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.

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.

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.