the Infoiilejinial Calculus. 339 



jm equation, the firft term of which contains only given 

 quantities, and the fecond only arbitrary quantities, which 



and that the new equation fhould be, 



( s ? , \ 4. ( _L 2j 2 j> : x + ay — xy \ _ 



\ y a — A'' \ _y (a — x) (za — ix — a) ' 



fTP y_\ I T'T 2y 2 . (RZ: MZ) +aR Z-xRZ\_ 



\y n — x) \ y (a-x).(ia — 2.x — MZ) )~ * 



and, in this laft manner, 1 fhall, in Future, take the liberty to write it. 

 If the value of a — x had been taken from the accurate equation 



(in article 0) — = — . r , we fhould then have had 



y 2a — 2x — ■ x 



2y , {a—x) = (2yi*y + y'f + yx 2 ) : x; 

 fo that, in rigid jlritlnefs, 'he new equation fhould be, 



/ s y \ /J (2yy + y'f- + y- 2 ) : ■<■ + ay — xy \_ 



\ y a — x) \ y {a — x) . (za — 2x — x) )~ 



Now, though j 2 and x 2 be infinitely lefs Jtill than their infinitely 

 fmall roots y and £, and confequ^ntly^y 2 -f yx 2 infinitely lefs than 

 2y y, and thus may be lately neglected ; yet it would not, perhaps, 

 have been amife, if the 'luihor, while he was exhibiting all his quan- 

 titie , had brought them alio into view. 



Some readers may be furpriled at the mention of quantities infi- 

 nitely lefs than infinitely fmall ones. But their wonder will ceafe 

 when they recollect, That if ~ny integer, or any fraction, be multi- 

 plied by a fraction, the product will be lefs than the multiplicand, 

 and will, in fact, be only luch a p^rt of the multiplicand, as the mul^ 

 tiplier is of unity. Thus, if '0000000000001 (prefixing cyphers ad 

 infinitum) be multiplied by any other interminably or inconceivably 

 fmall fraction, the product will be only fuch an inconceivably 

 fmail part of the already inconceivably fmall multiplicand, as is 

 expreffed by the inconceivably Imall multiplier): in other words, the 

 product (relatively to our conceptions} may be laid to be infinitely 

 le? than the infinitely fmall multiplicand. In like manner, xy may 

 reprelcnr a rectangle, whofe breadth x is infinitely fmall compared 

 wi'h it£ length y, which may be any finite line, or it may even bean 

 indefinitely or infinitely great line. But now fuppofe y alfo to be in- 

 finitely fmall, or to become y; then it is eafy to fee, that this fecond 

 rectangle xy, both whofe dimenfions are infinitely fmall, will be in- 

 finitely lefs than the firft rectangle xy, which has only one of its 

 dimenfions infinitely fmall. Thus alfo the fquares x 2 and y 1 will 

 be infinitely lefs than their infinitely fmall roots x and y, and the 

 cubis x'> and y s , than the fquares x z and y 2 , Sec. — Ncque enim novit 

 natura limit em. 



Were this the proper place, we might recommend thefe and many 

 analogous confiderations, both mathematical and metaph} fical, to 

 the ferious attention of certain gentlemen, who, without abating a 

 tittle from their high pretentions to accurate reafoning, Te uple not 

 to tell us. in very general terms, " that they cannot believe any 

 (thing which they cannot conceive or comprehend." — W. D. 



laft 



