24 NEWTON S PRINCIPIA. 



fluxions or differentials of all quantities, and, conversely, 

 lead to those for investigating or finding the fluents or 

 integrals of fluxional or differential expressions. 



A rectangle A M being generated by the side P M 

 moving along A P while the side N M moves along A N, 

 the movement or fluxion or differential of A M, or of 

 A P x P M, is P S + M O, part of the gnomon P S 



+ S 0, because the rectangle M V is evanescent compared 

 with the other two, and is to be rejected. Therefore the 

 differential ofAP x PM = PM x PT + NM 

 x N O, orPM x PT + AP x MR. Calling A P 

 = x, and PM = y, and P T = dx 9 and MR = dy, 

 we have the differential of xy = x dy -\-ydx. But if 

 the figure be a square, and A P = P M, or x = y, then 

 the differential is 2 x d x. So if we would find the diffe 

 rential of a parallelepiped whose sides are x, y, and z, 

 we shall in like manner find that it is x y d .z + x z d y 

 -f y z dx ; if # = 2, then it is 2 y x d x + x* d y ; and 

 if x = y = z } or the figure be a cube, it is 3 x 2 d x. 

 From hence, although the geometrical analogy serves us 

 no further (as there are only three dimensions in figures), 

 we derive by analogy the rule that the differential of 

 x m is m x m ~ l d x. Also there is no dimension of figure less 

 than unity ; but by the same analogy we obtain the dif 

 ferential of x~ m , or , namely, mx~ m - l dx, or r^, 



x x 



