3. Mathematics in the Maghrib during the Almohad epoch (12^th-13^th centuries)

The importance of the 12^th century in the political and economic history of the Maghrib has often been underlined by the specialists. It is thus reasonable to think that this importance extends to other spheres of social life. However, the cultural and scientific history of this period still remains, although some contributions exist, there is a vast unexplored field. To take the example of mathematics, one is obliged to note that, in the light of the research results, if one disregards two or three little algebraic or arithmetical poems that we will evoke later, only four works (of which one is incomplete) have come to us. It needs, however, to be specified that the discovery and the analysis of some of them took place only a few years ago.

The authors of these works are, in chronological order, al-Hassār (12^th century), Ibn al-Yāsamīn (d. 1204) and Ibn Mun‘im (d. 1228), to which one should add a fourth, al-Qurashī (d. 1184), who had as much influence as the others through his writings, which unfortunately have not come to us but on the contents of which we have some information.

The importance of these four scholars is due to several reasons : in the first place, and independently of their origins, they may be considered mathematicians of the Maghrib in the sense that all four seem to have lived there and published certain parts of their mathematical works, even if some of them were educated, totally or partially, in a city of Muslim Spain. In the second place, these are the first mathematicians of this part of North Africa of whom the contents of certain parts of their writings have come to us, permitting us thus to have direct information on important aspects of mathematical activity in this region. In the third place, one may consider the works written in this epoch as direct testimonies of two closely linked phenomena. The first is the diffusion, at a larger scale than before, of part of the Andalusian mathematical tradition towards North Africa, through the intermediary of the cities of the Maghrib, such as Sebta, Fez and Marrakech for the Extreme Maghrib, Bougie for the Central Maghrib, and Tunis for Ifriqiya. The second phenomenon is the redynamisation of the teaching and research activity in the Maghrib, in particular due to the patronage of the first Almohad caliphs and then of those dynasties who have divided their empire and who have made efforts to continue their patronage in favour of theologians, poets, writers and scholars.

3.1. The contribution of al-Qurashī

We know little about the life of Abū ‘l-Qāsim al-Qurashī, his education and his scientific production. This scholar, who was a native of Seville, in Spain, spent a period of his life in Bougie where he died in 1184. The biographers who evoked him consider him a specialist in Algebra and in the Science of Inheritance (next to his specialisation in certain religious sciences) [Zerrouki 1995: no. 5, 10-19].

In Algebra, al-Qurashī is known for his commentary on the book of the great Egyptian mathematician Abū Kāmil (d. 930). This commentary has not yet been recovered but its importance is confirmed by the historian Ibn Khaldūn (d. 1406) who considered it one of the best treaties written on the book of Abū Kāmil [Ibn Khaldūn 1967: 899], and the judgment of somebody who knows because Ibn Khaldūn had an excellent mathematical education, and one even attributes him a youth of writing on this discipline [Djebbar 1988b: 68-69]. Relative to the content of the treatise of al-Qurashī, we have found interesting fragments in the book of Ibn Zakariyā (d. 1404). These few elements permit us to say that this work of al-Qurashī was not a simple commentary on a famous treatise of algebra of its time. One finds here indeed some new elements, first of all, at the level of presentation, since al-Qurashī starts by presenting the objects and the operations of Algebra before explaining the solution of the canonical equations followed by the demonstration of the existence of the solutions of these equations. In these two chapters, one remarks that al-Qurashī distinguishes himself from his predecessors in the classification of the six canonical equations and in the demonstrations [Djebbar 1980: 8-10, 1986a: 106-7]. This work continued to be studied and taught in the Maghrib until the 14^th century, as confirmed by Ibn Zakariyā. One may even suppose that it is the importance of this book that led Ibn al-Bannā (d. 1321) to write, some decades later, his Kitāb al-‘us ūl wa-‘l-muqaddimāt fi-‘l-jabr [Book of the foundations and of the preliminaries in Algebra] [Djebbar 1990c].

In the domain of inheritance, al-Qurashī is known for having elaborated a new method based on the decomposition of the numbers in prime factors in order to reduce the fractions that intervene in the distribution of a given inheritance to the same denominator [Zerrouki 1995: no. 6]. His method was very quickly appreciated by the mathematicians who wrote the handbooks explaining it and by showing its usefulness through the presentation of concrete problems of inheritance. Among these mathematicians, one may cite al-‘Uqbānī (d. 1408) and al-Qalasādī (d. 1486) [Harbili 1996: no. 7]. Having said this, and in spite of the effectiveness of the method of al-Qurashī, the majority of the practitioners of Law continued, and still today continue, to utilise the ancient method [Laabid 1990].

