48 LIMITS AND FLUXIONS 



66. Newton's Analysis per cequationes numero 

 terminonim infinitas, which was sent on July 31, 

 1669, through Barrow to Collins, and which was 

 first published at London in 171 1, was reprinted in 

 the Commercium Epistolicuni. In this Analysis in- 

 finitely small quantities are used repeatedly, but the 

 word ' ' fluxion " and the fluxional notation do not 

 occur. In a letter to H. Sloane, who was then 

 Secretary of the Royal Society of London, written 

 in answer to a letter of Leibniz dated March 4, 17 1 1, 

 John Keill, professor of astronomy at Oxford, re- 

 counts the achievements of Isaac Barrow and James 

 Gregory, and says : " If in place of the letter 0, 

 which represents an infinitely small quantity in 

 James Gregory's Geometrice pars uìiiversaìis (1667), 

 or in place of the letters a ox e which Barrow em- 

 ploys for the same thing, we take the x or y of 

 Newton or the dx or dy of Leibniz, we arrive at the 

 formulas of fluxions or of the differential calculus."^ 

 Thus Keill, the would-be great champion of Newton, 

 instead of warning the reader against confusing 

 differentials and fluxions, himself comes dangerously 

 dose to conveying the erroneous idea that x and y 

 are infinitely small, the same as dx and dy. He 

 Comes so near to this as to be guilty of lack of 

 caution, if not of inaccuracy. 



More serious is a statement further on. The en- 



^ " Nam si prò Litera 0, quae in Jacobi Gregorii Parte Matheseos Uni- 

 versali quantitaleni infinite parvam repraesentat ; aut prò Literis a vel e 

 quas ad eandem designandam adhibet Barrovius ; ponamus x vel y 

 Newtoni, vel dx seu dy Leibnitii, in P'ormulas Fliixionum vel Calculi 

 Differentialis incidemus " (p. 112). 



