Abstracts of Symposium 15

Tradition and Innovation in Mathematics in Late Antiquity and the Middle Ages

Ahmed Djebbar
Les mathématiques grecques en Occident musulman: L’exemple des Eléments d’Euclide et de l’Introduction arithmétique de Nicomaque

Entre la fin du VIIIe siècle et le début du XIXe, Les Eléments d’Euclide (IIIe s. av. J.C.) et l’Introduction arithmétique de Nicomaque (IIe s.) ont été parmi les ouvrages scientifiques grecs les plus enseignés, étudiés, commentés et enrichis par les mathématiciens des pays d’Islam.
L’étude des sources arabes produites en Orient durant cette longue période ont montré que les prolongements de ces deux traditions ont été très importants dans le centre de l’empire musulman et en Asie Centrale. Mais, les travaux de ces dernières décennies ont révélé la circulation, en Occident musulman (Andalus et Maghreb), de différentes versions de ces deux ouvrages, ainsi que leur présence dans les manuels d’enseignement et dans des travaux de recherche.
En nous basant sur des sources inédites, dont une partie n’a encore fait l’objet d’aucune publication, notre communication apportera quelques éléments sur les différentes versions des Eléments et de l’Introduction arithmétique qui ont circulé en Andalus et au Maghreb. Elle donnera également des informations sur les écrits produits dans ces deux régions et qui contiennent de nouveaux développements théoriques.

Michalis Sialaros and Jean Christianidis
Rhetoric of Mathematics: The Case of Diophantus of Alexandria

Over the last few decades, historians of science have shown a growing interest in exploring the cultural framework of ancient mathematical texts—an extremely challenging task, given the scarcity of the extant sources. In this direction of research, a growing number of recent studies have hinted at the existence of a multifaceted interaction between ancient mathematical practice and the art of rhetoric, especially from the Late Antiquity onwards. The main aim of this paper is to further explore this idea by examining the case of Diophantus of Alexandria (c. late 3rd century AD) and, more specifically, the linguistical patterns and methodological approaches that appear in his most celebrated work, the Arithmetica. It is true that almost all research on the Arithmetica has been conducted with an eye towards its apparent aim; namely, to provide solutions to a number of particular arithmetical problems. Without neglecting the importance of these studies, we propose a new reading of the Arithmetica in accordance to Diophantus’ own exposition in the introduction of the treatise. There, Diophantus explicitly states that the principal aim of the Arithmetica was to teach the reader how to employ his general method (i.e. algebra) in order to solve arithmetical problems. Through this prism, we argue, the Arithmetica must be read also as a pedagogical treatise.

Dora Touliatou
Indeterminate analysis in the “heronian”corpus? A new reading of problems 24.1-13 of Heron’s Geometrica

Thirteen problems from the codex Constantinopolitanus palatii veteris no 1, a Constantinople manuscript of the tenth century which is our unique witness for Heron’s Metrica, now published and translated by J. L. Heiberg with comments by H. G. Zeuthen, are traditionally interpreted by the historians of mathematics as problems of indeterminate analysis (e.g. T.L. Heath and E. M. Bruins) or as problems equivalent with indeterminate equations (e.g. I. G. Basmakova). Both parties share the same methodological approach: the solving-algorithm which concludes the solution of each problem seems to follow an algebraic identity that functions on the one hand as an interpretive basis of the resolutory procedure, and on the other hand as “proof” of the algorithm. As a consequence, the algebra of that period acquires a demonstrative character. We will attempt to revisit this interpretation employing the newly contextualized historiographical category of “pre-modern algebra”, which has been proposed for describing the period of algebra’s history which precedes the period of “modern” algebra. More specifically, “pre-modern algebra” could be described as a numerical problem-solving method which has a specific technical vocabulary and syntax. By using problem 24.10, as a case study, we discuss the reconstruction of the solution given by T.L. Heath, as well as the ones given by Diophantus, al-Sulamῑ and al-Khwarῑzmῑ. The comparative analysis illustrates the fundamental characteristics of the two approaches. Through this prism we will try to look for elements of “pre-modern” algebra in the solution of Heron’s problem. The result of this inquiry is negative: the computational procedure is, apparently, arithmetical.

Ioanna Skoura
Computus ecclesiasticus in Byzantium

The knowledge of computus ecclesiasticus, namely, the knowledge of calculating Easter and other moveable feasts depended on it, was a need felt in the Christian World after the First ecumenical Council of Nicaea. Besides an accurate knowledge of both the solar and lunar cycles, the complex task of dating Easter required the ability to calculate dates from those cycles using calendrical algorithms. A tradition was then created of writing texts aiming at satisfying this need, which was, of course, felt also in Byzantium, in which this tradition was vivid, as it is attested by several eponymous and anonymous texts on computus ecclesiasticus written during the Byzantine period.
In this paper, after first giving a brief overview- classification, general characteristics, transmission- of texts on computus ecclesiasticus preserved from the Middle and Late Byzantine periods, I will discuss about the eponymous fourteenth-century texts of this kind. My focus will be on a newly-discovered eponymous text on computus ecclesiasticus. I will show the characteristics and the astronomical and arithmetic background of this text, which is representative of a tradition that has not received considerable attention by modern scholars.

