of Infinitesimals in England. 333 



to I/ + oq-r-p, the velocities being p and q. Those terms in 

 which remains are " infinitely less " than those in which it is 

 not, and are therefore " blotted out." And " those terms also 

 vanish in which o still remains, because they are infinitely little." 



The Method of Fluxions, translated by Colson* from Newton's 

 Latin, and published in 1736, written, it is supposed, at some 

 time in the period 1671-1676, is also strictly infijiitesimal. The 

 is now the infinitely small increment of the time, and the 

 fluxion, conceived and defined as a velocity, is its coefficient. 

 Thus X, y, &c. become x + xo, y + i/o, &c. : "but whereas o is sup- 

 posed to be infinitely little, that it may represent the Moments 

 of Quantities ; the Terms that are multiply'd by it will be nothing 

 in respect of the rest." (p. 25.) 



In the fiii'st edition of the Principia (1687) the description of 

 the fluxions is founded on infinitesimals, and in the second (1713) 

 this foundation is somewhat altered. In the first, moments are 

 infinitely small quantities ; in the second, it is not clear what else 

 they are. As in the following extract from the fii'st edition, with 

 its substitute in the second : — 



First Ed. (Book ii. Lemma ii.). Second Edition (ditto). 



"Cave tamenintellexerispar- "Cave tamen intellexeris par- 



ticulas finitas. Momenta, quam ticulas finitas. ParticulcB finit<e 



jirimumfinitce sunt magnitudinis, non sunt momenta sed quanti- 



desinunt esse momenta. Finiri tales ipsa ex momentis genitce. 



enim repugnat aliquatenus pei'- Intelligendasuntprincipiajam- 



petuo coram incremento vel de- jam nascentia finitarum mag- 



cremento. Intelligenda sunt nitudinum." 

 principia jamjam nascentia fini- 

 tarum magnitudinum." 



Through the difficulty of the phrases in both extracts this 

 much distinctly appears, that in the first edition the moments, 

 or momentaneous increments, are infinitely small quantities : 

 and this is what I assert. In the celebrated scholium which 

 follows, the first edition states that Leibnitz's system hardly 

 differed from Newton's except in words and symbols. The 

 second edition adds et Idea generationis quantitatum. But this 

 addition, as well as the alterations of phrase which are to sup- 

 port it, was made while the controversy was raging. It is only 

 from a subsequent and private source that we can be well assured 



* There is no doubt this work is Newton's : but, independently of in- 

 ternal evidence, if Colson were now eharjjed with a fraud, there would be 

 nothin}; to argue from, except the consent of Jones and Pemberton, as 

 given by silence. The work, dedicated to the former, purports to be the 

 one which would have been published by the latter, if Newton's death had 

 not prevented. 



Y2 



