Al-Kūhī's revision of Book I of Euclid's Elements

Abstract

The 10th-century mathematician Abū Sahl al-Kūhī, one of the best geometers of medieval Islam, wrote several treatises on the first three books of Euclid's Elements. We present an edition and translation of al-Kūhī's revision of Book I of the Elements, in which he altered the book's focus to the theorems and rearranged the propositions. The most dramatic of the changes is the complete absence of all geometric constructions, suggesting that al-Kūhī preferred instead to use Euclid's Data, and the related procedures of analysis and synthesis, for constructions. Other novel aspects include a proof of Euclid's fourth postulate, the use of the parallel postulate early in the work, and a different proof of the Pythagorean theorem. ▪