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Mathematics in the Medieval Maghrib: General Survey on Mathematical Activities in North Africa

Figure 1: Front covers of two books by Ahmed Djebbar: Une histoire de la science arabe (Paris, 2001) and L'âge d'or des sciences arabes (Paris, 2005).

The text that we will present here is the first part of a study which attempts to present the essential aspects of mathematical activity in North Africa since the 9^th century. The abundance of material has obliged us to report, in a future article, the study dedicated to mathematical activities in Egypt during the same period. The whole work, as soon as concluded, will constitute the second volume of the study "Recent research on the history of Mathematics in Africa: an overview", by A. Djebbar and P. Gerdes, whose first volume has already been published (Gerdes 1992: 3-32, 1994) [1].

This work is just a first outline that needs to be completed by the presentation of certain activities which abundantly used the different mathematical disciplines and were sometimes considered more important than mathematics. Essentially this refers to Astronomy, Astrology, and the Science of Succession Division or Science of Inheritance. We have satisfied ourselves by evoking certain of their aspects, or certain authors who are particularly distinguished, hoping to deal with them more fully in a later study.

However, so as to avoid eventual misunderstandings or ambiguities, it seems useful to make some remarks about the terminology that will be used in the different parts of this study.

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Figure 2: Professor Ahmad Djebbar in Sharjah, UAE, during the International Conference of the History of Science among the Arabs and the Muslim (March 2008). © FSTC and Salim Ayduz..

First of all we need to specify what we understand by "scientific tradition". As soon as we begin to deal with the contents of scientific, and more particularly mathematical activities, which take place in the context of the Arab-Islamic civilisation, it is not possible to speak of a specific tradition in the Maghrib (as opposed to that of Muslim Spain or that of the East). In fact we may speak of one overall tradition, that of Arabic mathematics - that is mathematics that has been thought about, written and taught in the Arabic language (also called Mathematics of the countries of Islam) -, which has developed in the East since the end of the 8^th century, and that has been partially transmitted to the cities of the Muslim West and of Central Asia, and later to southern Europe, by means of translations (essentially Latin and Hebrew). This tradition has been assimilated, revived and enriched by the scientific environments of the different countries of Islam that gave it, sometimes, certain specific mark at the level of this or that research and teaching orientation, as well as at the level of composition of the contents of the works, of the terminology, or of the classification of the studied disciplines. But, as far as we know, this internal process of differentiation has not lead to the emergence, at local or regional level, of a new mathematical tradition characterised by its own concepts and paradigms. The scientific traditions which are evoked in this study are such in an externalist sense that takes scientific practice in relation to its environment into account.

The same remark applies to the contents of the mathematics produced or taught in each of the five vast regions that constituted, in the Middle Ages, North Africa, that means the Extreme Maghrib, the Central Maghrib, the Oriental Maghrib that at that time was called Ifr¥qiyya, Egypt and the extended sub-Saharan zone of Muslim confession that went then by the name of BilŒd as-S'dŒn [Countries of the Sudan]. The analysis of the scientific texts which have come to us does not allow us to speak of regional specificities either at the level of content itself or at the evolution of its contents. On the contrary, we observe, following the epochs, notable differences between the regions, at the level of the number of mathematicians or of their works, of the vitality of this or that taught discipline, of the dynamism of the different scientific foyers of each of these regions.

It is also necessary to explain what we here understand by "Maghribian mathematicians". In bio-bibliographic treatises, in particular those written by oriental authors, one finds a certain number of scholars, poets and writers who are referred to as "Maghribian" without being born in the Maghrib. That is the case, for example, of scholars native to Muslim Spain. That is also the case of those whose parents are native of the Muslim West but who have grown up or have been educated in the East. In addition, there are persons whose origin is not Maghribian but who have played a role in the scientific activity of the Maghrib. Thus, in the case of mathematicians, it is necessary to specify that only one category is representative of the mathematical production of the Maghrib. This concerns those who lived there during a given period of their life and who, through their teaching or through their production, contributed to the development or the perpetuation of a local or regional mathematical activity. As an example of this type of scholar, one may cite al-Qatrawani (15^th century), who was born in Egypt and who lived for a certain time in Tunis where he wrote one of the books that we will mention later [Lamrabet 1981: 41-91; Djebbar 1986a: 118-19; Hadfi 1989, 1992: 138-39].

