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It is well known that Al-Khay\^am, for the first time in history, formulateda complete theory to solve third degree equations using the intersection ofgeometric curves and moreover solved the fourteen types of equations using thismethod. His solution for these equations was either using the intersection oftwo parabolas, a parabola and a circle, a parabola and a hyperbola, a circleand a hyperbola or two hyperbolas. His style was purely synthetic lacking anyanalysis 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 thechoice of these curves, knowing that R. Rashed studied this choice in all ofthe mentioned fourteen equations. As for Descartes, whose style is as syntheticas Al-Khay\^am's, we ask the same question concerning the solution of the thirddegree equations, searching for any possible resemblance between the twomethods of solution, both mathematicians having been motivated similarly ingeometric resolutions. We noticed that the choice of the fourteen curvescouples was made by Al Khayy\^am regarding a uniform calculation technique thathe applied systematically to each equation type among the fourteen. It hasappeared that the choice of the curves made by Descartes is the result ofdistinct calculation techniques, even though the projects of the twomathematicians are of the same essence, and their motivations are the same:finding the solution of third degree equations using the intersection ofconical curves and the equations of curves.

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