the Infimtefimal Calculus. 349 



ferent indefinite! v fmall quantities which anfwer to them. 

 Now the consideration of thefe lalt, at lead mentally, is ne- 

 ceffary to the apprehenfiuh of the law of continuity, which 

 determines the required ratio of the evanefcent quantities 

 which are their limits; and confequently it u effentially ne- 

 eerTary to keep them in view, and to characterize them by 

 expreffions which mav prevent them from being confounded. 

 46. Evanefcent quantities, which are the lubjeft of the 

 Infinitefimal Calculus, confidered in this new point of light, 

 are, it is true, entia ratmns, creatures of the undemanding. 

 But this does not hinder them from having mathematical 

 properties, and from being compared together, as well as 

 imaginary quantities in Algebra \ which have no better claim 

 to exiftence. For it is nut more certain, for example, that^ 



60 is = 20 + 40, than that /— a is = /— o X V f 



Now no perfon doubts the accuracy of the refults obtained 

 by the calculus of imaginary quantities; though they be 

 onlv the algebraic forms and emblems of abfurd quantities. 

 With much greater reafon are we prevented from r«jeftipg 

 evanefcent quantities, which are at leaft the limits of real 

 quantities, and are in intimate contaB, fo to fpeak, with their 

 exiftence. What fignifies it, indeed, Whether thefe eva- 

 nefcent quantities are, or are not, chimerical entities, if their 

 ratios be notfuch, and if thefe ratios alone interejlus? When, 

 therefore, we fubjeft infinitefimal quantities to calculation, 



* The author might have added points, lines, furfaces and folids M 

 geometry; for they too are entia variants; which have no exiftence in ex- 

 ternal nature. Pqjnts are merely the terms or limits of lines, lines of 

 furfaces, and fn.taces oTfoiidel juft as the prime ratios of nafcent. and the 

 Ultimate ratio, of emiefcem, quantities, are the limits of thofe ratios, wherf 

 the quantities are confidered as beginning, orcraftog, to be. Yet geometry 

 re ft, on this foundation of abftraQ entities, with perfed fecurity ; for th« 

 truth is, that, without abftraft ideas, fcience, Jhiaiy fo called, cannot 

 txift. And he who can conceive a mathematical point, a mere abftraft 

 toeus, a fomethiog without parts, an entity truly one and indivifible, 

 Which, being a creature of the intellect, entirely eludes the cognizance of 

 every fenfe ; I fay fuch a man will no fooner underftand Newton's doc- 

 trine of prime and ultimate ratios, than he will eftcera it s. legitimate 

 foundation of mathematical reafoning — W. D. 



Vol. VIIL Z a we 



