Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/104

 parallel, the triangle SBC will be equal to the triangle SBc, and therefore alo to the triangle SAB. By the like argument, if the centripetal force acts ucceively in C, D, E, &c. and makes the body in each ingle particle of time, to decribe the right lines CD, DE, EF, &c. they will all lye in the ame plane; and the triangle SCD will be equal to the triangle SBC, and SDE to SCD, and SEP to SDE. And therefore in equal times, equal areas are decrib'd in one immovable plane: and, by compoition, any ums SADS, SAFS, of thoe areas, are one to the other, as the times in which they are decrib'd. Now let the number of thoe triangles be augmented, and their breadth dimnihed in infinitum; and (by cor. 4. lem. 5.) their ultimate perimeter ADF will be a curve line: and therefore the centripetal force, by which the body is perpetually drawn back from the tangent of this curve, will act continually; and any decrib'd areas SADS, SAFS, which are always proportional to the times of decription, will, in this cae alo, be proportional to thoe times. Q. E. D.

The velocity of a body attracted towards an immovable centre, in paces void of reitance, is reciprocally as the perpendicular let fall from that centre on the right line that touches the orbit. For the velocities in thoe places A, B, C, D, E are as the baes AB, BC, CD, DE, EF, of equal triangles; and thee baes are reciprocally as the perpendiculars let fall upon them.

If the chords AB, BC of two arcs, ucceively decribed in equal times, by the ame body, in paces void of reitance, are compleated into a parallelogram ABCB and the diagonal BV of this parallelogram, in the poition which it ultimately acquires when thoe arcs are diminihed in infinitum, is produced