An Important Historical Milestone: The Classification of the Cubic Equations

Abstract

This article investigates the use of the history of mathematics as a pedagogical tool for the teaching and learning of mathematics, using the history of the cubic equation as a specific example.Cubic equations arise intrinsically in many applications in natural sciences and mathematics. For example, in physics, the solutions of the equations of state in thermodynamics, or the computation of the speed of seismic Rayleigh waves require the solutions of cubic equations. In mathematics, they are instrumental in solving the quartic equations, for in the process, these are reduced to cubic equations. The impossibility of trisecting an angle or doubling a cube using only a straightedge and compass is equivalent to solving some cubic equations. As the name implies, the cubic spline approximation, an important tool in numerical analysis, also entails working with cubic functions. Although cubic equations were explored by the ancient Babylonian, Greek, Chinese, Indian, and Egyptian scholars, it took the collective work of many well-known mathematicians such as Diophantus, Archimedes, Fibonacci, del Ferro, Khayyam, Tartaglia, Cardano, Viète, Descartes, and Lagrange to finally obtain a full solution. Our goal in this paper is to investigate one of the most formidable steps in this extensive and prolific history, namely the complete classification of the cubic equations by Omar Khayyam in eleventh century, who was the first scholar to classify cubic equation and hence facilitate a methodical and logical approach to obtaining a general solution.