Arithmetical Algebra in the Islamic History of Mathematics and Its Peak in the 9th/15th Century: Ibn al-Ha'im's al-Mumti

Abstract

Algebra, defined as a method to determine the unknown by means of what is known, given the link between the two, took its initial steps toward disciplinary status during the third/ninth century when al-Khwarizmi produced the first systematic study on the subject. Later Muslim mathematicians followed his lead due to this novel discipline's propensity for improvement and beneficial application. Thus they applied arithmetic to algebra to make it more practical and open and, as a result, derived great benefits from employing it in matters of inheritance, commerce, land surveys, architecture, and other areas. Roughly 550 years after its formation as a discipline, algebra reached its peak in the aforementioned areas. One of its most famous practitioners, Ibn had a lasting and widespread influence first with his commentary on Yasamini and then with his versified work al-Muqni` and its commentary al-Mumti'. However, the latter work eluded the researchers' attention - perhaps it was overshadowed by the former or lost among the other commentaries - despite its remarkable presentation of the entire conceptual and methodical repertoire of algebra as it was known at that time, not to mention its analysis of the problems and discussion of the philosophical implications in a long-lasting debate on Islamic mathematical history: Should algebra be arithmetical, geometrical, or both? Which track would be more conducive to improving the discipline so it could break new ground in the historical studies of mathematics? Thus, this article seeks to present the status of Ibn al-Mumti' fi sharh al-Muqni` in the history of mathematics, along with its outstanding features and mathematical analysis.