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

Sect. I (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 progreion; Therefore if the right lines Kk, Ll, Mm, Nn, &c. are produced o 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 $$\scriptstyle \frac12kq$$, or AC to $$\scriptstyle \frac12AK$$, that is as the force of gravity to the reitance in the middle of the firt time. And by the like reaoning the areas qKLr, rLMs, sMnt, &c. are to the areas qklr, rmls, smnt, &c. as the gravitating forces to the reitances in the middle of the econd, third, fourth time, and o on. Therefore ince the equal areas BAKq, qKLr, rLMs, sMNt, &c. are analogous to the gravitating forces, the areas Bkq, qklr, rlms, smnt, &c. will be analogou to the reitances in the middle of each of the times, that is (by uppoition) to the velocities, and o to the paces decribed to Take the turns of the analogous quantities, and the areas Bkq, Blr, Bms, Bnt, &c. will be analogous to the whole paces decribed and alo the areas ABqK, ABrL, ABsM, ABtM &c. to the times. Therefore the body, in decending. will in any time ABrL, decribe the pace Blr, and in the time LrtN the pace rlnt. Q. E. D. And the like demontration holds in acending motion.

Therefore the greatet 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 reiting force which oppoes it at the end of that time.

But the time being augmented in an arithmetical progreion, the um of that greatet velocity and the velocity in the acent, and alo their difference in the decent, decreaes in a geometrical progreion.