3.2. The contribution of al-Hassār

Until today we have not found either any biography of al-Hassār or any precise element concerning his origin nor his dates of birth and death. The only information at our disposal, provided either by Ibn Khaldūn or by later Maghribian mathematicians, is concerned with his mathematical production. It seems that this mathematician was equally well-known as a reader of the Koran and as an inheritance specialist, and that he had a high rank as he bore the title of Shaykh al-jamā‘a [Chief of the Community]. One may equally suppose that he lived for some time, or that he exercised his activities, at Sebta (in the Extreme Maghrib), taking into account the links he seems to have had with the mathematicians of this city.

Two writings of al-Hassār have survived. The first, entitled Kitāb al-bayān wa t-tadhkār [Book of proof and recall] is a handbook of calculation treating numeration, arithmetical operations on whole numbers and on fractions, extraction of the exact or approximate square root of a whole of fractionary number and summation of progressions of whole numbers (natural, even or odd), and of their squares and cubes. Despite its classical content in relation to the Arab mathematical tradition, this book occupies a certain important place in the history of mathematics in North Africa for three reasons: in the first place, and notwithstanding the development of research, this manual remains the most ancient work of calculation representing simultaneously the tradition of the Maghrib and that of Muslim Spain. In the second place, this book is the first wherein one has found a symbolic writing of fractions, which utilises the horizontal bar and the dust ciphers i.e. the ancestors of the digits that we use today (and which are, for certain among them, almost identical to ours) [Woepcke 1858-59: 264-75; Zoubeidi 1996]. It seems as a matter of fact that the utilisation of the fraction bar was very quickly generalised in the mathematical teaching in the Maghrib, which could explain that Fibonacci (d. after 1240) had used in his Liber Abbaci, without making any particular remark about it [Djebbar 1980 : 97-99; Vogel 1970-80]. Thirdly, this handbook is the only Maghribian work of calculation known to have circulated in the scientific foyers of south Europe, as Moses Ibn Tibbon realised, in 1271, a Hebrew translation.

The second work of al-Hassār is entitled al-Kitāb al-kāmil fī sinā ‘at al-‘adad [The complete book on the art of number]. Only the first part of this work, that consists of 117 folios, was recovered and identified in 1986 [Aballagh & Djebbar 1987: 147-58]. Its content returns to themes of part of the first book relative to whole numbers, by developing them, and presents new chapters like the one on the decomposition of a number in prime factors, one on the common divisors and multiples and also one on the extraction of the exact cubic root of a whole number. The second part of the work, that has not yet been recovered, but of which we have been able to reconstitute a part of the chapter titles, is dedicated to operations on fractions, to the summation of the different categories of whole numbers and to the exposition of the algorithms that allow for the calculation of perfect, deficient, abundant, and amicable numbers.

The natural question that has to be posed in relation to these two books concerns their link with a possible regional mathematical tradition, and also about the nature of these links. Unfortunately, and until today, no work of calculation produced in Spain or in the Maghrib before the 12^th century, has come down to us. But the reference of al-Hassār to two Andalusian works of the 11^th century, lost today, the Kitāb al-mu‘āmalāt [Book of the transactions] of az-Zahrāwī and the Mudkhal al-‘amal [Practical introduction] of Ibn as-Samh, allows us to say that, in one way or another, the arithmetical tradition of Muslim Spain was present in the Maghrib during the 12^th century. This presence will later be reinforced, both by the direct diffusion of Andalusian works treating Algebra, Geometry and Astronomy, and by the utilisation of the contents of the works of al-Hassār by the later Maghribian mathematicians. This is what Ibn Mun‘im and Ibn Ghāzī (d. 1513) will do, for example, who cite explicitly the passages of one or the other of the two books, or who refer to them specifically [Aballagh & Djebbar 1987: 152-53].

