Page:The method of fluxions and infinite series.djvu/28

4 8. In the same manner for $aa⁄x &minus; aab⁄x^{2} + aab^{2}⁄x^{3}$, &c. may be wrote

9. And thus instead of $&radic;\overline{aa &minus; xx}$ may be wrote $\overline{aa &minus; xx}|^$; and $\overline{aa &minus; xx}|^{2}$ instead of the Square of $aa &minus; xx$; and $1⁄\overline{abb &minus; y^{3}}|by + yy|undefined|^|undefined$ instead of $^{3}&radic;ab^{2} &minus; y^{3}⁄by + yy$: And the like of others.

10. So that we may not improperly distinguish Powers into Affirmative and Negative, Integral and Fractional.

11. The Quantity $aa + xx$ being proposed, you may thus extract its Square-Root.

$$ \begin{aligned} &aa + xx \left(a +\frac{x^2}{2a} - \frac{x^4}{8a^3} + \frac{x^6}{16a^5} - \frac{5x^8}{8a^7} + \frac{7x^{10}}{256a^9} - \frac{21x^{12}}{1024a^{11}},\ {\rm\&c}. \right. \end{aligned} $$

So that the Root is found to be $a + x^{2}⁄2a &minus; x^{4}⁄8a^{3} + x^{6}⁄16a^{5}$, &c. Where it may be observed, that towards the end of the Operation I neglect all those Terms, whose Dimensions would exceed the Dimensions of the last Term, to which I intend only to continue the Root,

suppose to $x^{12}⁄a^{11}$ Rh