Cubic Equations Through the Looking Glass of Sylvester

TL;DR

This paper explores Sylvester's approach to solving cubic equations, revealing its connection to Cardano's formula and illustrating the historical and mathematical significance of Sylvester's method.

Contribution

It clarifies Sylvester's method for solving cubic equations and links it to classical solutions like Cardano's formula, providing new insights into the algebraic structure.

Findings

Sylvester's approach reduces to expressing cubics as sums of two third powers.

The method relates to Cardano's formula involving third roots of unity.

Provides a historical perspective on solving cubic equations.

Abstract

One can hardly believe that there is still something to be said about cubic equations. To dodge this doubt, we will instead try and say something about Sylvester. He doubtless found a way of solving cubic equations. As mentioned by Rota, it was the only method in this vein that he could remember. We realize that in the generic case Sylvester's magnificent approach aimed at reduced cubic equations boils down to an easy identity expressing a cubic polynomial as a sum of two third powers of linear forms. This leads to Cardano's formula for cubic equations involving the third roots of unity.