Abdelmalek Bouzari
La géométrie des Coniques en Occident musulman

Apollonius de Perge (IIe av. J. C.) est l’un des grands géomètres grecs qui a profondément marqué, depuis l’Antiquité, le développement de la géométrie, tant en pays d’Islam, du IXe au XVe siècle, qu’en Europe à partir du XVIe siècle. Notre contribution portera sur la circulation de son Livre des Coniques dans les foyers scientifiques de l’Occident musulman.
Dans la première partie, nous montrerons brièvement comment grâce à la contribution de mathématiciens comme Eutocius (ca.501), les frères Banū Mūsā (IXe), Thābit Ibn Qurra (m.901) et Ibn Abī Hilāl (IXe), cette théorie a circulé puis a intervenu dans les mathématiques arabes.
Dans la seconde partie, nous verrons la présence des Coniques d’Apollonius dans la tradition mathématique d’al-Andalus à travers trois importants ouvrages. Le premier est celui d’al-Mu’taman (m. 1085) et est intitulé : le Kitāb al-istikmāl [Livre de la complétion]. Quant aux deux autres ouvrages perdus, ils sont attribués respectivement à Ibn Sayyd et Ibn as-Samh, tous deux du 11e s. Pour leur contenu, nous nous contenterons de quelques traces indirectes pour le premier et de quelques fragments en hébreu pour le second.

Athanasia Megremi
Problem solving tradition and Diophantine legacy in Greek Arithmetic: testimonies from the Anthologia Palatina

Scholia to the arithmetical epigrams of the Palatine Anthology, the famous collection of hellenistic, late antique and early medieval Greek epigrams, provide us with valuable information concerning the problem solving traditions within the broad spectrum of Greek Arithmetic. One of the methods of problem solving attested in this collection is the ‘way of Diophantus’. In this paper we will be investigating evidence from the Byzantine commentary tradition that present the transmission of the mathesis of Diophantus of Alexandria, in the context of Greek arithmetical tradition, in the Greek-speaking world. We will suggest that Diophantus’ heritage was present in intellectual milieus of the Greek-speaking world during the late antique and early medieval times.

Ezzaim Laabid
Les procédés mathématiques utilisés dans la résolution des problèmes des héritages en occident musulman (XIe-XVe s) : entre tradition et innovation

Le domaine des héritages est l’un des domaines de la tradition mathématique arabe qui ont contribué à la pérennisation de pratiques mathématiques très anciennes tout en s’ouvrant par moment aux pratiques mathématiques issues des recherches réalisées au sein de cette tradition. L’analyse de plusieurs textes sur ce sujet ont permis de déceler l’utilisation de plusieurs genres de procédés de résolution. On y trouve, en effet, des procédés hérités de l’antiquité et dont il est difficile de tracer le cheminement, et d’autres procédés qui semblent être le fruit d’innovations de leur époque. Par ailleurs, ces innovations ne se produisaient pas toujours sans susciter des réactions à leurs égards de la part de certains milieux conservateurs qui les refusaient.
Dans cette communication, nous tenterons de présenter les principales innovations qu’ont connues les procédés de résolutions dans ce domaine, et notamment en occident musulman entre le XIe-XVe siècle, et nous relatons aussi les réactions que ces innovations avaient suscitées.

Alain Bernard
Theon’s commentary on the Almagest, as series of problems

Ancient scientific commentaries, like Theon of Alexandria’s extensive commentaries on Ptolemy’s Almagest or Handy Tables (or both) are highly interesting testimonies in spite of their poor reputation. But their precise cultural and historical value is not so easy to evaluate.
More recently, the question of the worth of Theon’s commentary has been renewed through the recognition that the arabic-speaking philosopher al-Kindi grounded his « mathematical » philosophy of knowledge on the reading of the Almagest as well as on Theon’s commentary to it (E. Gannagé ). This is based on the ground breaking work initiated by J. Feke to delineate the coherency of Ptolemy’s philoosphical system (Feke 2018). In this context, I have contributed to the discussion by showing that Theon’s commentary on Ptolemy’s « philosophical preface » to the Almagest is structured around two main difficulties of the text, that Theon sets up to elucidate in accordance with the general principles of his own commentary. If we then name problems these exegetical difficulties brought out by Theon, the following general questions arise : to what extent is Theon’s commentary structured through such problems ? Of what nature are these problems ? Does the commentary as a whole constitue a series of such exegetical problems ? And then what is the meaning and structure of this series ?