Issues in the History of Mathematics Teaching in Arab

Issues in the History of Mathematics Teaching in Arab - CIMM

Paedagogica Historica
Vol. 42, Nos. 4&5, August 2006, pp. 629–664
Issues in the History of Mathematics
Teaching in Arab Countries
Mahdi Abdeljaouad
Taylor and
Paedagogica
10.1080/00309230600806880
CPDH_A_180636.sgm
0030-9230
Original
Stichting
2006
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[email protected]
LiviaGiacardi
Article
Paedagogica
(print)/1477-674X
Francis
Historica
2006LtdHistorica
(online)
George Makdisi’s The Rise of Colleges: Institutions of Learning in Islam and the West (Edinburgh:
Edinburgh University Press, 1981: 10) says: ‘with the advent of the madrasa, the institution inclusive of
the foreign sciences began to fade away, becoming extinct by the XIIth century’. In fact, the status of teaching
rational sciences in the Arab/Islamic Middle Ages was not as clear-cut as in this quote and requires more
elaborate and specific studies. When considering the history of teaching mathematics in Arab/Islamic countries, many issues must be closely examined, some of which will be discussed by highlighting similarities,
developments and contrasts, and by attempting to provide answers to a number of questions: Did mathematics have the same status in the organization of knowledge before the twelfth century and after? In which type
of institutions was mathematics taught? Who were the teachers of mathematics, what status did they have
in academe? Which mathematics subjects figured in the curricula? What textbooks, tools and methods were
used to teach mathematics? Our undertaking will be illustrated by a case study involving a student and a
teacher of mathematics from the eighteenth century: Mahmū d Maqdîsh (Tunisia).
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Introduction1
Influenced, like all of Europe during the Middle Ages, by the Hellenistic culture,
Arabic – or Islamic – civilization introduced a dichotomy concerning the statute of
mathematics: the dichotomy between ‘theoretical’ mathematics necessary to understand the world, and ‘practical’ mathematics used to solve the problems of everyday
life. For the Arabs, practical mathematics was a teaching subject intended for the
greatest number, whereas theoretical and argumentative mathematics were reserved
for specialists. This typology will be modified somewhat during this study, because
we will show that for reasons religious, philosophical, political or social, both
dominant mathematical contents were actually taught and major pedagogical methods underwent many changes. The pedagogical conceptions of the Arabs, the representations resulting from them and their practical applications conditioned the social
function assigned to the teaching of mathematics, the place of mathematics among
the various disciplines taught, the nature of the institutions where mathematics was
1
I wish to thank Béchir Lamine and Günter Seib for reviewing the English version of this paper.
ISSN 0030-9230 (print)/ISSN 1477-674X (online)/06/040629–36
© 2006 Stichting Paedagogica Historica
DOI: 10.1080/00309230600806930
630 M. Abdeljaouad
taught, the status of the teachers of mathematics, and the curricula and handbooks
used. We will not be able to treat all these subjects, because of the diversity of the situations concerned even from a geographical point of view, the Arab/Islamic world
being so vast and so diverse for the duration of the period covered by this study:
extending from the eight to the nineteenth century. Furthermore, the multiple
sources and references both old and recent are no help in finding interesting facts;
they often mention mathematics teaching only marginally and are thus of little value.
In this paper, we shall discuss the following issues: (1) sources, where to get
information; (2) what place did mathematics have according to the hierarchy of
Arabic knowledge?; (3) which kind of mathematical education was provided for children, adolescents and young adults?; (4) who taught mathematics?; (5) what kind of
mathematics textbooks were used?; (6) what pedagogy was applied for mathematics?;
(7) one case study: Maqdîsh (1742–1813).
Sources: Where to get Information
The teaching of mathematics may be explored using either primary sources or more
recent publications.
Primary Sources
Education is a topic covered by a number of ancient authors. Ibn Sahnūn2 (817–870),
Ibn al-Jazzār3 and al-Qābisı̄ 4 have each devoted a treatise to it in which they recommend providing children with a good grounding in Islamic religion, starting with
learning the Koran by rote. Based on a deeply traditionalist outlook, their prescriptions cover the subjects to be taught, the place devoted to the Arabic language and to
arithmetic, the desired behavior towards other students and towards parents, as well
as teacher remuneration. Review of such texts permits one to glean only a small
amount of information on the teaching of mathematics.
Encyclopedias5 also explored, from a more rationalist standpoint, issues posed by
the education of children and adolescents. Influenced by Greek philosophy, they
recommend that more prominence should be given to the sciences of Antiquity, that
is to philosophy, mathematics, physics and music.
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Ibn Sahnūn. Kitāb ādāb al-mucallimı̄n [A Book on the Conduct of Teachers]. Edited by
Mu.hammad al-’Arūsı̄ al Ma.twı̄ . Tunis: Dār al-Kutub as-Sarqı̄ ya, 1972.
3
Ibn al-Jazzār. Siyāsat as-sibiyān wa tadbı̄rihim [Child Education and Care]. Texte établi et presenté par Muhammad al-Hābib al-H ı̄ la. Beyrouth: Dār al-Gharb al-Islami, 1984.
4
Al-Qābisi, ar-Risāla al-mufassala li ahwāl al-mutacallı̄m wa ahkām al-mucallim wal mutacallim [A
Detailed Epistle on the Situation of Pupils, their Rules of Conduct and those of Teachers], edited
by Ahmed Khaled. Tunis: S.T.D., 1986.
5
Rasā ’il Ikhwā n as-Safā (Epistles of Purity), a collective work by anonymous authors from Basra
between the ninth and the tenth century, Kitā b ash-Shifā [The Book of Healing] by Ibn Sina (d.
1037), and al-Muqaddima (Prolegomena) by Ibn Khaldūn (d. 1406) are the most widely known.
2
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Paedagogica Historica
631
The Encyclopedia was an important vehicle for intellectual discourse in the Medieval/
Islamic world. While such treatises may not always be intellectually innovative or creative,
they are crucial for understanding the transmission and assimilation of (specially foreign)
knowledge into the Islamic intellectual milieu.6
Autobiographies, rihlas (scholarly travel accounts), and the private correspondence of
mathematicians, such as that of al-Qalasādi’s Rihla7 (d. 1486) or al-Kāshı̄ ’s Letters
to his Father8 (d. 1436), are invaluable sources as they often include priceless information regarding the names of professors, subjects of study and sometimes textbooks/
manuals used. Unfortunately, these can only be exploited if they belonged to a
specialist in mathematics or astronomy, as other scholars, jurists or diplomats, hardly
studied mathematics, or not at all, and, as a rule, did not give any account of their
own scientific training.9
Ancient biographies and bio-bibliographies10 help to place in perspective the status of
mathematics in training elites. They abound with data telling of the life or work of a
specialist in the sciences in general, and in mathematics in particular. Some of these,
such as Ibn Nadhı̄ m’s (d. 993), Sācid al-Andalusı̄ ’s (1029–1070), Ibn al-Qiftı̄ ’s
(1172–1248), Ibn Abı̄ Usaybica’s (1194–1270) or Ibn Juljul’s, devote a specific
chapter to mathematicians.
While introductions to mathematical books and textbooks11 may give indications
regarding the role of the discipline in society, one should not get over-enthusiastic
with regard to such sources, as the style of numerous authors tends to be conventional
and hagiographic.
For all ancient sources, Djebbar12 insists that they do not as a rule provide any
insights into how Arab mathematics was taught. They do, however, yield enough
corroborative evidence for a tentative, albeit fragmentary, representation of the teachers of mathematics, their status in society and, more generally, of how mathematics
was being taught.
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6
De Young, Gregg. “Euclidian Geometry in Two Medieval Islamic Encyclopaedias.” al-Masāq
14, no. 1 (2002): 47.
7
al-Qalasādi’s Rihla, edited by M. Abu l-Ajf ān. Tunis: S.T.D., 1978.
8
Kennedy, E. S. “A Letter of Jamshı̄ d al-Kāshı̄ to his Father.” Orientalia 29 (1960): 191–213.
See also: Bagheri M. “A Newly Found Letter of al-Kāshı̄ on Scientific Life in Samarkand.” Historia
Mathematica 24 (1997): 241–56.
9
From M’ghirbi, Salah. Les voyageurs de l’Occident musulman du XIIe–XIVe siècles. Doctoral thesis, Université Paris III, 1986.
10
Ibn Nadhı̄ m. al-Fihrist, translated by B. Dodge. New York: Columbia University Press, 1970;
al-Andalusı̄ , Sācid. Tabaqāt al-umam [Categories of Nations], French translation by R. Blachère.
Paris: Larousse, 1935.
11
For example the introduction to Sharh al-Urjuzā al-Yasminı̄ ya by Ibn al-Hā’im (d. 1412) edited
with commentaries in French by Mahdi Abdeljaouad. Tunis: ATSM, 2003, or to Sharh at-Talkhı̄s
by al-Qalasādi (d. 1486), edited and translated in French by Farès Bentaleb. Beyrouth: Dar alGharb al-Islami, 1999.
12
Djebbar, Ahmed. Une histoire de la science arabe. Paris: Editions du Seuil, 2001.
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632 M. Abdeljaouad
Contemporary Sources
More recent publications are concerned with education in medieval Islamic states;
numerous others deal with the history of mathematics, and some focus on mathematics
teaching in Arab countries.
History of Arab/Islamic Institutions of Education
Teaching and teachers of the az-Zaytouna Great Mosque in Tunis and of madrasa
(colleges of law), have been the focus of several academic studies by scholars such as
Abdelmoula,13 Ben Achour,14 Ben Mami,15 M. H. Bougamra,16 Makdisi,17 Shmays
āni18 or J. Petersen.19 While such investigations provide extensive information on the
history of institutions, their curricula, their teachers and students, as well as their
teaching methodologies, ‘they provide but scanty information on the teaching of
sciences’.20
To compensate for the inadequacy of the data collected in such publications, we
have turned to books on the history of mathematics.
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History of Mathematics
Upon looking into general publications on the history of mathematics, such as books
by George Sarton21 or René Taton,22 that predictably include an important chapter
on Arab mathematics, or reading authors specializing in the history of Arab mathematics such as Youschkevitch,23 Berggren,24 Rashed25 or Djebbar, you find that their
contributions, however important for understanding certain aspects of the teaching
of mathematics, are inadequate and biased, as they focus on the development of
13
Abdelmoula, Mohamed. L’Université Zaytounienne et la Société. Tunis: S.T.D., 1971.
Ben Achour, Mohamed Aziz. Les cUlamas à Tunis aux XVIIIe et XIXe siècle. Doctoral thesis,
Paris IV University, 1977.
15
Ben Mami, Mohamed el-Bèji. Madāris madı̄nat Tunı̄s [The Madrasas of the City of Tunis].
M.A. Dissertation, University of Tunis, 1981.
16
Bougamra, Mohamed Hichem. L’Enseignement de la langue arabe et de la littérature arabe à la
Nizamiyya de Bagdad. Doctoral thesis, University of Tunis, 1983.
17
Makdisi, George. The Rise of Colleges, Institutions of learning in Islam and the West. Edinburgh:
Edinburgh University Press, 1981.
18
Shmaysāni, Hassen. Madāris Dimashq fi al-casr al-’ayyoubi [Madrasas in Damascus under the
Ayubid dynasty]. Beyrouth: Dar al-Af āq al-Jadı̄ da, 1983.
19
Petersen, J. “Madrasa.” In Encyclopédie de l’Islam, vol. V, 1119–30.
20
According to Djebbar, Une histoire de la science arabe, 84.
21
Sarton, George. Introduction to the History of Science, 2 vols. Baltimore, MD: Williams &
Wilkins, 1927–1931.
22
Taton, René. Ancient and Medieval Sciences from the Beginning to 1450. New York: Basic Books,
1967.
23
Youschkevitch, A. P. Les mathématiques arabes (7e–15e siècles). Paris: Vrin, 1976.
24
Berggren, J. L. Episodes in the Mathematics of the Medieval Islam. Berlin: Springer, 1986.
25
Rashed, Roshdi. Histoire des scences arabes. Vol. 2. Paris: Editions du Seuil, 1997.
14
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mathematical concepts, and even more on highly creative mathematicians with recognized innovative ideas. In their publications, little room is devoted to teachers and to
teaching as such, as is most aptly expressed by Gregg De Young in the following
extract:
Too often, historians of mathematics have focused on such ‘mountain peaks’ of creativity
while ignoring what lies below them. They have tended to focus on innovations, specially
those that seem to resemble or lead to some part of modern mathematics (1).
… It is important that historians of mathematics look beyond the giants of the field, the
creative innovators, to examine the reception of ideas into the intellectual tradition. One
of the best means to this end is to examine the textbook tradition itself – it may be somewhat less exciting from a mathematical perspective, but it may also provide a more
balanced view of the history of mathematics.26
Dictionaries or Scientific Biographies
Many historians of mathematics, modern encyclopedias and biographical dictionaries
have studied Arab mathematics. Special mention should be made of l’Encyclopédie de
l’Islam, the Dictionary of Scientific Biography, as well as Carl Brockelman’s27 and
Sezgin’s28 bio-bibliographic studies, which provide a wide variety of extremely useful
data. With regard to Arab teachers of mathematics in general, mention should be
made of Tuqan29 and Rosenfeld-Ihsanôglu;30 on Andalusian mathematicians, Marie
Geneviève Balty-Guesdon’s doctoral thesis,31 on Maghrebinian mathematicians,
Driss Lamrabet,32 and on Tunesian mathematicians, Mohamed Mahfoudh33 and
Hmida Hedfi.34
Concluding Remarks on the Literature
Studying and writing the history of the teaching of mathematics in Arab/Islamic countries is a momentous task, in view of the period to be covered (from the seventh to the
nineteenth centuries), the wide variety of countries concerned, and the political,
26
De Young, Gregg. “The Khulāsat al-Hisāb of Bahā al-Dı̄ n al-’Amilı̄ and The Dar-i-Nizāmı̄ in
India.” Indian Society for History of Mathematics 1986, 8: nos 1–4: 1–2.
