26 DISSERTATION SECOND. [past h. 



tities or fluents of Newton correspond to the differentials and 

 the sums or integrals of Leibnitz ; and though the symbols 

 which denote fluxions are different from those used to ex- 

 press differentials, they answer precisely the same purpose. 

 The fluxionary and differential calculus may therefore be 

 considered as two modifications of one general method, 

 aptly distinguished by the name of the infinitesimal analysis. 



By the introduction of this analysis, the domain of the 

 mathematical sciences was incredibly enlarged in every di- 

 rection. The great improvement which Descartes had 

 made by the application of algebraic equations to define 

 the nature of curve lines was now rendered much more effi- 

 cient, and carried far beyond its original boundaries. From 

 the equation of the curve the new analysis could deduce the 

 properties of the tangents, and, what was much more diffi- 

 cult, could go back from the properties of the tangents to the 

 equation of the curve. From the same equation it was able 

 to determine the curvature at every point 5 it could measure 

 the length of any portion of the curve or the area corres- 

 ponding to it. Nor was it only to algebraic curves that those 

 applications of the calculus extended, but to curves transcen- 

 dental and mechanical, as in the instances of the catenaria, 

 the cycloid, the elastic curve, and many other.*. The same 

 sort of research could be applied to curve surfeces described 

 according to any given law, and also to the solids contained 

 by them. 



The problems which relate to the maxima and minima, 

 or the greatest and least values of variable quantities, are 

 among the most interesting in the mathematics ; they are 

 connected with the highest attainments of wisdom and the 

 greatest exertions of power ; and seem like so many im- 

 moveable columns erected in the infinity of space, to mark 

 the eternal boundary which separates the regions of possibi- 

 lity and impossibility from one another. For the solution of 



