Abū al-Jūd Muḥammad ibn al-Layth is one of the mathematicians of the 10th century who contributed most to the novel chapter on the geometric construction of the problems of solids and super-solids, and also to another chapter on solving cubic and bi-quadratic equations with the aid of conics. His works, which were significant in terms of the results they contained, are moreover important with regard to the new relations they established between algebra and geometry. Good fortune transmitted to us his correspondences with the mathematician and astronomer al-Bīrūnī. The questions they debated, and the answers they yielded, all offer us multiple in vivo perspectives on the research that was undertaken in that period. The reader would find in this article a critical edition and French translation of this correspondence, with historical and mathematical commentaries.

Scholars have long debated the possibility of a mystical or illuminationist strain of thought in Ibn Sīnā's body of writing. This debate has often focused on the meaning and contents of his partly lost work al-Mashriqiyyūn (The Easterners), also known as al-Ḥikma al-Mashriqiyya (Eastern Wisdom), mentioned by Ibn Sīnā himself as well as by numerous Western writers including Ibn Rushd and Ibn Ṭufayl. A handful of references to what is called Ibn Sīnā's “Oriental Philosophy” are also found in the Castilian and Hebrew works of the Castilian Jew Abner of Burgos (ca. 1270-ca. 1347), known after his conversion to Christianity as Alfonso of Valladolid. Although the content of these citations has not been identified, it has been proposed that they may preserve otherwise unknown passages from Ibn Sīnā's lost work. This study considers the references to Ibn Sīnā's so-called “Oriental Philosophy” within Abner's writings and concludes that rather than preserving lost passages from Ibn Sīnā's writing, Abner's references were drawn primarily from Ibn Ṭufayl and offer no support for the argument of a possible mystical or illuminationist strain in Ibn Sīnā's thinking.

In his encyclopedic book (al-Istikmāl), the mathematician of Saragossa, Ibn Hūd (d. 1085/478 H.), established by an intrinsic demonstration of spherical geometry, a remarkable theorem which generalizes the proposition III.11 from Theodosius’s Spherics and integrates the propositions III.23-25 from Menelaus’s Spherics. In this paper, we study this theorem and the demonstration of Ibn Hūd. The reader will find also some established and translated texts (Ibn Hūd, Ibn ʿIrāq, al-Ṭūsī) addressing the same theme.

Between 9th and 11th centuries, the geometrical transformations gave to the mathematicians a method more and more fertile, leading them to modify their modes of apprehension of the geometrical figures. This article aims to highlight al-Sijzī’s contribution to this change by setting two tasks: first, to precisely understand what al-Sijzī means by transformation (he is one of the first to recognize explicitly a method in their use); and secondly, to give an account of his research on geometrical invariants, obtained by a variation of some elements of a figure. The use of transformations and the search for invariants seem to be the two faces of the same tendency, that to break with an Euclidean manner to consider the figures in a way isolated and static for better exploiting the common properties which can link them. The article is completed by the edition and the translation of a small treatise devoted to invariants.

In the Greek/Indian period, it is noticeable that different radii were used in connection with the chord. This manner continued in the Indian period with the sine, i.e. different sine tables existed. But throughout the Arabic-Islamic period, there was stability in the radius (for the sine). At the time of al-Battānī new terms were introduced, not as functions of angles but as lengths, and again different tables for the same term. Here these terms were not bounded to the circle, and the term miqyās r (measure), which was variable, was used to express “the radius” related to these terms. The trigonometric “functions” at that time were treated as if they were two different groups. While at Abū al-Wafā’’s time, there was an advancement by introducing the new terms as functions of angles, and they were immediately bounded to the circle, and instead of having two circles in the same figure, a kind of unity appeared, and again there was stability in the value of r, and therefore only one table for each function, and thus the new functions started to appear more abstract than practical as the sine did before, and this unity remained fixed in the modern times.

We here present the Arabic text, with an English translation, of certain pages dedicated by al-Khayyām to the mathematical theory of music. Our edition is based on a manuscript extant in a library in Manisa (Turkey), and corrects the mistakes found in another transcription. Lastly, we compare the theory of al-Khayyām with other Arabic theories of Music, and with those coming from other traditions.

The medieval Islamic philosophers held a certain conception of the divine unity that assumes the necessary existent to be both one and simple. The oneness of the necessary existent meant that it is the only necessary existent and its simplicity meant that it admits no composition whatsoever – it is pure essence and its essence is necessary existence. In The Incoherence of the Philosophers al-Ġazālī presents, with elaboration, an exposition of the philosophers' conception of the divine unity, several arguments for its two components (i.e., oneness and simplicity), and his critique of these arguments. In this paper I focus on six of the arguments attributed to the philosophers. Following the textual evidence, I reconstruct these arguments and offer two possible interpretations of them. The first interpretation, which I call the many-argument interpretation, sees one of the arguments as employing the simplicity of the necessary existent to establish its oneness and the other five arguments as invoking oneness to establish simplicity. The second interpretation, which I call the one-argument interpretation, doesn't offer a new reading for the first argument but sees the other five arguments as defending the simplicity of the necessary existent based on its basic concept. I argue for the superiority of the one-argument interpretation.

The Persian philosopher Atīr al-Dīn al-Abharī (d. 1265), in his Revealing Thoughts, sketches out several different logics of propositions containing complex terms such as ‘writing man’. I will outline his results and the criticisms of them made by Naṣīr al-Dīn al-Ṭūsī (d. 1274) in his Setting the Scale for an Evaluation of “Revealing Thoughts”, and I will compare Abharī’s various logics of complex terms with modern treatments.

This article aims at presenting for the first time a central concept in al-Fārābī’s work that constitutes a keystone in understanding his thought, be it in its logical or political aspects. This concept is that of naqla, which, in terms of transmission and translation, its generic transcription can be rendered as ‘transfer’. The naqla is a notion that pertains to rupture in linguistic, logical or temporal continuities, and hints at confusing contiguities in the use of words, in demonstrations and in historical processes. This notion of naqla is at the centre of the preoccupations of al-Fārābī in his various domains of thinking. First of all, in terms of his linguistic reflection that consists of thinking about the transfer (naqla) of a given word in between its notions of first and second imposition. Then, in logic, the integration of the modes of reasoning of the theologians in Aristotelian syllogism, which passes by way of a mechanism of logical transference in the case of induction and the shift in paradigm. Finally, the Fārābian conception of intellectual history, as a transmission of knowledge, cannot be grasped in its fullest scope except through an understanding, not only of the common notion of naqla, but rather in terms of its particular Fārābian sense; namely as a concept that entirely renews the question of transmission and translation.