Viète, Descartes and the cubic equation

Abstract

An appreciation of the geometry underlying algebraic techniques invariably enhances understanding, and this is particularly true with regard to polynomials. With visualisation as our theme, this article considers the cubic equation and describes how the French mathematicians François Viète (1540–1603) and René Descartes (1596–1650) related the ‘three-real-roots’ case ( casus irreducibilis ) to circle geometry. In particular, attention is focused on a previously undescribed aspect, namely, how the lengths of the chords constructed by Viète and Descartes in this setting relate geometrically to the curve of the cubic itself.