the areas qklr, rmls, smnt, &c. as the gravitating forces to the reſiſtances in the middle of the ſecond, third, fourth time, and ſo on. Therefore ſince the equal areas BAKq, qKLr, rLMs, sMNt, &c. are analogous to the gravitating forces, the areas Bkq, qklr, rlms, smnt, &c. will be analogouſ to the reſiſtances in the middle of each of the times, that is (by ſuppoſition) to the velocities, and ſo to the ſpaces deſcribed to Take the turns of the analogous quantities, and the areas Bkq, Blr, Bms, Bnt, &c. will be analogous to the whole ſpaces deſcribed and alſo the areas ABqK, ABrL, ABsM, ABtM &c. to the times. Therefore the body, in deſcending. will in any time ABrL, deſcribe the ſpace Blr, and in the time LrtN the ſpace rlnt. Q. E. D. And the like demonſtration holds in aſcending motion.

Corol. 1. Therefore the greateſt velocity that the body can acquire by falling, is to the velocity acquired in any given time, as the given force of gravity which perpetually acts upon it, to the reſiſting force which oppoſes it at the end of that time.

Corol. 2. But the time being augmented in an arithmetical progreſſion, the ſum of that greateſt velocity and the velocity in the aſcent, and alſo their difference in the deſcent, decreaſes in a geometrical progreſſion.