Review of the Principia af Newton. 317 



the ellipse we have Q, : -5^, d being the distance ; in the cir- 

 cle we must have Q, : -p, for, as has been shown in the last 



sections, a body to preserve an equality of areas at different 

 distances, moving in a circle or equi-angular spiral must have 

 the central force acting on it in that ratio. 



Hence we conclude, that if a body by a central force de- 

 scribe a curve, it will describe the same curve moveable 

 about the center of forces by compounding with the proper 

 force in the immeveable orbit, another force, which is in the 

 inverse triplicate ratio of the distance. If this force be 

 added, the motion of the curve and that of the body tend 

 the same way. If it be substracted, they will be directed to- 

 wards contrary parts. In order to investigate the motion of 

 the orbit, or of the apsides, in virtue of the new or extrinsi- 

 cal force, Newton assumes a moveable elliptical orbit, and 

 calculates the ratio of the forces necessary for the movement 

 of a body in such an orbit, and the angular motions in the 

 fixed and moveable orbits. The formulas of the forces in 

 terms of the angular motion being obtained, the angu- 

 lar motion of the orbit may be found, and vice versa. 

 This angular motion is that of an ellipse, whose retaining 

 force varies in the inverse duplicate ratio of the distance, to- 

 gether with that of a circle, whose force varies in the inverse 

 triplicate ratio of the distance. The angular motion of other 

 orbits whose forces vary in other ratios, may be deduced from 

 the formula for that of the ellipse, if we suppose those orbits 

 not to vary much in their radii vectores from circles. For 

 circles can be described by forces varying according to any 

 law, and trajectories varying little from them in distance, are 

 little affected by variations of force depending on the dis- 

 tance, compared with the extrinsical force which produces 

 a revolution of the orbits themselves. If now the motion of 

 the body in the curve be similar to that of a body in a move- 

 able ellipsis, the force by which it is retained in its trajectory 

 must be analogous to the force by which a body is retained 

 in such an ellipse : for it is by analogous forces only, or 

 such as consist of corresponding proportional parts, that 

 similar curves are described. The analogous forces, as ex- 

 pressed by the formula, being compared, it will be seen what 

 part of the force retaining the body in the curve, is in excess, 



