4. Mathematical production in the Maghrib during the 14^th-15^th centuries

In the history of scientific activities in the Maghrib, the 14^th century constitutes a privileged moment both for the quantitative importance of the mathematical production that one may observe (in the light of the bibliographical testimonies and manuscripts which have survived), and equally for the content of this production and influence that it would have, during centuries, on the teaching of mathematics in the whole of North Africa and sometimes even in certain sub-Saharan regions.

According to the actual state of our knowledge concerning this period of the scientific history of the Maghrib, one may say that the majority of the mathematical production of this century is a return, in the form of commentaries, summaries or developments, to a part of what had already been discovered or assimilated during the preceding centuries. New contributions are indeed exceptional, which may only confirm, in the case of mathematics, the conclusions arrived at by Ibn Khaldūn, in his Muqaddima, when he evoked the decline of certain scientific activities during his epoch.

This remark gives more importance to the mathematician that we will present now, as he appeared at the same time as one of the last innovators of the great Arab mathematical tradition and as one of the initiators of a new tradition of teaching of mathematics, based on the commentary. This tradition would spread throughout the Maghrib and extend itself even to Egypt.

4.1. The contribution of Ibn al-Bannā

Ibn al-Bannā was born in Marrakech in 1256. There he grew up and acquired an excellent training in various domains. He also lived and taught for some time in Fez which became, after the fall of the Almohads, the capital of the dynasty of the Merinids, and which tried to rival, on an intellectual level, Marrakech, the only city which had the privilege of having been, for almost two centuries (1062-1248), the capital of the entire Maghrib, including vast sub-Saharan zones [Laroui 1970: 147-185]. The information that we possess shows that a strong scientific tradition was established in this city, but the research is not sufficiently advanced to be able to describe the content of this tradition, its links with that of al-Andalus, and the profile of the men of science who lived there.

Returning to Ibn al-Bannā, we have to state immediately that we find ourselves in the presence of the last Maghribian mathematician who was involved in research, to the extent that he tackled problems which were new for the epoch, and that he contributed with original solutions or advanced with new ideas. We have already mentioned his contribution to combinatorics which lies in the prolongation of research activities and preoccupations of Ibn Mun‘im. This contribution is not limited to what he has written in his Tanbīh al-albāb. One finds equally interesting elements in his Raf‘ al-hijāb [The lifting up of the veil] where he established the results that we mentioned earlier and where he lets us understand that in his epoch the problems of enumeration were not only related to the field of language. He also introduced a new trend in Algebra concerning the justification of the existence of solutions to the canonical equations of al-Khwārizmī and he would, according to the testimony of Ibn Haydūr, continue with a reflection on bases different from ten that would have been started by Ibn Mun‘im (in a work that has not come to us) [Djebbar 1980: 76-98; Aballagh 1988: 145-168, 517-543].

But, notwithstanding his exceptional qualities, noted by all biographers, the importance and prestige of Ibn al-Bannā did not only come from his mathematical works. In fact, our scholar distinguished himself from his Maghribian predecessors by the richness and diversity of his production. Based on the inventory that was made, at the time, by Ibn Haydūr, we have registered more than 100 titles of writings that are attributed to him, of which only 32 concern Mathematics and Astronomy [Djebbar & Aballagh 2001], the others being dedicated to disciplines very distant from each other, like Linguistics, Rhetoric, Astrology, Grammatics and Logic.

Among his scientific writings, those related to the Science of Calculation seem to have assured the scientific notoriety of Ibn al-Bannā. His works on the Science of Calculation, that have come to us and been analysed are Talkhīs [The abridgment], Arba‘ maqālāt fī l-hisāb [The four works on Calculation] and the Raf‘ al-hijāb [The lifting up of the veil]. It is indeed the Talkhīs that illustrates the trend and conceptions of Ibn al-Bannā, at the level of the ordering of the chapters, the conciseness, the rigor, the formulation and the absence of all symbolism, although, with respect to this last point, there existed already a practice, in his epoch, both in the domain of the fractions, in that of the algebra of the polynomials and in that of equations.

The factors that led Ibn al-Bannā to inaugurate this type of very condensed manual, in opposition to the great collections of the 12^th-13^th centuries that we mentioned earlier, are perhaps not strictly pedagogical. This might have been the consequence of the whole training of Ibn al-Bannā and in particular of his mystical guiding that favoured a certain esoterism in addition to the difficulty and the abstract character of mathematical notions and techniques. However it may be, this character of the Talkhīs will lead numerous later mathematicians to write more or less detailed commentaries of this manual.

