Page:The Analyst; or, a Discourse Addressed to an Infidel Mathematician.djvu/30

Propoſition, a certain Point is ſuppoſed, by virtue of which certain other Points are attained; and ſuch ſuppoſed Point be it ſelf afterwards deſtroyed or rejected by a contrary Suppoſition; in that caſe, all the other Points, attained thereby and conſequent thereupon, muſt alſo be deſtroyed and rejected, ſo as from thence forward to be no more ſuppoſed or applied in the Demonſtration." This is ſo plain as to need no Proof.

XIII. Now the other Method of obtaining a Rule to find the Fluxion of any Power is as follows. Let the Quantity x flow uniformly, and be it propoſed to find the Fluxion of xⁿ. In the ſame time that x by flowing becomes x + o, the Power xⁿ becomes ${\overline {x+o}}|^{n}$, i. e. by the Method of infinite Series $x^{n}+nox^{n-1}+{\frac {nn-n}{2}}oox^{n-2}+$ &c. and the Increments o and $nox^{n-1}+{\frac {nn-n}{2}}oox^{n-2}+$ &c. are one to another as 1 to $nx^{n-1}+{\frac {nn-n}{2}}ox^{n-2}+$ &c. Let now the Increments vaniſh, and their laſt Proportion will be 1 to $nx^{n-1}$. But it ſhould ſeem

that