In this paper, I explore general features of the “architecture” (relations of white space, diagram, and text on the page) of medieval manuscripts and early printed editions of Euclidean geometry. My focus is primarily on diagrams in the Arabic transmission, although I use some examples from both Byzantine Greek and medieval Latin manuscripts as a foil to throw light on distinctive features of the Arabic transmission. My investigations suggest that the “architecture” often takes shape against the backdrop of an educational (...) landscape. The constraints of the economic marketplace and cultural aesthetic ideals also appear to play a role in determining the “architecture” of both manuscripts and early printed editions. (shrink)

During the Middle Ages many Greek mathematical and astronomical texts were translated from Greek into Arabic and from Arabic into Latin. There were many factors complicating the study of them, such as translation from or into other languages, redactions, multiple translations, and independently transmitted scholia. A literal translation risks less in loss of meaning, but can be clumsy. This article includes lists of translations and a large bibliography, divided into sections.

The article deals with the Arabic sources of Chr. Clavius in Rome and the six different ways they were used by him in mathematics and astronomy. It inquires especially into his attitude towards al-Farghani, Thabit ibn Qurra, al-Bi[tdotu]ruji, Ibn Rushd, Mu[hdotu]ammad al-Baghdadi, Pseudo-Ibn al-Haytham, Jabir ibn Afla[hdotu], and Pseudo-al-[Tuotu]usi.

Les documents chinois, depuis le Iersiècle, indiens, depuis le Vesiècle, et arabes, depuis le IXesiècle, contiennent des procédures tabulaires similaires pour l'extraction de racines carrées et cubiques avec des systèmes de numération positionnels. Par ailleurs tant Jia Xian, astronome chinois du XIesiècle, qu'al-Samaw'al, mathématicien arabe du XIIesiècle, ont extrait des racines de degré plus élevé par la procédure dite de Ruffini-Horner. L'article tente de définir une méthode textuelle pour organiser ce corpus, en y distinguant des axes pertinents qui permettent de (...) dégager similarités et différences, d'un point de vue tant historique que conceptuel. Une première partie analyse trois états différents des descriptions d'algorithmes entre le Iersiècle et le XIesiècle en Chine, qui présentent chacun une stabilité historique certaine. La réécriture qui fait passer d'un état au suivant laisse émerger progressivement une uniformité dans les composantes de l'algorithme, laquelle culmine avec des procédures du type Ruffini-Horner. Des critères textuels font apparaître une affinité plus grande de certains algorithmes, tels ceux décrits par Kūshyār ibn Labbān (ca1000), avec les textes chinois qu'avec les textes indiens, plus proches, eux, des algorithmes décrits par al-Khwārizmī. Des critères de même nature lient, d'une part, les algorithmes de Jia Xian et d'al-Samaw'al, d'autre part les algorithmes de Kūshyār et d'al-Samaw'al. (shrink)

This study analyzes the content of 12th-grade mathematics textbooks and workbooks, based on their inclusion of mathematical discourse components. The mathematics textbooks and workbooks were used in a Saudi Arabian school, where students are transitioning from secondary education to university. The results revealed that Saudi Arabian school textbooks and workbooks did not appropriately include discourse components or discourse skills to help facilitate mathematical learning among students. Furthermore, these textbooks did not exceed level two of the four levels of inclusion. As (...) a result, the inclusion was insufficient in helping students meaningfully understand mathematical concepts, become active students, and develop successful community leadership. This implies that mathematics textbooks and workbooks should be revised to include mathematical discourse so that this inclusion is more student directed than teacher directed. (shrink)

Medieval Arabic algebra is a good example of an artificial language.Yet despite its abstract, formal structure, its utility was restricted to problem solving. Geometry was the branch of mathematics used for expressing theories. While algebra was an art concerned with finding specific unknown numbers, geometry dealtwith generalmagnitudes.Algebra did possess the generosity needed to raise it to a more theoretical level—in the ninth century Abū Kāmil reinterpreted the algebraic unknown “thing” to prove a general result. But mathematicians had no motive to (...) rework their theories in algebraic form. Because it offered no advantage over geometry, algebra remained a practical art in both the Islamic world and in Europe until the scientific uphevals of the 17th–18th centuries. (shrink)