27
Brockelmann, Carl. Geschichte der arabischen Litteratur. Leiden: Brill, 1943–1944.
28
Sezgin Fuat. Geschichte der arabischen Schrifttums, 5: Mathematik. Leiden: Brill, 1971.
29
Tuqan Qadri Hafidh. Turath al-cArab al-cIlmi. Nouvelle édition. Beyrouth: Dar ash-Shuruq,
1963.
30
Rosenfeld, B. A., and Ihsanoglu. Mathematicians, Astronomers and Other Scholars of Islamic
Civilisation and their Works (7th–19th c.). Istanbul : IRCICA, 2003.
31
Balty-Guesdon, M.-G. Médecins et hommes de sciences en Espagne musulmane (2e–5e / 8e–9e siècles.). Thèse de doctorat, Université de Paris III-Sorbonne nouvelle, Paris, 1992.
32
Lamrabet, Driss. Rabat, 1994.
33
Mahfoudh, Mohamed. Tarā jim al-mu’allif ı̄ n at-tū nusiyyı̄ n. Beyrouth: Dar al-Gharb al-Islami,
1981.
34
Hedfi, Hmida. ar-Riyadhiyat bi Ifriqiya. Mémoire de recherche, Faculté des lettres de Tunis,
1989.
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634 M. Abdeljaouad
economic, linguistic and cultural transformations recorded throughout the centuries.
As a matter of fact, very few research works concerning this subject matter have been
published. Most of them provide ‘a useful starting point, [but] also show how much
more research remains to be done’.35 Based on this brief review of the literature, we
shall introduce a number of other issues that have emerged from the investigation of
the history of teaching mathematics in the Arab/Muslim world.
What Status did Mathematics have According to the Hierarchy of
Arabic Knowledge?
In his doctoral thesis, Ahmed Abdesselem insists on religion as a prevalent factor in
Arab teaching at all levels: ‘mosques, medresas and zaouias are places where teaching
– whether considered superficially or in-depth – was related to worship and religion’.36 Gregg de Young (1986) confirms that Islamic education is essentially motivated by considerations of a religious nature: ‘Islamic education has always been
primarily religious education, in the sense that it is explicitly intended to preserve the
religious tradition from which Islamic communal life springs’.37 These views are
shared by a number of authors; Islamic education is even identified by Makdisi, in his
book on the history of madrasas, in which he considers the development of Arab
educational institutions to have been completely determined not only by the requirements of religious proselytizing, but also by the struggles between the various religious
movements, especially between Sunnites and Shiites: ‘the history of Islamic institutions of learning was inextricably linked with Islam’s religious history, and their development was linked with the interaction of the religious movements, legal and
theological’.38 Insisting as well on the religious factor as a dominant feature of all
aspects of political, economic and social life in Muslim countries, Brunschvig
confirms that ‘teaching is only stripped of its religious character to a small extent’.39
It follows that, for Arabs, the terms of ‘science’ and ‘scientist’ will cover the
widest range of knowledge, primarily religious and only incidentally secular. To
what extent should we start from this observation when studying the history of the
teaching of mathematics in Islamic countries? Could an analysis of the hierarchy of
knowledge, as presented by early Arab philosophers, and the consequences of this
hierarchy for teaching help in addressing this question? Note should be taken, with
Ahmed Djebbar, that the early Arab philosophers, faithful to the teachings of
Ancient Greece:
35
Djebbar, Ahmed. “Mathematics in al-Andalus and the Maghrib between the Ninth and Sixteenth Centuries.” in The Entreprise of Science in Islam: New Perspectives, edited by J. P. Hogendijk
and A. Sabra. Cambridge, MA: MIT Press, 2003: 334.
36
Abdessalem, Ahmed. Les historiens tunisiens des XVIIe, XVIIIe et XIXe siècle. Paris: C. Klincksieck,
1973: 73.
37
De Young, “The Khulāsat al-Hisāb”, 3.
38
Makdisi, The Rise of Colleges, xiii.
39
Brunschvig, Robert. La Berbérie orientale sous les Hafsides. Paris: Adrien-Maisonneuve, 1947: 352.
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… had received, or had insisted on receiving, a solid grounding in mathematics based on
the traditional four elements: the science of numbers, geometry, astronomy, and music;
these were later expanded to new topics inaugurated by the Arabs as early as the 8th
century: Indian calculus … algebra … and, finally, trigonometry.40
These philosophers included in their philosophical teachings a number of more or less
elaborate chapters on mathematics. Their books, or some sections within their books,
were later taught, and even reproduced in mathematical textbooks.
In his encyclopaedia Kitā b al-Shifā (Book of Healing), the great physician and
philosopher Ibn S ı̄ nā (980–1037) included:
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… as inseparable elements of his philosophical teaching a summary of the thirteen books
of Euclid’s Elements, another book on astronomy, a third developing the basis of the theory
of numbers, not to be confused with Euclid’s Books 7, 8, and 9, nor with Nicomachus’s
Introduction to Arithmetic, and finally a fourth book on music.41
Several centuries later, these same chapters of al-Shif ā were still taught in their
original form, reproduced or quoted in mathematical textbooks.
Was the prominent place of mathematics in knowledge, as advocated by early Arab
philosophers, reduced by the victorious assault against ‘foreign’ science waged by the
Sunnite orthodox theologians, after the capture of Baghdad by the Seljukide Turks in
1055 and the establishment of the madrasa al-Nizamiyya in 1066? To answer this question, we should first assess the effect of such assaults, as described by George Makdisi
upon explaining how non-religious topics survived for some time in the Muslim world
by retaining a specific status, often at the margins of the dominant educational system:
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A striking feature of Muslim education in the Middle Ages was the dichotomy between two
sets of sciences: the ‘religious’ and the ‘foreign’.
… By the time the traditionalist institutions had won the battle against those of rationalism
and absorbed them, they had also absorbed a great amount of what they had originally
opposed…. The exclusion meant that the study of ‘foreign sciences’ had to be pursued
privately; they were not subsidized in the same manner as the Islamic sciences and its
ancillaries. But there was nothing to stop the subsidized student from studying the foreign
sciences unaided, or learning in secret from masters teaching in the privacy of their homes
or in the Waqf institutions, outside of the regular curriculum.42
Gradually, the study of the rational sciences was restricted to specialists and depended
on the goodwill and mood of the respective ruler. The attitude of orthodox theologians and of the princes under their influence was, however, less clear-cut with regard
to mathematics, an art recognized as useful to religion and society. A minimum of
knowledge of arithmetic and geometry continued to be part of certain basic teachings;
the remaining mathematical topics were henceforth the monopoly of specialists who
in all cases cumulated both speculative activities and activities more acceptable to
society and eventually more lucrative.
40
Djebbar, Ahmed. “Quelques remarques sur les rapports entre philosophie et mathématiques
arabes.” Revue Tunisienne des études philosophiques no. 2 (1984): 6.
41
Ibid., 6–7.
42
Makdisi, The Rise of Colleges, 77–78.
636 M. Abdeljaouad
When the works of philosophers were blacklisted, the arithmetic and geometry
textbooks were not burnt, an omission that permitted the perpetuation of teaching
such subjects. As he put his final touches to his al-Muqaddima in 1377, Ibn Khald ūn
provided an eloquent testimonial to the place of mathematics in the hierarchy of the
knowledge taught in those days:
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These are the basic seven philosophical sciences. Logic comes first. Then comes mathematics, beginning with arithmetic, followed in succession by geometry, astronomy, and
music. Then comes physics and, finally, metaphysics. Subdivisions of arithmetic are
calculation, the inheritance laws, and business arithmetic.
Ibn Khald ūn insists on the part played by arithmetic and geometry in developing
intelligence:
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The crafts, especially writing and calculation, give intelligence to the person who practises
them…. Calculation is connected with [writing]. It entails a kind of working with numbers
… which requires much deductive reasoning. Thus, [the person occupied with it] gets
used to deductive reasoning and speculation, and this is what is meant by intelligence.
Geometry enlightens the intellect and sets one’s mind right. All its proofs are very clear
and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it
is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is
not likely to fall into error. In this convenient way, the person who knows geometry acquires
intelligence…. Our teachers used to say that one’s application to geometry does to the mind
what soap does to a garment. It washes off stains and cleanses it of grease and dirt.43
Astronomers, whose training required a solid grounding in Euclidian geometry, in the
geometry of cones and spheres, as well as in arithmetic and algebra, and whose skills
were highly prized by sultans, vizirs and lesser rulers, contributed towards keeping the
teaching of mathematics alive.
Specialists for how to distribute inheritances also played a crucial part in developing
the teaching of arithmetic and algebra, as a critical quote from Ibn Khald ūn’s alMuqaddima suggests:
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… Religious scholars in the Muslim cities have paid much attention to [the science of
inheritance laws]. Some authors are inclined to exaggerate the mathematical side of the
discipline and to pose problems requiring for their solution various branches of arithmetic,
such as algebra, the use of roots, and similar things. It is of no practical use in inheritance
matters, because it deals with unusual and rare cases.44
Which Kind of Mathematics Education was Given to Children, Adolescents
and Young Adults?
Is it possible to collect enough information on the status of teaching mathematics in
the education of children and adolescents? At what age did mathematics instruction
end? Is it possible to determine what was taught, where and for how long?
43
Ibn Khaldūn: The Muqaddima, An Introduction to History, edited and abridged by N. J. Dawood.
Bollingen Series. Princeton, NJ, 1989: 371–72; 331–32; 378–79.
44
Ibid., 347.
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Paedagogica Historica
637
The Basic Instruction of Children
In Arab and Islamic cities, the basic education of children, especially boys, does not
seem to have changed from the early days of Islam till fairly recently. Codified in the
tenth century by Ibn Sahnūn (817–870) in his book Kitā b adā b al-Mucallimı̄ n
(Deontological Rules for the Teachers) and later by al-Q ābisi (tenth century), it is
given by masters (mu’addib) either in public places of learning (often mosques) or in
private places of learning (al-kutt ā b), or in the learner’s home. Most individual testimonials indicate that such studies began at about the age of five, continued for a
period of approximately five years and were essentially devoted to learning the Koran
by rote.45 Other subjects may have been taught in the course of these studies;
however, the selection of such subjects has been an issue much debated among Arab
teaching specialists. This selection varied widely, ranging from the extremely
categorized to the most extravagant. Ibn Khald ūn (d. 1406) succinctly presented the
problem in his al-Muqaddimma:
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Instructing children in the Qur’an is a symbol of Islam. Muslims have, and practise, such
instruction in all their cities, because it imbues the heart with a firm belief in Islam and its
articles of faith….
In his Rihlah, Judge Abu Bakr b. al-cArabi makes a remarkable statement about instruction…. He places instruction in Arabic and poetry ahead of all other sciences, as in the
[Andalusian] method…. From there the student should go on to arithmetic and study it
assiduously, until he knows its basic norms. He should then go to the study of the Qur’an
because with his previous preparation, it will be easy for him.46
Two traditions emerge from biographies and textbooks: the first could be considered ‘dogmatic’ and the second ‘enlightened’. As Ibn Khald ūn himself suggests,
adhering to the dogmatic tradition is the actual practice of masters, whether in the
Maghreb or in the Orient, little room having been devoted to calculating. Ibn Sı̄ nā
(d. 1037) and ash-Shirāzi (d. 1193) indeed advocated the inclusion of rudiments of
writing and reading in the initial training of children; neither of them, however,
mentions mathematics. In Morocco, during the Almohad dynasty (1130–1269),
basic education was compulsory for both girls and boys but as it was clearly ideologically oriented, it relied on specific aspects of doctrine requiring literacy in the Arabic
or the Berber language. There was no room for calculation.47
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45
In the Orient, Ibn Sı̄ nā (980–1037) considers that a child who is capable of learning should
attend his first classes. He himself attended until the age of ten the classes of a mu’addab in Bukh
āra. Ibn Hazm al-Andalusi (994–1064) suggests that learning should start as soon as the child is able
to understand and to answer, i.e. at the age of five. Ibn al-cArabi (1076–1148) indicates that he himself studied the Koran until he was nine years old. Ibn Radhouane (d. 1061) declares in his autobiography that ‘he had given himself to the master’ from the age of six. (Based on Najjar, Brahim, and
Béchir Zribi. al-Fikr attarbawi cinda l-carab. Tunis: ad-Dar at-tūnusiya li-n nashr, 1985.)
46
Ibn Khaldūn, The Muqaddima, 423–24.
47
Based on Mannouni, Mohamed. al-cUlūm wa’l-ādāb wa’l-funūn calā ahd al-muwahhidı̄ne
[Sciences, Letters and Arts in Almohad’s Time]. Rabat: Dar al-Maghrib, 1977: 28.
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638 M. Abdeljaouad
The Instruction of Adolescents
While the initial instruction of children was codified, because it was considered
crucial for the training of faithful believers, the education of adolescents was not institutionalized. The elite and the educated encouraged their children to pursue their
own education, either by providing them with masters in their homes, or by having
them attend classes in mosques or in other local public places. As a rule, the desired
training was elementary and comprehensive, focusing on religious subjects, and was
provided by generalist teachers. Such studies could take between five and ten years
and lead to a number of trades such as mu’addab (kuttā b master) or preacher in a
small rural community. All testimonials tend to show that this training devoted some
space to a number of chapters on arithmetic deemed necessary to solving inheritance
issues (far ā ’idh).
