3次方程式の還元不能な場合

Abstract

When a real cubic equation has three distinct real roots, it is called the casus irreducibilis or the case of irreducible. To find the roots of such equations using Cardano's formula, it is necessary to extract cube roots of imaginary binomials. In this paper we review remarkable works from Cardano to Euler on the solution of cubic equations in the casus irreducibilis and on the extraction of the cube roots of imaginary binomials. In it we give new proofs of Girard's method of constructing three real roots of a cubic equation and of Newton's method of extracting cube roots of imaginary binomials.