the Infinlttjlmal Calculus, 237 



24. But, from o and i being the limits of infinitefimal 

 quantities, it by no means follows that thefe quantities them- 

 felves are chimerical entities. On the contrary, it appears 

 from the definition (article 19) that an infinitely finall quan- 

 tity is the difference between two very efficient quantities, 

 namely, between any auxiliary quantity whatfoever and its 

 limit. 



25, Hence it follows farther, that every infinitely fmall 

 quantity may be confidercd as the difference between two 

 auxiliary quantities, which have the fame third affigned 

 quantity for their limit. For, let A and Y be two different 

 auxiliary quantities, whofe limit is the fame third quantity 

 A. I fay, that A" — Y is an infinitely fmall quantity. For, 

 fince the limit, or ultimate value of A' is A, and that of Y 

 is alfo A, it follows that the ultimate value of A — I^will be 

 A — A, or o. Therefore the limit of A + (A'— Y) \%A\ 



earth mny be reckoned very fmoll, compared with the diftance of the fun, 

 and as relative o, compared with that of any fixed ftar; and converlely, 

 the diftance of the fun is very great, and that of a fixed ftar J, or infinite 

 compared with the earth's diameter. — Again ; fnppofing a rectangle to be 

 continually Uiminiflied by the parallel motion of one of its fides, the law 

 of continuity tells us that it will come to o, or that the moving fide will 

 arrive at the fixed fide of the rectangle ; fo that o here is not ablblute 

 nothing, but it is a line, which however may be looked upon as o, if com- 

 pared with the rectangle ; and the rectangle as 1. or infinite, if compared 

 with the line. In like manner, a point and a rectangle are as o, or no- 

 thing, when refpedtivcly compared with a line and a parallelopiped ; and 

 thefe laft as *, 01 infinite, when ref t ectively compared with the two firft. 

 Thus we fee that the lenfe of the o is always to be determined by the fub- 

 jeet into which it is introduced. See Emerfon's P. S. to the 2d edition of 

 his Algebra, where he replies to the Reviewers, who had mifunderftood 

 the very elegant foluuons of his 73d and 74th problems. 



The ift, 3d, 6th, and 7th corollaries to the former of thefe problems 

 will be ufeful in this place. 



" Cor. 1. If o multiply any finite quantity, the product will be 

 nothing. 



" Cor. 3. If a finite quantity be divided by o, the quotient is infinite. 



* Cor. 6. Adding or fubtracting any finite quantities to or from an in- 

 finite quantity, makes no alteration. 



" Cor. 7. Therefore, in any equation where are fome quantities in- 

 finitely lefs than others, they may be thrown out of the equation. 



W. D. 

 and 



