The project aims to carry an in-depth investigation on the algorithmic features of some deductive patterns in pre-modern mathematics. The project will be mainly focused on Greek mathematics, but perspectives will be opened on such mathematical cultures as Arabo-Latin, Babylonian. Chinese, Egyptian and Indian, where algorithms played a major role in shaping deductive patterns. Task 1) Chinese mathematical texts (Andrea Bréard) Research will focus on the treatment of finite arithmetical series in Zhu Shijie's The Jade Mirror of Four Primordials and Qing-dynasty commentaries written after its rediscovery at the beginning of the 19th century. Systematically deductive aspects by which Zhu Shijie constructs a system of finite arithmetical series will be studied. Task 2) Greek mathematical texts (Fabio Acerbi, Nathalie Gasiglia, Myriam Hecquet, Bernard Vitrac) Task 2.1) Deductive structures in Greek mathematics. A wide ranging study of the interactions between deductive patterns in mathematical proofs, the ancient method of analysis and synthesis, and ancient Greek logic, in particular Stoic logic. A software will be produced to analyze the 'deductive density' of a Greek mathematical text. Task 2.2) Edition of Greek mathematical texts. i) The Metrica by Hero of Alexandria. Their edition in the Heronian Opera omnia is unsatisfactory. ii) The so-called Geometrica and Stereometrica. These 'treatises' have been edited in the Heronian Opera omnia, but they are in fact philological patchworks made up of disparate treatises attested in manuscripts of the metrological corpus. iii) The De polygonis numeris by Diophantus of Alexandria. It has been edited in the Diophantine Opera omnia, but the edition is not satisfactory. iv) The anonymous Prolegomena to the Almagest. The entire text still lies unedited. Task 3) Arabic and Latin mathematical texts (Marc Moyon, Mohamed Aballagh) The possible filiations between the Arab tradition of al-Andalus and the Latin texts written from the 12th century are studied: the pre-algebraic practices will be privileged with the Arab texts of misâha [measurement] and the Latin Practica Geometria. Latin texts of interest: 1) The Liber mensurationum by Abû Bakr. This treatise has been edited, but still lies not translated. 2) The Liber Saydi Abuothmi and the Liber Aderameti. Only a (non critical) edition and a light commentary do exist. 3) The Liber embadorum by Abraham Bar Hiyya. On the Arabic side, the seminal role of the work of the Maghreb mathematician Ibn al-Banna al-Murrakushi (1256-1321) in the development of logistic and algebra in 14th century Maghreb mathematics will be studied, through edition and translation of Ibn Zakariya al-Gharnati's Commentary on Ibn al-Banna's Talkhis Task 4) Babylonian and Egyptian Mathematical texts (James Ritter) Research is proposed in the following two directions: 1) The mathematical corpus in the case of Egypt has been systematically explored from the algorithmic point of view. A similar near exhaustive analysis needs to be carried out for the Mesopotamian mathematical texts. In both cases the structural role of subalgorithms, branchings and so forth need to be made explicit and their logical and pedagogical roles understood. 2) The similarity between the algorithmic structure of the mathematical texts and those from other disciplines has only been touched upon up to now. New examples have recently been uncovered in unexpected areas: glass-making, harp-tuning. Task 5) Egyptian Demotic mathematical texts (Annette Imhausen) As Egyptian mathematics has traditionally been seen as having reached the height of its development in the early Middle Kingdom, it is planned to extend the scope of the studies of Egyptian mathematics by taking up the Demotic mathematical problem texts, which are written in an algorithmic style. In 1972 R. Parker claimed to discern a Mesopotamian influence in some of the problems. Examining their procedures for solving problems in terms of an algorithmic analysis makes it possible to achieve a detailed comparison of the respective mathematical methods they used. Task 6) Sanskrit mathematical texts of the Indian subcontinent (Agathe Keller, Catherine Singh) A comparative study of mathematical texts from the 5th to the 12th century will be undertaken, examining what characterizes the way they express algorithms. The goals are: 1) editing and translating portions of the commentaries to the Âryabhatîya, a 5th century astronomical text with a mathematical chapter. 2) editing and translating portions of a commentary (ca. 864) to the Brâhmasphutasiddhânta, an astronomical treatise critical of the Âryabhatîya, with a mathematical chapter. 3) editing and translating portions of the commentaries to two practical mathematical treatises during this period, the Patiganita and the Ganitasârasangraha. 4) Setting up a close comparison of the modes of expressing and explaining algorithms in these different set of texts.