The government of the State of Ohio determined to design several mandatory programs in regular classes to help NRC’s female in mates to prepare themselves upon their release back to the community; one of the programs is Islamic teaching. It is a chance to insert and implement the innovated peaceful Islamic propagation beside the conventional one, especially in western countries like the United States of America. The chance for an innovated dakwah is open upon observing that a lot of non- (...) Muslimah in mates were curious about Islam in an inter-religious meeting. The Arabic script classis not a part of mandatory NRC programs, it is voluntary for non- Muslimin mates designed and implemented by the author. They learn to write and pronounce the Arabic alphabet from “alif’ to ‘ya’ within six weeks. This free class is to support the government programs in order to reduce recidivism among in mates, and upon completing this class they are encouraged to attending the Islamic teaching class. This paper is qualitative research using a sociology approach. The data collection for this research is using participative observation, voluntary interview, and documentation; the collected data were analyzed in the descriptive-qualitative method. The effectiveness and efficacy of the research should be measured in a quantity of non- Muslimah in mates who gradually and peacefully convert to become Muslimah after attending the Arabic script class. The benchmark of effectiveness should be affirmed in mathematic formula i.e. the effectiveness is equal to the number of Muslimah converts divided by the total participant time 100; it is x 100 = 33.33%. Among 9 non- Muslimah participants of free Arabic script class,3 of them converted to be Muslimah. (shrink)

RésuméDans les Sphériques, Ménélaüs ne démontre pas l'importante proposition III.5, mais propose seulement une esquisse de démonstration. Une fois le livre des Sphériques traduit en arabe, les mathématiciens, à partir de la fin du IXe siècle, ont voulu en donner une démonstration complète. Le développement de la géométrie sphérique a permis à Ibn ʿIrāq de parvenir au but. Un premier article a été consacré à sa contribution. Deux mathématiciens du XIIIe siècle – Naṣīr al-Dīn al-Ṭūsī et Ibn Abī Jarrāda – (...) ont repris la démonstration de cette même proposition, à partir de Ménélaüs et des acquis d'Ibn ʿIrāq. C'est le sujet du présent article. (shrink)

RésuméC'est le premier d'une série d'articles comportant quatre textes composés entre le XIeet le XIIIesiècle, qui traitent de la proposition 5 du livre III desSphériquesde Ménélaüs. Le premier article comprend des commentaires historiques et mathématiques de l'œuvre d'Ibn ʿIrāq en géométrie sphérique et une édition critique des deux textes qu'il a consacrés à la rectification de la proposition III.5, ainsi que la traduction de ces deux textes. Le second article propose une édition critique des textes de Naṣīr al-Dīn al-Ṭūsī et (...) d'Ibn Abī Jarrāda sur le même sujet, cette fois aussi avec une traduction et des commentaires historiques et mathématiques. (shrink)

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, 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. Résumé Nous présentons ici le texte arabe, avec traduction anglaise, de certaines pages consacrées par al-Khayyām à la (...) théorie mathématique de la musique. Notre édition se fonde sur un manuscrit conservé dans une bibliothèque de Manisa et corrige les erreurs d'une autre transcription. Nous comparons enfin la théorie d'al-Khayyām avec d'autres théories arabes de la musique, et avec certaines théories issues d'autres traditions. (shrink)

This is the first critical edition of the first and second Epistles of the Brethren Purity--the Rasa 'il--in Arabic with a fully annotated English translation. It presents technical and epistemic analyses of mathematical concepts and their metaphysical bases, and an overview of the mathematical sciences within Islamic intellectual milieu.

Arabic contributions to medicine, mathematics, astronomy, and other fields have been extensively studied, yet Arabic logic has never been systematically investigated. In this book, Nicholas Rescher sheds new light on the major philosophical contribution of Arab logicians. He provides a historical account of the evolution of Arabic logic, from its inception in the early ninth century through the sixteenth century, when these tenets gained wide acceptance. The book also includes a bio-bibliography of 170 Arabic logicians, and a discussion of (...) the place of each within the overall scheme of developments. (shrink)

