LAW OF THE UNIVERSE. 



257 



the case of the inverse squares of the distances, and gives the 

 general equation to the conic sections with singular elegance), 

 are all derivable from the Sixth Proposition of the First Book, 

 it is eminently probable that Newton had first tried for a 

 general solution by those means, and only had recourse to 

 the one which he has given in the Forty-first Proposition 

 when he found those methods unmanageable. This would 

 naturally confirm him in his plan of preferring geometrical 

 methods ; though it is to be observed that this investigation, 

 as well as the inverse problem for the case of rectilinear 

 motion in the preceding section, is conducted more analyti- 

 cally than the greater part of the Principia, the reasoning of 

 the demonstration conducting to the solution and not follow- 

 ing it synthetically. 



A is the height from which a body must fall to acquire the 

 velocity at any point D, which the given body moving in 

 the trajectory V I K (sought by the investigation) has at the 

 corresponding point I ; D I, E K, being circular arcs from 

 the centre C, and C I = C D and C K = C E. It is shown 

 previously that, if two bodies whose masses are as their 

 weights descend with equal velocity from A, and being acted 

 on by the same centripetal force, one moves in V I K and the 

 other in A V C, they will at any corresponding points have 



the same velocity, that is at equal distances from the centre G. 

 So that, if at any point D, D b or D F be as the velocity at D 



s 



