This paper presents an edition of al-Hasan ibn al-Ḥasan ibn al-Haytham's (d. 1040) treatise, Qawl fi samt al-qibla bi-al-ḥisāb (on finding the azimuth of the Qibla by calculation) with translation and commentary. In it Ibn al-Haytham provides a universal method for finding the direction of the qibla at any location on the surface of the earth by using spherical trigonometry and accurate calculation. Ibn al-Haytham's computational solution has not been studied before, and it has often been confused with another work of his in which he uses an analemma construction to solve the problem of the qibla graphically. As a result of this confusion, contemporary scholars have mistakenly attributed the first universal solution of the qibla problem to Jamshīd al-Kāshī (15th century), some four centuries after the introduction of this method by Ibn al-Haytham. The present treatise represents an important juncture in the history of the development of mathematics of the qibla, and sheds more light on the contributions of one of the most important scientists of medieval Islam.

Averroes defended philosophy by returning to the true Aristotle. For this purpose, Aristotle's book “On the Heaven,” in which he explained the eternity, uniqueness and movement of the universe, occupied a place of special importance. But the Aristotelian philosopher had a hard time holding his own in the face of contradictions within the book and with respect to Aristotle's later works. In his early Compendium, later Paraphrase, and final Long Commentary of De Caelo, Ibn Rushd continued the efforts of the Hellenistic commentators in order to integrate all the elements of his doctrine into a unified system, to harmonize his early cosmology with his later Metaphysics – the early doctrine of natural movement of the elements, and of the self-moving star-souls (a Platonic element), with the doctrine of potency and actuality and the theory of the First Mover – and to uphold his models of homocentric planetary spheres against the mathematical paradigm of Ptolemaic astronomy. By insisting throughout on demonstrative arguments based on rational principles, he asserted the philosophers' claim to irrefutable truth.

Ibn Rušd devoted a certain number of works to Aristotle's Prior Analytics. In a series of opuscules written over a period of twenty years and following upon his Middle Commentary on Aristotle's Prior Analytics, he faced a problem particular to the modal syllogism – that of the mood of the conclusion in mixed syllogisms.

The problem can be stated as follows: At the beginning of the Prior Analytics, Aristotle established a formal deductive principle – that of universal attribution (Pr. An. I.1.24b26–30). Applied to the modal syllogism, this principle is inadequate as stated. It is too general to be applied in a univocal manner in all modal syllogisms. To preserve a sense of coherence in Aristotle's declarations, the commentators had to interpret it. Presenting the interpretations of the commentators, primarily al-Fārābī and Alexander, on the basis of al-Fārābī's Large Commentary on Aristotle's Prior Analytics, Averroes criticizes them. Applied according to Alexander's interpretation, the principle of universal attribution is valid only for modal syllogisms one of whose premises is necessary and the other assertoric; according to al-Fārābī's interpretation, it is verified only when the minor premise is possible. Averroes proposes two preliminary solutions. Either this formal deductive principle must be applied differently according to the modal differences of the minor premises in mixed syllogisms (first solution) or would be used in two ways, generally or in keeping with each mood (second solution). These solutions are not satisfactory, for they call into question the unity and universality of the principle of universal attribution as established by Aristotle. What is the utility, Averroes asks, of a principle which does not hold for all modalities or does not apply to all the premises when the Pr. An. ought to furnish formal and universal principles of deduction? And why did Aristotle define the principle of universal attribution without distinguishing its application according to each of the three modal premises? Returning at the end of his career to a literal exegesis of Aristotle's propositions and without harkening back to the earlier solutions, he proposes a theory of making the terms modal (fourth solution) in order to save Aristotle's declarations with respect to the principle of universal attribution and the mood of the conclusion of mixed syllogisms (Pr. An. I. 9.30al5–20). Though formally inadequate, this solution, which had a continued history, proposes a new way of looking at the classification of modal propositions.

