Proposition XLVII. Theorem XV.
Suppoſing the centripetal force to be proportional to the diſtance of the body from the centre; all bodies revolving in any planes whatſoever will deſcribe ellipſes, and compleat their revolutions in equal times; and thoſe which move in right lines, running backwards and forwards alternately, will compleat their ſeveral periods of going and returning in the ſame times.
For letting all things ſland as in the foregoing propoſition, the force SV with which the body Q revolving in any plane PQR is attracted towards the centre S, is as the diſtance SQ; and therefore becauſe SV and SQ, TV and CQ are proportional, the force TV with which the body is attracted towards the given point C in the plane of the orbit is as the diſtance C. Therefore the forces with which bodies found in the plane PQR are attracted towards the point C, are in proportion to the diſtances equal to the forces with which the ſame bodies are attracted every way towards the centre S; and therefore the bodies will move in the ſame times, and in the ſame figures in any plane PQR about the point C, as they would do in free ſpaces about the centre S; and therefore (by cor. 2. prop. 10. and cor. 2.