Ibn Sı̄ nā (d. 1037) in the Orient and Ibn Hazm (d. 1064) in Andalusia explicated
this phase of adolescent instruction as follows:
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Ibn Sı̄ nā suggested that subsequent to basic instruction in the kuttāb, the child
should move on to specialized training. He recommended starting by deciding to
what calling we wish to direct the child. Hence, if we prepare him to be a secretary,
we teach him, in addition to language, correspondence and speech writing. Some
children are thus directed to the science of reckoning, others to geometry, and others
still to medicine.48
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The curriculum suggested by the multi-talented theologian, poet and scientist of
Cordoba, Ibn Hazm al-Andalusı̄ (993–1064), whose iconoclastic views were
contested both by Malekite men of religion and by the authorities, is ambitious if not
extreme, as it advocates introducing the teaching of the religious sciences and metaphysics only after the child has been initiated in writing and reading Arabic, grammar
and poetry, all of which are considered as taking priority:
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Once he has mastered grammar and language, the adolescent should tackle the science of
numbers and start mastering multiplication, division, addition, subtraction, and denomination. Thereafter he will learn part of the science of areas (misāha) and will study
arithmetika, i.e. the science exploring the nature of numbers. He should read Euclid’s book
in a manner that allows for comprehending the text by addressing its objectives and
understanding its significance; it is a lofty science which allows for knowing the situation
of Earth and its area, the organisation of the stars, their movements, their centers, and their
distances; it also allows for reviewing evidence of all results…. Studying the Almageste will
enable him to predict eclipses, determine distances between countries, calculate time,
tides…. Studying the geometry of areas (handasa) will help him attract waters, elevate
weights, draw buildings and design machines.49
]am
[rac
This ambitious program was in fact aimed at training specialists, targeting only a very
small minority. It is exactly similar to the program followed by Ibn Sı̄ nā or Ibn alc
Arabı̄ themselves, as will be seen shortly.
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49
Based on Najjar and Zbidi, al-Fikr attarbawi cinda l-carab, 130–33.
Ibid., 135–41.
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Paedagogica Historica
639
Ibn S ı̄ n ā (d. 1037) states indeed that in his adolescence his father had him attend
the classes on Indian arithmetic taught by a greengrocer50 in Bukharah. Then he
assigned to him a professor entrusted with teaching him notions of logic, Euclidian
geometry and astronomy. However, he quickly surpassed his professor’s skills and
studied Euclid’s and Ptolemy’s books on his own. When he was 16, he started studying philosophy and medicine.
Ibn Radhw ā ne (d. 1061) indicates that, after a course of general studies, he began
to study philosophy and medicine at the age of 14. He became a great physician in
Cairo.
al-Khā zinı̄ (c.1115) was a slave boy of Byzantine origin owned by a treasurer of the
court of Marv. His master gave the young man the best possible education in mathematics and philosophical disciplines (caql ı̄ yya).51
Ibn al-cArabı̄ (d. 1148 in Seville) finished his initial education at the age of nine. His
father then provided him with three teachers: one to consolidate his knowledge of the
Koran, the second to master the Arabic language, and the third to learn calculation.
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Thus, at the age of 16, I had studied, among other subjects:
- In the science of numbers: transactions, algebra, and the rules of inheritance.
- Euclid’s books and ensuing aspects, up to the proposition on secants.
- The 3 zı̄ jes in astronomy and the study of the astrolab.52
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i[]arc
as-Samaw’al Ibn Yahya al-Maghribı̄ (d. 1180), born into an educated Jewish family
of Baghdad, was a successful doctor and a creative mathematician. He converted to
Islam in 1163. From his biography, we learn that:
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He started by studying Hebrew and Torah up to the age of thirteen. He then took up the
study of medicine and the exact sciences. He started to learn mathematics, beginning
with Hindu computational methods, zı̄ jes (astronomical tables), arithmetic and misāha
(surveying), then progressing to algebra and geometry. Since scientific study had
declined in Baghdad, as-Samaw’al was unable to find a teacher to instruct him beyond
the first books of Euclid’s Elements and was therefore obliged to study independently. He
finished Euclid, then went to the algebra of Abu Kāmil, the Badic of al-Karājı̄ , and the
Arithmetic of al-Wāsitı̄ …. By the time he was eighteen, as-Samaw’al had read for himself
all the works fundamental to the study of mathematics and had developed his own mode
of thinking.53
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Ibn Khald ūn (d. 1406), in his autobiography, does not explicitly state that he was
trained in mathematics in his adolescence. Only after 1348, i.e. at the age of 16, he
attended the classes of ‘the eminent master in the rational sciences: al-’ Ā bil ı̄ for three
years’.
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50
To be related to what is said about Fibonacci who, as a child, in the late twelfth century, is said
to have studied Indian computation methods from a merchant in Bougie (in Algeria).
51
From Hall, Robert E. “al-Khāzinı̄ .” In Dictionary of Scientific Biography, edited by C. C.
Gillispie. Vol. 7. New York: Scribner’s, 1973: 335–36.
52
Today the proposition on secants is called ‘Menelaus’s Theorem’. Based on Najjar and Zbidi,
al-Fikr attarbawi cinda l-carab, 127.
53
Anbouba, Adel. “al-Khāzinı̄ .” In Dictionary of Scientific Biography, edited by C. C. Gillispie.
Vol. 12. New York: Scribner’s, 1975: 91–95.
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640 M. Abdeljaouad
Al-Qalas ā d ı̄ (1412–1486) lists, in his Rihla (account of study travels), the names
of his masters and the disciplines he studied under their aegis. In Seville, his
Andalusian native city, he attended the classes of six masters: the first taught him the
Koran and Sunna, as well as arithmetic as developed in al-Maq ā lat al-’Arbaca by ibn
al-Bannā (d. 1321). The fourth master taught him the laws of inheritance and, again,
al-Maq ā lat al-’Arbaca and the Talkhı̄ s by ibn al-Bannā. He ended this phase of his
education at the age of 24 when he went to Tlemcen (c.1436), where he enhanced his
knowledge of both the religious sciences and mathematics.54
Ibn Ghā zı̄ al-Makn ā sı̄ (1437–1513) spent his adolescence in his native city,
Meknès, where he was trained, among other subjects, in the Arabic language, in the
laws on sharing inheritance and in arithmetic.
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The Education of Young Adults
Where does adolescence end and adulthood begin? Should this section not have the
heading ‘The status of mathematics in higher education’? We know that, in the
Middle Ages, higher education could last many years, and that students wishing to
receive specialized instruction in mathematics had to organize study travels, Rihla, to
find specialists. There was no age limit but, at each stage of his journey, the candidate
had to demonstrate that he had the scientific skills needed to attend the master’s
courses. As a rule, courses in mathematics were given in the Great Mosque in the
capital, and sometimes in madrasas or in private homes. This was essentially the
master’s choice, and depended on his status within the academic community. This is
why biographers mention courses in far ā ’idh and his ā b in the Valenzia mosque in the
twelfth century, taught by Abu Bakr ibn Ghuzzayy (d. 1187). Ibn al-Bann ā (d. 1321)
taught all his courses in the Marrakesh mosque, and received only special students at
home. Ibn Rammah (fourteenth century), in his old age, asked some of his disciples
to give lessons on his behalf in the Kairouan great mosque every morning, courses in
theology and Islamic law, and to spend the rest of the day teaching grammar, far ā ’idh,
and hisā b. On Friday afternoons, the master gathered his assistants to discuss and
address problems.55 Ibn Majdı̄ (1365–1447) and Sibt al-Maridı̄ nı̄ (1423–1506)
taught mathematics at the Great Mosque of al-Azhar in Cairo.
The teaching of mathematics could also take place in a madrasa. Thus, in 1363, the
King Hammu II (1359–1388) built a madrasa in Tlemcen, for which he appointed
Abu cAbdallah ash-Sharif (d. 1370) to teach Euclidian geometry. Also in Tlemcen,
al-Qalasād ı̄ (d. 1486) attended, for several years, the classes of Ahmad Ibn Zaghu
(1380–1441) held in the madrasa al-Yacqubiya. Ibn Zaghu taught the religious sciences in winter and mathematics and far ā ’idh in summer. In the madrasa alAtt ā ryn in Fez, cAbd ar-Rahmān al-Lujā’i (d. 1369) commented on books written by
his master, Ibn al-Bannā, on mathematics and astronomy.
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54
Al-Qalasādi: Sharh Talkhı̄s a’māl al-hisāb, edited by Farès Bentaleb; Arabic and French edition. Beyrouth: Dar al-Gharb al-Islami, 1999: 24.
55
Based on Brunschig, La Berbérie orientale sous les Hafsides.
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Paedagogica Historica
641
Gregg de Young points out that the attitude of those responsible for madrasas with
regard to mathematics varied according to whether they belonged to the Hanafite or
Sh ā ficite rite:
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Hanafi madrasa have more room in their curriculum for the study of the rational sciences
– including mathematics – than did Sh ā fii mad ā ris.
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For az-Zarnuji, a Hanafi writer of the late twelfth century, mathematics appears to fall
within the knowledge necessary to allow one to fulfill his station in life, either clearly less
important than either medicine or jurisprudence. Al-Bubak āni in sixteenth-century Sind
called mathematics a ‘permissible’ choice for specialization – only slightly above ‘blamable’
studies.56
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The publications we were able to consult neither confirm nor deny this observation
by De Young.
Madrasas attached to astronomical observatories were a privileged place for teaching mathematics, as they doubtless fostered encounters among the most eminent
scientists, surrounded by their disciples and their assistants, and thus created an ideal
atmosphere for learning the rational sciences.
Jamsh ı̄ d al-Kāsh ı̄ (d. 1436), in a letter to his father, gives some evidence of scientific life in Samarkand’s madrasa :
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… at Samarqand presently the champions of the learned are gathered together, and teachers who hold classes in all sciences are at hand, and the students are all at work with the
art of mathematics. Of these, four people have apportioned among themselves the explanation of the Ashkāl at-Ta’sı̄ s, and one is at work on an explanation of the tajnı̄ s al-hisāb,
and another wrote a treatise on a geometric proof for the rule of double false position.
Qadi-Z āde ar-R ū mi, who is the most learned of all, has written a commentary on the treatise of Jaghmı̄ ni and a commentary on the Ashkā l at-Ta’s ı̄ s….
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57
Every few days, His Majesty the Sultan would be present in the study circle, and when
he was there, the study of mathematics would be given priority…. One of the examinations
of the students is this, that every one who enters the study circle is taken unaware as to
what problem will come up, and the people of the madrasa was eloquent in repeated
investigation of it.58
What personal role scientists like Nas ı̄ r ad-D ı̄ n at-T ūs ı̄ (d. 1274), Qutb ad-D ı̄ n ashShirāz ı̄ (d. 1311) and Jamsh ı̄ d al-Kāsh ı̄ (d. 1436), directors of the observatories in
Marāgha, Tabr ı̄ z and Samarkand, had in the teaching of mathematics remains to be
determined. More generally, did the major translators of classical Greek and Indian
works and the eminent mathematicians and astronomers attached to the courts of
kings and princes, or to libraries and observatories, devote any of their time to
teaching? How, otherwise, are we to explain the fact that we can find textbooks,
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56
DeYoung, Gregg. “The Khulāsat al-Hisāb of Bahā al-Dı̄ n al-’Amilı̄ and The Dar-i-Nizāmı̄ in
India.” Indian Society for History of Mathematics 8, nos 1–4 (1986): 5.
57
Ulug Beg (1394–1449), ruler of Samarkand, founded the Samarkand observatory in 1425,
which was directed by Jamshid al-Kashi. Rosenfeld and Ihsanôglu, Mathematicians, 277–78.
58
Translation by E .S. Kennedy, in Orientalia 29, no. 2 (1960): 194.
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642 M. Abdeljaouad
commentaries and abstracts, traditionally used for teaching, within their considerable
mathematical output?59
Who Taught Mathematics?
It is difficult to obtain a definitive answer to this question from the data, as BaltyGuesdon saw when she tried to identify Andalusian mathematicians. Some of the
difficulties she encountered can be generalized to other regions, for example:
1. Securing information on given scholars, related to mathematics and figuring in
some bio-biographical ancient works requires patient cross-examination of other
sources. And, in spite of all precautions taken, it is still difficult to sketch a
convincing profile of students, or of teachers of mathematics.
2. Designation of their activities may be misleading and is often imprecise. For
example, the terms riy ā ’dha, his ā b or tacal ı̄ m were used to designate mathematics. Handasa may signify ‘theoretical geometry’, ‘practical geometry’ or ‘surveying. ‘Furthermore, individuals are often introduced as multi-specialists, as for
example is seen in the following excerpts from Rosenfeld’s and Ihsanôglu’s brief
bibliographies:
n° 343: Apparently a mathematician.
n° 347: Mathematician, astronomer and physician.
n° 351: Mathematician.
n° 362: Cryptographer, mathematician and astrologer.
n° 363: Philosopher, mathematician, astronomer and physician.
n° 365: Knowledgeable in inheritance (al-far ā ’idh) and arithmetician (h ā sib).
n° 369: Physician and astrologer.
n° 370: Mathematician, astronomer, knew linguistics and law well.
n° 420: Mathematician, astronomer and a great Persian poet.
n° 458: Scholar of mathematics and logic, also an astronomer.
n° 487: Mathematician and physician.
3. On encountering such designations of multi-specialists, Balty-Guesdon assumes
that the first qualification is the principal one, the others being indications of
serious interest in the further fields: n° 347 in the above list could be considered
to be a mathematician who was also an astronomer, while n° 369 was a doctor
who has some works in mathematics. Balty-Guesdon settles for a definitive
profile of the respective scholar, however only after meticulously checking other
data. In the above list, n° 458, designated as ‘a scholar of mathematics and
logic’ published eight mathematical treatises, none of which can be considered
as a textbook; while posterity remembers n° 487, al-Samaw’al al-Maghribı̄
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59
This question is particularly relevant for mathematicians and astronomers listed in Rosenfeld
and Ihsanôglu. Mathematicians under n° 41: al-Khwarizmi, n° 256: abu’l-Waf ā, n° 277: al-Kūhi, n°
281: Maslamā al-Majrı̄ tı̄ , n° 296: al-Sijzı̄ , 299: Ibn cIrāq, n° 309: al-Karājı̄ , n° 328: Hasan ibn alHaytham, n° 348: al-Birunı̄ , n° 420: Omar al-Khayyām, n° 635: ash-Shirāzı̄ , n° 845: al-Qushji, n°
873: Sibt al-Māridı̄ nı̄ .