It has long been considered that Arabic algebra scarcely left any traces in mathematical literature of Hebrew expression. Thanks to the unpublished sources we have discovered, and to an attentive examination of already-known texts, one can no longer subscribe to such a judgement. The evidence we examine in this first article sheds light on the circulation, in erudite Jewish circles, of Arabic algebraic knowledge in Spain, Italy, Provence, and Sicily, between the 12th and the 14th centuries. The Epistle on number (...) by the Castillian astronomer Isaac ben Salomon al-A[hudot]dab was written in Sicily at the end of the 14th century, and based on the Talkhi[sudot] a'mal al-[hudot]isab of Ibn al-Banna' (1256-1321). That part of the Epistle that is devoted to algebra follows the tradition of al-Karaji. It offers, for the first time in Hebrew, a rational presentation of arithmetical operations extended to algebraic expressions. (shrink)

Representations of the heavens in various levels of detail can be found in a number of branches of Arabic literature. One particular genre, the hay'a texts, has as its purpose a full though non-mathematical discussion of the arrangement of the celestial orbs; hay'a writers are particularly sensitive to the philosophical requirements which all systems must meet. The pivotal work in this genre, On the Configuration, was written by Ibn al-Haytham. Later writers continued to produce works in the spirit of On (...) the Configuration. In the east, al-Tusi and his followers developed new models; in the west, a group of thinkers tried to rediscover the models which, so they thought, were the ones endorsed by Aristotle himself. (shrink)

In an influential article, A. I. Sabra identified an intellectual trend from twelfth and thirteenth-century Andalusia which he described as the ‘‘Andalusian revolt against Ptolemaic astronomy.” Philosophers such as Ibn Rushd , Ibn Tufayl , and Maimonides objected to Ptolemy’s theories on philosophic grounds, not because of shortcomings in the theories' predictive accuracy. Sabra showed how al-Bitrūjī's Kitāb al-Hay'a attempted to account for observed planetary motions in a way that met the philosophic standards of those philosophers and others. In Nūr (...) al-‘ālam , the subject of this article, Joseph ibn Joseph ibn Nahmias endeavoured to improve upon al-Bitrūjī’s models. Levi Ben Gerson's Hebrew writings on astronomy criticized al-Bitrūjī, but Ibn Nahmias did not mention them. Nūr al-‘ālam deserves attention, too, because it is the first Arabic text on theoretical astronomy by a Jewish author to come to light. In the body of this article, I will describe and analyze Ibn Nahmias’ theory, from Nūr al-‘ālam , for the motion of the sun. (shrink)

This paper investigates the objections that were raised by the philosopher ‘Abd al-La&tdotu;īf al-Baghdādī against al-&Hdotu;asan ibn al-Haytham’s geometrisation of place. In this line of enquiry, I contrast the philosophical propositions that were advanced by al-Baghdādī in his tract: Fī al-Radd ‘alā Ibn al-Haytham fī al-makān, with the geometrical demonstrations that Ibn al-Haytham presented in his groundbreaking treatise: Qawl fī al-Makān. In examining the particulars of al-Baghdādī’s fragile defence of Aristotle’s definition of topos as delineated in Book IV of the (...) Physics, which was rejected on mathematical grounds by Ibn al-Haytham, a special attention is also given to highlighting the systemic distinctions between the entities that are studied within the speculative physical doctrines of common sense and immediate experience, and the postulated ‘objects’ of scientific and mathematical research. (shrink)

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition. Bibliography.

It was a strange posthumous fate that awaited the Englishman Robert of Ketton : he was to be both best known and most strenuously criticized for a work that he surely viewed as a sideline to his own interests and career. By trade Robert was a Latin translator of Arabic scientific and mathematical works, one of those remarkable twelfth-century men who, as his contemporary Petrus Alfonsi put it, were willing “to traverse distant provinces and withdraw into remote regions so as (...) to be able to attain fuller knowledge of the astronomical art.” Like other contemporary translators, such as the Italian Gerard of Cremona , Robert had left his birthplace—in his case Rutlandshire—and traveled to Spain in order to have access to the great works of Arabic astronomy and mathematics that could be found in the still very Arab Iberian Peninsula. By 1141 we find him and his close friend Herman of Carinthia studying and translating scientific works together in northeastern Spain, and in later years his astronomical interests and ecclesiastical responsibilities led him to travel widely in Spain and England. As a scientific translator, he is perhaps best known for his Liber algebrae et Almucabola, a Latin version of al-Khwarazmī's great manual of algebra, Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-al-muqābalah. (shrink)

