Arithmetic, Induction, and the Algebra of Polynomials : Al-Samaw'āl and his "Splendid Book of Algebra"

Al-Samaw'āl (with the complete name Al-Samaw'āl Ibn Yahyā Al-Maghribī), born around 1130 in Baghdad, Iraq, is best known in the History of Mathematics for his seminal work the Al-Bāhir fī Al-Jabr (literally The Splendid Book of Algebra) which he composed at the prodigious age of nineteen. In this work, following the Euclidean tradition, Al-Samaw'āl put together and advanced many key algebraic rules formulated by his predecessors, notably Al-Khwārizmi, Ibn Turk, Ibn Qurra, Al-Kūhī, Al-Uqlīdīsī, Abū'l-Wafā, Al-Karajī, Ibn Aslam, Al-Sijzī, Ibn Al-Haytham, Qustā Ibn Lūqā, and Al-Harīrī. Al-Bāhir is a large work and consists of four sections. Section one provides an account of operations on polynomials in one unknown with rational coefficients, section two deals essentially with second-degree equations, indeterminate analysis, and summations, section three concerns irrational quantities, and section four presents the application of algebraic principles to a number of problems. In this thesis, we give close attention to book four from the second section in which Al-Samaw'āl discusses mathematical relations which amount to the binomial theorem and the Pascal triangle and lays out a table of binomial coefficients and demonstrates how to generalise the entries in the table for any desired value. Our main contribution is a complete translation of the Arabic text, the first time this has been done in a European language. We will then provide a detailed mathematical commentary and offer a careful analysis of the status and impact of this work in the History of Mathematics, paying due attention to the historical context in which it was produced