Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/276



Suppoing the centripetal force to be proportional to the ditance of the body from the centre; all bodies revolving in any planes whatoever will decribe ellipes, and compleat their revolutions in equal times; and thoe which move in right lines, running backwards and forwards alternately, will compleat their everal periods of going and returning in the ame times.

For letting all things land as in the foregoing propoition, the force SV with which the body Q revolving in any plane PQR is attracted towards the centre S, is as the ditance SQ; and therefore becaue 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 ditance C. Therefore the forces with which bodies found in the plane PQR are attracted towards the point C, are in proportion to the ditances equal to the forces with which the ame bodies are attracted every way towards the centre S; and therefore the bodies will move in the ame times, and in the ame figures in any plane PQR about the point C, as they would do in free paces about the centre S; and therefore (by cor. 2. prop. 10. and cor. 2.