Published online by Cambridge University Press: 01 March 2008
In order to find positive solutions for third-degree equations, which he did not know how to solve for roots, ‘Umar al-Khayyām proceeds by the intersections of conic sections. The representation of an algebraic equation by a geometrical curve is made possible by the choices of units of measure for lengths, surfaces, and volumes. These units allow a numerical quantity to be associated with a geometrical magnitude. Is there a trace of this unit in the mathematicians to whom al-Khayyām refers directly in his Algebra? How does this unit enable the measurement of quantities and rational and irrational relations? We find answers to these questions in the commentaries to Books V and X of the Elements.