138 NEWTON S PEINCIPIA. 



I P 2 



proportional to I H x I q, is as -p - pro 5 and if the attrac 



tion upon the particle P is as the surface directly, and 

 the square of the distance inversely, or p~ri? that attrac 



tion will be as -^ - ^a But; if the force acting in the 

 PpxPS 



line PI is resolved into one acting in P S and another acting 



in SD, the force upon P will be as p-|, or (because of the 



p 



similar triangles P I Q, P Sp) as ~. The attraction, 



therefore, of the infinitely small curvilinear surface formed 



P 1 



by the revolution of I H is as p - - or as-- ; that 



.L p x 



is inversely as the square of the distance from the centre 

 of the sphere. And the same may be shown of the sur 

 face formed by the revolution of KL, and so of every part 

 of the spherical surface. Therefore the whole attraction 

 of the spherical surface will be in the same inverse du 

 plicate ratio. 



In like manner, because the attraction of a homogeneous 

 sphere is the attraction of all its particles, and the mass 

 of these is as the cube of the sphere s diameter D, if a 

 particle be placed at a distance from it in any given 

 ratio to the diameter, as m. D, and the attraction be 

 inversely as the square of that distance, it will be directly 



as D 3 , and also as 7 ^rjp, and therefore will be in the sim 



ple proportion of D, the diameter. Hence if two similar 

 solids are composed of equally dense matter, and have an 

 attraction inversely as the square of the distance, their at 

 traction on any particle similarly placed with respect to 

 them will be as their diameters. Thus, also, a particle 