The Ikhwan al-Safa (Brethren of Purity), the anonymous adepts of a tenth-century esoteric fraternity based in Basra and Baghdad, hold an eminent position in the history of science and philosophy in Islam due to the wide reception and assimilation of their monumental encyclopaedia, the Rasa'il Ikhwan al-Safa (Epistles of the Brethren of Purity). This compendium contains fifty-two epistles offering synoptic accounts of the classical sciences and philosophies of the age; divided into four classificatory parts, it treats themes in mathematics, logic, (...) natural philosophy, psychology, metaphysics, and theology, in addition to didactic fables. The Rasa'il constitutes a paradigmatic legacy in the canonization of philosophy and the sciences in mediaeval Islamic civilization, as well as having shown a permeating influence in Western culture. This is the fifth volume in a series presenting the very first critical edition of the Rasa'il in its original Arabic, complete with the first fully annotated English translation. Epistle 52: On Magic is the last in the corpus, and presents the short version of the 'Epistle on Magic, Incantations, and the Evil Eye'. Within its pages the authors of the epistle argue for the legitimacy of magic and the other occult sciences, and seek confirmation of their views in authorities as diverse as Plato, the Qur'ân, the Torah, the astrologer Abû Ma'shâr, and the mysterious sect of pagan star-worshippers known as the Sabians of Harrân. As in all other volumes of the series, this volume provides the first critical edition of the Arabic text, together with an English translation and a substantial range of notes. An extensive introduction highlights the unique relevance of this treatise within the framework of the Brethren's encyclopaedia itself, and to the history of science in general. (shrink)

Duhem has generally been understood to have maintained that the major Greek astronomers were instrumentalists. This view has emerged mainly from a reading of his 1908 publication To Save the Phenomena. In it he sharply contrasted a sophisticated Greek interpretation of astronomical models (for Duhem this was that they were mathematical contrivances) with a naive insistence of the Arabs on their concrete reality. But in Le Système du monde, which began to appear in 1913, Duhem modified his views on Greek (...) astronomy considerably; his more subtle understanding included the recognition that many Greeks subordinated mathematical astronomy to physical theory. But he could not completely repudiate his earlier views about Greek astronomy in part because his extreme nineteenth century prejudices led him to continue to insist on a clear-cut demarcation between Greek and Arabic astronomy. The inevitable result is a certain unevenness in the Système and some glaring inconsistencies. (shrink)

Like his Greek and Arab predecessors, Avicenna’s research in mathematics concerned the development of methods of exposition, proof procedures and analytical tools. But Avicenna belonged to a new era of mathematics, and the question that this paper seeks to examine is how Avicenna applied this new mathematical knowledge in his philosophy. A treatment of the subject of Avicenna and mathematics cannot, then, confine itself to generalities, important as these are, but must start from the degree of knowledge he had (...) of the various branches and the use he actually made of them. I shall restrict myself here to a number of examples which seem to me to possess evidential value and which all reveal a new conception of the connections between philosophy and mathematics. It will turn out that what Avicenna in fact did was to develop an analytical philosophy of mathematical concepts. (shrink)

This paper deals with the exact constructions of the regular heptagon in Greek and Arabic geometry, which are preserved in a number of mainly unpublished Arabic manuscripts. Appended are editions of the Arabic texts and English translations of Propositions 17 and 18 of the “Book of the Construction of the Circle, Divided into Seven Equal Parts”, attributed to Archimedes, and of the “Book on the Construction of the Heptagon in the Circle and the Division of the Rectilineal Angle into Three (...) Equal Parts” by the 10th century geometer Al-Sijzi. (shrink)

