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Moses ibn Tibbon’s Hebrew Translation of al-Hassar's Kitab al Bayan

Jeremy I. Pfeffer (Hebrew University of Jerusalem)


The Hindu (or Indian) number notation that employs nine numerals (digits) in decimal positions and a zero (ṣifr in Arabic and צפר in Hebrew) to indicate an empty position, had made its way to the Maghreb (Islamic North Africa) by the 10th century. Abu Bakr al-Hassar is believed to have taught mathematics during the 12th century in the city of Ceuta (Septa) on the north coast of Morocco. His most important contribution to mathematics was the introduction of a composite fractional notation in which the numerator and denominator are separated by a horizontal bar (Miller). Just two of al-Hassar’s works are known to have survived: the Kitāb al-Bayān wa-l-tadhkār (Book of Proof and Recall), in which he described this new notation and showed how it facilitates complex arithmetical calculations involving fractions, and the Kitāb al-Kāmil fi sinā’at al-adad (The Complete Book on the Art of Number), of which only the first volume is extant (Aballagheta and Djebbar, 147-158; Djebbar; Selin, 614-615). The earliest surviving copy of the first work was written in Arabic in Baghdad in the year 1192. In all, seven manuscript copies in Arabic and three in Hebrew are known, with the three in Hebrew written in the 15th century.

Al-Hassar is one of the earliest Western Arabic mathematical authors of whom a work has survived. The earliest extant Arabic arithmetic text is a 12th century copy of the Kitāb al-Fusūl fī al-Ḥisāb al Hindī written by Abū al Ḥasan al Uqlīdisī in Damascus in 952 CE. The work was translated into English and annotated by A. S. Saidan in The Arithmetic of Al-Uqlidisi in 1978.

In this article, we focus on the mathematics contained in al-Hassar’s Kitāb al-Bayān wa-l-tadhkār, as it appears in the translation from Arabic to Hebrew by Moses ibn Tibbon, who was born in Marseille, France, in about 1190 and died in 1283.

Jeremy I. Pfeffer (Hebrew University of Jerusalem), "Moses ibn Tibbon’s Hebrew Translation of al-Hassar's Kitab al Bayan," Convergence (May 2017), DOI:10.4169/convergence20170501