11 6 PHILOSOPHICAL TRANSACTIONS. • [aNN0 1714. 



An Account of a Book entitled, Commercium Epistolicum Collinii et Aliorum, 

 De Analyst promota ; published by order of the Royal Society, concerning the 

 Dispute between Mr. Leibnitz and Dr. Keill, about the Right to the Invention 

 of the Method of Fluxions, by some called the Differential Method. N° 342, 

 p. 173. 



Several accounts having been published abroad of this Commercium, all of 

 them very imperfect^ it has been thought proper to publish the following 

 account. 



This Commercium consists of several letters and papers in the custody of the 

 Royal Society, here put together in order of time, and either copied, or trans- 

 lated into Latin, from such originals as are mentioned in the title of each letter 

 and paper; and a numerous committee of the Royal Society was appointed to 

 examine the sincerity of the originals, and compare with them the copies so 

 taken. It relates to a general method of resolving finite equations into infinite 

 ones, and applying these equations, both finite and infinite, to the solution of 

 problems by the method of fluxions and moments. We will first give an 

 account of that part of the method which consists in resolving finite equations 

 into infinite ones, and by that means squaring curvilinear figures. By infinite 

 equations are meant such as involve a series of terms converging, or approach- 

 ing to the truth nearer and nearer, in infinitum, so as at length to differ from 

 the truth by less than any given quantity, and if continued in infinitum, to 

 leave no difference. 



Dr. Wallis, in his Opus Arithmeticum, published A. C. 1657, cap. 33, prop. 



68, reduced the fraction by a continual division into the series a + ak -|- 



AR^ + AR^ + AR^ 4- &C. 



Viscount Brounker squared the hyperbola by this series 

 _1__L. J_ 4. J_- 4. -i- -f &c. that is by this, l— ^.-j-^— «. + 4— ' -|- 1 _ . 



-|- &c. conjoining every two terms into one. And the quadrature was published 

 in the Philosophical Transactions for April 1 668. 



Mr. Mercator, soon after, published a demonstration of this quadrature by 

 Dr. Wallis's division, and soon after that, Mr. James Gregory published a 

 geometrical demonstration of it. And these books were a few months after 

 sent by Mr. John Collins to Dr. Barrow at Cambridge, and by Dr. Barrow 

 communicated to Mr. Newton, now Sir Isaac Newton, in June 1669, then a 

 fellow of Trinity College. In return. Dr. Barrow sent to Mr. Collins a tract of 

 Mr. Newton's, entitled Analysis per Aquationes numero terminorum infinitas. 

 This is the first piece published in the Commercium, and contains a general 



