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

portion. 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 ſubduct the Root and the Power reſpectively, we ſhall 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 Diviſor 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 ſuppoſed that x flows, that x hath a real Increment, that o is ſomething. And I have proceeded all along on that Suppoſition, without which I ſhould not have been able to have made ſo much as one ſingle Step. From that Suppoſition 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 Suppoſition contrary to the firſt, i. e. I will ſuppoſe that there is no Increment

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