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

The Epistles of the Brethren of Purity' is an encyclopedic compendium, probably composed in tenth-century Iraq by a society of adepts with Platonic, Pythagorean, and Shi'i tendencies. Its 52 sections ('epistles') are divided into four parts (Mathematics, Natural Philosophy, Sciences of the Soul and Intellect, and Theology). The current volume provides an edition, translation, and notes to Epistle 3 ('On Astronomia'), which forms one of the 14 sections on Mathematics. The content is a mixture of elementary astronomy and astrology, but (...) it is not a beginner's textbook. Rather, the purpose is to use examples from those disciplines to provide spiritual, moral, and soteriological guidance. Thus the Epistle uses the argument from design to show the necessity for a Creator who made the harmonious universe; this wondrous design is then employed by the authors as a model, providing humans with a paradigm for proper ethical, political, and even economic conduct; and the study of Astronomia helps the soul achieve ultimate happiness as it seeks to throw off the shackles of this mundane world and oppressive body in favour of the purity of the celestial realm.0Although by no means typical of Islamic astronomical literature, Epistle 3 of the Brethren of Purity gives a window into a fascinating and intriguing group operating during the early period of Islam who sought to continue and adopt one of the esoteric strands of Hellenistic philosophy within an Islamic context, meshing astronomy, astrology, Platonic-Pythagorean philosophy, Quranic and Biblical quotations, and anecdotes from the lives of the Abrahamic prophets. (shrink)

The square of opposition has never been drawn by classical Arabic logicians, such as al-Fārābī and Avicenna. However, in some later writings, we do find squares, which their authors call rather ‘tables’ (sing. _lawḥ_). These authors are Shihāb al-Dīn al-Suhrawardī and Muhammed b. Yūsuf al-Sanūsī. They do not pertain to the same geographic area, but they both provide squares of opposition. The aim of this paper is to analyse these two squares, to compare them with each other and with the (...) traditional square of opposition drawn by Apuleius and Boethius, with regard to their particular structure and their vertices and to evaluate their validity. We will show that both squares are different from each other and from the traditional square both with regard to their vertices, and to the arrangements of the oppositional relations. In addition, while al-Suhrawardī uses the notion of matter modalities to define the oppositions, al-Sanūsī relies rather on al-Rāzī’s and al-Khūnajī’s distinctions. Both authors agree, however, on the fact that affirmatives have an import while negatives do not. For this reason, their squares are valid from a logical viewpoint. As diagrams, these squares are nevertheless entirely original, due to the particular arrangements of their oppositional relations. Our analysis shows also that the later author does not seem to know the square drawn by the earlier one and that both authors do not seem to know Apuleius’ or Boethius’ squares. (shrink)

The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the mathematical (...) practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of “geometric judgments” from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) “space” should be revisited for the purposes of life sciences. (shrink)

The forma fluens/fluxus formae debate concerns the question as to whether motion is something distinct from the body in motion, the flow of a distinct form identified with motion , or nothing more than the successive states of the body in motion, the flow of some form found in one of Aristotle's ten categories . Although Albertus Magnus introduced this debate to the Latin West he drew his inspiration from Avicenna. This study argues that Albertus misclassified Avicenna's position, since Albertus (...) could not conceptualize motion at an instant, whereas it is claimed here this was the very position Avicenna adopted. The paper includes an overview of Albertus's discussion and a brief survey of the Avicennan sources upon which Albertus drew. The heart of the paper treats Avicenna's analysis of motion at an instant. Avicenna's general argument was that since spatial points have no extremities, nothing in principle prevents a moving object from being at a spatial point for more than an instant, understood as a limit. It is then argued that Avicenna had the philosophical machinery to make sense of a limit, albeit not in mathematical terms, but in terms of an Aristotelian potential infinite. (shrink)

