skct. i.] DISSERTATION SECOND. 23 



near, that is, nearer than within any assigned difference. 1 

 There was, however, as yet, no calculus adapted to these 

 researches, that is, no general method of reasoning by help 

 of arbitrary symbols. 



But we must go back a step, in point of time, if we would 

 trace accurately the history of this last improvement. Des- 

 cartes, as has been shown in the former part of this outline, 

 made a great revolution in the mathematical sciences, by 

 applying algebra to the geometry of curves ; or, more gene- 

 rally, by applying it to express the relations of variable quan- 

 tity. This added infinitely to the value of the algebraic 

 analysis, and to the extent of its investigations. The same 

 great mathematician had observed the advantage that would 

 be gained in the geometry of curves, by considering the 

 variable quantities in one state of an equation as differing in- 

 finitely little from the corresponding quantities in another 

 state of the same equation. By means grounded on this he 

 had attempted to draw tangents to curves, and to determine 

 their curvature ; but it is seldom the destination of Nature 

 that a new discovery should be begun and perfected by the 

 same individual ; and, in these attempts, though Descartes 

 did not entirely fail, he cannot be considered as having been 

 successful. 3 



At last came the two discoverers, Newton and Leibnitz, 

 who completely lifted up the veil which their predecessors 

 had been endeavouring to draw aside. They plainly saw, 

 as Descartes indeed had done in part, that the infinitely 

 small variations of the ordinate and absciss are closely con- 

 nected with many properties of the curve, which have but 

 a very remote dependence on the ordinates and abscissas 

 themselves. Hence they inferred, that, to obtain an equa- 

 tion expressing the relations of these variations to one an- 



1 Note D, at the end. 2 Dissert. Second, Part I. p. 18. 



