Proposition LXV. Theorem XXV.
Bodies, whoſe forces decreaſe in a duplicate ratio of their differences from their centres, may move among themſelves in ellipſis; and by radii drawn to the foci may deſcribe area's proportional to the time very nearly.
In the laſt propoſition we demonſtrated that caſe in which the motions will be performed exactly in ellipſes. The more diſtant the law of the forces is from the law in that caſe, the more will the bodies diſturb each others motions; neither is it poſſible that bodies attracting each other mutually according to the law ſuppoſed in this propoſition ſhould move exactly in ellipſes unleſs keeping a certain proportion of diſtances from each other. However in the following caſes the orbits will not much differ from ellipſes.
Case 1. Imagine ſeveral leſſer bodies to revolve about ſome very great one at different diſtances from it, and ſuppoſe abſolute forces tending to every one of the bodies, proportional to each. And becauſe (by cor. 4. of the laws) the common centre of gravity of them all is either at reſt or moves uniformly forward in a right line, ſuppose the leſſer bodies ſo ſmall that the great body may be never at a ſenſible diſtance from that centre; and then the great body will, without any ſenſible
