the Infinitefimal Calculus. 35 l 



47 . The theory which has been thus expounded eafily 

 funnn.es anfwers to all the objeaions which have been made 

 againft the Infinitefimal Analyfis, the principle of winch te- 

 veral geometricians have thought faulty, and capable ol lead- 

 in. to erroneous conclufions. But thofe gentlemen nave been 

 overwhelmed, if the expreffion mav be allowed with a mul- 

 titude of prodigies, and with the fplendour of the numerous 

 truths wherewith this principle has teemed. 



Thefe objeaions may be reduced to this. The quantities 

 faid to be infinitely fmall, are either abfolute nulht.es, or 

 they are not; for it is ridiculous to fupppfe, that there .ex.lt 

 entities, which hold a middle place between quantity and 

 nullity Now if thev are abfolute nullities thc.r companion 

 leads to nothing, for'the ratio of o to o is no more a than it 

 is b, or any other quantity u hatever. If, on the ether hand, 

 thev are not nullities, but real quantities, thev cannot with, 

 out error be neglecled, as the rules of the Lniimtef.mal Ana- 



lvfis prefcribe. ■ . . 



The anfwer is fimple. So far from its not being logical to 

 confider infinitely fmall quantities, either as real beings or 

 as nothings, they mav, on the contrary, be treated at pjea- 

 fure, either as nullities or as true quantities. For they who 

 wifh to confider them as nullities, may anfwer, that what they 

 call infinitely fmall quantities are not any nullities taken at 

 random, but nullities affigned by the law of continuity jrtuch 

 determines their relation*; that among all the relations or 

 which thefe quantities are fufceptible as nulht.es, they only 

 confider thofe which are determined by this law of conti- 

 nuity ; and, in a word, that thefe relations are not vague and 

 arbitrary, becaufe the law of continuity does not affign leveral 

 different relations between the differentials, tor example, of 

 the abfciffe and ordinate of a curve, when thefe ditkrenlials 

 Tanifl), but one only, which is that of the fubtangent to the 



ordinate. m . 



On the other hand, they who regard infinitely fmall quan- 

 tities as true ones, may anlwcr, that what they call an infi- 

 nitely fmall quantity is one which is arbitrary and independant 

 on the qualities propofed ; that, therefore, without luppofing 

 * Set the Note on article 22. 



