322 Prof. De Morgan on the Early History 



Dr. Jvirin under the name of Philalethes Cantabrigiensis, and, 

 finally, Pemberton himself. Now though Robins again and 

 again affirms Newton's authorship, and though Pemberton ap- 

 pears in the controversy expressly to state his own personal 

 knowledge of Newton's meaning and motives on certain points, 

 and to support Robins in the view he had taken of them, yet 

 Pemberton does not in any way contradict what Robins had 

 advanced as to the authorship in question. Dr. Jurin, though 

 not knowing on what foundation the paper in question is ascribed 

 to Newton, affirms that, whether written by Newton or not, it 

 was no doubt agreeable to his sentiments " as having been some 

 years afterwards republish'd in Latin with his consent and ap- 

 probation." {Rep. of Lett. July 1736, p. 51.) This addition 

 is here made more for completeness* than for any need which 

 there is of it : so far as I can judge, the conclusion of my last 

 paper will remain undisputed. 



My present subject is the early histoiy of the principle of the 

 (bflPerential calculus in England: I mean the principle of infi- 

 nitely small quantities, as distinguished from that of prime and 

 ultimate ratios or of limits. By the time the excitement of the 

 great controversy had subsided, the continental mathematicians 

 were well accustomed to the symbols of Leibnitz in connexion 

 with the infinitesimal principle ; the English to the symbols of 

 Newton in connexion with both principles. But as to how the 

 matter stood in England previous to the controversy, there has 

 not been much inquiry. I here propose to collect notices of 

 some points which the historian of fluxions will find to require 

 his attention. 



Up to the year 1704, and so far as algebraical calculus was 

 concerned, Newton himself used infinitely small quantities; and 

 nothing else in any document yet published. The prime and 

 ultimate ratios, or limits, appear in the Principtia, but are aban- 

 doned in those places in which fluxions are alluded to. I pi'oceed 

 to establish these assertions in detail. 



In Newton's earliest papers published by Rigaud {Hist. Essay 

 on Princ. App. pp. 20-24) the velocities are only diff"erential 

 coefficients : when A. changes from .r to a'-|-o, B changes from y 



twice reprinted in the present eentiu y ; first, in one edition of Berkeley's 

 works piddished by Priestley, London, 1820, 3 vols. 8vo; secondly, in 

 another, published by Tegg, London, 1843, 2 vols. 8vo, edited by G.N. 

 Wright. 



* It shoidd be mentioned that at the time when Robins made the asser- 

 tion, there was living a person who must have known the truth as well as 

 Pemberton. I mean "William Jones, the original possessor of CoUins's 

 papers, the first jmblishcr of some of the most material, a member of 

 the Coram. Epist. Committee, a personal friend of Newton, and the pos- 

 sessor of the best mathematical library of the dav. Jones did not die till 

 1749. 



