Elements of Algebra

262 ELEMENTS SECT. IV. the second, it is 8; and, according to the third, it is 10. The following statement shews, that all these will answer the conditions of the question: Number of merchants - - - - - 7 8 10 Each contributes $40x$ - - - - - 280 320 400 In all they contribute $40x^2$ - - - - 1960 2560 4000 The original stock was - - - - - 8240 8240 8240 The whole stock is $40x^2+8240$ - - 10200 10800 12240 With this capital they gain as much per cent as there are partners } 714 864 1224 Each takes from it - - - - - - 70 80 100 So that they all together take $10x^2$ - 490 640 1000 There remains therefore - - - - - 224 224 224 CHAP. XII. *Of the Rule of Cardan, or of Scipio Ferreo.* 734. When we have removed fractions from an equation of the third degree, according to the manner which has been explained, and none of the divisors of the last term are found to be a root of the equation, it is a certain proof, not only that the equation has no root in integer numbers, but also that a fractional root cannot exist; which may be proved as follows. Let there be given the equation $x^3 - ax^2 + bx - c = 0$, in which, $a, b, c$, express integer numbers. If we suppose, for example, $x = \frac{3}{2}$, we shall have $\frac{27}{8} - \frac{9}{4}a + \frac{3}{2}b - c = 0$. Now here, the first term alone has 8 for the denominator; the others being either integer numbers, or numbers divided by 4 or by 2, and therefore cannot make 0 with the first term. The same thing happens with every other fraction.