NEWTON S PRINCIPIA. 135 



Sixty-sixth Proposition (although they come later in the 

 Principia), and the latter of those corollaries, naturally lead 

 to the subject of the next two sections, the one upon the 

 attraction of spherical bodies, the other upon that of bodies 

 not spherical. 



i. The attraction exerted by spherical surfaces and by 



hollow spheres is first considered. If P be a particle si 

 tuated anywhere within A B D C, and we conceive .two 

 lines A D, B, C, infinitely near each other drawn through 

 P to the surface, and if these lines revolve round a P b, 

 which passes from the middle points a and b, of the small 

 arcs D C, and A B, through P, there will two opposite 

 cones be described ; and the attraction of the small circles 

 D C, A B upon P, will be in the lines from each point 

 of those circles to P, of which lines C P, D P, are two 

 from one circle, and A P, B P, two from the other circle. 

 Now this attraction of the circle C D is to that of the circle 

 A B, as the circle C D to the circle A B, or as C D- to A 

 B 2 (the diameters), and by similar triangles C D 2 : A B 



:: P C 2 : P A 2 . But by hypothesis, the attraction of C D 

 is to that of A B as A P 2 : P C 2 ; therefore the attraction 

 of D C is to the opposite attraction of A B as A P 2 , to 

 P C 2 , and also as P C 2 to A P 2 , or as A P 2 x P C 2 to A P 2 



x P C 2 , and therefore those attractions are equal ; and 



K 4 



