Page:The Analyst; or, a Discourse Addressed to an Infidel Mathematician.djvu/32

22. Therefore as x becomes x + o, xn will become $$\overline{x+o}|^n$$: that is, according to the Method of infinite Series, $$x^n + nox^{n-1} + \frac{nn-n}{2}oox^{n-2}+$$ &c. And if from the two augmented Quantities we ubduct the Root and the Power repectively, we hall have remaining the two Increments, to wit, o and $$nox^{n-1} + \frac{nn-n}{2}oox^{n-2} + $$ &c. which Increments, being both divided by the common Divior o, yield the Quotients 1 and $$nx^{n-1} + \frac{nn-n}{2}ox^{n-2} +$$ &c. which are therefore Exponents of the Ratio of the Increments. Hitherto I have uppoed that x flows, that x hath a real Increment, that o is omething. And I have proceeded all along on that Suppoition, without which I hould not have been able to have made o much as one ingle Step. From that Suppoition it is that I get at the Increment of x^n, that I am able to compare it with the Increment of x, and that I find the Proportion between the two Increments. I now beg leave to make a new Suppoition contrary to the firt, i. e. I will uppoe that there is no Increment of