A Significant Step Toward the Development of Algebra

Mathematics of the Islamic medieval world is still not sufficiently studied. As a result, a goldmine of Islamic medieval books and materials lie unexplored. One manuscript that certainly deserves attention is al-Bāhir fi’l-Jabr (The Shining Treatise on Algebra) of al-Samaw’al ibn Yahya al-Maghribi, a twelfth century mathematician. Al-Bāhir fi’l-Jabr is a manuscript written in Arabic and has never been translated except for a few excerpts in French. The purpose of this study was to explore the mathematical and pedagogical contribution of al-Samaw’al through an analysis of al-Samaw’al’s mathematical techniques and methods in al-Bāhir fi’l-Jabr. Moreover, the treatise provides a precise description of the “arithmetization of algebra”, and gives an accounting of the original ideas of another mathematician, al-Karaji, whose original documents have been lost. To develop a comprehensive picture of al-Samaw’al’s mathematical techniques and methods in al-Bāhir fi’l-Jabr (and his contribution to algebra in particular) this research has been based mainly on a careful analysis of al-Samaw’al’s manuscript in MSS Aya Sofia numbered 2718 (116ff), stored in the Suleymaniye library (Istanbul, Turkey). This study of al-Bāhir fi’l-Jabr focuses on an overview of how al-Samaw’al dealt with signed numbers, exponents and polynomial operations. Furthermore, this study describes the al-Samaw’al’s “method of the tables,” certain algorithms he employed, as well as his work on the binomial theorem, binomial coefficients, and the tabular arrangement known today as Pascal’s triangle. Most importantly, the study attempts to show the pedagogical approaches of al-Samaw’al.