82 NEWTON S PRINCIPIA. 



n (in some respects the simplest of all, being 



taken in respect of d t constant, and which is integrable 

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

 gives the general equation to the conic sections with sin 

 gular elegance), are all derivable from the Sixth Propo 

 sition of the First Book, it is eminently probable that he 

 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 un 

 manageable. This would naturally confirm him in his plan 

 of preferring geometrical methods ; though it is to be ob 

 served that this investigation, as well as the inverse pro 

 blem for the case of rectilinear motion in the preceding 

 section, is conducted more analytically than the greater 

 part of the Principia, the^reasoning of the demonstration 

 conducting to the solution and not following it synthe 

 tically. 



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 cor 

 responding points have the same velocity, that is at equal 

 distances from the centre C. So that, if at any point D, 

 D b or D F be as the velocity at D of the body moving 

 in A V C, D b or D F will also represent the velocity at 

 I of the body moving in V I K. Then take D F=y as 

 the centripetal force in D or I (that is, as any power of 

 the distance D C, or a x, V C being a, and C D, x) 



