Reading Book II within arithmetical-algebraic practices. The case of al-Karajī with a continuation in al-Zanjānī”

In Arabic treatises on algebra, Book II of Euclid's Elements quickly became a traditional work of reference, especially for justifying quadratic equations. However, in many of these treatises we find a representation of Euclid's notions that deviates from the original Euclid. In this article, I focus on the way in which propositions of Book II were understood and reported by al-Karaǧī (11th c.) in two of his algebraic writings. Inspired by the variety of arithmetical practices of his time, al-Karaǧī transposed these Euclidean propositions from geometrical objects to numbers and applied them to an algebraic context. This allowed him to combine various argumentative strategies deriving from different fields. Building upon al-Karaǧī's work, al-Zanǧānī (13th c.) no longer needed to mention Euclid and instead conceived of a justification of quadratic equations (the cause of the equation) which is completely internal to algebra. These case studies provide evidence for the use of the Elements as a toolbox for the development of algebra. More importantly, they shed further light upon a typical feature of medieval mathematics, namely the existence of a plurality intrinsic in the name Euclid.