Page:The Analyst; or, a Discourse Addressed to an Infidel Mathematician.djvu/56

scure and illogical. But if we fairly delineate the Area and its Increment, and divide the latter into two Parts BCFD and CFH^[1] and proceed regularly by Equations between the algebraical and geometrical Quantities, the reaſon of the thing will plainly appear. For as as xⁿ is equal to the Area ABC ſo is the Increment of xⁿ equal to the Increment of the Area, i. e. to BDHC; that is, to ſay, $nox^{n-1}+{\frac {nn-n}{2}}oox^{n-2}+\&c.=BDFC+CFH$ . And only the firſt Members, on each Side of the Equation being retained, $nox^{n-1}=BDFC$: And dividing both Sides by o or BD, we ſhall get $nx^{n-1}=BC$. Admitting, therefore, that the curvilinear Space CFH is equal to the rejectaneous Quantity ${\frac {nn-n}{2}}oox^{n-2}+\&c.$ and that when this is rejected on one Side, that is rejected on the other, the Reaſoning becomes juſt and the Concluſion true. And it is all one whatever Magnitude you allow to BD, whether that of an infiniteſimal Difference or a finite Increment ever ſo great. It is therefore plain, that the ſuppoſing the rejectaneous

alge-

1. See the Figure in Sect. 26.