Practical Geometries in Islamic Countries: the example of the division of plane figures Marc Moyon To cite this version: Marc Moyon. Practical Geometries in Islamic Countries: the example of the division of plane figures. Kronfellner Manfred; Barbin Évelyne; Tzanakis Costa. History and Epistemology in Mathematics Education, Verlag Holzhausen GmbH, pp.527-538, 2011, 978-3-854932-08-6. hal-00933041 HAL Id: hal-00933041 https://hal.archives-ouvertes.fr/hal-00933041 Submitted on 19 Jan 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PRACTICAL GEOMETRIES IN ISLAMIC COUNTRIES The Example of the Division of Plane Figures Marc MOYON IREM de Lille & CHSE Lille 1 / UMR “Savoirs, Textes, Langage”, Université de Lille 1, Bâtiment P5bis, Bureau 168, 59655 Villeneuve d’Ascq Cedex [email protected] ABSTRACT The division of plane figures is a geometrical chapter developed in numerous works written in Arabic. In the extension of Greek practices, this chapter is also found in original developments in Islamic countries. The aim of the presentation is to show the diversity from several books of the Muslim Orient and Occident from the 9th century until the 14th century. This diversity is first based on the multiple origins of the problems. They are linked, among others, to the practices of craftsmen, architects, or jurists. For example, jurists had to decide on the sale or the sharing of fields. To divide a geometrical figure in a certain number of similar figures is an important problem for the decorators who embellished palaces, madrasas, and other mosques and mausoleums. Moreover, these problems are illustrated in some writings of eminent geometers. This diversity also expresses itself by the wealth of procedures of construction and resolution for which the whole mathematical knowledge is included. 1 Introduction. It is unfortunate that the history of scientific activities in Islamic countries is so poorly known to Europeans and this history repeats itself. Today, some European scholars (certainly ideologically tainted) neglect the original developments of science in the Islamic era and even deny their appropriation by Christian Europe from the 12th century. Even though studies on the advent and development of scientific activities in Islamic countries are still incomplete or deficient, our goal in this paper is to give some general and major features of their history. Then, we discuss the interactions that have been established between scientific practices and several social, cultural, political, and religious aspects. 2 Sciences in Islamic Countries. 2.1 Generalities on the Scientific Development in Islamic Countries. From Hegira (632), the Islamic countries, that is to say all regions dominated and unified by a single religion – Islam – gradually correspond to a huge Empire. It extends from the Pyrenees to Timbuktu (from North to South), and from Samarkand to Saragoza (from East to West). In this controlled and pacified geographical area, Islamic law is the canon law for the society. All roads, in particular those of trade, are free. In the history of scientific practices in Islamic countries, three major periods can roughly be distinguished1. The first one is a period of appropriation of the knowledge of the Ancients (Syrian, Persian, Sanskrit, or Greek, of course). Scientific activities benefit from the fluidity of 1 For further details, see Djebbar, A., 2001, L’âge d’or des sciences arabes, Paris: Seuil. movement of men and books. The local scientific practices (weights and measure, calculation of inheritance, decorative art, astrology, for example) are not neglected and the scholarly knowledge consolidates them and introduces rational approaches. Arabic, the language of the Qur’ân, is needed as the language of scientific communication2. An important movement of translation grows, from the 8th century until the mid-10th century, to become acquainted with the whole knowledge of the Ancients3. From the first conquests of new territories, Islam held a leading role for the sciences. The second period of the history of scientific practices in Islamic countries runs from the 9th century to the 12th century. It is the so-called “Golden Age of sciences in Islamic countries”, corresponding to a period of scientific creation and development. When the knowledge from the Ancients was assimilated, scientists from Islamic countries taught, commented, and surpassed them. Many scientific foyers emerged, both in the East and in the West, such as in Baghdad, Samarkand, Cairo, Cordoba, but also in Kairouan, Nishapur, and Marrakesh. In addition to improving results inherited from the Ancients, innovations are observed, for example, in medicine, astronomy, and mathematics. New disciplines emerged such as trigonometry, algebra, and combinatorics in mathematics. These