The thirteenth-century Hebrew texts that discuss salinity all ultimately go back to Aristotle's treatment of the subject in the Meteorology. However, in these Hebrew texts the question of what exactly makes the sea salty is answered in diverging ways. The oldest of them, the Otot ha-Shamayim (1210), being the Hebrew translation of the Arabic paraphrase of the Meteorology, proposes various causes of the sea's salinity, to wit, the dry exhalation, the action of heat, and the admixture of an earthy substance. This is due partly to Aristotle's own ambiguity, and partly to the fact that his Greek commentators interpreted his words in different ways. Two later encyclopedias, the Midrash ha-Hokhma (c. 1245) and the De'ot ha-Philosofim (c. 1275?) base their expositions of salinity on Ibn Rushd, whose two commentaries on the Meteorology contain various theories. The first encyclopedia opts for the action of heat as the major cause in producing saltiness, whereas the second attempts to explain in which way the various causes are interrelated by advisedly combining Ibn Rushd's accounts.

The importance of the Treatise on equations by Sharaf al-Dīn al-Ṭūsī (12th century) has been brought to our attention by R. Rashed (1974, 1986), who underlined the analytical aspects of this essentially algebraic work. Following Rashed, this article concentrates on one of these analytical concepts, namely the maximum of a polynomial expression f(x) of degree 3. The purpose is to clarify the techniques that led al-Ṭūsī, when computing the maximum of f(x), to systematically display algebraic equations equivalent to f(x) = 0. By demonstrating that al-Ṭūsī's essentially algebraic proofs were also based on analytical procedures, we show that the presence of these equations was not fortuitous, but resulted from a correct understanding of the maximum of f(x). The accompanying geometrical representations were primarily for illustration.

This article claims that Averroes wrote his Middle Commentary on the De anima after he composed both his Short and Long commentaries. A close comparison of the two texts proves that he had the Long commentary before him when composing the Middle. This has implications both for the development of Averroes' doctrine of the intellect, and for understanding Averroes' style of composing commentaries.

What follows is a detailed study of Averroes's Decisive Treatise based on a new English translation of the text. It seeks to reveal the steps of the exposition, the quality of the reasoning used therein, and the general goal. The study adheres closely to the framework established by Averroes at the outset: namely, that this is an inquiry conducted from the perspective of what is permitted by the divine law of Islam. And it takes seriously the proposition that one can seek for an intelligent and intelligible connection between that law and the results reached through philosophical investigation. Thus, Averroes's ultimate conclusion – that the divine law of Islam and philosophy or wisdom are in basic agreement – is seen to make sense insofar as both seek to provide for the well-being of all members of the community. It makes sense, that is, as an understanding of what governance entails.

The Treatise on Equations of Sharaf al-Dīn al-Ṭūsī (2nd half of the 12th century) is in the tradition of ‛Umar al-Khayyām (d. 1131). However, it has two special features. First, it contains a full discussion of the existence of a solution for third-degree equations, which al-Ṭūsī establishes by proving that the conic curves that represent this solution effectively intersect – a proof based on an intuitive notion of connexity. Secondly, al-Ṭūsī develops algorithms for the numerical resolution of these third-degree equations. The first stage of one of these algorithms follows a procedure which is akin to the so-called method of Newton's polygon.

This article presents the Arabic text of al-Qūhī's short geometric treatise, The Book of the centres of the tangent circles on lines, by the method of analysis (Xe century), with translation and a mathematical commentary. In this treatise al-Qūhī solves by analysis an ordered set of eight problems where the goal is to locate, on a given line, the centre of a circle which is tangent to two given elements, which may be points, straight lines or circles. For example, in the first problem he wanted to find the centre of a circle which contains two given points. In fact, al-Qūhī establishes the locus of the solutions, and this locus can be a straight line or a conic. The treatise is reminiscent of one of the purposes of al-Qūhī's work on the perfect compass (an instrument to draw conics by moving continuously), namely to assign to the conics the status of regular lines.