g4^ Carno't on the Theory of 



For example, in the cafe before adduced, as long as Ro 



does not coincide with MP, the fraction — R -y is greater than 



TP 



; nor do thefe fractions become equal, till MZ and RZ 



y 



MZ 

 are reduced to nothing. It is true, that then, -^tt is as much' 



TP o . 



tqua! to any other quantity as to — ; becaufe — is a 



quantity altogether arbitrary ; but, among all the different 



M7 TP . 



values which ^- may be fuppofed to have, - — is the only 



one which is fubje&ed to the law of continuity and deter- 

 mined by it. For, if a curve were conftru&ed, whofe abfeiffe 



doclrire of prime and ultimate ratios, and of the whole method of 

 fluxions. That great man, in the concluding fcholium of Sett. i. B. i. 

 of the Princ'tpia, has thefe words': " Ohjeilio efl, &c." " It may be ob- 

 jected that evane'feent quantities have no ultimare proportion ; for that, 

 before they vanifli, that proportion cannot be the laft, and after they have 

 vanifhed, it is nothing. But, by the fame argument, when amoving body 

 flops at a certain place, it may be faid that it has no ultimate velocity, for 

 that, before the body reaches that place, the velocity is not the ultimate 

 velocity, and when it has reached the place, the velocity is nothing. The 

 anfvver is eafy ; for by the ultimate velocity is meant the velocity of the 

 body, neither before it reaches its laft place, nor after it has reached it, 

 but that velocity with which it aftually reaches it ; that is, the very velo- 

 city with which the body attains its laft place, and comes to reft. In like 

 manner, by the ultimate ratio of evanefcent quantities is to be underftood, 

 the ratio of thofe quantities, neither before, nor after they vanifli, but /Af 

 ratio with tvhich they •vanijb. And thus alfo the prime ratio of nafcent 

 quantities is that ratio with which they firft ftart into exiftence," &c. 

 Though but a humble and diftant follower of Newton, 

 §>uem lotige fequor, et 'vrjiigio. promts adoro, 

 I fee nothing that could hinder him from " thinking he could exprefs " 

 the fundamental principle of this doftrine by fuch language. For my 

 own part, I muft frankly fay, that the fcholium whence it is quoted, con- 

 veys, or fuggefts, that principle more clearly to my mind, than all that out 

 ingenious author and others have written on the fubjett. But we do not 

 all fee things, with equal clearnefs, in the fame point of view. Some of 

 my fuperiors in genius and knowledge have a different opinion of that 

 fcholium, and of the reft of Newton's fluxionary doctrine, as delivered by 

 himfelf, and even as explained Ly Ditton, Simpfon, and others. To fuch 

 I would recommend the prefent perfpic'uous tratt. — W. D. 



Was 



