skst. i] DISSERTATION SECOND. 25 



did not come up to the perfect measure of geometrical pre- 

 cision. The analysis, thus constituted, necessarily divided 

 itself into two problems ; — the first is, — having given an 

 equation involving two or more variable quantities, to find 

 the equation expressing the relation of the differentials, or 

 infinitely small variations of those quantities ; the second is 

 the converse of this ;■ — having given an equation involving 

 two or more variable quantities, and their differentials, to 

 exterminate the differentials, and so to exhibit the variable 

 quantities in a finite state. This last process is called inte- 

 gration in the language of the differential analysis, and the 

 finite equation obtained is called the integral of the given 

 differential equation. 



Newton proceeded in some respects differently, and so as 

 to preserve his calculus from the imputation of neglecting or 

 throwing away any thing merely because it was small. In- 

 stead of the actual increments of the flowing or variable 

 quantities, he introduced what he called the fluxions of those 

 quantities, — meaning, by fluxions, quantities which had to 

 one another the same ratio which the increments had in their 

 ultimate or evanescent state. He did not reject quantities, 

 therefore, merely because they were so small that he might 

 do so without committing any sensible error, but because he 

 must reject them, in order to commit no error whatsoever. 

 Fluxions were, with him, nothing else than measures of the 

 velocities with which variable or flowing quantities were 

 supposed to be generated, and they might be of any magni- 

 tude, providing they were in the ratio of those velocities, or, 

 which is the same, in the ratio of the nascent or evanescent 

 increments. 1 The fluxions, therefore, and the flowing quan- 



1 " I consider mathematical quantities in this place not as con- 

 sisting of small parts, but as described by a continued motion. 

 Lines are described and thereby generated, not by the apposition 

 of part?, but by the continued motion of points, superfices by the 

 motion of li>!e«, ,, &c. — Quadrature of Curves. Introduction. 



4 



