Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/13

Sect. I continually proportional. Therefore if out of an equal number of particles there be compounded any equal portions of time, the velocities at the beginning of thoe times will be as terms in a continued progreion, which are taken by intervals, omitting every where an equal number of intermediate terms. But the ratio's of thee terms are compounded of the equal ratio's of the intermediate terms equally repeated; and therefore are equal. Therefore the velocities, being proportional to thoe terms, are in geometrical progreion. Let thoe equal particles of time be diminihed, and their number increaed in infinitum, o that the impule of reitance may become continual; and the velocities at the beginnings of equal times, always continually proportional, will be alo in this cae continually proportional. Q. E. D.

And, by diviion, the differences of the velocities, that is, the parts of the velocities lot in each of the times, are as the wholes: But the paces decribed in each of the times arc as the lot parts of the velocities, (by Prop. 1. Book 2.) and therefore are alo as the wholes. Q. E. D.

Hence if to the rectangular aymptotes AC, CH, the Hyperbola BG is decribed, and AB, DG be drawn perpendicular to the aymptote AC, and both the velocity of the body, and the reitance of the medium, at the very beginning of the motion, be expred by any given line AC, and after ome time is elaped, by the indefinite, line DC; the time may be expres'd by the area ABGD, and the pace decribed in that time by the line AD. For. if that area, by the motion of the point D, be uniformly increaed in the ame manner as the time, the right line DC will decreae in a geometrical ratio in the ame manner as the velocity, and the parts of the right AC, decribed in equal times, will decreae in the ame ratio.