324 Pi'of. De Morgan on the Early History 



of the meaning of the second edition, when we compare it with 

 the first. Newton, writing to Keill in May, 1714, says "moments 

 are infinitely little parts." And further, " wherever prickt letters 

 represent moments and are without the letter o, this letter is 

 always understood." (Edleston, p. 176.) 



The treatise on the Quadrature of Curves, written* long before 

 it was published, made its appearance, to a sufficient extent for 

 our purpose, in 1693, in the Latin edition of Wallis's Algebra 

 ( Op. vol. ii. pp. 390-396) . That this was substantially a contribu-- 

 tion of Newton's, was obvious from the beginning : but it was 

 not known, until Rigaud found it [Hist. Essay on Princ. p. 22), 

 that a note made by Wallis in his own copy points out from 

 p. 390 line 18 to p. 396 line 19 as being Newton's own ivorcls 

 [the first word of all, darissimus, no doubt excepted, as also the 

 parenthetic description of David Gregory, nunc mens coUeya dig- 

 nissimus] . The famous proposition Data aquatione quotcunque 

 &c. occurs in this extract ; and is of course repeated in the Qua- 

 dratura Ciirvarum, at the end of the Optics, in 1704. The two 

 publications of the proposition agree sentence for sentence, and 

 clause for clause, but not word for word. The following com- 

 parisons will prove my assertion, both as to the first adoption, 

 and suljsequent abandonment, of uncloaked infinitesimals. 



1693. 1704. 



"quantitas infinite parva .... ''quantitas admodum parva. . . 

 Et ha; quantitatcs proximo tern- Et si quantitates fluentes jam 

 poris momento per accessum sunt z, y, et x, ha3 post mo- 

 incrementorum momentane- mentum temporis inerementis 

 Drum evadent z-\-o's . . . . suis o'z, o'y, ox auct?e, evadent 



z + oz .... 



" Terminos multiplicatos per " Minuatur quantitas o in 



tanquam infinite parvos dele, infinitum, et neglectis terminis 

 et manebit tequatio . . . ." cvanescentibus restabit. , . ." 



It will now, I think, be very clear that Newton commenced 

 with the infinitesimal system in as absolute a form as did Leib- 

 nitz, so far as infinitely small quantities of the first order are 

 concerned. Further than these he did not go ; and the early 

 distinction between the systems of the two is this, that Newton, 

 holding to the conception of the velocity or fluxion, used the in- 

 finitely sinallf increment as a means of determining it; while, 



* " The book of Quadratures is ancient, many things being cited out of 

 it by me in my Letter of 24 Octob 1676." (Newton to Keill; Edleston, 

 p. 176.) 



+ It is also to be noticed that Leibnitz and the BeruouUis demand the 

 method of exhaustions, or something equivalent, whenever an objection is 

 raised to infinitesimals. They do not face a human enemy with small shot ; 

 they only use it to kill game. 



