Note Sur Le Choix Des Courbes Fait Par Al-Khayyam Dans Sa Resolution Des Equations Cubiques Et Comparaison Avec la Methode De Descartes

TL;DR

This paper investigates the geometric methods used by Al-Khayyam and Descartes to solve cubic equations, analyzing their choices of curves and comparing their motivations and techniques in the context of historical mathematical practices.

Contribution

It provides a detailed analysis of the geometric curve choices of Al-Khayyam and Descartes, highlighting systematic versus varied approaches in their solutions of cubic equations.

Findings

Al-Khayyam used a uniform technique for all equation types.

Descartes employed different calculation techniques for each case.

Both mathematicians aimed to solve cubic equations via conic intersections.

Abstract

It is well known that Al-Khay\^am, for the first time in history, formulated a complete theory to solve third degree equations using the intersection of geometric curves and moreover solved the fourteen types of equations using this method. His solution for these equations was either using the intersection of two parabolas, a parabola and a circle, a parabola and a hyperbola, a circle and a hyperbola or two hyperbolas. His style was purely synthetic lacking any analysis which may lead to deducing any motives for his choice of the curves. Our article is centered on a pointed detail which is the justification of the choice of these curves, knowing that R. Rashed studied this choice in all of the mentioned fourteen equations. As for Descartes, whose style is as synthetic as Al-Khay\^am's, we ask the same question concerning the solution of the third degree equations, searching for any possible resemblance between the two methods of solution, both mathematicians having been motivated similarly in geometric resolutions. We noticed that the choice of the fourteen curves couples was made by Al Khayy\^am regarding a uniform calculation technique that he applied systematically to each equation type among the fourteen. It has appeared that the choice of the curves made by Descartes is the result of distinct calculation techniques, even though the projects of the two mathematicians are of the same essence, and their motivations are the same: finding the solution of third degree equations using the intersection of conical curves and the equations of curves.