94 LIMITS AND FLUXIONS 



certainly open to the logicai objection raised by 

 Berkeley. Eighteenth-century mathematicians did 

 not attach due importance to this point. 



115. The existence of infinitesimals (infinitely 

 small quantities) was denied by Berkeley, but, it 

 would seem, not denied by Jurin and Walton. AH 

 three finally abjured the philosophy which permits 

 their being dropped because so small. It is well 

 known that many mathematicians of prominence 

 have believed in the reality of such quantities. 

 From Leibniz to Lagrange ali Continental writers 

 of note used them. Lagrange headed a small 

 school that was opposed to them, when he pub- 

 lished his Fonctions analytiques. There followed 

 a reaction against Lagrange. De Morgan once 

 remarked : '*Duhamel, Navier, Cournot, are pure 

 infinitesimalists. Some of them say an infinitely 

 small quantity is one which may be made as small 

 as you please. This is an evasion ; but they do not 

 mean that dx is finite. . . . By-the-way, Poisson 

 was a believer in the rea/ùy of infinitely small 

 quantities — as I am."^ 



"... For myself, I am now fixed in the faith 

 of the subjective reality of infinitesinial quantity. But 

 what an infinitely small quantity is, I know no 

 more than I know what a straight line is ; but I 

 know it is ; and there I stop short. But I do not 

 believe in objectively realised infinitesimals." 



^ Life of Sir William Rowan Hamilton, by Robert P. Graves, 

 voi. iii, pp. 572, 580. Consult also De Morgan's article, "'On 

 Infinity ; and on the Sign of Equality," in Trans, of the Cambridge 

 Phil. Society^ voi. xi, Cambridge, 187 1 [read May 16, 1864]. 