The encyclopedic character of the production of Ibn al-Bannā may have contributed to the social status of the mathematician who was honored by the Merinid power, which led him to leave Marrakech in order to instal himself for a time in Fez, at the invitation of the sultan of the epoch. This eminent position, from which he benefited in the Merinid capital, could only reinforce the authority that he had already acquired through his scientific works. One may also suppose that it was this double status, both scientific and social, that helped him to solve the problems that preoccupied his contemporaries, and which led him to publish an original book whose contents might be related, because of certain aspects, to Ethnomathematics. We are speaking about Tanbīh al-albāb that we have mentioned already on several occasions.

The first part of this little book contains the precise mathematical answers to questions that touch very varied domains of everyday life, like the composition of medicaments, the calculation of the drop of irrigation canals, the arithmetical explanation of a verse of the Koran concerning inheritance, the determination of the hour of the third daily prayer, the explanation of frauds linked to instruments of measurement, the enumeration of delayed prayers which have to be said in a precise order, the exact calculation of legal tax in the case of a delayed payment, etc. [Djebbar & Aballagh 2001]. The second part, which belongs to the already ancient tradition of ludical and cultural mathematics, joins a collection of little arithmetical problems presented in the form of poetical riddles [Djebbar 1995b].

Before Ibn al-Bannā, mathematicians had also solved problems related to everyday life, but often the concrete aspect of the problem was no more than a dressing-up of arithmetical or abstract algebraic exercises. This does not happen in the case of the 17 problems exposed by Ibn al-Bannā. Having said this, and notwithstanding the originality of his book, Ibn al-Bannā was not the first to have solved problems specific to the Islamic city of the Middle Ages. To stay in the Maghrib, one may point out the case of a mathematician from Ifriqiya, al-Jītālī (d. 1305) who wrote a work entitled Kitāb maqāyīs al-jurūh [Book of the measures of wounds] in which he describes the method for measuring all types of wounds and the way to calculate the indemnities demanded by law as compensation to be paid to the wounded person [Djebbar 1995a]. We think that the two examples that we just mentioned are only one aspect of a rich tradition that deserves systematic research, both in mathematical writings and in the works of other disciplines like Law and Astrology.

4.2. The continuators of the tradition of Ibn al-Bannā

At the level of the great orientations of mathematical activity during the Middle Ages, Ibn al-Bannā appeared as the starting point of a whole tradition that extends to the different regions of North Africa and as far as Egypt, and which remained in Muslim Spain. This tradition is that of commentaries. There were thus more than fifteen relatively important works dedicated to the explanation or to the development, and sometimes even to the critique, of his little manual at-Talkhīs. Certain of these commentaries were written by mathematicians of the extreme Maghrib: those of al-Misrātī (14^th century), al-Muwāhidī (14^th century), Ibn Haydūr and of Ibn Ghāzī. One commentary was written by an Andalusian, Ibn Zakariyā, and some were written by either mathematicians of the Central Maghrib, like al-‘Uqbānī (d. 1408), al-Habbāk (d. 1463) and Ibn Qunfudh, or by those from Ifriqiya, like al-Qalasādī, or by Egyptians, like Ibn al-Majdī (d. 1447) and Ibn al-Hā’im (d. 1412) [Djebbar & Aballagh 2001].

So far, none of these commentaries has been the object of a complete analysis, but a comparative study of their contents reveal both quantitative and qualitative differences. At the quantitative level, one finds short commentaries, like that of al-Misrātī, a student of Ibn al-Bannā, who contents himself with explaining the definitions and the algorithms by means of examples without ever leaving the cadre of the Talkhīs. At the other extreme, one finds true treaties for which the content of the Talkhīs seems to be just a pretext or a guideline to permit their authors to expose in their own way, while sometimes criticizing Ibn al-Bannā severely, the themes dealt with by him and others he had deliberately abandoned. Among these treaties which significantly enrich the history of different mathematical disciplines, one may cite Tamhīs [Profound study] of the Maghribian Ibn Haydūr and Hāwī ‘l-lubāb [The collection of the marrow] of the Egyptian Ibn al-Majdī.

At the qualitative level, these commentaries are distinguishable one from the other by the utilisation or not of arithmetical and algebraic symbolism and by the recourse or not to the explanation or to the critique of certain definitions, to the demonstration of the propositions evoked by Ibn al-Bannā and to the justification of the validity of the algorithms that he exposed. One representative commentary of this category of works, whose authors did not confine themselves to the illustration of their objectives by means of examples, is that of al-‘Uqbānī, a mathematician of the Central Maghrib who lived and taught in Tlemcen. It seems that this might be one of the last Maghribian works containing demonstrations using the propositions of Euclid’s Elements or instruments borrowed from works prior to the Arab mathematical tradition [Harbili 1996: no. 7].