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Paedagogica Historica
643
(d. 1175), as a creative mathematician who wrote no less than 14 books,
although he was actually a doctor of medicine and not a teacher. Often
biographical notes fail to indicate explicitly whether a scholar was actually a
teacher, or whether he had been granted Ij ā zat at-tadr ı̄ s (a license to teach). On
the other hand, should we not rule that every mathematician who had disciples
can be considered a teacher?
4. Should we assert that any mathematician who wrote mathematical treatises –
special commentaries, or books – was necessarily a teacher? Makdisi answers this
in the affirmative:
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In the Middle-Ages, writing books was a function of teaching connected with an oral
process of teaching, including dictation and note-taking. Books were meant for students;
they were the direct result of the teaching process.60
Some doubts may be appropriate with regard to famous doctors not renowned as
teachers, such as Adonı̄ m (d. 960) who wrote a textbook on Indian arithmetic in
Qayrawān, or Ab ū as-Salt (d. 1135), the author of a book on geometry written in
Mahd ı̄ ya. Did the hydrologist Ibn Shabbāt (d. 1282) really teach the content of his
treatise on geometry in Tozeur?
Should we differentiate between farā dhı̄ -hā sib from hā sib-farā dhı̄ (specialists in
inheritance and/or calculators)? The first would be specialists versed in the legal
aspects of inheritance with some knowledge of arithmetic, while the second would be
mathematicians capable of solving inheritance problems.
In her thesis, Balty-Guesdon presents some important results concerning mathematicians in Andalusia from the eighth to the eleventh centuries:
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- Ten per cent of the biographical notes are about scholars related either directly
or indirectly to mathematics.
- Mathematics was more frequently mentioned as a subject of study than as a
professional skill.
- Among 147 scholars designated as mathematicians, 55 were at the same time
specialists of farā ’idh [inheritance], the others were either astronomers, or
geometers, or specialists for commercial transactions as well.
- Teaching far ā ’idh was probably the major channel used to impose consideration for the teaching of arithmetic, and as a consequence, to develop teaching
mathematics in Andalus.61
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Adopting Balty-Guesdon’s approach, we examined Tuqan’s 150 biographical notes
concerning mathematicians and astronomers and found that 115 of these scholars
qualified as mathematicians, among whom only 34 were explicitly described as teachers while 105 had published at least one book on mathematics. We also noted that
some had worked in royal courts and in prestigious institutions, while others had had
a large number of disciples.
60
61
Makdisi, The Rise of Colleges, 74.
Balty-Guesdon, Médecins et hommes de sciences, 403.
644 M. Abdeljaouad
Table 1 North African Scholars with some Formation in Mathematics
Described as a teacher
No
textbook
Author of a
mathematical
textbook
No
textbook
Author of a
mathematical
textbook
25
49
5
11
4
10
14
16
2
12
27
101
3
19
0
14
18
48
6
20
Qualifications
Mathematics and farā’idh
Mathematics and/or astronomy,
medicine
Mathematics and other disciplines
Total
]am
[rac
Not described as a teacher
Upon examining biographical notes of 170 North African scholars listed by Driss
Lamrabet from n° 301 to n° 469 and living from the ninth to the fifteenth century,
we noted that 101 among these had had some instruction in mathematics, or practiced an activity related to the discipline. Table 1 summarizes our findings.
Nine of 33 mathematics teachers listed were specialists in inheritance science, and
21 were versed in rational sciences. Furthermore, not all teachers are on record for
having written a book, but 20 out of 68 authored some mathematics treatise, although
they are not described as teachers. We should like to suggest that these results should
be taken with a grain of salt, because scrupulous analysis of other sources as recommended by Balty-Guesdon is required prior to drawing definitive conclusions. It
appears that such methodical and systematic studies concerning the profiles of mathematics teachers working in different Arab/Islamic regions have yet to be undertaken.
How did a scholar become a teacher of mathematics? The literature on Arab education reveals that some factors like patronage, family tradition, a great master’s fame
or source of income seem to have been the major incentives to becoming a teacher.
How far did such factors determine a mathematics instructor’s choice of career?
(a) Patronage
The status of mathematics in society can be perceived from the extent of public intervention and financing, although the role of patronage seems, in fact, to have been
more decisive, as reported by Françoise Micheau:
No library has been founded, no hospital built, no astronomical observation made,
without financing from some rich patron: caliph or sultan, vizier or emir, wealthy notable
or mighty court man, looking for prestige, interested in knowledge, or displaying his
generosity. Somehow, the most important Arab scientific institution [in the Middle Ages]
was patronage.62
62
Micheau, Françoise. “Les institutions scientifiques dans le Proche-Orient médiéval.” In
Histoire des sciences arabes, edited by R. Rashed. Vol. 3. Paris: Le Seuil, 1997: 233.
Paedagogica Historica
645
Some of these patronages excelled:
●
Bayt al-Hikma [House of Wisdom] at Baghdad, held in high favor by Caliph alMa’m ūn who reigned from 813 to 833.
The courts of Cordoba’s Caliphs cAbd ar-Rahmān III (912–961) and al-Hakam II
(961–976), who encouraged the teaching of the rational sciences.
The court of Zaragoza’s ruler, al-Mu’taman ibn H ūd, who reigned from 1081 to
1085. Note that al-Mu’taman was himself a mathematician and published an innovative textbook on Euclidian geometry.
Maragha astronomical observatory, built in 1259 by the Mongol Khan Hulag ū,
and headed by Nas ı̄ r ad-D ı̄ n at-T ūs ı̄ (1201–1274).
Samarkand astronomical observatory, founded in 1425 by the Tim ūrid ruler
Ul ūgh Beg, who was himself an astronomer. The observatory was headed by
Jamsh ı̄ d al-Kāsh ı̄ (d. 1436) and an annex madrasa was directed by Qādhi Zāde
ar-R ūmi (c.1440).
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am
[]acr
m
i[]acr
am
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am
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u
[m]acr
Such institutions were generously financed, and they attracted the very elite of astronomers, mathematicians and other scholars – Arabs, Muslims and foreigners alike.
They received prestigious visitors as guests on their own in their quest for knowledge,
holding conferences for the purpose of presenting original results and new discoveries, followed by discussion and debate. For the adepts of the rational sciences, these
were places where could be found extensive libraries containing books to be read and
copied, as well as masters capable of commentary. For most of these institutions,
mathematics was considered to be a propaedeutic knowledge necessary for all the
theoretical sciences and practical arts. In the preceding section, we presumed that
these institutions may have been institutions of special learning that attracted the best
minds from afar – students and scholars who usually were able to draw on long years
of specialization in mathematics already.
(b) Family Tradition
Upon their retirement; teachers were usually succeeded by their most brilliant
disciples, or by the best qualified scholars available. Could the successor be one of the
teacher’s own descendants, as suggested by some items reported by Makdisi?63
Biographers indeed report on some well-known families of mathematicians and
astronomers, such as the Banū M ūsā family in Baghdad, whose members worked as
translators and mathematicians at Bayt al-Hikma, or the Thabit ibn Qurra (d. 901)
ancestry which boasted sons and grandsons, all of them teachers of mathematics and
astronomy,64 and the family of Abu’l-Waf ā al-Buzjāni (d. 998): Abu’l-Waf ā had
himself received instruction in mathematics from his uncles, and taught mathematics
um
[]rca
u
[m
]acr
am
[]acr
a[m]acr
63
64
am
[]acr
a[m]acr
Makdisi, The Rise of Colleges, 170–71.
Rosenfeld and Ihsanôglu, Mathematicians, n° 103, n° 169, n° 174, n° 251, n° 252 and n° 253.
646 M. Abdeljaouad
to his own son cUmar.65 It is also known that Sadr ad-D ı̄ n at-T ūs ı̄ succeeded his
father, Nas ı̄ r ad-D ı̄ n at-Tus ı̄ (d. 1274), as head of the Maragha observatory.66
In eighteenth-century Tunisia, the ash-Sharf ı̄ ’s family of Sfax produced many
scholars knowledgeable in mathematics and astronomy, some of them settling in
Cairo, where they taught at al-Azhar, while most of the others studied in Cairo but
returned to teach in Sfax in a madrasa founded by the ruler of Tunis for Muhammad
ibn al-Mu’addab ash-Sharf ı̄ (d. 1744). On scrutinizing Rosenfeld and Ihsanôglu’s
1500 notes, however, we discovered that, except for the families indicated above, no
more than a dozen cases of father and son are listed as mathematicians or astronomers, or as teachers of mathematics; these being:
m
i[]acr
m
i[]acr
m
i[]acr
u
[m
]acr
m
i[]acr
m
i[]acr
m
[i ac]r
m
[i ac]r
●
●
●
●
●
●
●
●
●
●
●
Yus ūf ibn al-Daya and his son Ahmad.67
Ishāq ibn Karnib (d. 878) and his son Abu’l-cAla.68
Muhammad as-Sijz ı̄ (tenth–eleventh century) and his son Abu Sacid.69
Ibn as-Saffār (d. 1035) and his son Muhammad.70
Ahmad al-Wāsitı̄ (d. 1046) and his son cIsa.71
Athı̄ r ad-D ı̄ n al-Abhāri (d. 1263) and his son Muhammad.72
Ibn al-Hā’im (d. 1412) and his son Muhammad.73
Qādi Zāde ar-R ūm ı̄ (c.1440) and his son Hasan Chelebi.74
Ahmad Lah ūr ı̄ (d. 1649) and his son Lutfalallah and his grandson Muhammad
(eighteenth century).75
Ibrāhim al-Jahhāf (d. 1655) and his grandsons al-Husayn and Hasan (d. 1716).76
Hasan al-Jabart ı̄ (d. 1774) and his son cAbd ar-Rahmān (d. 1774).77
u
[m
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am
[]acr
m
i[a]cr
am
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am
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m
i[]acr
m
i[]acr
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am
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am
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am
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am
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u
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u
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am
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m
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m
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m
i[]acr
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[]acr
We do not claim these to be definitive results, but it can at least be said that they are
puzzling, and need to be scrutinized by further investigation. Our tentative conclusion is that the influence of family tradition was not decisive.
(c) A Master’s Fame
When reporting on a scholar, biographers are inclined to tie him to his professors,
especially if the latter acquired fame. Lines of continuity can be established for
65
Ibid., n° 256 and n° 256.
Ibid., n° 606 and n° 610.
67
Ibid., n° 80 and n° 119.
68
Ibid., n° 123 and n° 153.
69
Ibid., n° 292 and n° 296.
70
Ibid., n° 310 , n° 312, n° 319, n° 358 and n° 363.
71
Ibid., n° 382.
72
Ibid., n° 595 and n° 615.
73
Ibid., n° 783.
74
Ibid., n° 808 and n° 834.
75
Ibid., n° 1106, n° 1178 and n° 1273.
76
Ibid., n° 1124, n° 1280 and n° 1279.
77
Ibid., n° 1367 and n° 1381.
66
Paedagogica Historica
647
teachers of mathematics (and sometimes for those of astronomy or of inheritance
sciences):
●
Maslama ibn al-Majr ı̄ t ı̄ (d. 1008) was head of the Andalusian mathematicians of
Cordoba. No less than five of his disciples became well-known teachers of mathematics, astronomy or far ā ’idh. His student Ibn Saff ār (d. 1035) also had many
students who became teachers of mathematics. Such lines of succession can be
traced for many generations in al-Andalus and in al-Maghrib.78
Sharaf ad-D ı̄ n at-T ūs ı̄ (d. 1213) was Kamāl ad-D ı̄ n ibn Y ūnis (1156–1242)
teacher. Kamāl ad-D ı̄ n, who settled in his hometown, Mossul in Iraq, taught arithmetic, algebra, Euclidian geometry, conics and astronomy. His disciples became
famous as mathematicians and teachers, as Theodorus of Antochia (thirteenth
century) and cAbdallatif al-Baghdādi (1162–1213). The best known among them
is Nas ı̄ r ad-D ı̄ n at-T ūs ı̄ (1201–1274).79
Ibn al-Bannā80 (d. 1321) of Marrakech had a large number of disciples who in turn
became teachers of teachers of mathematics in Southern Spain and in North
Africa. This line of continuity can be traced to Ibn Khald ūn (d. 1406), who taught
mathematics in Tunis around 1374.
Qādhi Zāde ar-R ūmi81 (c.1440), the head of Samarkand madrasa, taught geometry
and astronomy to students who worked and taught in Asia and in the Ottoman
Empire.
Bahā ad-D ı̄ n al-cAmil ı̄ 82 (d. 1622) started a long line of teachers of mathematics
in Ispahan.
m
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m
i[]acr
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]arc
●
m
i[]acr
u
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]acr
am
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am
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m
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am
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m
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m
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●
●
am
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am
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(d) The Incomes of Mathematics Teachers
Few mathematics teachers enjoyed regular emoluments or pensions paid by the
sovereign or by some wealthy donor. While this was the case of those working in
prestigious institutions, or in a royal or governor’s court, or for those teaching as a
full-time professor of law or as an assistant in a madrasa, all other mathematics
instructors had to procure their livelihood by various channels. In fact, this was the
case not only for teachers of mathematics but for all teachers, as reported by Makdisi:
‘[Abu Shama] chief complaints were that it was no longer possible as an honest man
to make a living teaching in the colleges’.83
How did mathematics teachers earn their living? Many mathematicians are known
to have instructed young boys, sons of wealthy merchants or of notables. Others
would teach arithmetic in kutt ā b to very young boys, charging their parents fees.
