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

46 and illogical. But if we fairly delineate the Area and its Increment, and divide the latter into two Parts BCFD and CFH and proceed regularly by Equations between the algebraical and geometrical Quantities, the reaon of the thing will plainly appear. For as as xn is equal to the Area ABC o is the Increment of xn equal to the Increment of the Area, i. e. to BDHC; that is, to ay, $$nox^{n-1} + \frac{nn-n}{2}oox^{n-2} + \&c. = BDFC + CFH$$. And only the firt Members, on each Side of the Equation being retained, $$nox^{n-1} = BDFC$$: And dividing both Sides by o or BD, we hall 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 Reaoning becomes jut and the Concluion true. And it is all one whatever Magnitude you allow to BD, whether that of an infiniteimal Difference or a finite Increment ever o great. It is therefore plain, that the uppoing the rejectaneous alge-