3.3. The contribution of Ibn al-Yāsamīn

The third representative of the Magrebian mathematical tradition of the 12^th century is relatively better known than the two preceding ones. According to his biographers, his mother, whose first name was Yāsamīn [Flower of jasmine], was black (a colour that he inherited) and his father was a native of the Banū ‘l-Hajjāj Berber tribe [Ibn Ibn Sa‘īd 1945: 42]. One knows also that he lived for a time in Seville where he probably perfected himself in Mathematics, before returning to the Maghrib and installing himself in Marrakech which was then the capital of the Almohad empire. For a long time, this mathematician was only known due to a minor work of 52 lines, his Urjūza fī l-jabr [Poem on Algebra] [Djebbar 1986a]. It is very possible that it was the success of this poem that incited him to write another one of the same type, that deals with the roots of numbers, and perhaps even a third one that summarizes the method of false position and of which a copy is attributed to him [Shawqi 1987; Zemouli 1993; Kouidri 1996]. But the bio-bibliographical sources do not say anything about these last two poems.

These same sources are equally silent regarding a work of the same author, entitled Talqīh al-afkār bi rushūm hurūf al-ghubār [Fecundation of the spirits with the symbols of the dust ciphers]. This work is much more important than the three poems, both at the quantitative and the qualitative level. In fact, we are dealing with a book of more than 200 folios that deals equally with the classical chapters of the Science of Calculation and certain chapters of Geometry relative to the calculation of areas. It is indeed, among the works of the Muslim West that have come down to us, the only one that links these two subjects. Its importance is equally due to the nature of his materials and mathematical instruments which make it an original book and certainly also significant to this period of transition where three mathematical practices run in parallel before flowing together into the same stream: that of the East, of Muslim Spain and of the Maghrib [Zemouli 1993].

As an example, one may indicate the following elements that contribute to the originality of the work and to its anchorage in the great Arabic mathematical tradition of the 9^th-11^th centuries: in Arithmetic, and in contrary to the Magrebian tradition that will continue from the 14^th century, Ibn al-Yāsamīn treats first multiplication and division before addition and subtraction. This procedure that one will find later with Ibn Zakariyā, seems to have its origin in the mathematical practice of Muslim Spain [Guergour 1992].

In the domain of fractions, the remarks and suggestions of Ibn al-Yāsamīn, relative to the reading of certain expressions, show that, besides the fraction bar, the rest of the symbolism was not yet definitively fixed at his epoch. This book is also the oldest, to our knowledge, that contains both the objects and the operations of Algebra which permit the writing and solution of equations or abstract manipulation of polynomials. One observes indeed two things regarding this symbolism: first the silence of Ibn al-Yāsamīn about this practice, as if he used instruments introduced by other mathematicians before him, and further the total resemblance of the symbols intervening in his book with those that one finds in the works of the 14^th-15^th centuries, as those of Ibn Qunfudh (d. 1407), of al-Qalasādī (d. 1486) or of Ibn Ghāzī. If this fact was confirmed by other testimonies, one would then be in the presence of a much older symbolic practice than one believed [Woepcke 1854b; Djebbar 1980: 41-54; Guergour 1990; Zemouli 1993].

Regarding the presence of Geometry in a work on the Science of Calculation, this does not constitute an exceptional fact with regard to the Arab mathematical tradition taken as a whole, in the measure where similar chapters (i.e. treating problems of metric geometry) have already been included in works edited in the East, such as at-Takmila fī ‘l-hisāb [The complement in calculation] of al-Baghdādī (d. 1037) [Sa‘īdan 1985] or al-Kitāb al-Kāfī [The sufficient book] of al-Karajī (d. 1029) [Chelhoub 1989].

Notwithstanding this richness of the Talqīh of Ibn al-Yāsamīn, we have not found any explicit reference to this book in later Maghribian mathematical writings. This might be explained in two ways at least: the first reason would be a rupture of tradition whose cause has to be looked for outside the scientific environment of the epoch. This hypothesis is not improbable taking into account, first of all, the personality of Ibn al-Yāsamīn and his controversial behaviour which was strongly criticised by some of his contemporaries, and taking into account also his close links with the Almohad power that was subsequently violently contested at the political, ideological and cultural levels [Julien 1969: 92-131; Laroui 1970: 162-206]. The second reason, which is also probable and which might have reinforced the first one, is to be found in mathematical practice after Ibn al-Yāsamīn, that strongly bore the stamp of the mathematicians of Marrakech of whom we will speak later. One would be thus in the presence of the same phenomena of absorption that has already been observed earlier in the East, with the first Arab arithmetical writings of the 9^th century, then in Muslim Spain with the writings of the 10^th century. But, in the actual state of research, it is not possible to decide in favor of the one or the other hypothesis.

