parallel, the triangle SBC will be equal to the triangle SBc, and therefore alſo to the triangle SAB. By the like argument, if the centripetal force acts ſucceſſively in C, D, E, &c. and makes the body in each ſingle particle of time, to deſcribe the right lines CD, DE, EF, &c. they will all lye in the ſame 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 deſcrib'd in one immovable plane: and, by compoſition, any ſums SADS, SAFS, of thoſe areas, are one to the other, as the times in which they are deſcrib'd. Now let the number of thoſe triangles be augmented, and their breadth dimniſhed 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 deſcrib'd areas SADS, SAFS, which are always proportional to the times of deſcription, will, in this caſe alſo, be proportional to thoſe times. Q. E. D.
Cor. 1. The velocity of a body attracted towards an immovable centre, in ſpaces void of reſiſtance, is reciprocally as the perpendicular let fall from that centre on the right line that touches the orbit. For the velocities in thoſe places A, B, C, D, E are as the baſes AB, BC, CD, DE, EF, of equal triangles; and theſe baſes are reciprocally as the perpendiculars let fall upon them.
Cor. 2. If the chords AB, BC of two arcs, ſucceſſively deſcribed in equal times, by the ſame body, in ſpaces void of reſiſtance, are compleated into a parallelogram ABCB and the diagonal BV of this parallelogram, in the poſition which it ultimately acquires when thoſe arcs are diminiſhed in infinitum, is produced