Summary In terms of control of composition, the fabrication of money was arguably the most demanding of all pre-Industrial Revolution metallurgical practices. The calculations involved in such control needed arithmetical computations involving repeated multiplications and divisions, not only of integers but also of mixed numbers. Such computations were possible using Roman numerals, but with some difficulties. The advantages gained by employing arithmetic using Indo-arabic numerals for alloying calculations would have been the same as for other types of commercial calculations. A (...) method of calculating the proportions of two coinage alloys of different compositions that were needed to produce a third alloy of specified intermediate composition, eventually known as the rule of alternate alligation, was however a specific requirement in the minting of money. This requirement made a particular demand on medieval mathematics, as a solution was not possible using arithmetic alone. Furthermore, the problem became indeterminate when more than two components were available for mixing. All the necessary methods of calculation were described by Fibonacci in his Liber abbaci. These methods are to be found in a sequence of practical arithmetics that stretch through the trattati d'abbaco of the Italian city states during the fourteenth and fifteenth centuries, to the first printed books of Continental authors such as Pacioli. Of the earliest sixteenth-century printed English books on practical arithmetic, The Grounde of Artes by Robert Recorde contained the first advance in the methodology of alternate alligation since Fibonacci. In the seventeenth century, formal proof of the rule of alternate alligation was given by Kersey in his practical arithmetics and in his treatise on algebra. The topic was also addressed by several authors of academic algebras. It was in this century that the problem of indeterminacy in the solution of alternate alligation involving the mixture of more than two components, was finally resolved. English mint accounts provide little direct information on the alloying calculation carried out, and nothing on the methods used. However, the continued description of such calculations in other mint documents, merchants notebooks, and in goldsmiths' handbooks spanning four centuries imply that the calculations described in practical arithmetics were used in practice. Monetary matters had high economic and political profiles during the whole of the time span covered in this study. The influence of such considerations pervades the history of the technology of minting and the mathematics on which it called, and is discussed in context. (shrink)

In terms of control of composition, the fabrication of money was arguably the most demanding of all pre-Industrial Revolution metallurgical practices. The calculations involved in such control needed arithmetical computations involving repeated multiplications and divisions, not only of integers but also of mixed numbers. Such computations were possible using Roman numerals, but with some difficulties. The advantages gained by employing arithmetic using Indo-arabic numerals for alloying calculations would have been the same as for other types of commercial calculations. A method (...) of calculating the proportions of two coinage alloys of different compositions that were needed to produce a third alloy of specified intermediate composition, eventually known as the rule of alternate alligation, was however, a specific requirement in the minting of money. This requirement made a particular demand on medieval mathematics, as a solution was not possible using arithmetic alone. Furthermore, the problem became indeterminate when more than two components were available for mixing. All the necessary methods of calculation were described by Fibonacci in his Liber abbaci. These methods are to be found in a sequence of practical arithmetics that stretch through the trattati d'abbaco of the Italian city states during the fourteenth and fifteenth centuries, to the first printed books of Continental authors such as Pacioli. Of the earliest sixteenth-century printed English books on practical arithmetic, The Grounde of Artes by Robert Recorde contained the first advance in the methodology of alternate alligation since Fibonacci. In the seventeenth century, formal proof of the rule of alternate alligation was given by Kersey in his practical arithmetics and his treatise on algebra. The topic was also addressed by several authors of academic algebras. It was in this century that the problem of indeterminacy in the solution of alternate alligation involving the mixture of more than two components, was finally resolved. English mint accounts provide little direct information on the alloying calculations carried out, and nothing on the methods used. However, the continued description of such calculations in other mint documents, merchants notebooks, and in goldsmiths' handbooks spanning four centuries imply that the calculations described in practical arithmetics were used in practice. Monetary matters had high economic and political profiles during the whole of the time span covered in this study. The influence of such considerations pervades the history of the technology of minting and the mathematics on which it called, and is discussed in context. (shrink)

This is the first critical edition of a fascinating medieval work on music, written in Iraq in the tenth century. It is accompanied by an English translation and full annotation. The Epistle examines not just the technical, scientific, and mathematical aspects of music, but its cosmic, psychological, and spiritual dimensions.

This volume is a collection of essays in honour of Professor Mohammad Ardeshir. It examines topics which, in one way or another, are connected to the various aspects of his multidisciplinary research interests. Based on this criterion, the book is divided into three general categories. The first category includes papers on non-classical logics, including intuitionistic logic, constructive logic, basic logic, and substructural logic. The second category is made up of papers discussing issues in the contemporary philosophy of mathematics and logic. (...) The third category contains papers on Avicenna’s logic and philosophy. Mohammad Ardeshir is a full professor of mathematical logic at the Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran, where he has taught generations of students for around a quarter century. Mohammad Ardeshir is known in the first place for his prominent works in basic logic and constructive mathematics. His areas of interest are however much broader and include topics in intuitionistic philosophy of mathematics and Arabic philosophy of logic and mathematics. In addition to numerous research articles in leading international journals, Ardeshir is the author of a highly praised Persian textbook in mathematical logic. Partly through his writings and translations, the school of mathematical intuitionism was introduced to the Iranian academic community. (shrink)

