Al-Khayyām’s Scientific Revision of Algebra

Geometers in medieval Islam found algebra to be a useful tool in solving problems, but the Greek restriction of “number” to positive integers, adhered to by al-Khayyām (and presumably others), is at odds with the more liberal inclusion of fractions and irrational roots in traditional numerical Arabic algebra. By regarding the known and unknown numbers of traditional algebra as the homogeneous measures of continuous magnitudes (and not as magnitudes themselves), al-Khayyām was able to give a rigorous foundation for the art. This revision helps us understand al-Khayyām’s main work on
algebra in the context of geometrical and numerical problem solving, and it leads us to a reassessment of his generalized notion of “number” explained in his commentary on Euclid.