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

8. In the same manner for aa/x − aab/x² + aab²/x³ , &c. may be wrote

9. And thus instead of √aa − xx may be wrote aa − xx|^1/2; and aa − xx|² instead of the Square of aa − xx; and abb − y³/by + yy|^1/3 instead of ³√ab² − y³/by + yy: And the like of others.

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

Examples of Reduction by Extraction of Roots.

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 + x²/2a − x⁴/8a³ + x⁶/16a⁵ , &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 x¹²/ a¹¹