Summary Syriac translations and Syriac scholars played an important role in the transmission of the sciences, including the mathematical sciences, from the Greek to the Arabic world. Relatively little, unfortunately, remains of the translations and original mathematical works of earlier Syriac scholars, but some materials have survived, and further glimpses of what once existed may be gained from works of later authors. The paper will provide an overview of the earlier materials that have survived or are known to have existed. (...) This will be followed by an account of some of the later materials, which will include an example illustrating how further earlier materials can be recovered from the writings of later authors, and a mention, by way of an epilogue, of an instance where a Syriac scholar was involved in a major historical and scientific event in the West, namely the reform of the calendar by Gregory XIII. (shrink)

The article deals with the Arabic sources of Chr. Clavius in Rome and the six different ways they were used by him in mathematics and astronomy. It inquires especially into his attitude towards al-Farghānī, Thābit ibn Qurra, al-Bi[tdotu]rūjī, Ibn Rushd, Mu[hdotu]ammad al-Baghdādī, Pseudo-Ibn al-Haytham, Jābir ibn Afla[hdotu], and Pseudo-al-[Tuotu]ūsī.

This book highlights the emergence of a new mathematical rationality and the beginning of the mathematisation of physics in Classical Islam. Exchanges between mathematics, physics, linguistics, arts and music were a factor of creativity and progress in the mathematical, the physical and the social sciences. Goods and ideas travelled on a world-scale, mainly through the trade routes connecting East and Southern Asia with the Near East, allowing the transmission of Greek-Arabic medicine to Yuan Muslim China. The development of science, first (...) centred in the Near East, would gradually move to the Western side of the Mediterranean, as a result of Europe's appropriation of the Arab and Hellenistic heritage. Contributors are Paul Buell, Anas Ghrab, Hossein Masoumi Hamedani, Zeinab Karimian, Giovanna Lelli, Marouane ben Miled, Patricia Radelet-de Grave, and Roshdi Rashed. (shrink)

The thesis deals with a section of the major philosophical summa by Avicenna (Ibn Sīnā, d. 1037), namely the Book of the Healing (Kitāb al-Šifāʾ). The summa is structured into four parts, devoted to Logic, Natural Philosophy, Mathematics and Metaphysics; the section at stake is Avicenna’s reworking of Porphyry’s Isagoge (Kitāb al-Madḫal, i.e. “Book of the Introduction”) opening the section of Logic of the Šifāʾ. The thesis is articulated into three main parts, namely (i) an edition of the Arabic text (...) of Avicenna’s Madḫal, (ii) an English translation and (iii) a systematic commentary; these three main parts are preceded by an Introduction (0) divided into two halves: (0a) a General Introduction focused on the doctrinal major innovations of the work and (0b) an Introduction to the Edition of the text, which offers a first comprehensive study of its direct and early indirect tradition and explains the criteria of the edition. (i) The only previous edition of Avicenna’s Madḫal was the one printed in Cairo in 1952 (then reprinted in Tehran in 1983 and in Beirut in 1993), based on ten manuscripts. The huge manuscript tradition of Avicenna’s Madḫal – and, more in general, of Avicenna’s Book of the Healing – is still a matter of investigation. According to the provisional results of the bibliographical research conducted in the present work, it can be estimated that the manuscript tradition of the Madḫal amounts to at least 119 certain witnesses and 14 possible additional witnesses (as explained in the Introduction to the edition, 0b). The text of the Cairo edition, far from being representative of the whole textual tradition, was reconstructed on the basis of ten manuscripts that were not selected on the basis of philological criteria and whose reciprocal stemmatic relations were not clear. Although a critical edition based on all the 119 certain witnesses of the work exceeds the scope of the present work, the edition provided in the thesis takes into account a larger number of manuscripts than the Cairo edition did and, most importantly, selects them by means of a preliminary collation made on portions of the text. The purpose of this preliminary collation is making the selection of the manuscripts employed less arbitrary and providing a reconstruction of their stemmatic relations (in the Introduction, 0b). Overall, 32 manuscripts among the 68 inspected were employed in the edition here proposed, 21 of which were systematically collated and 11 were eliminated as codices descripti. The present edition also assumes as a witness the twelfth-century Latin translation of the work, which is, at the present moment, among the earliest extant witnesses of the text. Hopefully, this editorial work allowed us to improve the text of the Cairo edition in a number of points that are crucial to the correct understanding of Avicenna’s argumentation. (ii) As this project started, complete translations of Avicenna’s Madḫal in modern languages were only available in Turkish and Russian. Chapters of the work had also been translated in other languages, but an English translation of the whole work was still a desideratum. A complete English translation is provided in the thesis, based on the newly-established Arabic text of the work and aiming at preserving the major possible consistency in rendering the relevant philosophical terms. As a complement to the translation, an apparatus of notes is offered to facilitate the immediate understanding of the text. (iii) The third section of the thesis consists in a systematic commentary on Avicenna’s Madḫal, which is meant to support a deeper understanding of the text in its entirety. The commentary deals with both historical and philosophical aspects; more in detail, it includes an analysis and an identification, when possible, of Avicenna’s sources, and underlines the points of major philosophical interest of the work, which are dealt in a more comprehensive manner in the General Introduction (0a). (shrink)

