In order to find positive solutions for third-degree equations, which he did not know how to solve for roots, ‘Umar al-Khayyām proceeds by the intersections of conic sections. The representation of an algebraic equation by a geometrical curve is made possible by the choices of units of measure for lengths, surfaces, and volumes. These units allow a numerical quantity to be associated with a geometrical magnitude. Is there a trace of this unit in the mathematicians to whom al-Khayyām refers directly in his Algebra? How does this unit enable the measurement of quantities and rational and irrational relations? We find answers to these questions in the commentaries to Books V and X of the Elements.

In the fifth book of his De aspectibus, the medieval Latin version of Ibn al-Haytham’s Kitāb al-Manāẓir, Alhacen undertakes to determine precisely where a given ray of light will reflect to a given center of sight from a variety of convex and concave mirrors based on circular sections. As applied specifically to convex and concave spherical mirrors, this problem exercised several seventeenth-century thinkers, Christiaan Huygens foremost among them, and in that context it soon became known as ‘‘Alhazen’s Problem.’’ By current standards, Alhacen’s solution (or solutions) of this problem is deficient in comparison to that of Huygens and later mathematicians. It is my purpose in this paper to show by reconstruction that in fact, far from deficient in any absolute sense, Alhacen’s approach to this problem was remarkably ingenious and elegant in its conceptual simplicity.

Na‘īm ibn Mūsā’s lived in Baghdad in the second half of the 9th century. He was probably not a major mathematician. Still his Collection of geometrical propositions – recently edited and translated in French by Roshdi Rashed and Christian Houzel – reflects quite well the mathematical practice that was common in Thābit ibn Qurra’s school. A relevant characteristic of Na‘īm’s treatise is its large use of a form of inferences that can be said ‘algebraic’ in a sense that will be explained. They occur both in proofs of theorems and in solutions of problems. In the latter case, they enter different sorts of problematic analyses that are mainly used to reduce the geometrical problems they are concerned with to al-Khwārizmī’s equations.

This article proposes a reconstitution of the philosophical tenor of al-Fārābī's lost treatise On Changing Beings (Fī al-Mawǧūdāt al-mutaġayyira). It is shown that this work is not only a response to book VI of John Philoponus' Contra Aristotelem, but that its real issues can only be grasped in the context of the author's metaphysical system. Although, for al-Fārābī, genuine demonstrations proceed from the cause to the caused, thus following the order of being, it will be explained how he also admits a strictly physical proof of the simple fact, independently of its cause, and that the physical demonstration of the eternity of the world pertains to this type of proof. This physical proof is specifically directed against the Kindian doctrine of creation.

Al-Fārābī, in the Kitāb al-Ḥurūf, is apparently the first person to maintain that existence, in one of its senses, is a second-order concept [ma‘qūl thānī]. As he interprets Metaphysics Δ7, ‘‘being'' [mawjūd] has two meanings, second-order ‘‘being as truth'' (including existence as well as propositional truth), and first-order ‘‘being as divided into the categories.'' The paronymous form of the Arabic word ‘‘mawjūd'' suggests that things exist through some existence [wujūd] distinct from their essences: for al-Kindī, God is such a wujūd of all things. Against this, al-Fārābī argues that existence as divided into the categories is real but identical with the essence of the existing thing, and that existence as truth is extrinsic to the essence but non-real (being merely the fact that some concept is instantiated). The Ḥurūf tries to reconstruct the logical syntax of syncategorematic or transcendental concepts such as being, which are often expressed in misleading grammatical forms. Al-Fārābī thinks that Greek more appropriately expressed many such concepts, including being, by particles rather than nouns or verbs; he takes Metaphysics Δ to be discussing the meanings of such particles (comparable to the logical constants of an ideal language), and he takes these concepts to demarcate the domain of metaphysics. This explains how al-Fārābī's title can mean both ‘‘Book of Particles'' and ‘‘Aristotle's Metaphysics.''

In his Short Treatise and his Commentary on the Peri hermeneias, al-Fārābī offers two different but related accounts of indefinite terms and the propositions that contain them. In both works he presents a series of different senses that an indefinite term may have, commencing with a sense in which such a term would be equivalent to a privative term, and concluding with a sense in which it would determine the logical complement of the corresponding definite term. I offer an interpretation of this series according to which they reveal the workings of a fine analytical mind and a subtle teacher.

Ancient Peripatetics and Neoplatonists had great difficulty coming up with a consistent, interpretatively reasonable, and empirically adequate Aristotelian theory of complete mixture or complexion. I explain some of the main problems, with special attention to authors with whom Avicenna was familiar. I then show how Avicenna used a new doctrine of the occultness of substantial form (whose roots are found in Alfarabi) to address these problems. The result was in some respects an improvement, but it also gave rise to a new set of problems, which were later to prove fateful in the history of early modern philosophy.

Al-Fārābī’s title of ‘‘Second Teacher’’ after Aristotle is well-warranted. Al-Fārābī’s work serves to illuminate the writings of the ‘‘First Teacher’’ in interesting and overlooked ways that go beyond the parameters of Aristotelian logic. Credence is lent to this assessment through the analysis of a specific topic, namely, authentic prophetic vision. At first glance, this seems like a strange assertion to make given Aristotle’s apparent skepticism and indifference regarding the topic of prophecy. However, as this paper will show, there is a latent theory of prophetic vision in the extant texts of Aristotle that al-Fārābī recognizes and completes in a way that is, in large part, faithful to and consistent with the corpus of Aristotle as we read it today. In the end, al-Fārābī provides insight into how a properly Aristotelian theory of authentic prophetic vision could be realized.

In Avicenna's expositions of his theory of the temporal origination of the human rational soul, its ḥudūth, one meets difficulties in understanding of what he actually means. Some of the expressions used are left unexplained and one has to extract their meaning from discussions given in a different context. There are also ambiguities in his use of such terms as al-‘aql al-kulliyy (the universal intellect) and al-nafs al-kulliyya (the universal soul). Although in one place he makes it clear that these expressions refer to concepts that exist only in the mind, distinguishing them from ‘aql al-kull (the intellect of ‘the whole [universal]') and nafs al-kull (the soul of ‘the whole [universel]'), the distinction is not uniformly observed. In a number of his works the term ‘‘universal'' is used to refer to both the celestial intellect and the celestial soul. There is also an ambiguity in his statements about the role the celestial soul plays in the emanation of the human rational soul. In some discussions he seems to hold that the rational human soul emanates from both the celestial intellect and the celestial soul. The Metaphysics of al-Shifā' (The Healing) suggests a resolution of this ambiguity. At the same time, there are statements in this work that are left unexplained and one has to look for their explanation in other books of The Healing. Thus questions do arise regarding the details of Avicenna's theory of the temporal origination of the human rational soul. His general exposition of his theory, however, remains comprehensible, its pivotal position within his entire philosophical system clear.

Galen’s Commentaries on the Hippocratic Epidemics constitute one of the most detailed studies of Hippocratic medicine from Antiquity. The Arabic translation of the Commentaries by Ḥunayn ibn Isḥāq (d. c. 873) is of crucial importance because it preserves large sections now lost in Greek, and because it helped to establish an Arabic clinical literature. The present contribution investigate the translation of this seminal work into Syriac and Arabic. It provides a first survey of the manuscript tradition, and explores how physicians in the medieval Muslim world drew on it both to teach medicine to students, and to develop a framework for their own clinical research.

Muhsin Sayyid Mahdi, James Richard Jewett Emeritus Professor of Arabic at Harvard University, died on July 9 in Brookline, Massachusetts, after a long series of illnesses.