Scholars are reported to have taught arithmetic to youths before specializing in
a[m
]arc
78
Ibid., n° 783.
Ibid., n° 541, n° 576, n° 564 , n° 568 and n° 606.
80
Ibid., n° 696.
81
Ibid., n° 808.
82
Ibid., n° 1058.
83
Makdisi, The Rise of Colleges, 171.
79
648 M. Abdeljaouad
astronomy or in medicine, or even in other fields. Makdisi says that teachers had to
require fees from their students, and some of them would charge one-quarter of one
dirham for every verse dictated from a grammar textbook written in verse.84 Some
mathematicians are known to have been professional calculators, consulted by
lawyers and judges to help them solve difficult problems of inheritance or conflict
between merchants. Ibn al-cArab ı̄ (d. 1148) describes the urgent need for this kind
of specialist: ‘Calculating with small numbers is easy, but with complicated and
fractional numbers on has to think for a long period of time and, he might even have
to consult a specialist who has to be paid a high price’.85 Many professors of mathematics were at the same time judges, muftis, time-keepers, calendar specialists or
teachers in fields related to these professions. Others would have more technical
vocations: astronomers, astrologers, geometers (muhandis), examiners of measures
and weights, geographers, tax officers, bookkeepers, copyists of manuscripts, booksellers, ….
m
i[]acr
What Kind of Mathematics Textbooks were Used?
The field of Arab/Islamic mathematics has been the object of a large number of publications with a special focus on the achievements of Arab mathematicians and on
transmission and circulation of knowledge.86 Editions, translations and analysis of
many extant mathematics manuscripts, most of them used as textbooks, are now
accessible, but the literature is so extensive that it cannot be summarized in this
paper. We intend, however, to present some problems that could be of interest for the
history of mathematics instruction.
Can mathematics textbooks be classified according to some hierarchy? Can any
specific mathematics curricula be isolated?
A Hierarchy of Mathematics Textbooks
Arab textbooks can be a general introduction to some field, or extremely specialized.
While intended to be elementary, some textbooks contain original subject matter apt
to change problem-solving completely. This is true for all textbooks introducing
Indian arithmetic, and for Omar al-Khayyam’s works on algebra. Some other textbooks are long dissertations reproducing and commenting on the works of predecessors. Presenting the many textbooks that he authored, al-Karājı̄ (953–1023) wrote:
am
[]acr
84
m
i][acr
Ibid., 161.
From Balty-Guesdon, Médecins et hommes de sciences, 398.
86
Recent selected bibliographies on Islamic mathematics can be found in print such as Rosenfeld
and Ihsanôglu, Mathematicians, and on the Internet, such as those published by Hogendijk, Jan P.
Available online at: htpp://www.math.uu.nl/people/hogend/Islamath.html; INTERNET, or by
Oaks, Jeff. Available online at: htpp://facstaff.uindy.edu/∼oaks/Biblio/IslamicMathBiblio.htm; INTERNET.
85
Paedagogica Historica
649
‘Some were intended for beginners, some for intermediate and others for graduate
students’.87
Specialized treatises containing high-mathematics (ummah ā t) will often be evoked
and sometime quoted, but their transmission diminished and their copies tended to
disappear. Among these ummah ā t, Arabic translations of Euclid’s Elements, Apollonius’s Conics, and Archimedes’s Sphere and Cylinder were considered compulsory for
those who specialized in theoretical geometry, in astronomy and in philosophy. The
arithmetical chapters of Euclid’s Elements and Nicomachus’ Introduction to Arithmetic,
as summarized by Ikhwān as-Safā or by Ibn Sı̄ nā, were part of the curriculum for
students in philosophy. Every serious student learning algebra had to refer to works
of al-Khwārizmı̄ and of Abu Kāmil. All these basic standard writings were studied,
cut into parts, confronted with newer results, supplemented by original theorems and
eventually replaced by new treatises. These new writings would be abridged into brief
texts, which were again commented upon in more voluminous books that were,
however, themselves abridged.
For these manuscripts, Arab authors established a hierarchical order of types:
mabsūt (expanded), mutawassit (intermediate) and mukhtasar (abridged).
Expanded books contain complete theories, with all the necessary propositions
and proofs, and illustrated by a large number of examples. ‘They are useful for
anyone who studies a new field’.88 As-Sinjāri and Ibn al-Hā’im put Abu Kāmil’s
Kitā b al-Kā mil into this category. Ibn al-Hā’ı̄ m and Hājji Khalif ā (d. 1657)
followed suit for al-Karāj ı̄ ’s Kit ā b al-Fakhri.
Intermediate books are ‘those in which propositions and expressions are in equilibrium. They can be used by all kind of readers’.89 Ibn al-Bannā’s Kit ā b Usūl al-Jabr
and al-Karāj ı̄ ’s al-Badic are placed within this category by Ibn al-Hā’ ı̄ m, although the
latter handbook was intended for advanced students by its author, while the former
was meant for beginners.
Both expanded and intermediate textbooks were copied, and read by students who
studied on their own, calling their professor for assistance when needed.
Abridged textbooks or epitomes have ‘terse expression. They are useful for those who,
finishing their studies, wish to recall the main propositions, and also for the intelligent
beginner capable of catching concepts through concise expression’.90
In their article on epitomes, A. Arazi and H. Ben Chamai indicate that these books
‘s’adressaient à des publics de spécialistes et à des lettrés pressés d’en apprendre le plus
possible, dans le temps le plus court’.91
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am
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m
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This definition is reported in Adel Anbouba’s introduction to: al-Karājı̄ . Kitāb al-Badic fi’l
hisāb, Arabic edition and French commentaries by Adel Anbouba. Beirut: Université Libanaise,
1964: 29, note 96. He is quoting as- Sinjārı̄ , Irshā d al-qasid, Beirut, 1900: 19.
88
As-Sinjārı̄ in Irshā d, as recalled by Hājji Khalifa, Kitāb kashf az-zunun can ‘asami al-kutub wal
funun, Istanbul, new edition, vol. I, 1941; vol. II, 1943.
89
Ibid.
90
Ibid.
91
Arazi, A., and H. Ben Chamai. “Mukhtasar.” In Encyclopédie de l’Islam. Vol. VII. Leiden: E. J.
Brill, 1993: 536–38.
87
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a[rca
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m
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m
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650 M. Abdeljaouad
The geometry section of Ibn S ı̄ nā’s ash-Shifa is considered by its author to be an
epitome of Euclid’s Elements in which he ‘strives for an economical text in terms of the
number of words used while removing some of the standard components of a geometric demonstration’.92 In his introduction to Kit ā b al-Usul fi’l-Jabr wal muq ā bala, Ibn
al-Bannā insisted on the fact that ‘taking into account its volume, this is a short book
resembling epitomes, and taking into account the great quantity of knowledge it
contains, it may replace any expanded book’.93 For As-Sinjār ı̄ , al-Karāj ı̄ ’s al-K ā fi fi
c
ilm al-his ā b is an abridged script, and, according to ibn al-Hā’ ı̄ m, this is also true for
ibn Fall ūs’ Nis ā b al-hibr, and for Ibn al-Yāsam ı̄ n’s didactical poem al-Urjūza fi al-jabr
wal muq ā bala. ‘Ibn Yāsam ı̄ n’s expression, said he, is so delightful that many memorized it, and the propositions that it contains are so elaborate that many had to explain
them’.94 In the same vein, introducing his own arithmetical epitome: Kashf al-asr
ā rcancilm hur ūf al-ghub ā r, Al-Qalasād ı̄ wrote: ‘extract this concise and self sufficient
book avoiding lengthiness from my book titled “Kashf al-Jilbāb”. Some students will
find in it what they need, and more learned persons can use it as a memory aid’.95
As it can be seen from the preceding excerpts, abridged books essentially had the
function of memory aids for scholars who had already studied a subject in detail,
sometimes for intellectuals who sought to get a general idea of a given field, and for
very gifted students desiring to become initiated in new subject matter. Usually, as
soon as an expanded book had been written, its author would write an abridged text
summarizing what he considered essential in this work. According to this pattern,
abridged books were not supposed to be used as textbooks for beginners but the pedagogy based on rote learning turned out to change the role of epitomes. According to
this new methodology, students would write down some section of an abridged book
or of an Urjuza dictated by their instructor on a wooden slate, learn it by heart, then
wipe the slate clean and recite it next day in class. Memorizing prescribed sections of
an abridged book, usually consisting of commentaries, was mandatory for attending
courses on an advanced level.
This new method was sharply criticized by Ibn Khald ūn (d. 1406):
m
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]
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The large number of abbreviated textbooks available on scholarly subjects is detrimental
to the process of instruction…. These abbreviated textbooks express all the problems of a
given discipline, and the evidence supporting them, in a few concise words that are loaded
with meaning. This procedure is detrimental to good style and makes things difficult to
understand.
Scholars often approach the principal learned works of the various disciplines, which are
rather voluminous, with an intention to interpret and explain. They abridge these to make
it easier [for students] to acquire knowledge of them. This has a corrupting influence upon
92
De Young, “Euclidean Geometry”, 51–52.
Saı̄ dan, A.S. Arab edition of Ibn al-Bannā’s Kitā b al-Jabr wal muqā bala, Koweit: Al-Majlis alwatani li at-thaqā fa wal funūn wal adab, 1986: 505.
94
Ibn al-Hā’im. Sharh al-Urjūza al-Yā sminı̄ ya, Arabic edition and French commentaries by Mahdi Abdeljaouad. Tunis: ATSM, 2003: 57.
95
Souissi, Mohamed. Bilingual edition in Arabic and French of al-Qalasādi’s Kashf al-asrār can
ilm hurūf al-ghubā r. Tunis: Maison Arabe du Livre, 1988.
93
m
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m
a][rca
]am
[rac
]am
[rac
]am
[rac
u
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[rac
u]m
[rac
m
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]am
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m
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m
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u
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Paedagogica Historica
651
the process of instruction and is detrimental to the attainment of scholarship. For it
confuses the beginner by presenting the final results of a discipline to him before he is
prepared. This is a bad method of instruction. It also involves a great deal of work for the
student. He must study the abridgement’s wording carefully, which is complicated to
understand because it is crowded with ideas, and try to find out on that basis what the
problems of the given discipline might be. Hence, the texts of such abbreviated textbooks
are found to be difficult and complicated. A good deal of time must be spent on trying to
understand them.
… [More extensive and lengthy works] contain a great amount of repetition and lengthiness, but both are useful for the acquisition of perfect habituation. When there is little
repetition, inferior habituation is the result. This is the case with abridgements.96
In spite of Ibn Khald ūn’s critique, this pedagogical pattern had flourished in North
Africa and in Egypt since the thirteenth century, and traces were observed in Tunisia
even at the beginning of the twentieth century. Some abridged handbooks and
rhymed prose texts (urzūza) became popular:
u
[m
]acr
u[m
]a
rc
●
Al-Urjūza fi’l jabr wa’l-muqā bala [Poem on Algebra and al-muqabala], written
by Ibn al-Yāsam ı̄ n (d. 1204).97 It is a rhymed prose with 55 lines, introducing
different terms used in algebra and standard resolutions of all six canonical
equations.
Talkhı̄ s ‘acmā l al-hisā b (Concise Exposition of Arithmetic Operations) written by
Ibn al-Bannā (d. 1321).98 It is a short rhetorical presentation of operations on
numbers and fractions, proportions and algebra.
An-Nuzhat fi’l-hisā b [Delight of Arithmetic], written by Ibn al-Hā’ ı̄ m (d. 1312).99
It is a short introduction to Indian arithmetic.
al-Wası̄ lā fi cilm al-hisā b [Means in the Science of Arithmetic] written by Ibn alHā’ ı̄ m.100 It is an epitome of the author’s al-Macuna fi cilm al-hisā b al-hawā ’I
(Guidebook for the Science of Mental Reckoning), a book intended for
merchants and scribes, with many solved problems and not using Indian arithmetic.
Al-Muqnic fi’l jabr wa’l-muqā bala [Sufficient on Algebra], written by Ibn al-Hā’ ı̄ m.101
It is a rhymed treatise supposed to replace Ibn al-Yāsam ı̄ n’s Urjū za.
a
]m
[a
rc
am
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●
m
i][arc
m
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a[m
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a[m
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am
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●
●
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m
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am
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am
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Commentaries
A commentary (Sharh) can be of two kinds: either it contains explanations of definitions and propositions, justifications for algorithms, proofs of theorems, numerical
examples and solved problems, or it insists on linguistic, syntactical, stylistic and
96
Ibn Khaldūn, The Muqaddima, 415–6.
Rosenfeld and Ihsanôglu, The Mathematicians, n° 521. At least 13 commentaries on this poem
are actually known.
98
Ibid., 696. At least 12 commentaries on this poem are actually known.
99
Ibid., n° 783, M7. At least 16 commentaries on this handbook are actually known.
100
Ibid., n° 783, M8. At least four commentaries on this book are actually known.
101
Ibid., n° 783, M9. At least four commentaries on this poem are actually known.
um
[]rca
97
652 M. Abdeljaouad
epistemological aspects of the work’s wording. Al-Qalasādi’s Sharh Talkhı̄s ‘Acm ā l alhis ā b belongs to the first type; while many nineteenth-century commentaries are of
the second type, they are in fact of no help for effective calculations.102
The decay of mathematics seems to have been brought about as an ominous and
regrettable consequence of this succession of abridged works, commentaries on them,
commentaries on commentaries, epitomes of such commentaries, and commentaries
on such epitomes, ….
