Descartes' Method for Constructing Roots of Polynomials with 'Simple' Curves

Introduction

Throughout the history of mathematics, people have searched for algorithms for solving algebraic equations of various degrees. Solutions to certain quadratic equations (understood geometrically) were known as early as 1600 BCE in Babylon, though it wasn’t until Cardano’s Ars Magna in 1545 that algebraic solutions to cubic and quartic equations (described in words and via specific numerical examples) were finally published. In 1824 Abel proved the impossibility of finding an algebraic solution for equations of degree five or higher for the general case.

Though general algebraic solutions to fifth and sixth degree equations are not possible, there has been a tradition of graphically solving equations that has spanned thousands of years. The Greeks had methods of solving quadratic equations with intersecting circles and lines and of solving certain cubics with intersecting conics by 200 BCE.

In 1637, as an appendix to his Discours de la Méthode, René Descartes (of “I think, therefore I am” fame) published ‘The Geometry’ (‘La Géométrie’), which included in its third and final part methods of graphically solving cubic, quartic, quintic, and even sextic equations through the intersection of different curves. What was remarkable about his methods is that the curves were much ‘simpler’ than expected.

This article will explore and analyze Descartes’ methods.