The beginnings of Archimedean mathematics in Arabic: Banū Mūsā (Q2717213)

This article attempts to delineate the Archimedian legacy by drawing distinctions between the original Greek Archimedes, the much later Arabic, and then the Latin Archimedes. As far as it can be referred to as a tradition, the Archimedian tradition had many dimensions, some stemming from legends such as Archimedes's ability to set fleets on fire way out at high seas with the use of reflecting mirrors, and his famous determination of the amount of gold in the crown of the king, both generating generations of discussion on optics and specific weights, and both producing very original results of their own during Arabic/Islamic times. Then there was the mathematical tradition a good portion of which no longer extant in the Greek texts and can only be deduced from the Arabic translations of ancient Greek texts, or from Latin translations of the same Arabic. Some of the texts in the latter tradition were simply inspired by the Archimedian works while others still were known to have been written by Archimedes even as far back as late antiquity.NEWLINENEWLINENEWLINEThe article under review is devoted to the Arabic mathematical Archimedian tradition which the author claims to have started with a text composed by the famous three brothers, collectively called Banu Musa. The text in question is \textit{The Book of the Knowledge of the Measurement of Plane and Spherical Figures}, whose Latin translation is edited together with an English translation in \textit{M. Clagett}'s, \textit{Archimedes in the Middle Ages}, Vol. 1, p. 223f (1964; Zbl 0129.24804), and which is obviously written in the Archimedian spirit although explicitly claimed at the end to be the work of Banu Musa themselves, except for the determination of the ratio between the circumference of a circle and its diameter which they explicitly attribute to Archimedes, and the determination of two magnitudes in between two others so that the four would be in the same ratio which they attribute to Menelaus. The rest of the article is devoted to a systematic examination of this very claim in order to determine how much of the work is Archimedian and how much of it is due to Banu Musa, to conclude that although much of the original results reached by Banu Musa, independently of Archimedes, were attributed by modern historians of mathematics to the students and followers of Banu Musa instead of Banu Musa themselves, those very results constituted the actual beginnings of Arabic Archimedian studies, but now set on new grounds unknown to Archimedes. Some of the methods proposed by Banu Musa were to have their influence on European mathematics even as late as the seventeenth century and thereafter. NEWLINENEWLINENEWLINEThe second more important, and much less known, result emanating from this study is the degree to which mathematicians of ninth-century Baghdad were creatively engaged with the inherited Greek tradition, criticizing it, emending it, completing it, and in many instances outpacing it, just at the same time as that Greek tradition was being translated into Arabic.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00025].