142 



NEWTON S PKINCIPIA. 



R 



( = M) equal to P S. r&quot; 1 y 2 , and let B T) = z, and the curve 

 B N A will be described, and the differential area N D dn will 

 be ndz (by construction) PS. r n ~ l y* 1 dx\ consequently 

 u dz will be the attractive force of the differential solid E e 

 /F; and fudz will be that of the whole body or sphere 

 A E B, therefore the area A N B = f udz is equal to the 

 whole attraction of the sphere. 



Having reduced the solution to the quadrature of 

 A N B, Sir Isaac Newton proceeds to show how that 

 area may be found. He confines himself to geometrical 

 methods; and the solution, although extremely elegant, 

 is not by any means so short and compendious as the 

 algebraical process gives. Let us first then find the 

 equation to the curve A N B by referring it to the rec 

 tangular coordinates D 1ST, AD. Calling these y and x 

 respectively, and making P A = b, AS (the sphere s 



f 

 radius) = a and P S, or a + b, for conciseness, = -. Then 



PE = 



(b 



2 ax- 