]m
ir[ac
am
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a[m]arc
a[m
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A Further Characterization of Mathematics Textbooks
Examining Rosenfeld’s and Ihsanôglu’s 1500 bibliographical notes, we gathered
further information on textbooks (Table 2).103
Although we had some difficulties in characterizing these textbooks, particularly
because of imprecise data and overlapping categories, we should like to add some
remarks that might lead to further research:
1. Ummahat (fundamental treatises) and expanded books were not used directly
as textbooks; some of their sections were copied back and commented on.
Commentaries were themselves abridged and their epitomes used as textbooks.
(2) Concerning teaching Indian arithmetic for beginners, Ibn al-Bannā’s Talkhı̄ s and
Ibn al-Hā’ ı̄ m’s an-Nuzha were used as textbooks to be memorized, teachers
commenting on them.
3. ‘Comprehensive’ arithmetic textbooks were considered as an introduction to the
fundamentals of mathematics. They usually begin with an introduction to Indian
arithmetic, continued with calculation techniques on positive integers and fractions, and with determining the unknown quantities by using proportions, or ‘the
method of two errors’, or algebra. Then came a section on practical geometry,
and a section containing a list of everyday problems with solutions obtained by
different methods.
4. The fact that so many textbooks refer to the Euclidian tradition is somehow
misleading, since only some of them dealt only with Euclidian geometry,
others only with Euclidian arithmetic, and some only with proportions and
irrationals. At the end of the thirteenth century, as-Samarkandi renewed
Euclidian geometrical demonstrative methods with his Ashkā l at-Ta’sı̄s, a
textbook containing 35 propositions taken from Euclid’s Elements (34 from
books I and II and the first proposition of book VI). As-Samarqandi’s Ashkā l
at-Ta’sı̄s was ‘widely read and copied for centuries…. It was also the subject
am
[]acr
m
i[]acr
a[m
]arc
]m
ir[ac
a[m
]arc
]m
ir[ac
From Souissi, Mohamed. Tadrı̄s ar-Riyā dhiyā t bil-cArabı̄ya, Actes du 3e Colloque maghrébin
d’histoire des mathématiques arabes. Tipaza, 1990: 34.
103
Only one textbook is recorded here even if its author had written several in the same field. We
did not take into account treatises that were not clearly intended for teaching purposes, such as letters or epistles.
102
m
]i[rac
]am
[rac
]am
[rac
m
]i[rac
Paedagogica Historica
653
Table 2 Types of Arabic Mathematical Textbooks
alKhwā rizm ı̄
d. 850
am
[]acr
Rosenfeld and Ihsanôglu (2003) →
Hisā b
(Calculus)
a[m]acr
1–255
am
[]acr
u[m
]acr
m
i[]acr
alK ā sh ı̄
d. 1430
alAmil ı̄
d. 1622
802
–1057
18
1057
–1500
26
am
[]acr
m
i[]acr
m
i[]acr
10
256
–419
7
420
–605
14
606
–801
5
Sexagesimal
Hawa’i
1
0
1
4
0
2
1
4
6
4
4
1
Indian
7
7
4
13
30
16
Reckoning
]am
[rac
N. atAbu’lalT ū s ı̄
Waf ā Kayy ā m
d. 998 d. 1131 d. 1274
m
i[]acr
Textbook titles
not explicit.
Sibt-al-Mārid ı̄ nı̄ 1
Ab ū’l-Waf ā2 al-Kar āj ı̄ ’s al-Kafi3 Ibn al-Bannā’s atTalkhı̄ s
Ibn al-H ā’ ı̄ m’s anNuzha.
Ibn al-Khawwām4
Al-Naysabūr ı̄ 5
Al-cAmil ı̄ ’s Khulā sat6
Al-Kwārizm ı̄ ’s Jabr7 Ibn al-Y āsam ı̄ n’s
Urj ūza
Ibn al-Hā’ ı̄ m’s alMuqnic
308
As-Samarkand ı̄ 8 al-Mu’taman9- atT ūs ı̄ 10 - Q ādh ı̄ Z āde11
Essential for astronomy
Essential for astronomy
Essential for astronomy
m
a][rca
u[m
]acr
m
]i[arc
m
]i[rca
a[m]acr
am
[]acr
m
i[]acr
am
[]acr
m
]i[a
rc
am
[]acr
comprehensive
0
4
1
9
7
43
m
i[]acr
am
[]acr
]u
m
[rca
m
]i[arc
a[m
]arc
m
i[]acr
Algebra
13
7
6
12
9
12
am
[]acr
m
i[]acr
am
[]acr
m
i[]acr
u]m
[rac
am
[]acr
Sub-total
Euclidian
31
25
30
13
27
8
44
10
74
15
102
10
Conics
Spheres
Trigonometry
Surveying
Sub-total
Total
4
7
1
11
48
79
3
10
3
5
34
64
2
1
1
5
17
44
1
4
0
4
19
63
0
2
4
7
28
102
0
1
1
6
18
120
Handasa
(Geometry)
m
i[]acr
u
[m
]acr
1
m
i[]acr
m
i[]acr
am
[]acr
m
i[]acr
am
[]acr
164
472
Rosenfeld and Ihsanôglu, (2003), n° 873, Sibt-al-M ā rid ı̄ n ı̄ (1432–1494): Raqā ’iq al-haqā ’iq fi hisā b ad-daraj wa’d-daqā ’iq [Subtilities of Truths on
Arithmetic of Degrees and Minutes].
2
Ibid.: n° 256, Ab ū’l-Waf ā al-Buzjā n ı̄ (940–978): Kitā b fi mā yahtā ju ilayhi al-kuttā b wa’l-cummā l min cilm al-hisā b [A Book about what is Necessary
for Scribes, Dealers, and Others from the Science of Arithmetic].
3
Ibid.: n° 309, al-Karā j ı̄ (d. ca 1025): al-Kā f ı̄ fi cilm al-hisā b [Sufficient Book on the Science of Arithmetic].
am
[]acr
u[m
]arc
a[m]acr
am
[]acr
am
[]acr
m
i[]acr
a[m
]arc
m
i[]acr
a[m
]arc
m
i[]acr
a[m
]arc
m
i[]acr
am
[]a
rc
m
[ia]cr
a[m
]arc
a[m
]arc
a[m
]arc
a[m
]arc
am
[]a
rc
am
[]a
rc
a[m
]arc
am
[]a
rc
Ibid.: n° 657, Ibn al-Khawwā m (1245–1325): al-Fawā ’id al-Bahā ’iyyā fi’l-qawā cid al-hisā biyyā [Notable Uses of Arithmetic Rules].
5
Ibid.: n° 686, al-Naysabūr ı̄ (13th – 14th): ar-Risā la ash-Shamsiyya fi’l-hisā b [Solar Treatise on Arithmetic].
4
a[m
]arc
am
[]acr
um
][rca
6
am
[]acr
8
a[m
]arc
a[m
]arc
a[m
]arc
a[m
]arc
m
i[]acr
a[m
]arc
m
i[]acr
am
[]a
rc
am
[]a
rc
m
i[]acr
a[m
]arc
Ibid.: n° 655, as-Samarkand ı̄ (2d half of 13th c.): Ashkal at-Ta’sı̄ s [Propositions of Substantiation].
m
]i[arc
m
i[]acr
9
a[m
]arc
Ibid.: n° 1058, Bahā ad-D ı̄ n al-cAmil ı̄ (1547–1622): Khulā sat al-hisā b [Essence of Arithmetic].
Ibid.: n° 41, Al-Kwā rizm ı̄ (780–850): al-Kitā b al-Mukhtasar fi’l-Jabr wal muqā bala [Abbreviated Book on the Reckoning of Algebra and Muqabala].
am
[]acr
7
a[m
]arc
a[m
]arc
m
i[]acr
Ibid.: n° 391, al-Mu’taman ibn Hūd (ca 1081–1085): Kitā b al-Istikmā l [Book of Improvement].
Ibid.: n° 606 , Nas ı̄ r ad-D ı̄ n at-T ūs ı̄ (1201–1274): Tahrı̄ r Kitā b usul al-handasa li-Uqlidis [Exposition on ‘Results of Geometry’ of Euclid].
a[m
]arc
um
][rca
10
m
i[]acr
11
m
i[]acr
u[m
]arc
m
i[]acr
a[m
]arc
m
]i[arc
a[m
]arc
Ibid.: n° 808 , Qā dhi Zā de ar-R ūm ı̄ (ca 1440): Sharh ashkā l at-Ta’sı̄ s [Commentary on ‘Propositions of Substantiation’].
am
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[]acr
u[m
]arc
m
i[]acr
a[m
]arc
m
]i[arc
654 M. Abdeljaouad
of a remarkable number of commentaries and supercommentaries of
commentaries’.104
Some Mathematics Curricula
What kind of curricula were prescribed to students learning mathematics? This
question is difficult to answer, since the subject matter taught was usually the sole
responsibility of the teacher. Nonetheless, we tried to trace some successions of
textbooks used in mathematics instruction in biographies of eminent scholars, or in
lists of textbooks established by famous teachers. At the end of this section, we shall
describe two curricular reforms, one in eighteenth-century India, and the second in
nineteenth-century Tunisia.
Curricula established through examining autobiographies:
(a) In fourteenth-century Tlemcen.105
- Urjūzat Ibn al-Yāsam ı̄ n (d. 1204);
- Talkhı̄ s ‘am ā l al-his ā b of Ibn al-Bannā (d. 1321);
- Kit ā b al-usūl wa’l-muqaddim ā t fi’l-jabr of Ibn al-Bannā;
- Rafc al-hij ā b of Ibn al-Bannā;
- First ten books from Euclid’s Elements.
(b) In fourteenth-century Tunis, Abu cAbdallah Muhammad al-Ansār ı̄ (d. 1488),
wrote that he had used Abu Bakr al-Wunshar ı̄ sı̄ ’s mathematics and farā ’idh
courses in the morning, and other lessons later for several years.106 The textbooks
studied were:
- al-Hassār’s small book,107 completed several times;
- Urjūzat Ibn al-Yāsam ı̄ n;
- Talkhis ‘am lā al-his ā b of Ibn al-Bannā, studied twice;
- Kit ā b al-jabr of Ibn Badr (thirteenth century);108
- Kit ā b al-far ā ’idh of al-H ūf ı̄ (d. 1192).
(c) In sixteenth-century Meknès, Mohammad Ibn al-Qādh ı̄ (d. 1573) taught the
following texts to his son Ahmad:109
- al-Hassār’s two textbooks;
- Sharh Muniyat al-hisāb of Ibn Ghāz ı̄ (d. 1513);110
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]m
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a[m
]arc
a[m
]arc
m
i[]acr
a[m
]arc
am
[]acr
u[m
]a
rc
a[m
]arc
a[m
]arc
am
[]acr
am
[]acr
am
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m
i[]acr
m
i[]acr
a[m
]arc
m
]i[acr
am
[]acr
u[m
]a
rc
am
[]acr
a[m
]arc
m
i[]acr
a[m
]arc
am
[]acr
a[m
]arc
a[m
]arc
a[m
]arc
u
[m
]acr
m
i[]acr
am
[]acr
m
i[]acr
am
[]acr
am
[]acr
am
[]acr
104
m
i[]acr
De Young, Gregg. “The Ashkā l at-Ta’sı̄ s of al-Samarqandi, a Translation and Study.”
Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften 14 (2001): 57–117.
105
According to Harbili, Anissa. L’enseignement des mathématiques à Tlemcen au 14ème siècle à
travers le commentaire d’al-Uqbā ni (m.811/1408). Thèse de magistère en histoire des mathématiques.
Alger: ENS Kouba, 1997.
106
According to Hedfi, ar-Riyadhiyat bi Ifriqiya.
107
Rosenfeld and Ihsanôglu, The Mathematicians, n° 532. M1.
108
Ibid., n° 587.
109
According to Lamrabet, Introduction à l’histoire des mathématiques maghrébines, n° 482, 139.
110
Rosenfeld and Ihsanôglu, The Mathematicians, n° 913.
a][marc
a[m
]arc
m
]i[arc
Paedagogica Historica
655
- Kitā b al-farā ’idh of al-H ūf ı̄ ;
- Some parts of Euclid’s Elements.
(d) In seventeenth-century Cairo: cAli al-Umi as-Safāqus ı̄ (d. 1789):111
a[m
]arc
a[m
]arc
u
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]acr
m
[iac]r
am
[]acr
-
m
i[]acr
Urj ūzat Ibn al-Yāsam ı̄ n;
First two chapters of al-Qalasād ı̄ ’ Sharh Talkhı̄ s Ibn al-Bannā ;112
Ibn Ghāz ı̄ ’ Sharh Muniyat al-hisā b;
Ibn al-Hā’ı̄ m’ Sharh an-Nuzhā ;
Ibn al-Hā’ı̄ m’ Sharh al-Was ı̄ la.
u[m
]a
rc
am
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m
i[]acr
am
[]acr
am
[]acr
m
i[]arc
m
i[]acr
a[m
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a[m
]arc
m
i[]acr
am
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m
]i[acr
am
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m
]i[acr
A Curriculum Established by Examining Sibt al-M ā ridinı̄ ’s Publications
a[m
]arc
]m
ir[ac
Muhammad Sibt al-Mārid ı̄ n ı̄ (1423–1506), a timekeeper of the al-Azhar Mosque in
Cairo, wrote no less than 50 treatises on astronomy (sine quadrants, sundials, astronomical tables and prayer times). He must have been a teacher of mathematics, since
he authored at least 23 mathematics textbooks. His prime interest being timekeeping,
he wrote an expanded book on sexagesimal arithmetic: Raq ā ’iq al-haq ā ’iq fi his ā b addaraj wa’l-daq ā ’iq [Subtleties of Truths on Arithmetic of Degrees and Minutes]. His
teaching of this subject is attested by his writing three different epitomes on it.113
Most of the other courses taught at al-Azhar Mosque by Sibt al-Māridini are based
on Ibn al-Hā’ı̄ m’s works on mathematics and inheritance:
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m
i[]acr
a[m
]arc
a[m
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am
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am
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●
●
m
]i[acr
On heritance sciences: nine handbooks.
