6 DISSERTATION SECOND. [fart 11. 



series, the terms of which were all known. He farther 

 remarked, that, in the first of these series, the equation to 

 the circle itself might be introduced, and would occupy 

 the middle place between the first and second terms of 

 the series, or between an equation to a straight line and 

 an equation to the common parabola. He concluded, 

 therefore, that if, in the second series, he could interpo- 

 late a term in the middle, between its first and second 

 terms, this term must necessarily be no other than the area 

 of the circle. But when he proceeded to pursue this very 

 refined and philosophical idea, he was not so fortunate ; and 

 his attempt toward the requisite interpolation, though it did 

 not entirely fail, and made known a curious property of the 

 area of the circle, did not lead to an indefinite quadrature of 

 that curve. 1 Newton was much more judicious and success- 

 ful in his attempt. Proceeding on the same general princi- 

 ple with Wallis, as he himself tells us, the simple view 

 which he took of the areas already computed, and of the 

 terms of which each consisted, enabled him to discover 

 the law which was common to them all, and under which 

 the expression for the area of the circle, as well as of in 

 numerable other curves, must needs be comprehended. 

 In the case of the circle, as in all those where a frac- 

 tional exponent appeared, the area was exhibited in the 

 form of an infinite series. 



The problem of the quadrature of the circle, and of so 

 many other curves, being thus resolved, Newton immedi- 

 ately remarked, that the law of these series was, with a 



1 The interpolation of Wallis failed, because he did not em- 

 ploy literal or general exponents. His theorem, expressing the 

 area of the entire circle by a fraction, of which the numerator 

 and denominator are each the continued product of a certain 

 series of numbers, is a remarkable anticipation of some of Eu- 

 ler's discoveries, Calc. bit. Tom. I. cap. 8. 



