118 PHILOSOPHICAL TRANSACTIONS. [aNNO 1714. 



)()72, Mr. Collins wrote of it to this effect: " I sent a copy of the Logarith- 

 motechnia to Dr. Barrow at Cambridge, who immediately sent me some papers 

 of Mr. Newton's, from which, and others formerly communicated to the Dr. 

 by the author, it appears that that method was invented some years before by 

 the said Mr. Newton, and applied universally ; so that by means thereof in any 

 proposed curvilinear figure, defined by one or more properties, the quadrature 

 or area of the said curve may be found, accurately, if possible, but if not, yet 

 infinitely near; as also the evolution or length of a curve, the centre of gravity 

 of a figure, solids generated by its rotation, and their superficies may be obtained 

 without any extraction of roots. After Mr. Gregory understood that this 

 method, used by M. Mercator in his Logarithmotechnia, and applied to the 

 quadrature of the hyperbola, and also improved by the said Mr. Gregory him- 

 self, was now made universal, and applied to all sorts of figures, he, after a 

 great deal of close application, discovered the same method; and both Mr. 

 Newton and Mr. Gregory propose to improve it; but the latter does not think 

 it fair to anticipate Mr. Newton, the first inventor." And in another letter to 

 Mr. Oldenburg, to be communicated to M. Leibnitz, dated June 14, 1676, 

 Mr. Collins adds: " Such is the excellency of this method, that being so uni- 

 versal it stops at no difficulty; and I believe that Mr. Gregory and others are 

 of opinon, that whatever was known before of this matter, was but like the 

 glimmering light of the dawn when compared to the brightness of noon-day." 



This tract was first printed by Mr. William Jones in 17 10, who found it 

 among the papers, and in the hand-writing of Mr. John Collins, and collated 

 it with the original, which he afterwards borrowed of Mr. Newton. It contains 

 the above-mentioned general method of analysis, showing how to resolve finite 

 equations into infinite ones, and how by the method of moments to apply equa- 

 tions, both finite and infinite, to the solution of all problems. It begins where 

 Dr. Wallis left off, and founds the method of quadratures on three rules. 



Dr. Wallis published his Arithmetica Infinitorum in the year l655, and by 

 the 59th proposition of that book, if the abscissa of any curvilinear figure be 

 called Xj and m and n be integer numbers, and the ordinates erected at right 



n n *"+" 



angles be X" , the area of the figure shall be -— - x n • And this is assumed 

 by Mr. Newton as the first rule on which he founds his quadrature of curves. 

 Dr. Wallis demonstrated this proposition by steps or induction in many parti- 

 cular cases, and then collected all the propositions into one by a table of the 

 cases. Whereas Mr. Newton reduced all the cases into one, by a power with 

 an indefinite index, and, at the end of his compendium, demonstrated it at 

 once by his method of moments, he being the first who introduced indefinite 



