268 ATTRACTION OF BODIES. 



PS.r"" 1 ^, and let BD = z, and the curve BNA will be 

 described, and the differential area N Ddn will be ndz = (by 

 construction) PS . r"~ l y*dx ; consequently udz will be the 

 attractive force of the differential solid Ee/F; and fudz 

 will be that of the whole body or sphere A E B, therefore the 

 area ANB = C udz is equal to the whole attraction of the 

 sphere. 



Having reduced the solution to the quadrature of ANB, 

 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 rectangular coordinates DN, AD. Calling these 

 y and x respectively, and making PA = &, A S (the sphere's 



f 

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



-;r 8 ; PE= 



a x 



x ; and D N =y = (by construction) 



1 ^ 

 the attractive force of the particles being supposed as the -th 



n 



power of the distance, or inversely as (6 s +/#)*. This 

 equation to the curve makes it always of the order . 



m 



If then the force is inversely as the distance, A N B is a conic 

 hyperbola ; if inversely as the square, it is a curve of the fifth 

 order ; and if directly as the distance, it is a conic parabola ; 

 if inversely as the cube, the curve is a cubic hyperbola. 



The area may next be determined. For this purpose we 



have / y d x = C^ 2ax ~ x *\^. Let 2 (a/ + 6 s ) = h, this 

 J 2(6"+/*)~ 



