328 Prof. Dc Morgan on the Early Histonj 



except slightly, in one scholium, the name of Leibnitz does not 

 appear. 



De Moivre took his idea of fluxions direct from the Principia, 

 and {Phil. Trans. 1695, No. 216) uses the infinitely small mo- 

 ments. His fluxion of an area, for instance, is an infinitely 

 small rectangle. Halley, in his paper on logarithms in the same 

 number, uses infinitely small rntiunculce and different iola; in a 

 manner which must have astounded every beginner who ever 

 saw the reprint in Sherwin's Logarithms. 



Of the elementary writers, Harris and Hayes (1702 and 1704), 

 I have spoken in the Companion to the Almanac for 1852 (p. 15). 

 Both used infinitely small quantities. Hayes adopts infinitesi- 

 mals of infinitesimals : and as he happened to publish his work 

 in the very year in which Newton declared himself against all 

 infinitely small quantities, his book was neglected, in spite of its 

 merit. His list of the writers to whom he is indebted includes 

 Wallis, Barrow, Newton, Leibnitz, De UHopital, the Bernoullis, 

 Craig, Cheyuc, Gregory, Tschirnhauss, De IMoivrc, Fatio, "\''a- 

 rignon, Nicuv\ entiit, Carre : those in Italics being those to whom 

 he considers himself particularly indebted. 



Cotes, in 1701 (Edleston, p. 196), makes the term fluxion 

 interchangeable with differentiola , and considers it as infinitely 

 small. Cotes was then an undergraduate, and, as is pretty clear 

 from his enumeration of results, a reader of transactions and 

 other original papers, and also, perhaps, of De L'Hopital. 



Cheyne, in his Fluxionum Methodus inversa, London, 1703, 

 4to (pp. iv -r 128), refers to and follows Newton's contribution to 

 Wallis. By such suppositions as x=\, he shows that he inter- 

 prets this symbol, taken apart from o, in the same manner as 

 Newton. There is an additional leaf, published in 1704, entitled 

 Addenda et adnotanda in libro Georgii Cheyncei, which I have 

 never seen. De Moivre in his Animadversiones in D. Georgii 

 Cheijnai Tractatum .... London, 1704, 8vo (pp. xiv-fl29) fol- 

 lows the plan of Cheyne. 



Lastly, Fatio de Duillicr, in his memorable paper on the line 

 of shortest descent, Linece brevissimi descensus investigatio .... 

 London, 1699, 4to (pp. 24 and plate), uses fluxions as infinitely 

 small quantities. 



Thus it appears that, up to 1704, all the writers who used 

 the new calculus, used infinitely small quantities ; that all M'ho 

 used x, except Newton and Cheyne, interpreted it as an infinitely 

 small quantity; and that Newton himself, though he never 

 varied in his meaning of ,r, used xo and not x, admitting that 

 he sometimes omitted o, which has the force of dt. 



In 1704, Newton, in the Qtiadratura Curvarum, renounced 

 and abjured the infinitely small quantity ; but he did it in a 