On arithmetic: six textbooks on Indian arithmetic (two of them are commentaries
on Ibn al-Hā’ı̄ m’s short works, one is an introductory textbook and three are
epitomes of this book), and the last is on hawa’i hisab.114
am
[]acr
Sharh al-Lumca fi cilm al-his ā b li Ibn al-H ā ’ ı̄ m;
Tuhfat al-’ahb ā b fi cilm al-his ā b;
Talkhı̄s at-Tuhfa;
Khul ā sa fi’l-his ā b, written in 1495;
Al-Muqaddima fi cilm al-his ā b;
Irsh ā d at-Tull ā b ila’l-wasila fi’l-his ā b [li Ibn al-H ā ’ı̄ m]. The only textbook on
mental calculus.
●
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]i[acr
a[m
]arc
a[m
]arc
a[m
]arc
m
i[]arc
a[m
]arc
]m
ir[ac
a[m
]arc
a[m
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a[m
]arc
a[m
]arc
a[m
]arc
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]arc
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]arc
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rc
On algebra: six textbooks (three of them are commentaries on Ibn al-Hā’ı̄ m’s short
works, one is a commentary on Ibn al-Hā‘ı̄m’ Sharh al-Urjuza al-Yasmin ı̄ ya,
followed by two epitomes of it. The last is an introduction to algebra.115
am
[]acr
]m
i[a
rc
am
[]acr
m
]i[acr
]m
ir[ac
- Sharh al-Muqnic fi’l-jabr wa’l-muqabā la li Ibn al-Hā ’ı̄ m. It is one of two
commentaries on al-Muqnic of Ibn al-Hā’ı̄ m;
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]arc
a[m
]arc
am
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111
Ibid., n° 564, 158.
Ibid., n° 532. M1.
113
Ibid., n° 873, M1 (293), M2, M3 and M4 (294).
114
Ibid.: M8, M12, M13, M18, M23 and M7.
115
Ibid.: M5, M6, M9, M10, M11 and M25.
112
m
]i[acr
m
i[]arc
656 M. Abdeljaouad
- Sharh al-Mumtic li Ibn al-H ā ’ ı̄ m;
- Sharh Sharh al-urjūza al-Yasminı̄ ya, followed by two epitomes;
- Nis ā b al-jabr wa’l-muq ā b ā la.
a[m
]arc
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]a
rc
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i[]arc
a[m
]arc
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]arc
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]arc
From the data consulted, we were not able to establish a chronology for the periods
of teaching with these books. We conjecture, however, that Sibt al-Mārid ı̄ nı̄ started
teaching Ibn al-Hā’ı̄ m with short works, and later taught and wrote his own textbooks.
am
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m
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Two Official Curricula
It is difficult, from available data, to show which mathematics curricula were official.
Two of them were published, the Dars-ı̄ -Niz ā mı̄ , in eighteenth-century India, and
Khayreddin’s reform of az-Zaytuna Mosque at the end of the nineteenth century in
Tunisia. Although both were tentative efforts at renewing curricula, they are actually
an indication of how completely all modern mathematics was ignored, and how
totally isolated the Arab/Islamic scientific community was.
(a) The Dars- ı̄ -Niz ā mı̄ , in India (eighteenth century). Nizāmudd ı̄ n b. Qutbudd ı̄ n (d.
1748) attempted to reform the Lucknow (India) educational system, emphasizing the
teaching of the rational sciences, and excluding a large number of religious and legal
subjects. For mathematics, the proposed syllabus, known as the Dars-ı̄ -Nizā mı̄ ,
called for the study of:
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The Khul ā sat al-his ā b by Bahā ad-D ı̄ n al-c Ā mil ı̄ (d. 1622);
Al-Maq ā la al-ul ā li Tahr ı̄ r Uql ı̄ dis by Muhammad Barakāt (eighteenth century);
Tasr ı̄ h al-afl ā k by Bahā ad-D ı̄ n al-c Ā mil ı̄ (An introduction to astronomy);
Ar-Ris ā la al-Qushj ı̄ ya by cAli al-Qushj ı̄ (d. 1474);
Al-Bab al-’Awal li Sharh al-Jaghmı̄ nı̄ , a commentary by al-Jurjān ı̄ (d. 1413) on alJaghm ı̄ n ı̄ ’s abridged book on astronomy (d. 1221).
a[m
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am
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m
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A
[m
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m
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m
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A
[m
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The Dars-i-Nizām ı̄ aimed not merely at static preservation of past knowledge, but at
assisting the student in mastering the fundamentals of this knowledge so that he would
be broadly prepared to continue his studies in a variety of disciplines. At the same time,
this curriculum was criticized for failing to give adequate space to studies such history,
geography, literature, …. The Dars- ı̄ -Nizam ı̄ became the backbone of traditional
madrasah education in the Indian Islamic community from its first publication.116
(b) Az-Zayt ūna in Tunis (end of the nineteenth century). Higher education in the nineteenth century was the monopoly of the az-Zaytuna Mosque in Tunis. To be admitted, the student had, among other things, to have learnt by heart the didactic poem
ad-Durra al-baydh ā fi ahsan al-funūn wa’l-ashy ā ’ [White Pearl on the Better of Science
of Arithmetic and Inheritance] by cAbd ar-Rahmān al-Akhdhari (1510–1575).
Unchanged for centuries, the educational system generated an elite incapable of
meeting the imminent threat of French colonization.117
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116
De Young, “The Khulāsat al-Hisāb”, 7.
The French Protectorate in Tunisia was established in 1881.
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117
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[]arc
Paedagogica Historica
657
Among the 217 courses taught in 1871, only five were reserved for the commentaries of fara’idh and mathematics rhymed poems. In 1875, the Tunisian Prime
Minister Khayreddine decided to regulate pedagogy and the contents of courses at
the az-Zayt ūna mosque, and to structure the new curricula into three cycles: lower,
intermediary and superior, composed of 28 disciplines, with textbooks recommended for each.
For mathematics, 10 works were prescribed:
u
[m
]acr
Lower cycle:
- An-Nukhba al-his ā biyya;
- Sharh ad-Durra al-baydh ā fi’l- far ā ’idh wa’l-his ā b.
a[m
]arc
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]arc
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]arc
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]arc
Intermediate cycle:
- al-Murshida of Ibn al-Hā’ ı̄ m;
- Sharh Talkhı̄ s ‘acm ā l al-his ā b of al-Qalasād ı̄ ;
- Sharh Ashk ā l at-Ta’sı̄ s, a commentary by Qād ı̄ -Zāde ar-R ūm ı̄ on Ashk ā l atTa’sı̄ s of as-Samarqandi in geometry;
- Sharh al-Jaghmı̄ nı̄ , a commentary by Qād ı̄ -Zāde ar-R ūm ı̄ on al-Jaghm ı̄ n ı̄ ’s
abridged book on astronomy.
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am
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m
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m
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m
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Superior cycle:
- Sharh Muniyat al-hisā b, a commentary by Ibn Ghāz ı̄ on his own poem in arithmetic and algebra;
- Sharh Talkhı̄ s ‘acm ā l al-his ā b li Ibn al-Bann ā by al-Masrāt ı̄ ;
- Sharh at-Tadhkira, a commentary on as-Sayy ı̄ d on astronomy;
- Tahr ı̄ r at-Tusı̄ li maq ā lat Uql ı̄ dis, an important textbook on Euclidian geometry.
a]m
[rac
m
i[]arc
a[m
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a[m
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a[m
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m
i[]acr
am
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m
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m
i[a]cr
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m
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This new curriculum, which in fact was aimed merely to preserve traditional knowledge and methods, was severely criticized by az-Zaytuna’s professors, who hindered
its effective implementation. Mathematics instruction continued to be based on rote
learning of urjūzas, and on commentaries on linguistic and stylistic aspects of their
wording, preventing any effective practice of arithmetic or geometry.118
u[m
]a
rc
What Pedagogy for Mathematics?
Insights into Arab pedagogy can be found in some recently published works, few of
them specific for mathematics instruction. We should like to pinpoint three aspects
that seem to have had a noteworthy influence on mathematics instruction: (1) memorizing; (2) note-taking; (3) dust board.
118
Refer for example to Abdeljaouad, Mahdi. “L’enseignement des mathématiques en Tunisie
au 19ème siècle.” Cahiers de Tunisie 41–42, nos 151–154 (1986): 247–63, or to Souissi, Tadr ı̄s arRiyā dhiyā t bil-cArabı̄ ya.
]m
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The Status of Memorizing in Mathematics Instruction
Makdisi had noted two important features in Arab education: ‘The development of
the memory is a constant feature of medieval education in Islam…. Memorizing,
not meant to be unreasoning rote learning, was [to be] reinforced with intelligence
and understanding’.119 The entire process of learning had then been organized so
as to take into account memorizing as the most important pedagogical means:
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During the lesson, students were seated around the teacher in a halqa (a circle of
study) to listen to the recitation of the day’s course by a professor’s assistant, then
to listen to some commentaries by the professor.
Going back to his room, the student had to learn the course by heart; he eventually
wrote it down in a notebook, so it may serve as a reference.
Some of the student’s senior classmates would help him repeat the lesson many
times, to make him firm in recalling it.
To be able to understand the lesson, the student was supposed to study his professor’s commentaries on the subject matter by himself, analyze it and prepare for
being quizzed by his professor, or even for asking questions.
As examples of this type of teaching, we should like to mention al-Kāshı̄ ’s testimony
in a letter to his father, and Ibn al-Hā’ı̄ m’s recommendations to his students:
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… sometimes in the madrasa between [the king Ulug Beg] and one of the students, who
asks about a problem from any science, there may be such mutual refutation and give and
take as cannot be described. This is because he decreed and directed that until a scientific
problem penetrates his mind it is not established, and obsequious flattery should not be
indulged in and, if sometimes someone accepts blindly he embarrasses him by saying you
are treating us as ignorant. And, for the sake of examination of the problem, he may intentionally insert a mistake into the middle of the argument. As soon as anyone accepts it, he
reproaches and shames him.120
This method of teaching, as reported by al-Kāshı̄ , was applied in one of the most
prestigious state institutions: the madrasa of Samarkand. Elsewhere, students had
to find good professors in order to be able to listen to their lessons and study their
commentaries. An advanced student had to copy mathematical treatises, and to
devote time to studying them on his own at home. Many textbooks were written
for that purpose, particularly those on ‘comprehensive’ arithmetic; they contain a
great number of numerical examples and of solved problems, and sometimes even
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120
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Makdisi, The Rise of Colleges, 99 and 103.
Kennedy, “A Letter of Jamsh ı̄ d al-Kāsh ı̄ to his Father”, 205.
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some open problems.121 In his algebra textbook, Sharh al-Urjūza al-Yasminı̄ yya,
ibn al-Hā’ı̄ m gives advice to his students:
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When [the author of the poem] says ‘follow the scheme carefully’, he expects that the calculator takes some precautions and tries to avoid errors when computing, so that he gets
exact results. For that purpose, he has to test the accuracy of the results using adequate
rules set for verification of this type of reckoning…. Do not trust the apparent easiness of
this [numerical] example and its clearness, which might let you think that you have
mastered the five steps of the algorithm…. Do not hope that I will detail all the steps
needed to resolve each of the problems proposed….
These are examples of different kinds proposed to the reader in order to illustrate this
proposition. I did not increase the number of examples, however, to make the student
bored and tired, but [their number is sufficient] for mastering the technique and exercising
one’s mind.
It is evident that [what I have presented here] can be generalized by analogy. Try to generalize, and do not content yourself with memorization, only reproducing the cases treated.
What I have presented is in fact applicable to all situations.122
Ibn al-Hā’ı̄ m’s demands are examples of good teaching. However, he himself was
compelled to write concise résumés of his longer textbooks, and to teach on the basis
of his abridged versions. Since the dominant method of teaching was based solely on
the learning of abbreviated textbooks or didactical poems by heart, professors
commented less and less on mathematical aspects of the contents, and more and
more on their terminological and stylistic points of view. In fact, the feature of
‘memorizing’ undermined the goal of ‘comprehension and understanding’, and
ignorance of mathematics replaced creativity and expertise.
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Note-Taking in Mathematics Instruction
Many Arab scholars said in their autobiographies that they used to record their own
professor’s commentaries on the day’s lesson on an erasable board, in order to learn
the contents during the night, and then to wipe the board so that its was ready for use
next day. Advanced students, however, were expected to take notes on paper, as
suggested in the following text:
Once the lesson has been learned by heart, the student should write it down from memory.
The written record of the lesson is to serve as a reference when recall fails…. Committing
subject matter to writing was considered most important in the process of learning.123
121
Such as al-Fawa’id al-Baha’iyya fi’l qawa’id al-hisabiyya of ibn al-Khawwam (d. 1324). At the
end of this textbook the author proposes 33 open problems of the Diophantine type. He says: ‘I do
not pretend that I can establish the proof of their impossibility, but I say only that I cannot solve
them. Any one who can do it has competences that I don’t have’ (from Abdeljaouad and Hedfi,
“Vers une étude des aspects historiques et mathematiques des problèmes ouverts d’ibn al-Khawwā
m”, in Histoire des mathématiques arabes, Actes du 1er colloque maghrébin sur l’histoire des mathématiques arabes, Alger 1986.)
122
Ibn al-Hā ’ ı̄ m, Sharh al-Urjūza al-Yasmin ı̄ yya, 77–79, and 135.
123
Makdisi, The Rise of Colleges, 104.