3.4. The contribution of Ibn Mun‘im

Ahmad Ibn Mun‘im was born in Dénia (on the east coast of Spain, near Valencia), but he spent a great deal of his life in Marrakech. In his epoch, he was considered one of the best specialists in Geometry and Number Theory. At the age of 30, he started to study Medicine which he practised in Marrakech in parallel to his activities of teaching and research. In Mathematics, Ibn Mun‘im would have published numerous works, dealing with such diverse subjects as Euclidean Geometry, Calculation, construction of magic squares, Number Theory and Combinatorics. [Ibn ‘Abd al-Malik 1973: VI, 59-60]. But only one of his writings has come down to us. It concerns Fiqh al-his b [The science of Calculation] whose title does not reflect the diversity and the richness of content [Djebbar 1980, 1983a; Lamrabet 1981].

Before speaking of this content, one must notice the importance of the book for the history of mathematical activities in Muslim Spain. In fact, the author refers specifically to Andalusian scholars by citing their names, the titles of certain of their writings, and sometimes even passages from these writings. Thus it is due to him that we learn about the existence of the mathematician Ibn Tāhir and that we discover another aspect of the activities of Ibn Sayyid, an important geometer of the 11^th century. We have known for a few years, due to the testimony of his pupil, the philosopher Ibn Bājja (d. 1138), that this mathematician explored two very important areas of Geometry: that of the intersections of solids whose bases are curves different from conic sections, and that of curves obtained by the projection of these intersections of solids on a given plane [Djebbar 1984a: 84-91; Alaoui 1983: 84-87]. Unfortunately, the geometrical works of Ibn Sayyid are lost and we do not know if they have circulated in the Maghrib. Ibn Mun‘im does not speak of these works since his book does not treat Geometry, but he specifically informs us about the contribution of Ibn Sayyid to the study of figurate numbers.

It is also due to Ibn Mun‘im that we may claim the presence, in the Maghrib from the 12^th century onwards, of the book of al-Mu‘taman (d. 1085), entitled Kitāb al-istikmāl [The book of perfection] which is essentially dedicated to Geometry but with a first chapter on Number theory [Djebbar 1984a; Hogendijk 1986]. It seems indeed that this important work was transmitted from the Almohad capital. Later, Maimonides (d. 1204) taught the content of the book of al-Mu‘taman in Cairo, and possibly also in Fez where he stayed for a time. During the 13^th-14^th centuries, certain chapters of the work continued to be studied in the extreme Maghrib, in particular by Ibn al-Bannā who refers to it explicitly in his Risāla fī t-taksīr [Epistle on the calculation of areas] and, a little later, by Ibn Hayd’r (d. 1413) [Djebbar 1986b].

With regard to the content of the Fiqh al-hisāb, its analysis shows that it does not always concern a simple return to techniques and earlier mathematical results from the Andalusian tradition or transmitted by it. In fact in it one finds new trends and results whose origin is perhaps to be found in the activities of the Almohad capital or in the preoccupations of its intellectual environment. More precisely, one discovers in it, alongside the classical chapters on arithmetical operations, others like one on the study of figurate numbers, one on the determination of amicable numbers and, in particular, one on enumeration of all words of a language utilising a given alphabet. It is indeed here that Ibn Municd’s most important contribution lies. He does not explain the reasons that motivated him to the study of this type of problem and which led him to dedicate a chapter of 19 pages that contains the important combinatorial propositions and trends which would be rediscovered in Europe only in the 16^th and 17^th century, in particular by Cardano (d. 1576), Mersenne (d. 1648), Frénicle (d.1675) and Pascal (d.1662) [Djebbar 1983a: 25-44]. However one may reasonably suppose that it was the redynamisation of Arabic linguistic and grammatical activities in Marrakech that brought the problems of enumeration to the order of the day. On the other hand, it is reasonable to think that his way of solving these combinatorial problems by utilising the concrete model of threads of silk of different colours, finds its origin in the industrial or commercial environment of Marrakech.