Having said this, the detailed analysis of the most important chapters of these commentaries permits us to make further remarks concerning both the nature of mathematics taught in the Maghrib in this epoch and its level:

In the first place, one notes that the level of the mathematics that is exposed is not lower than that of the previous period, but one does not recover certain themes that had been taught from the 10^th century, such as the extraction of the approximate cubic root of a number, or the computation of new pairs of amicable numbers. This phenomenon was already perceptible in the work of Ibn al-Bannā and it was only to extend from the 14^th century onwards.

In the second place, one does not notice any new contribution in these commentaries, either on the theoretical plane or at the level of the applications of earlier ideas and techniques. The most meaningful novelty is situated at the level of written expression with the progressive use of a relatively elaborate symbolism. The use of this symbolism which, as already mentioned, made its appearance in the writings of al-Hassār and Ibn al-Yāsamīn, was frozen or curbed during the entire 13^th century and the first half of the 14^th century. At least no mathematician of this epoch used it in the writings that came down to us. Indeed, even after this period, this symbolism does not seem to have been used by all commentators. In the case of the Extreme Maghrib, one finds it with al-Muwāhidī and Ibn Ghāzī, but not with al-Misrātī and Ibn Haydūr. In the Central Maghrib, only Ibn Qunfudh and al-‘Uqbānī used it [Djebbar 1980; Guergour 1990; Harbili 1996: no. 7]. In Ifriqiya, one finds this symbolism with al-Qatrawānī [Lamrabet 1981: 141-143; Djebbar 1986a: 118-120] and, later, with al-Qalasādī [Woepcke 1858-59]. Outside the Maghrib, the same symbolism is utilised by the Egyptian Ibn al-Majdī, although it is absent in the works of his compatriot Ibn al-Hā’im [Djebbar 1980: 48]. In relation to this, it would be interesting to see if this contradiction does not reflect two conceptions of teaching mathematics or, at least, of what came down of these two conceptions to the teachers of this epoch.

A final remark on the commentaries of the 14^th-15^th centuries concerns the wording and composition of their content and the style that is used. At this level, two trends are perceptible: the first is characterised by their stereotypical formulations which correspond to the usual style of mathematics, and the second, more rhetorical, is of a higher cultural level in the measure where the authors prolong and complete their mathematical explanations with grammatical, literary or philosophical commentaries. This trend contributes nothing on the mathematical level, but it informs us about some aspects of the culture of the epoch and the progressive intervention of this culture in certain technical domains.

In the domain of Algebra, different sources inform us that the works of Ibn Badr [Sanchez-Perez 1916], al-Qurashī, Ibn al-Yāsamīn and Ibn al-Bannā continued to be taught in cities of the Maghrib such as Fez, Tlemcen, Sebta and Tunis. However only the poem of Ibn al-Yāsamīn seems to have been in favour with the commentators, both in the Maghrib and in Egypt. Among the commentaries that circulated from the 14^th entury onwards, one may cite those of Ibn Qunfudh and al-Qalasādī in the Maghrib [Guergour 1990; Zemouli 1993] and those of al-Māradinī (d. 1501) and Ibn al-Hā’im (d. 1412), in Egypt [Shawqi 1987; Souissi 1983b].

Having said this, and if one excludes only one commentary of Raf‘ al-hijāb realised by Ibn Haydūr, no other work on Calculation or on Algebra in the 12^th-14^th centuries motivated the Maghribian commentators. Indeed until today, we have not found any mention of an eventual commentary of the great works on Calculation of al-Hassār and of Ibn al-Yāsamīn or of the book of Ibn Mun‘im. Similarly we have still not discovered any Maghribian commentary on the classics of Arab Algebra of the 9^th-10^th centuries, which nevertheless circulated in Andalusia and in the Maghrib.

The explanation of this phenomenon is not simple: one may look for the reasons either in a lowering of the general standard of teaching, or in the stopping of research activity, or in the lack of interest in the theoretical aspects of scientific disciplines. These causes are in fact linked one to another, and all can be traced to factors beyond scientific activity itself which the great Maghribian historian of the 14^th century, Ibn Khaldūn, had already, in his epoch, understood the effects on society as a whole and, from there, on the dynamism of scientific and cultural activities of the entire Muslim West.

Whatever the cause, one notes that difficult Maghribian works, or those having that reputation, are left by the commentators or are only used to better clarify the explanation of this or that passage of the Talkīs of Ibn al-Bannā. One notes also that there is, from this epoch onwards, a change in the scientific production of the Muslim West, even at the level of references to the works. One continues certainly to evoke Euclid, Nicomachus and al-Khwārizmī, but it is foremost the authors of the Maghrib or of Muslim Spain who are cited in the commentaries of the 14^th-15^th centuries that have come down to us.

By Professor Ahmed Djebbar, to be continued…