Elements of Algebra

268 ELEMENTS SECT. IV. $x = \frac{3}{2} \pm \sqrt{(\frac{9}{4} - \frac{16}{4})} = \frac{3}{2} \pm \sqrt{-\frac{7}{4}} = \frac{3 \pm \sqrt{-7}}{2}$; which are the other two roots, but they are imaginary. 749. It was, however, by chance, as we have remarked, that we were able, in the preceding example, to extract the cube root of the binomials that we obtained, which is the case only when the equation has a rational root; consequently, the rules of the preceding chapter are more easily employed for finding that root. But when there is no rational root, it is, on the other hand, impossible to express the root which we obtain in any other way, than according to the rule of Cardan; so that it is then impossible to apply reductions. For example, in the equation $x^3 = 6x + 4$, we have $f=6$ and $g=4$; so that $x = \sqrt[3]{(2+2\sqrt{-1})} + \sqrt[3]{(2-2\sqrt{-1})}$, which cannot be otherwise expressed *. * In this example, we have $\frac{1}{27}f^3$ less than $\frac{1}{4}g^2$, which is the well-known *irreducible case*; a case which is so much the more remarkable, as the three roots are then always real. We cannot here make use of Cardan's formula, except by applying the methods of approximation, such as transforming it into an infinite series. In the work spoken of in the Note, Art. 40, Lambert has given particular Tables, by which we may easily find the numerical values of the roots of cubic equations, in the irreducible, as well as the other cases. For this purpose we may also employ the ordinary Tables of Sines. See the Spherical Astronomy of Mauduit, printed at Paris in 1765. In the present work of EULER, we are not to look for all that might have been said on the direct and approximate resolutions of equations. He had too many curious and important objects, to dwell long upon this; but by consulting l'Histoire des Mathematiques, l'Algebre de M. Clairaut, le Cours de Mathematiques de M. Bezout, and the latter volumes of the Academical Memoirs of Paris and Berlin, the reader will obtain all that is known at present concerning the resolution of Equations. F. T. For a clear and explicit investigation of the method of solving Cubic Equations, by the Tables of Sines, \ldots the reader is also referred to Bonnycastle's Trigonometry; from which the following formulæ for the solution of the different cases of cubic equations are extracted. 1. $x^3 + px - q = 0$. Put $\frac{q}{2} (\frac{3}{p})^{\frac{3}{2}} = \tan. z$, and $\sqrt[3]{(\tan. (45^\circ - \frac{1}{2}z))} = \tan. u$; Then $x = 2\sqrt{\frac{p}{3}} \times \cot. 2u$. Or, putting Log. $\frac{q}{2} + 10 - \frac{1}{2} \log. \frac{p}{3} = \log. \tan. z$, and