Elements of Algebra

234 ELEMENTS SECT. IV. This formula contains a general rule for finding all the possible polygonal roots of given numbers. For example, let there be given the XXIV-gonal number, 3009: since $a$ is here $= 3009$ and $n = 24$, we have $n - 2 = 22$ and $n - 4 = 20$; wherefore the root, or $x = \frac{20 + \sqrt{(529584 + 400)}}{44} = \frac{20 + 728}{44} = 17$. CHAP. VIII. *Of the Extraction of the Square Roots of Binomials.* 669. By a binomial* we mean a quantity composed of two parts, which are either both affected by the sign of the square root, or of which one, at least, contains that sign. For this reason $3 + \sqrt{5}$ is a binomial, and likewise $\sqrt{8} + \sqrt{3}$; and it is indifferent whether the two terms be joined by the sign $+$ or by the sign $-$. So that $3 - \sqrt{5}$, and $3 + \sqrt{5}$ are both binomials. 670. The reason that these binomials deserve particular attention is, that in the resolution of quadratic equations we are always brought to quantities of this form, when the resolution cannot be performed. For example, the equation $x^2 = 6x - 4$ gives $x = 3 + \sqrt{5}$. It is evident, therefore, that such quantities must often occur in algebraic calculations; for which reason, we have already carefully shewn how they are to be treated in the ordinary operations of addition, subtraction, multiplication, and division: but we have not been able till now to shew how their square roots are to be extracted; that is, so far as that extraction is possible; for when it is not, we must be satisfied with affixing to the quantity another radical sign. Thus, the square root of $3 + \sqrt{2}$ is written $\sqrt{3 + \sqrt{2}}$; or $\sqrt{(3 + \sqrt{2})}$. 671. It must here be observed, in the first place, that the --- \* In Algebra we generally give the name binomial to any quantity composed of two terms; but Euler has thought proper to confine this appellation to those expressions, which the French analysts call quantities partly commensurable, and partly incommensurable. F. T.