324- darrtot on the Theory 0/ 



Advertifement by the Author. 



Sortie years ago, the author of the following reflections re- 

 duced them to the form in which he now prefents them to 

 the public. He is at prefent charged with bufinefs fo very 

 important as to forbid his refuming his former ftudies ; buty 

 as every circumftance announces that the mathematical fai- 

 ences are about to foar to a new elevation, it is believed that 

 fome advantage may refult from the publication of a memoir 

 in which the theory of the differential calculus is difcuffed at 

 large, and with precifion, and in which are united the different 

 points of view in which that theory has been confidered. 



Suhjec? of the following Dijfertation. 



1. No difeovery ever produced fo happy and fo fudden a 

 revolution in the mathematical fciences as the infinitefimal 

 analyfis ; nor halh any improvement furniflied us with fuch 

 fimple and efficacious methods of arriving at a knowledge of 

 the laws of nature. By decompofing, fo to fpeak, magnitudes 

 into their conftituent elements, that analyfis feems, as it 

 were, to have detected their internal ftru&ure and organifa- 

 tion. But as all extremes elude the cognifance of the fenfes 

 and the powers of the imagination, we have been able to 

 form but an imperfect idea of thofe elements, a Angular kind 

 of beings, which muft fometimes be treated as real quanti- 

 ties, fometimes as abfolute nullities, and which feem, by 

 their equivocal properties, to hold a middle place between 

 magnitude and zero, between exiftence and nothing *. 



Happily this difficulty hath not interrupted the progrefs of 

 the difeovery. There are certain primitive ideas which are 

 always fomewhat clouded in the imagination, but whofe firfic 



'■' I here fpeak in conformity with the vague ideas which we commonly 

 entertain of infinitefimal quantities, when we have not beer, at the pains 

 to examine their nature. But in truth, nothing can be more fimple than: 

 the notion of infiniTefimals. Indeed, to fay that a quantity is infinitely 

 fmall, is precifely to affirm, that it is the difference of two magnitude* 

 which have for their limit the fame third magnitude, and nothing more is 

 meant. The idea, then, of an infinitefimal quantity is not more difficult 

 to conceive than that of a limit ; but it has befides, as is univerfally a'* 

 lowed, the advantage of leading to a much more fimple theory. 



confequences, 