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A large number of mathematical manuscripts still extant show that most have been
annotated on their margins and interlined. One finds teachers’ commentaries,
students’ remarks, solutions of problems and numerical examples not treated in the
text written down on the folios in all directions. In his paper on the Dar-I-Nizāmı̄ in
India, De Young (1986) noted that many manuscripts
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… were produced with exceptionally widely spaced lines and extraordinarily large margins.
[That seems] to show that this was a conscious action on the part of the copyists. It seems
likely that this style of copying was intended to facilitate the taking of notes.
De Young also suggests that examining annotations on a manuscript can help determine which parts of it have been effectively studied:
Typically, the annotations in these manuscripts will occur in the first quarter or third of
the volume, then suddenly cease. It would seem that then, as now, instruction did not
always cover the entire textbook assigned.124
One fascinating eighteenth-century manuscript, written in Istanbul, throws further
light on the function of margins: its writer added 300 mathematical expressions or
solutions of problems represented with North African algebraic symbols in the
margins.125
The Dust Board
The Arab users of Indian arithmetic associated with it the takht, a kind of dust board
described in the following terms by al-Hassār, one textbook author of the twelfth
century:
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It is called ‘Ghubār’ or Indian. They have given it this name because they used at the very
beginning a wooden lawha (board) on which thin dust was spread. Then the apprentice
reckoner would take a small stick whose form is that of a stiletto, draw the ciphers on the
dust and execute the intended calculations. Once the work was done, he would wipe up
the dust and store it. Efficiency of this tool stems from the fact that one can execute
calculations without having to constantly use ink, board and wiping out [of ink]. They
used dust instead of ink.126
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It should be noted that the text quoted refers to two kinds of wooden boards, the first
one is the takht or dust board imported from India and used for a long time in Arab
countries, and the second one is the lawha which is still being used in Kutt ā b around
Islamic countries. With a cane stick dipped in ink, one writes on a lawha covered with
soft argil, and at the end of the work it is wiped with water.
Using the takht as a computing tool, the author of a textbook would write down a
rhetorical version of his course illustrated by results taken from the dust board. He
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De Young, “The Khul āsat al-His āb,” 10.
Abdeljaouad, Mahdi. “Le manuscrit de Jerba, une pratique des symboles algébriques maghrébins en pleine maturité. Actes du 7ème colloque maghrébin sur l’histoire des mathématiques arabes.”
Marrakesh, 2002.
126
Ibid., 21.
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would usually precede any use of Indian ciphers by the expression ‘the image of the
result is as follows …’. J. L. Berggren (1986) explains clearly the process involved in
this transposition:
In the text of his book, Kushyār writes out, in words, all the names of the numbers, and it
is only when he is actually exhibiting what is written down on the dust board that he uses
the Hindu-Arabic ciphers. A reason for this may be that explanations were considered as
text and therefore written in words, like any other text. The examples of what was written
on the dust board, however, may have been viewed as illustrations, much like a diagram
in a geometrical argument, and they were there to show what the calculator would actually
see on the dust board.127
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Most famous Arabic mathematicians used takht and advocated its use for all types of
computing. This is the case of Al-Karāji and as-Samaw’al for operations on polynomial expressions represented by tables drawn on the dust board, Omar al-Khayyām
and Sharaf ad-D ı̄ n at-T ūsı̄ for calculating solutions of third-degree equations, and of
North African specialists in algebra who represented equations, and operations on
fractions, irrationals and polynomial expressions, by symbols drawn on the dust
board.
Associating stick and dust board with reckoning governed the techniques – based
on erasing intermediate results – used in Indian arithmetic and later in Arabic algebra.
For example, in early Arab/Indian arithmetic textbooks, multiplication of two
numbers was presented as a sequence of images copied from the takht, inserted in a
rhetorical discourse explaining each step of the calculation.128 It was also the case
later for multiplication of two polynomial expressions, as noted by J. L. Berggren:
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As-Samaw’al’s procedure is obviously intended to be used on the dust board, where
erasure is easy but space is at a premium, and it proceeds by a series of charts. It adapts
easily, however, to paper, where erasure is not easy but space is ample.129
However, not all Arabic mathematicians were satisfied with these techniques, and
from the outset they tried to replace them by methods using paper, pen and ink only,
as shown in al-Qalasād ı̄ ’s plea in the last chapter of his textbook on Indian arithmetic:
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In this book we state all that is done by Hindu schemes not with takht or erasure, but with
ink, pen and paper. This is because many a man hates to expose the takht between his
hands when he finds the need to use this art of calculation, for fear of the misinterpretation
of the attendants or whoever may see it. It belittles him, for it is seen between the hands of
the misbehaved who earn their living by astrology in the streets. Moreover, he who calculates on it finds it so difficult to reconsider what he has calculated to the extent that in most
cases he repeats it, not to mention the exposure of the content to the blowing wind which
changes the figures, apart from making the fingers dirty, over and above other things which
distort orderliness.
127
Berggren, Episodes in the Mathematics of the Medieval Islam, 32.
This appears clearly in Kushyā r ibn Labbān’s “Usul al-hisā b al-hind ı̄ .” See A translation with
introduction and notes by Martin Levey and Marvin Petruck. Madison: University of Wisconsin
Press, 1965.
129
Berggren, Episodes in the Mathematics of the Medieval Islam, 115.
128
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In addition to all that we have said, what we here suggest is simpler and quicker than the
arithmetic of the takht. Of it we shall show what will be appreciated and considered as a
novelty by all who see it.130
By the end of the thirteenth century, textbook authors advocated the sole use of
paper, pen, and ink for all calculations.131
We have presented some elements of three aspects of Arabic pedagogy here. Much
work, however, is left for a more accurate study of the process of teaching and learning mathematics in Arab/Islamic countries in the Middle Ages.
Case Study: Mahmûd Maqdîsh
A famous Tunisian historian, Mahm ūd Maqd ı̄ sh (1742–1813), was also the author
of a mathematics textbook. His case is interesting in that it offers irrefutable evidence
of the decline of the teaching of mathematics in the late eighteenth century in Tunisia.
Maqdı̄ sh was born in Sfax, a merchant and agricultural city on the Tunisian
seaboard. Despite his modest origins, he nonetheless pursued the long educational
process leading to professorship. After his initial training, he left Sfax and tried to
attend the courses of the al-Zayt ūna Great Mosque in Tunis but was unable to stay
due to a lack of funds. He then settled in Jerba, in the al-madrasa al-Jimmanı̄ ya,
created in 1703 by Ibrāhı̄ m al-Jimman ı̄ , ‘an islet of Malekite worship and education
within a predominantly Kharejite environment’.132 Maqd ı̄ sh indicated that accommodation in those days was free of charge, and that 270 students attended classes.
His description of how lessons were delivered is edifying:
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Students will prepare at night, reading under the supervision of the more advanced
among them two paragraphs from the Mukhtassar, one early in the night and the second
at dawn. Between the two sessions they had some rest and were awakened by the
shaykh. Each of the two lessons thus learned were then taught by two masters in two
successive sessions…. Within nine months, the book was finished, the text being read
three times in one day.133
In Jerba, Maqd ı̄ sh received his first grounding in mathematics of which he gives an
account in these terms:
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I attended shaykh Alı̄ b. Shāhid al-Muniy ı̄ ’s classes. Among other books he taught us
Kashf al-astār can cilm hur ūf al-ghubār. Once we had finished the first two chapters, our
professor did not tackle the chapter on irrationals and said: ‘I will not go any further’. I
then told him: ‘Our wish is to finish the book’. He retorted: ‘That is where our shaykh Sidı̄
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130
A l-Uqlı̄ dis ı̄. The Arithmetic of al-Uql ı̄ disı̄ , translated and annotated by A. S. Sa ı̄dan. Dordrecht–Boston: Reidel, 1978: 247.
131
For example, the Risā la ash-Shamsı̄ yya fi’l-hisā b [Treatise on Arithmetic Dedicated to Shams
ad-D ı̄ n] of al-Naysabur ı̄.
132
Abdesselem, Ahmad. Les historiens tunisiens des XVIIe, XVIIIe et XIXe siècles. Paris: Librairie C.
Klincsieck, 1973: 68.
133
Loc. cit.
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Ibrā hı̄ m stopped’. Our teacher stopped his course at this point and did not go any
further.134
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After finishing his studies in Jerba, Maqd ı̄ sh went to Cairo, where he attended the
teachings of the al-Azhar professors in various subjects. We know the names of two
of his professors in the rational sciences: Ahmad ad-Damanh ūri and Hasan Jabarti,135
but nothing is known about the course of studies pursued. Throughout his stay in
Cairo, Maqd ı̄ sh had to provide for his own living and for that of his family left behind
in Sfax, copying and selling valuable manuscripts.
Once he returned to his hometown, Maqd ı̄ sh devoted all his time to teaching and
continued to make his living as a copyist and seller of manuscripts. As a professor, he
was appreciated and many of his disciples acquired fame as well. He wrote numerous
epistles on fiqh (positive law) and a quite original history book that became the cause
of his own fame. He insisted, however, that the first book he had ever written was on
mathematics: Icanat dhawi ‘l-istibsā r cala Kashf al-astā r can cilm hurūf al-ghubā r136
[Helping those who Scrutinize in Opening the Store on the Science of Ghubār
Figures]. It is in fact a commentary of the abridged textbook that the Jerba professor
was unable to finish several years earlier. In the introduction to this book, we can read
the author’s recriminations against his Tunisian teachers of mathematics:
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… the traces of the science of reckoning are rubbed out, its secrets lost…. Those who teach
these days do it rashly and incorrectly…. They choose the shortest abstract by al-Qalasād
ı̄ …. I was among those who tried to study, but I had only myself to rely on…. The teachings we were given in the subject were scattered by the winds, words spoken at night were
forgotten in the morning as that which is not recorded in a book is not fixed in the mind
and does not integrate thoughts.137
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Although accessible on the Internet, on the Tunis National Library website,
Maqdı̄ sh’s textbook has not been published and analyzed yet. In any case it
stands as evidence of a final effort at revitalizing a mathematical culture that slunk
into its own shell, totally impervious to the fantastic developments taking place to
the North of the Mediterranean, in Europe.
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Conclusion
When we started this work, we had in mind some problems concerning Arab/Islamic
education that we hoped to look into: the status of mathematics within Arabic
knowledge, institutions where mathematics were taught, the teachers of mathematics
fulfilling this task, the kind of textbooks they used and the methods of teaching. On
134
This anecdote is told in Mahmud Maqd ı̄ sh’s own book on history, when he presents the teachers
in the madrasa al-Jimmanı̄ ya in Jerba. Maqd ı̄ sh, Mahmud. Nuzhat al-anzar fi caja’ib at-tawarikh wal
‘akhbar, edited by M. Mahfoudh and A. Zouari. 2 vols. Beirut: Dar al-gharb al-islami, 1988: 446–47.
135
Jabarti was a great teacher of astronomy; he died in 1774. Cf. Rosenfeld and Ihsanôglu, The
Mathematicians, n° 1367.
136
By Al-Qalasadi; Rosenfeld and Ihsanôglu, The Mathematicians, n° 865.
137
From the biobibliographical notice written by Mahfoudh. Tarajim al-muallifin at-tunusiyyin,
vol. IV, 358–59.
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the finishing line, we are left with a great number of unanswered questions, which
shows an urgent need for more analysis of all the accessible sources, one focusing on
mathematics instruction and learning.
While presenting the different issues, we felt the necessity to do more work and to
use new types of investigative tools, some of the statistical type and some borrowed
from history, sociology and ethnography, in order to obtain answers to certain
questions:
1. Did mathematics take any place in the instruction of specialists other than those
working in the inheritance fields or in the astronomical subjects?
2. Can we obtain more precise information on the training of teachers whose major
field of teaching was mathematics?
3. Did the great mathematicians working in royal courts and official institutions,
whose books were creative and innovative, have a direct role in mathematics
instruction?
Several other problems should be investigated, notably the open questions concerning the education of the mathematicians in Spain and in North Africa, enumerated in
the conclusion of Djebbar’s recent paper. These questions can easily be widened to
the other Arab and Islamic regions, in particular the following:
… Why did the science of calculation become the main element of mathematics (in all
aspects) in the Maghreb in the post-Almohad period? [This] question involves the position
of mathematics as a whole, and the possible negative influences of the environment on
scientific activity.138
The last haunting questioning that accompanied this work concerns the search for an
acceptable explanation for how a number of nations that were able methodically and
ingeniously to develop so many fields of mathematics – Indian arithmetic, algebra,
geometry, trigonometry, … – lost all interest in furthering their knowledge and learning
new sciences, stopped their study trips (Rihla) and retired into their own shell, repeating the teaching of ancient and for the most part obsolete subject matter. Sonja
Brentjes’s recent paper is a good lead in the search for some answers to this question.139
Finally, one should not neglect the important field of research concerning the introduction of modern European mathematics and sciences into the Arab educational
system in the nineteenth century, as developed recently, for example, by Pascal
Crozet.140
138
Djebbar, Ahmed. “Mathematics in al-Andalus and the Maghrib between the Ninth and Sixteenth Centuries.” In The Enterprise of Science in Islam: New Perspectives, edited by J. P. Hogendijk
and A. Sabra. Cambridge, MA: MIT Press, 2003: 333.
139
Brentjes, Sonja. “On the location of the ancient or ‘rational’ sciences in Muslim educational
landscapes (AH 500–1100).” Bulletin of the Royal Institute for Inter-Faith Studies 4, no. 1 (Spring–
Summer 2002): 47–71.
140
Crozet, Pascal. “La trajectoire d’un scientifique égyptien au 19ème siècle: Mahmûd al-Falaki
(1815–1885).” In Entre Réforme sociale et mouvement national, Identité et modernisation en Egypte
(1882–1962), edited by Alain Roussillon. Cairo: CEDEJ, 2002.

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