developments were widely promoted by several factors including the patronage of Princes and foremost that of the Caliph himself, by various social demands or by the non- intervention of religion in scientific practices. The third period is not a period of decline as it has often been characterized. Original scientific researches continued to occur (in mathematics and astronomy) but they were more isolated in several parts of the Islamic area, both in the West with the Maghreb and Andalus and in the East with actual Iran. This period is characterized, from the late 12th century, by the disappearance of Arabic as the only language of scientific discourse. Indeed, three other languages competed with Arabic: Persian in the East and Hebrew and Latin in the West. The men of science in Islamic countries, whatever their profile, took advantage of collective management and its contingencies to guide and deepen part of their production. We now discuss a few aspects of the “Islamic culture: (taken in its widest meaning) that are considered as an impetus of scientific research in Islamic countries. 2.2 Mathematics and Cultural Aspects in Islamic Era. To be encouraged and developed, any scientific knowledge needs institutional support and cultural values. Science in Islamic countries is no exception. First of all, institutionally, we mention the strong political will of many successive Caliphs to support scientific research and teaching. Then, culturally speaking, several direct or indirect evidences guarantee the existence of interaction between the scholarly science and the know-how of artists and crafstmen. We shall even see that some of these evidences give the proof of a certain stimulation of scholars by or for craftsmen. The Umayyadds (661-750) and the first Abbasids (especially between 750 and 850) expressed the same desire by supporting scientific activities. The Caliph al-Ma‘mûn (813- 2 Thus, due to an abuse of language, science in Islamic countries is sometimes called « Arabic science ». The term « arabic » must have been understood in the meaning of the language used to write and teach the science. It does not refer to geographic, cultural or religious origins. 3 Gutas, D., 1998, Greek Thought, Arabic Culture, New York: Routledge. 833) is probably the more significant. To provide the library Bayt al-hikma [House of Wisdom] in Baghdad, he interceded with Leo V (813-820), Emperor of Byzantium, to obtain books on philosophy and science. He also supported delegations of scholars in Asia Minor and Cyprus to bring books written in Greek. He organised the measurement of the diameter of Earth. He gave assignments to scientists in order to determine the geographical locations of various events described in the Qur‘ân. This same Caliph encouraged al-Khwârizmî to “compose a short book on algebra and muqabala4”, namely Mukhtasar fî hisâb al-jabr wa l-muqâbala [Compendium on calculating by completion and reduction], which can be regarded as the official birth of Algebra as a new field of knowledge. This kind of patronage was still present at least until the first half of the 15th century. This is confirmed by the astronomer and mathematician al-Kâshî (d. 1429) in his correspondence with his father. Member of the scientific staff of Ulûgh Beg (1394-1449) in Samarqand, one of the most important scientific foyer of the Muslim Orient, he wrote on the construction of an mihrâb5 according to the wishes of the Sultan : “His Majesty [once] said: ‘‘We would like to make a hole in the wall of a mihrâb in such a way that the sun may shine through that hole for a short while at the afternoon [prayer] time both in summer and in winter. That single hole must be round from inside, but from the outside it must be in such a way that sunshine cannot pass through it at times other than the afternoon [prayer time]. This [royal wish] had been [already] expressed before my arrival, and nobody had been able to realize it; [but] when I came [here], I did this also6.” This last evidence allows us to evoke the idea that some of the scientific production (research and teaching) can be considered as replies to individual or collective societal needs. These responses are directed to several practitioners such as architects, craftsmen decorators and, according to al-Khwârizmî himself, “inheritance, legacies, partition, law- suits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals (...) are concerned7”. The duality between the scholar and the practitioner is not widely known but is nevertheless real. It is illustrated by three distinguished scholars of Islamic countries who provide evidence on meetings between mathematicians and craftsmen, which can be considered as a forum to discuss methods for designing ornamental patterns in several materials (wood and tile, for example).