20 DISSERTATION SECOND. [part i, 



ought to preponderate. They had both the skill and talent 

 which fitted them for this, or even for more difficult re- 

 searches. 



The other properties of this curve, those that respect 

 its tangents, its length, its curvature, &c. exercised the 

 ingenuity, not only of the geometers just mentioned, but of 

 Wren, Wallis, Huygens, and, even after the invention of 

 the integral calculus, of Newton, Leibnitz, and Bernoulli. 



Roberval also improved the method of quadratures in- 

 vented by Cavalleri, and extended his solutions to the 

 case, when the powers of the terms in the arithmetical 

 progression of which the sum was to be found were frac- 

 tional ; and Wallis added the case when they were nega- 

 tive. Fermat, who, in his inventive resources, as well as 

 in the correctness of his mathematical taste, 1 yielded to 

 none of his contemporaries, applied the consideration of 

 infinitely small quantities to determine the maxima and 

 minima of the ordinates of curves, as also their tangents. 

 Barrow, somewhat later, did the same in England. After- 

 wards the geometry of infinites fell into the hands of Leib- 

 nitz and Newton, and acquired that new character which 

 marks so distinguished an era in the mathematical sciences. 



2. Algebra. 



It was not from Greece alone that the light proceeded 

 which dispelled the darkness of the middle ages; for, with 

 the first dawn of that light, a mathematical science, of a 

 name and character unknown to the geometers of antiquity, 



i He also was very skilful in the geometrick analysis, and 

 seems to have more thoroughly imirihed the spirit of that i»ge- 

 nious invention than any of the moderns before Halley. 



