XXXVIII.—Some Mathematical Researches

Extract

It is well known, that whenever the three roots of a cubic are all real, the solution of the equation by Cardan's rule becomes illusory. This is the more remarkable, because, à priori, one might have expected that the rule would only fail when the roots were imaginary. Numerous researches have been made by mathematicians on this subject; but they have not succeeded in removing this obstacle; and the only mode of finding the roots of a cubic, when all three are real, has been, by successive approximations, or the use of trigonometrical tables, or (in the case of one root being a whole number), by tentative methods and trials (which often succeed without much difficulty, when the coefficients of the equation are small numbers).