98 



NEWTON S PRINCIPIA. 



book, in which the sun s disturbing force on the moon s 

 motion is investigated. But respecting that proposition, 

 it is wholly inaccurate to say that he there makes any 

 hypothesis or assumption of the proportion between the 

 disturbing force and the moon s gravity; for he deduces the 

 proportion of 1 to 178f , (or which is nearly the same thing, 

 2 to 357,) from the duplicate ratio of the periodic times, and 

 deduces it as a consequence of the Seventeenth corollary to 

 the Sixty-sixth proposition of the First book, which corol 

 lary comes easily from the Second corollary of the Fourth 

 proposition of the First book. It is, therefore, wholly 

 impossible to represent that position as a mere assumption 

 to suit the observation of the moon s actual variations. 



ii. The next subject of consideration is the motion of bo 

 dies along given surfaces, not in planes passing through the 

 centre of forces, to which case our inquiries have hi 

 therto been confined. 



Let a body move in any plane, in a trajectory, by a force 



directed towards a centre out of that plane, and we are 

 to examine its motion under two heads, as we did the mo 

 tion of a body when the centre was in the plane of the tra 

 jectory ; that is, first, the curve described by the given force; 



