In his Sphaerica, Menelaus did not prove Proposition III.5 which is particularly important. He only gave an outline of a proof. Once the Sphaerica were translated into Arabic, mathematicians, starting from the end of the 9th century on, took up this proof. That was made possible to Ibn ʿIrāq thanks to the development of spherical geometry. A first paper contained the history of his contribution. Two other mathematicians, from the 13th century – Naṣīr al-Dīn al-Ṭūsī and Ibn Abī Jarrāda – worked out again the proof of the proposition with the help of Menelaus' book and of the new acquisitions of Ibn ʿIrāq. This is the subject of this second paper.

Galen's On the Elements according to Hippocrates is an important source for physical doctrines circulating in late antiquity. The variety of atomistic doctrines that Galen brings into the discussion, as well as his arguments aimed at refuting them, were closely studied by the early kalām atomists. Of particular interest are the summaries of this text, which seem to have been written many centuries after Galen; some of them are products of early Islamicate culture. In this paper, we present an edition, translation, and study of a hitherto unknown summary, extent in a unique manuscript where it is attributed to Ḥunayn ibn Isḥāq. Like the other summaries, it presents a doxography of physical doctrines, not always repeating precisely what Galen says, but rather reflecting the discourse of its period.

Kalām atomism stood in opposition to the Aristotelian natural philosophy of the falāsifa. In the Physics of the Shifā', Ibn Sīnā undertook a detailed refutation of kalām atomism through several arguments. These arguments elicited a muted response from al-Ghazālī, whose commitment to kalām was minimal at best. A more forceful response seems to have been offered by al-Shahrastānī but its details remain sketchy due to the lack of surviving sources. Fakhr al-Dīn al-Rāzī, whose intellectual development went through a phase of commitment to Avicennism and thereby a vigorous endorsement of Ibn Sīnā's anti-atomist arguments followed by a phase of a critical engagement with Avicennism, provides a detailed rebuttal to Ibn Sīnā's arguments in addition to constructing novel arguments in defense of kalām atomism.

The extant manuscript tradition of the Kitāb al-Kašf provides evidences of three phases of redaction and revision of the treatise. This study aims to illustrate the relations between the two Arabic versions and the anonymous fourteenth century Hebrew translation, taking into account the additions and modifications that differentiate them. I conclude that the Hebrew translation represents an intermediate stage of reworking, attesting important additions to the text as well as philosophical changes, especially in arguments concerning the creation of the world at the beginning of the treatise. However, it precedes the next revision phase of the text, which leaded to the modifications of the last Arabic version, in the chapters dealing with the thorny questions of divine corporeality and direction. The analysis of additions and arguments in the matter of content, context and reasoning helps to understand the redaction phases' history and the way the treatise was transmitted.

In the Planisphaerium, Ptolemy deals with a method of representing a sphere onto a plane, according to principles which were compatible with the stereographic projection. But this projection will only become a real mathematical concept in the 9th century, when al-Farghānī will demonstrate the fundamental property of this projection. It is only in the 10th century that the first general theory of projections of spheres was worked out by al-Qūhī and Ibn Sahl. This paper addresses the reasons which allow to claim that Ptolemy did not conceive of projection and that this conception was based on the Apollonius' definition of the conical surface.