326 Biographical Account of [Nov. 



and by him they were made known to various other British 

 mathematicians. He "likewise entered into a correspondence 

 with Collins and Oldenburg, and by them was induced to write 

 several lung letters to Mr. Leibnitz, in which he gave an histo- 

 rical detail of the way that he was led to some of his most con- 

 siderable discoveries. All these letters were afterwards published 

 in the Commerclum Epislolicum. The correspondence also 

 between James Gregory and Collins, published in the same 

 book,, throws considerable light upon the order and time of 

 Newton's mathematical discoveries. One of his first discoveries 

 struck him while perusing Wallis's Arithmetic of Infinites, about 

 the year 1663. Wallis had shown the method of finding the 

 quadrature of all curves, the ordinates of which are expressed 

 by (1 — x 2 )" 1 , x being the abscissa, supposing m a whole 

 number, either positive, or negative, or zero ; and that when 

 m was respectively 0, 1, 2, 3, 4, &c, the areas corresponding to 

 the abscissa x were respectively x ; x — 4-x 3 ; x — fx 3 + -|-x 5 ; 

 x — fx 3 + f x 5 — \x~' ; &e. ; and he showed, that if a number 

 could be interpolated between x and x — ±x 3 in the second 

 series, corresponding to the interpolation of -A- in the first series 

 between and 1, that this number would represent the quadra- 

 ture of the circle. But Wallis could not succeed in making this 

 interpolation ; it was left for one of the first steps of Newton, in 

 his mathematical career. Newton arranged the terms of the 

 second series given above, under each other in order, and exa- 

 mined them as follows : — 



x 



X — 5-JC 3 + %x* 



X — Ix 5 + fx 5 — |x 7 



X — AX 3 -f- fx b — -}X 7 + ±X 9 



On considering this table, Newton observed, that the first terms 

 are all x ; that the signs are alternately positive and negative ; 

 that the powers of x increase by the odd numbers ; that the co- 

 efficient of the first term is 1 ; that the co-efficient of all 

 the other terms are fractions ; that the denominators of these 

 fractions are always the indices of x, in the respective 

 terms ; that the numerators in the second terms are the 

 ordinary numbers ; in the third terms, the triangular num- 

 bers : in the fourth terms, the pyramidal numbers ; &c. These 

 observations made him master of the laws that regulated the 

 whole of the series. Hence he concluded, that having to 

 devetope in general (I — x s ) m , the series of numerators for the 

 respective fractions in the different terms must be 1 ; m ; 



»M . m — 1 m . m - 1 . in — 2 r . . 



- 1~-| — ') ■ 2 3 ) & c -> * or these are the expressions 



