54 Carnol ' an, the Theory^ of 



and therefore A^TP = '^j-*, an equation, which, as it eon- 



ti^tins none, other than affigiieii quantities, cannot but be 

 rigorouliy accurate, a. e. i. 



The fame method by analogous rcafbnings, may be ap- 

 plied to the quadrature of all other curves, and it is eafy to 

 extend it to their rectification, as well as to the inveftigatiou 

 of folids. 



Conclujkn^ 



(68.) Thefc few examples may fufficc to convey fome 

 idea of the fpiritof the Infinitelinial Analyfis. In vain will 

 its oppofers object:, that to admit errors, as we do, by em- 

 ploying imperfe61: equations, is to ruin mathematical cer- 

 tainty. For how can thefe errors be dangerous, when we- 

 are pofleflied of infallible methods of eliminating them, 

 and of marks by which we know with certainty that they 

 have difappeared ? Shall we renounce the immenfe advan*' 

 t^ges which this calculus affords, for fear of deviating, for an 

 inflant, from the rigorous procedure of elementary geometry?' 

 Or, fliall we prefer a thorny foot-path in which it is {o dif- 

 ficult to avoid being bew^ildered, to the plain and eafy road by 

 which this analyiis conduiSs us to difcoveries ? Of the former 

 defcription, we fhall find the JNIethod of Limits to be, if we 

 fhould wifh exclufively to employ it. For they who wifli to 

 profcribe the notion of infinitefimal quantities, are reduced 

 to the neccffity of fupplying their place by common Algebra, 

 which prefents numberlefs difficulties, or obliged to ufe the 

 words irifinite ^wA infinitely Jmall, even w^hile they arc de- 

 crying them, and treating as chimerical the very things of 

 which they arc the fymbols. We ufe thefe terms, fay they, 

 Di\ly in a figurative fenfe. But I afk. Whether a figurative 



* Had our ingenious author proceeded here to fubftitute for -P, 

 its value (by^he equation of the parabola) -—^ he woi»Id hare h^i 



Of 



that is, the parabolic fegment JMP, and confeqiiently the tvhofe 

 parabola, U tijo thinls of a rc^angk of thefanif bafe a?id dltitude. — 



And 



