Elements of Algebra

CHAP. VIII. OF ALGEBRA. 361 CHAP. VIII. *Of the Method of rendering the Irrational Formula, $\sqrt{(a + bx + cx^2 + dx^3)}$ Rational.* 112. We shall now proceed to a formula, in which $x$ rises to the third power; after which we shall consider also the fourth power of $x$, although these two cases are treated in the same manner. Let it be required, therefore, to transform into a square the formula $a + bx + cx^2 + dx^3$, and to find proper values of $x$ for this purpose, expressed in rational numbers. As this investigation is attended with much greater difficulties than any of the preceding cases, more artifice is requisite to find even fractional values of $x$; and with such we must be satisfied, without pretending to find values in integer numbers. It must here be previously remarked also, that a general solution cannot be given, as in the preceding cases; and that, instead of the number here employed leading to an infinite number of solutions, each operation will exhibit but one value of $x$. 113. As in considering the formula $a + bx + cx^2$, we observed an infinite number of cases, in which the solution becomes altogether impossible, we may readily imagine that this will be much oftener the case with respect to the present formula, which, besides, constantly requires that we already know, or have found, a solution. So that here we can only give rules for those cases, in which we set out from one known solution, in order to find a new one; by means of which, we may then find a third, and proceed, successively in the same manner, to others. It does not, however, always happen, that, by means of a known solution, we can find another: on the contrary, there are many cases, in which only one solution can take place; and this circumstance is the more remarkable, as in the analyses which we have before made, a single solution led to an infinite number of other new ones. 114. We just now observed, that in order to transform the formula, $a + bx + cx^2 + dx^3$, into a square, a case must be presupposed, in which that solution is possible. Now, such a case is clearly perceived, when the