4^ Carn'it 6nih Tbeory of 



the dilTciTT^tialF of tbefccond order, and hence will refult \\\^ 

 differentials of the third order; from the differ entiatio7i of thefc 

 lad will refult thofe of the fo^irth order, and fo on. Thus 

 eiddj, or d^y will he the third difference of t and ddddy, or 

 dy, the differential of the fourth order, he. Now, after 

 what has been faid on the generation of diffferentials of the 

 firff: and fccond orders, there will be no diflUculiy in compre- 

 hending the production of the Superior orders. I fliall there- 

 fore only obferve, that it confills in attributing, for each new 

 order, a new auxiliary value to each of the variable quantities, 

 and fuch, that not only each of ithefe new values may differ 

 infinitely little from that which precedes it, but that the fame 

 thing niav take place between their dilTerentials', the diflTeren- 

 tials of their differentials, and fo forth. 



52. To differentiate a quantity is to affiign its diflferential ; 

 tliat is, W X, for example, be anv fun<Sli6n whatever of ;v, to 

 ditlerentiateit, is to aflign thequanlity by which that fun6lion 

 will be increafcd, by fuppoling the increment of x to be dx\ 



To integrate, or to fu?n, a differential, on the contrary, is to 

 return from that differential to the quantity which produced 

 it ; and this laft quantity is called the Integral or S/iw of the 

 propofed differential *. For example, x is the integral or funi 

 of J.r, and to integrate, or to fum, dx is nothing more than 

 to ailign that quantity, ^, which is its fum or integral. 



We have feen that, in calculation, the differential of a 

 qiiantity is expreff^ed by that fame quantity, with the cha- 

 radler d prefixed. Reciprocally it has been agreed to exprefs 

 the integral or funi of any differential by the fame differen- 

 tial, preceded by the chara6tery; that is, y'J.v, for example, 

 lignifies the fum o^ dx; fo that we have evidently.r '=y<:/.r. 



53. The Calculi called Differential and Integral, confti- 

 tute the art of difcovering any ratios and relations whatever, 

 cxiffing between propofed quantities, by the help of their 

 differentials. The name Differential Calculus is properly- 

 applied to the art of invelligating the ratios, or relations of 

 differential quantities, and afterwards to eliminate them by 

 the ordinary rules of Algebra ; and the name of Integral 

 Calculus to the art of integrating or eliminating thefe fame 



* Or what \Ke call the Fluent of the propofed Ftr/xm* — W, D. 



• differential 



