30 LIMITS AND FLUXIONS 



by an equation, such ^i?, x^—2a^y-\- zzx—yyx-\-zyy — 

 z'^—o. ''If the body A, with the velocity /, 

 describe. the infinitely Httle line o in one moment, 

 in the same moment B, with the velocity q, will 

 describe the line oq / /," and the body C, with the 

 velocity r, will describe the line or / /. So that, 

 if the described lines be x, y and z "in one 

 moment," they will be x-\-o, y-\-oq / /, z + or j p 

 '*in the next." He finds that the relation of the 

 velocities /, q, r, in the above example, is '^pxx + 

 pzz —pyy — 2aaq — 2yxq + 2zyq + 2zxr -\-yyr •— ^zzr = o. 

 In proving his rules for differentiation, Newton 

 divides by o^ and in the resulting expression observes 

 that ''those terms in which o is, are infinitely less 

 than those in which it is not. Therefore, blotting 

 them out, there rests " the relation sought. The 

 notation by dots, " pricked letters," occurs on a leaf, 

 dated May 20, 1665, which has never been printed. ^ 



49. It is evident that Newton permitted twenty- 

 eight years to pass between the time of his first 

 researches on fluxions and 1693, the date when the 

 earliest printed account of his notation of fluxions 

 appeared from his pen in the Latin edition of Wallis's 

 Algebra. Moments and fluxions are mentioned in 

 his Principia^ as has been shown by our extracts. 



50. Of importance in the interpretation of the 

 meanings of "moment" in the second edition of 



^ S. P. Kigaud, op. cif.f Appendix, p. 23. Consult also the remarks 

 on this passage macie by CJ, Ene^tròm in Bibliotheca maihe>natica, 

 3. F., Bd. II, Leipzig, 1910-191 1, p. 276, and Bd. 12, 1911-1912, 

 p. 268, and by A. Wiiiing in Bd, 12, pp. 56-60. See also A catalogne 

 of the l'ortsinouth collection of books aìid papers^ written by or belonging 

 to Isaac Newton, Cambridge, 1889. 



