VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. Hg 



indices of powers into the operations of analysis. Also, by the 108th proposi- 

 tion of the said Arithmetica Infinitorum, and by several other propositions 

 which follow it ; if the ordinate be composed of two or more ordinates, taken 

 with their signs -|- and — , the area will be composed of two or more areas, 

 taken with their signs + and — respectively. And this is assumed by Mr. 

 Newton as the second rule on which he founds his method of quadratures. 



And the third rule is, to reduce fractions and radicals, and the affected roots 

 of equations into converging series, when the quadrature does not otherwise 

 sncceed; and by the first and second rules to square the figures, whose ordi nates 

 are the single terms of the series. Mr. Newton, in his letter to Mr. Olden- 

 burg, dated June 13, 1676, and communicated to Mr. Leibnitz, taught how 

 to reduce any power of any binominal into a converging series, and how by 

 that series to square the curve, whose ordinate is that power. And being desired 

 by Mr. Leibnitz to explain the origin of this theorem, he replied in his letter, 

 dated Oct. 24, 1676, that a little before the plague, which raged in London in 

 the year 1 665, on reading the Arithmetica Infinitorum of Dr. Wallis, and 

 considering how to interpolate the series .a?, ^ — ^^, x — ^ -f- -^v^, x — ^ -|- 



^x' —\x\ &c. he found the area of a circle to be a? — ^ — —^ — ^^- — ^211 --&c. 



o 7 ^ 



And, by pursuing the method of interpolation, he found the theorem above- 

 mentioned ; and by means of this theorem he found the reduction of fractions 

 and surds into converging series, by division and extraction of roots; and 

 then proceeded to the extraction of affected roots. And these reductions are 

 his third rule. 



When Mr. Newton had, in this compendium, explained these three rules, 

 and illustrated them with various examples, he laid down the idea of deducing 

 the area from the ordinate, by considering the area as a nascent quantity, grow- 

 ing or increasing by continual flux, in proportion to the length of the ordinate; 

 and supposing the abscissa to increase uniformly in proportion to time. And 

 from the moments of time he gave the name of moments to the momentaneous 

 increases, or infinitely small parts of the abscissa and area, generated in moments 

 of time. The moment of a line he called a point, in the sense of Cavallerius, 

 though it be not a geometrical point, but a line infinitely short; and the moment 

 of an area, or superficies, he called a line, in the sense of Cavallerius, though 

 it be not a geometrical line, but a superficies infinitely narrow; and when he 

 considered the ordinate as the moment of the area, he understood by it the 

 rectangle under the geometrical ordinate and a moment of the abscissa, though 

 that moment be not always expressed. " Let abd, pi. 4, fig. 2, says he, be any 

 curve, and ahkb a rectangle, whose side ah or kb is unity; and suppose the 

 right line dbk, moving uniformly from ah, to describe the areas abd and ak ; 



