NEWTON S PRINCIPIA. 105 



and impelled by a single original force of projection, as E 

 and M, their centre of gravity being G ; it is clear that 



if M moves a very small space to m by the attraction of 

 E, so will E move to e by the attraction of M, and the 

 two triangles E G e and M G m will be similar in all 

 respects ; for the lines M G, m G and EG, e G are 

 proportional, because the segments of the lines E M and 

 e m are always in the same proportion, G being the centre 

 of gravity, and those segments, therefore, inversely as the 

 masses of E and M. Therefore the. curves which the 

 bodies describe round the centre of gravity will be entirely 

 similar. In like manner they will describe similar curves 

 each round the other, and the radius vector of each from 

 the other, as well as from the centre of gravity, will describe 

 areas proportional to the times. It follows from this and 

 from what was before shown respecting centripetal forces, 

 that the two bodies will move in concentric ellipses round 

 one another and round their common centre of gravity, if 

 the centripetal force is as the distance, and that each 

 will describe one or other of the conic sections, having 

 the other, or the common centre of gravity, in the focus, 

 if the centripetal force is inversely as the square of the 

 distance. In like manner, because of the ratio between 

 the squares of the periodic times and the cubes of the dis 

 tances, it may be shown that if T be the periodic time of 

 the bodies moving round their centres of gravity, and t 



