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

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Sect. I
of Natural Philoſophy.
3

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 thoſe times will be as terms in a continued progreſſion, which are taken by intervals, omitting every where an equal number of intermediate terms. But the ratio's of theſe terms are compounded of the equal ratio's of the intermediate terms equally repeated; and therefore are equal. Therefore the velocities, being proportional to thoſe terms, are in geometrical progreſſion. Let thoſe equal particles of time be diminiſhed, and their number increaſed in infinitum, ſo that the impulſe of reſiſtance may become continual; and the velocities at the beginnings of equal times, always continually proportional, will be alſo in this caſe continually proportional. Q. E. D.

Caſe 2. And, by diviſion, the differences of the velocities, that is, the parts of the velocities loſt in each of the times, are as the wholes: But the ſpaces deſcribed in each of the times arc as the loſt parts of the velocities, (by Prop. 1. Book 2.) and therefore are alſo as the wholes. Q. E. D.

Corl. Hence if to the rectangular aſymptotes AC, CH, the Hyperbola BG is deſcribed, and AB, DG be drawn perpendicular to the aſymptote AC, and both the velocity of the body, and the reſiſtance of the medium, at the very beginning of the motion, be expreſſd by any given line AC, and after ſome time is elapſed, by the indefinite, line DC; the time may be expreſs'd by the area ABGD, and the ſpace deſcribed in that time by the line AD. For. if that area, by the motion of the point D, be uniformly increaſed in the ſame manner as the time, the right line DC will decreaſe in a geometrical ratio in the ſame manner as the velocity, and the parts of the right AC, deſcribed in equal times, will decreaſe in the ſame ratio.