We know that Ibn Mun‘im had students and that one of them, al-Qādhī ash-Sharīf (d. 1283-84) wrote a book, still not recovered, entitled al-Qānūn fī l-hisāb [The Canon in Calculation]. But we do not know if this pupil incorporated the combinatorial results of his teacher in this book, or if he contented himself by teaching them to his own pupils and in particular to the most brilliant among them, the famous Ibn al-Bannā. However that may be, we find in the aforementioned work combinatorial preoccupations similar to those that led to the new results. In fact, in his little work, entitled Tanbīh al-albāb [Advertisement to the intelligentsia], Ibn al-Bannā evokes explicitly one of the methods of Ibn Mun‘im, that of the arithmetic triangle, to enumerate all words which are possible to pronounce when one utilises the 28 letters of the Arab alphabet. But he does not stop here because he introduces three original contributions in this area: the most important concerns the announcement and the demonstration, for the first time to our knowledge, of the formula of factorials giving the combinations of n letters of a given alphabet taken p at a time, without utilising the arithmetic triangle, a result that will be established once again by Pascal three centuries later. In the second place, he establishes, as far as possible, the relations that exist between the figurate numbers of Nicomachus, the combinations of n objects taken p at a time and the sums of certain progressions of whole numbers. Finally, Ibn al-Bannā utilises the techniques or the trends of combinatorial type to solve certain problems outside mathematics and which lead to enumerations with constraints, as for example the determination of the number of possible readings of a given phrase, taking into account the rules of the Arab grammar, or of the number of prayers that a Muslim has to say to compensate for the forgetting of a certain number among them [Djebbar 1980: 67-98].

In the 14^th century, we do not find new results in this domain among the commentators of Ibn al-Bannā, but simple references to the already established results, sometimes accompanied by comments. This is done by Ibn Haydūr in his book at-Tamhīs fī sharh at-Talkhīs [The in-depth commentary of the Abridgment] and in his handbook of calculation [Guergour 1992]. One finds equally, and this is also important for the characterisation of the Magrebian mathematical tradition, the continuation of a certain combinatorial practice both inside and outside the mathematical field [Djebbar 1980: 99-112]. We do not yet have a detailed knowledge of this practice after the 14^th century, but the little that we do know leads us to believe that we have here a domain that should still be the object of attentive study as it has not as yet revealed all its secrets. Two precise examples allow us to assert this: the first concerns a rapid allusion to the enumeration of magical squares of a same type, of which one finds a trace in a manuscript of Euclid’s Elements copied in the Maghrib. It is for the first time, to our knowledge that this problem is evoked in such an explicit way. Unfortunately, no reference to an earlier work on these enumerations is evoked by the copyist of the manuscript [Ms. Rabat, Hasaniya no. 1101: f.1a]. The second example concerns the commentary on the book of al-Qalasādī Kashf al-asrār ‘an hurūf al-ghubār [The unveiling of the secrets relative to the dust ciphers] realised by Tufayyāshī (d. 1914), a theologian of the central Maghrib who lived and taught in Beni Izguen. In the last chapter of his commentary, he discusses certain combinatorial aspects making implicit references to the work of Ibn al-Bannā [Ms. Bibl. Béni Izguen: 445-50]. In our opinion, this evocation is not casual and could only be explained by the persistence of this combinatorial tradition that was born or established starting with the original contributions of Ibn Mun‘im and Ibn al-Bannā and that then was sustained in the Maghrib by manuals and teaching.

In conclusion to this chapter, we have to say that the relatively detailed presentation of the known writings of al-Qurashī, of al-Hassār, of Ibn al-Yāsamīn and of Ibn Mun’im does not mean that these writings were the only ones to have circulated and to have been studied in the 12^th-13^th centuries. The information we have at our disposal, shows the contrary. In fact, for the same period, one might evoke mathematicians sometimes as important as those of whom we spoke, like al-Qādhī ash-Sharīf, the pupil of Ibn Mun’im in Marrakech, al-Qal‘ī (d. 1271) who lived equally in Bougie and who taught the Science of Inheritance [Hafnawi 1982: 495-96], as also Ibn Ishāq at-Tūnusī (d. after 1218) known for his works in Astronomy [King 1988]. Unfortunately, the mathematical writings of these scholars have not come to us and we cannot speculate on their contents. At most one may judge their importance from certain quotations of their writings or from certain testimonies on their stature as men of science.

By Professor Ahmed Djebbar, to be continued…