Elements of Algebra

CHAP. XII. OF ALGEBRA. 263 735. As in those fractions the roots of the equation are neither integer numbers, nor fractions, they are irrational, and, as it often happens, imaginary. The manner, therefore, of expressing them, and of determining the radical signs which affect them, forms a very important point, and deserves to be carefully explained. This method, called *Cardan's Rule*, is ascribed to *Cardan*, or more properly to *Scipio Ferreo*, both of whom lived some centuries since*. 736. In order to understand this rule, we must first attentively consider the nature of a cube, whose root is a binomial. Let $a + b$ be that root; then the cube of it will be $a^3 + 3a^2b + 3ab^2 + b^3$, and we see that it is composed of the cubes of the two terms of the binomial, and beside that, of the two middle terms, $3a^2b + 3ab^2$, which have the common factor $3ab$, multiplying the other factor, $a+b$; that is to say, the two terms contain thrice the product of the two terms of the binomial, multiplied by the sum of those terms. 737. Let us now suppose $x = a + b$; taking the cube of each side, we have $x^3 = a^3 + b^3 + 3ab(a+b)$: and, since $a + b = x$, we shall have the equation, $x^3 = a^3 + b^3 + 3abx$, or $x^3 = 3abx + a^3 + b^3$, one of the roots of which we know to be $x = a + b$. Whenever, therefore, such an equation occurs, we may assign one of its roots. For example, let $a = 2$ and $b = 3$; we shall then have the equation $x^3 = 18x + 35$, which we know with certainty to have $x = 5$ for one of its roots. 738. Farther, let us now suppose $a^3 = p$, and $b^3 = q$; we shall then have $a = \sqrt[3]{p}$ and $b = \sqrt[3]{q}$, consequently, $ab = \sqrt[3]{pq}$; therefore, whenever we meet with an equation, of the form $x^3 = 3x\sqrt[3]{pq} + p + q$, we know that one of the roots is $\sqrt[3]{p} + \sqrt[3]{q}$. Now, we can determine $p$ and $q$, in such a manner, that both $3\sqrt[3]{pq}$ and $p + q$ may be quantities equal to determinate numbers; so that we can always resolve an equation of the third degree, of the kind which we speak of. 739. Let, in general, the equation $x^3 = fx + g$ be proposed. Here, it will be necessary to compare $f$ with $3\sqrt[3]{pq}$, and $g$ with $p + q$; that is, we must determine $p$ and $q$ in * This rule when first discovered by Scipio Ferreo was only for particular forms of cubics, but it was afterwards generalised by Tartalea and Cardan. See Montucla's Hist. Math.; also Dr. Hutton's Dictionary, article Algebra; and Professor Bonnycastle's Introduction to his Treatise on Algebra, Vol. 1. p. xii—xv.