The perfect compass, used by al-Qūhī, al-Sijzī and his successors for the continuous drawing of conic sections, reappeared after a long eclipse in the works of Renaissance mathematicians like Francesco Barozzi in Venice. The resurgence of this instrument seems to have depended on its interest to solve new optico-perspective problems. Having reviewed the various instruments designed for the drawing of conic sections, the article is focused on the sole conic compass. Theoretical and empirical applications are detailed. Contrarily to the common (...) thesis of an independant discovery, various elements suggest a direct descent between the birkār al-tāmm of the Arabic mathematical tradition and the Italian conic compass. Then we present the most probable transmission hypothesis involving: 1° Ibn Yūnus and his disciples of Mosul, 2° Sultan Malik al-Kāmil in Damas, 3° Master Theodore and Frederick II at the court of Sicily, 4° Andalò di Negro and Robert of Anjou in Naples, 5° Lorenzo della Volpaia, Vinci, Sangallo and Michelangelo in Florence, 6° Ausonio, Contarini, Thiene and Francesco Barozzi in Venice. (shrink)

The transformation of mathematics from ancient Greece to the medieval Arab-speaking world is here approached by focusing on a single problem proposed by Archimedes and the many solutions offered. In this trajectory Reviel Netz follows the change in the task from solving a geometrical problem to its expression as an equation, still formulated geometrically, and then on to an algebraic problem, now handled by procedures that are more like rules of manipulation. From a practice of mathematics based on the (...) localized solution we see a transition to a practice of mathematics based on the systematic approach. With three chapters ranging chronologically from Hellenistic mathematics, through late Antiquity, to the medieval world, Reviel Netz offers an alternate interpretation of the historical journey of pre-modern mathematics. (shrink)

This volume presents the first edition of the Arabic translation, by Hunayn ibn Ishaq, of Galen's Critical Days (De diebus decretoriis), together with the first translation of the text into a modern language. The substantial introduction contextualizes the treatise within the Greek and Arabic traditions. Galen's Critical Days was a founding text of astrological medicine. In febrile illnesses, the critical days are the days on which an especially severe pattern of symptoms, a crisis, was likely to occur. The crisis was (...) thought to expel the disease-producing substances from the body. If its precise timing were known, the physician could prepare the patient so that the crisis would be most beneficial. After identifying the critical days based on empirical data and showing how to use them in therapy, Galen explains the critical days via the moon's influence. In the historical introduction Glen Cooper discusses the translation of the Critical Days in Arabic, and adumbrates its possible significance in the intellectual debates and political rivalries among the 9th-century Baghdad elite. It is argued that Galen originally composed the Critical Days both to confound the Skeptics of his own day and to refute a purely mathematical, rationalist approach to science. These features made the text useful in the rivalries between Baghdad scholars. Al-Kindi (d.c. 866) famously propounded a mathematical approach to science akin to the latter. The scholar-bureaucrat responsible for funding this translation, Muhammad ibn Musa (d. 873), al-Kindi's nemesis, may have found the treatise useful in refuting that approach. The commentary and notes to the facing page translation address issues of translation, as well as important concepts. (shrink)