With respect to the others, one may classify them into two categories: the first regroups all who were natives of the Maghrib and who have left it in order to install themselves in another region of Africa (such as Egypt or the Sub-Saharan regions of the continent). Among the representatives of this category, one finds al-îasan al-MurrŒkush¥, one of the greatest specialists of astronomy in the 13^th century, who lived essentially in Egypt [Sédillot 1834-35, 1844; Souissi 1982; Murrakushi 1984].

The second category regroups all those who were born in or were natives of the Maghrib but who were, in reality, educated in the cities of the East or of Muslim Spain, and who spent the largest part of their life there. One of the most eminent representatives of this category is as-Samawth-14^th centuries [Rashed & Ahmad 1972; Rashed 1984].

As these mathematicians or astronomers have not participated, in any way, in a scientific activity in the Maghrib and as these have not been vectors of this activity, we have not judged it useful to mention their different contributions here.

At the end of this review of ten centuries of mathematical activities in the Maghrib, it seems us useful to make some remarks of a general nature concerning the content of these activities and what has been transmitted to other regions of Africa.

The first remark concerns certain non-Arab computational survivals. The information presented shows that mathematical practice in the Maghrib inscribes itself essentially in the Arab tradition. But, this does not mean that the practice was unique. One notes in fact, in the area of computation, and that seems to be a peculiarity of the Extreme Maghrib, the survival, from the pre-Islamic epoch, of a computational practice that uses symbols called the ciphers of Fez. These symbols distinguish themselves from the ghubar (dust) ciphers, i.e. today's ciphers, by their number and form. The persistence of this practice was such that mathematicians like Ibn al-Banna in the 14^th century, and others less prestigious, wrote manuals to explain its principles and use [Djebbar 1987: 239-40].

The second remark concerns the place of the Maghrib in the mathematical tradition of the Muslim West. It is clear, in the light of the information that we have presented, that it was essentially the cities of the Extreme Maghrib, in particular Sebta, Fez and Marrakech, which took the relay from Muslim Spain in mathematical activity from the 12^th century onwards until the end of the 14th. After this period there was a greater intervention of two other scientific poles: Tlemcen in the Central Maghrib and Tunis in Ifriqiya. The reasons why this relay did not take place before the 12^th century are not easy to determine with certainty. Among the reasons is probably the proximity of Muslim Spain and its dynamism during the 9^th-11^th centuries, which regularly attracted the elite who were first trained in the cities of the Maghrib and who then installed themselves provisionally or definitively in a city of al-Andalus. As examples, one may cite the case of al-Wahrani (ca. 1037) for the Central Maghrib [Suter 1900: no. 251], al-Kal‘i (d. 1111) for Ifriqiya [Djebbar 1988b: 64] and Ibn Yasin (9^th century) for the Extreme Maghrib [Suter 1900: no. 106]. This phenomenon was able to influence the constitution of scientific foyers of high level negatively in the Maghribian metropoles.

Other probable causes are to be found in the economic sphere where the Maghrib appeared, until the Almoravid epoch, much more as a relay than a pole attracting wealth and know-how. From the 12^th century onwards, it seems likely that diverse factors were united to allow the Maghrib to become a pole. But, it is also from this epoch onwards that Arab scientific activity taken as a whole started to show signs of running out of breath and slowing-down in the metropolis of other regions. Having said this, the 12^th-13^th centuries have not as yet revealed all their secrets and it is very likely that this short period during which the Maghrib realised its political unity has been, in the field of mathematics, still more fruitful than is apparent from the few works that came down to us and that we havepresented briefly.

The third and last remark concerns the role of the Maghrib in the diffusion of the Arab mathematics. This diffusion is first effected, it seems, towards southern Europe in a direct manner as a consequence of translations, such as the presently unique example of the Hebrew translation of the Kitab al-bayan of al-Hassar, or in an indirect manner by the assimilation of local teaching realised in Arabic followed by the elaboration of manuals or treaties in Latin or Hebrew. The most famous example that illustrates this phenomenon and is still little studied, is that of Fibonacci. As he says himself, this Italian scholar was trained, when very young, in Bougie, one of the Maghribian scientific poles of the 12^th century and, later, he reproduced, in his Liber Abbaci, certain aspects of the Magrebian mathematical tradition and in particular with respect to the symbolism and computation of fractions [Vogel 1970-80: 605-609].

