After having reformulated optics, Ibn al-Haytham conceived of an analogous project for astronomy. This has just been revealed by an important book by the mathematician which has never been studied until now. Ibn al-Haytham's reform consists in excluding all cosmology, and in developing a systematic study of a celestial kinematics that has been completely geometrized. In turn, the realization of such a reform demanded innovative research in infinitesimal geometry. In this article, an attempt is made to present this new geometry, as well as the mathematical means invented to elaborate it.

This paper investigates the objections that were raised by the philosopher ‘Abd al-Laṭīf al-Baghdādī (d. ca. 1231 CE) against al-Ḥasan ibn al-Haytham’s (Alhazen; d. after 1041 CE) geometrisation of place. In this line of enquiry, I contrast the philosophical propositions that were advanced by al-Baghdādī in his tract: Fī al-Radd ‘alā Ibn al-Haytham fī al-makān (A refutation of Ibn al-Haytham’s place), with the geometrical demonstrations that Ibn al-Haytham presented in his groundbreaking treatise: Qawl fī al-Makān (Discourse on place). In examining the particulars of al-Baghdādī’s fragile defence of Aristotle’s definition of topos as delineated in Book IV of the Physics, which was rejected on mathematical grounds by Ibn al-Haytham, a special attention is also given to highlighting the systemic distinctions between the entities that are studied within the speculative physical doctrines of common sense and immediate experience, and the postulated ‘objects’ of scientific and mathematical research.

Until the end of the 14th century, the sources of Hebrew mathematical writings were almost exclusively in Arabic. This was particularly true of texts that contained elements of algebra or algebraic developments. The testimonies we present and analyze here are due to Jewish authors living in Italy, primarily in the 15th century, who made use of the most varied sources, in addition to Arabic: in Castilian, in Italian, and perhaps in Latin. These testimonies constitute both an indication, and a product, of the circulation of Arab algebraic traditions in Renaissance Italy. Simon Moṭoṭ’s book on The Calculation of Algebra stems from the Italian tradition of ‘‘treatises on the abacus’’. Mordekhay Finzi of Mantua is the author of a Hebrew version of the great work on algebra by Abū Kāmil (9th century), as well as of a version, distinct from the preceding, of the Arabic scholar’s introductory exposition. Beginning in 1473, Finzi also translated from Italian to Hebrew the important treatise on algebra by Maestro Dardi of Pisa (1344). We also indicate some 16th century continuations of Hebrew mathematical production, which contain algebraic developments.