32 LIMITS AND FLUXIONS 



approximation. In resolving Ouestions or investi- 

 gating truths I use ali sorts of approximations 

 w^'' 1 think will create no error in the conclusion 

 and neglect to write down the letter o, and this do 

 for making dispatch. But vvhere x, j>, j/, y are put 

 for fluxions without the letter o understood to make 

 them infinitely little quantities they never signify 

 differences. The great Mathematician ^ therefore 

 acts unskilfuUy in comparing prickt letters with 

 the marks dx and dy^ those being quantities of a 

 different kind. " 



Remarks 



53. The extracts from Newton's writings demon- 

 strate the following : — 



(i) At first Newton used infinitesimals (infinitely 

 small quantities), as did Leibniz and other mathe- 

 maticians of that age. As early as 1665, when 

 Newton was a young man of twenty-three, he used 

 them and speaks of " blotting them out."^ He 

 uses infinitesimals in the Principia of 1687^ and in 

 his account of the quadrature of curves in Wallis's 

 Algebra of 1693, where Newton speaks of himself 

 in the third person.^ It is worthy of emphasis, in 

 contrast to Leibniz, that Newton uses only infinitesi- 

 mals of the first order. Moreover, as De Morgan 

 remarked long ago,^ ''the early distinction between 

 the systems of the two is this, that Newton, 

 holding to the conception of the velocity or fluxion, 



' John Bernoulli. See Edleston, op. cit., p. 171. ~ See our § 48. 



3 See our §§ io, 13, 16, 18. ^ See our §§ 21, 26. 



^ De Morgan, Philosophtcal Alagaztne, 4 S., voi. iv, 1852, p. 324. 