However according to our information, the transmission of mathematical writings of the Maghrib essentially went in two other directions. The first is that of the East, more precisely Egypt, where the writings of al-Hassar, Ibn al-Yasamin and Ibn al-Banna circulated or were the object of commentaries. This fact is confirmed by the testimony of the encyclopedist of the 14^th century, Ibn al-Akfani [Ibn al-Akfani 1990], and by the commentaries of the Talkhis. The content and the modalities of this transmission have still not been the object of an in-depth study, but a comparative analysis of the documents that exist might clarify more about this little known phenomenon.

The third direction in which the transmission went is that of the Sub-Saharan Africa. This phenomenon is, unfortunately, still less known than the two preceding ones. It seems that it was the Extreme Maghrib that was the principal relay for this transmission that essentially concerned the Science of Computation and Astronomy. We have a first confirmation of this through the manuscripts that are today in the Ahmad Baba library in Timbuktu. Among these manuscripts, only one, treating computation, is attributed to a scholar from the region. It concerns Ahmad Babir al-Arawani, a mathematician native from Arawani (Mali) who lived after the 16^th century, as he refers in his writing to an arithmetical poem, ad-Durra al-baydha' [The white pearl] of a mathematician of the Central Maghrib called al-Akhdhar (d. 1575) [Ms. Bibl. Ahmad Baba no. 3027]. The other manuscripts are either mathematical poems, like those of as-Samlali and ar-Rasmuki, both the Extreme Maghrib [Lapousterle 1990], or also astronomical writings, like the Kitab tarhil ash-shams [The book of the movement of the sun] of Ibn al-Banna.

We cannot appreciate the importance of this diffusion on the basis of the rare elements that are in our possession, and a number of questions concerning scientific activity South of the Sahara remain to be answered and probably so for some time. However, if one takes into account the information that came down to us in respect of the cultural history of this region [Batily 1989; Dramani-Issifou 1982; Niane 1975; El-Fasi & Hrbek 1990; Niane 1975a, 1975b, 1985], it seems reasonable to think that the circulation of scientific writings was more important than the documents accessible today suggest. Likewise one may suppose that students native of Sub-Saharan zones, perhaps had the possibility to move themselves to the North, for example to undertake a pilgrimage to Mecca or for other reasons, as happened with Ahmad Baba at-Tambukti (d. 1627). One or some other of these circumstances would have given these students the opportunity to perfect themselves under the guidance of known professors, before themselves becoming teachers or authors of works, as shown by the example of al-Arawani that we mentioned earlier, and that of al-Katsinawi (d. 1741), a scholar native of Katsina (Nigeria) who lived for a time in Cairo and who specialised in the construction of magic squares [Zaslavsky 1973: 138-151; Kani 1992b: 17-36; Gerdes 1992: 17; Sesiano 1994].

However, we cannot evaluate objectively either the content of the scientific production or the nature and intensity of the exchanges between the men of science of the different regions of North Africa, until new investigations of as yet inaccessible documents concerning all aspects of the intellectual life of the cities of these regions are carried on. Having said this, and waiting for the discovery and comparative analysis of these documents, we hope that the few elements we have presented already allow us to convince the reader of the importance of the scientific heritage of this part of Africa.

[1] See also the volume published recently: Mathematics in African History and Cultures: An Annotated Bibliography by Paulus Gerdes and Ahmed Djebbar, African Mathematical Union, 2004 (430 pp., ISBN 978-1-4303-1537-7). New updated and extended edition, 2007 (Chief Editor).

* Professor at the University of Sciences and Technologies Lille I in Lille, France, and scholar at the Laboratoire Paul Painlevé, UMR 8524 (Centre National de la Recherche Scientifique and Lille I University, France). The article was published originally as Ahmed Djebbar, "On Mathematical Activities in North Africa since the 9^th Century. First part: Mathematics in Medieval Maghrib", AMUCHMA Newsletter (Newsletter of the Commission on the History of Mathematics in Africa, published by Universidade Pedagógica in Maputo, Mozambique), No. 5, September 1995; online at: http://www.math.buffalo.edu/mad/AMU/amu_chma_15.html#3. In the present version, copy editing, images and captions were added, as well as a thorough update of the bibliography, by www.MuslimHeritage.com (Mohammed Abattouy, Chief Editor).