Published online by Cambridge University Press: 24 December 2009
A discussion of the mathematical manuscripts of Mauretania requires an overall introduction to this almost completely unexplored literature.
Its foundations must first be established. Between the years 1978 and 1985 a project financed initially by the Ministry of Foreign Affairs (BRD), then by the Deutsche Forschungsgemeinschaft, was entrusted to Dr. Rainer Oßwald and myself by the Mauretanian Ministry of Cultural Affairs. Its objective was the collection and protection of the selected Arabic manuscript literature of Mauretania.
1 The most recent and most comprehensive introduction so far to the development of Mauretanian literature, especially with emphasis on the social and economical framework of the intellectual life of the Moors, is the study of Oßwald, Rainer: Die Handelsstädte der Westsahara. Die Entwicklung der arabisch-maurischen Kultur von Šinqīt, Wādān, Tīšīt und Walāta (Berlin, 1986)Google Scholar. For further bibliographical sources see the review of Norris, H. T., BSOAS, LI, 2 1988, 337–40.Google Scholar
2 Short descriptions of the manuscripts are to be found in: Rebstock, Ulrich, Sammlung arabischer Handschriften in Mauretanien: Kurzbeschreibungen von 2–239 Manuskripteinheiten und Indices (Wiesbaden: Harrassowitz, 1989)Google Scholar. The Katalog der arabischen Handschriften in Mauretanien, bearbeitet von Ulrich Rebstock, Rainer Oßwald und A. Quid ‘Abdalqādir (Beiruter Texte und Studien, 30, Beirut, 1988)Google Scholar provides a fuller insight into the first 100 manuscripts of this collection. The manuscript numbers cited in this text refer to the numbering in the Sammlung. Since most of the original texts are not paginated and their number of folios on the microfilms not easily identified, a pagination per pages will be used.
3 Brockelmann, Carl: Geschichte der arabischen Litteratur, I–II (Weimar/Berlin, 1898–1902)Google Scholar. Supplementbände I–III (Leiden, 1937–1942)Google Scholar = GAL, S II, 337–8; Suter, Heinrich: Die Mathematiker und Astronomen der Araber (Amsterdam, 1981 [Leipzig, 1900–2]), 186–7Google Scholar; Matvievskaya/Rosenfeld, : Matematiki i Astronomy musulmanskogo srednevekovya i ix trudy (Moscow, 1983), I–IIIGoogle Scholar (= MM), II, 533; but all of them give Meknes as the last station of his life.
5 The Abjad-numerals, i.e. the ‘ABCD-numerals’, were used for counting with letters (ḥisāb aljummal) before the introduction of the ‘Indian’ ciphers and for a long time after. This notation was common in various fields of astronomical practice, see EI (1st edition), s.v. Abdjad.
6 MS No. 1789, p. 11/3 f.
7 MM, II, 533Google Scholar. M. Sawīsī's edition of the Bughya (Halab 1403/1983) is based on the copies from Tunis and Rabat. Since MS No. 226 differs considerably from this edition, I will cite from the Mauretanian manuscript.
8 cf. for example MS No. 695, pp. 68, 70 with MS No. 226, pp. 170 f.
9 MM, II (No. 444), 510–2Google Scholar; Suter, , Mathematiker, 180–2Google Scholar; GAL, II, 266, S II, 378Google Scholar; Djebbar, A., Enseignement et recherche mathématiques dans le Maghreb des xiiie–xvie siècles (Publications mathématiques d'Orsay, No. 81–2, Orsay, 1981), 144passimGoogle Scholar. A complete list of his works is listed in Bīblīughrāfīya al-Qalaṣādī (Manshūrāt dār al-kutub al-watanīya, Bāgha, 1976).Google Scholar
11 Woepcke, Franz, Etudes sur les mathématiques arabo-islamiques (Frankfurt, 1986), II, 1–2.Google Scholar
13 MM II (No. 339), 443–6Google Scholar: died c. 721/1320; Suter, , Mathematiker, pp. 162–4Google Scholar, n. 81: starb gegen 740/1339–40; cf. Djebbar, A., Enseignement, 140Google Scholar; Souissi, : ‘Ibn al-Bannā’’, Talkhīṣ a'māl al-ḥisāb. Texte élabli, annoté et traduit par Mohamed Ṣouissi (Tunis, 1969), 17–18.Google Scholar
15 cf. p. 62.
16 See, for example, the chart for the dissemination of factors, called jirbāl, in Kashf, 26, reproduced in Idrāk, 58.Google Scholar
18 MS No. 226, p. 129. al-'Uqbānī is not mentioned among the teachers of Ibn Ghāzī listed in Bughya (ed. Sawīsī), pp. w-y.
19 MS No. 438, p. 1.
20 The Simlāla, a sub-tribe of the Guezzoūlā, lived from the tenth century onwards along the southern flank of the Moroccan Atlas. Already al-Ghazzālī knew about their extraordinary contribution to the intellectual life of the Maghrib. ‘al-Samlāl’ would be a very uncommon nisba for a man from a tribe called Simlāla.
22 see EI (2nd edition), s.v. al-ḥurūf.
25 cf. No. 101, p. 29 with No. 489, p. 65/14.
27 cf. Woepcke, Franz, ‘Traduction du traité d'arithmétique d'Aboúl Hasan Alī Ben Mohammed Alkalsādī’, in Franz Woepcke, Etudes sur les mathématiques arabo-islamiques (Frankfurt, 1986), II [pp. 1–63], pp. 257–60Google Scholar with MS No. 101/1, pp. 28/7–33.
28 MS No. 101/1, p. 28/8.
30 MS, Berlin No. 5970, p. 13/–1 f.; Brentjes, A., ‘Untersuchungen zum Nichomachus Arabicus’ (presentation given at Oberwolfach 30.4.1987), paper: p. 8Google Scholar; the expanded version is published under the title ‘The first perfect numbers and three types of amicable numbers in a manuscript on elemantary [sic] number theory by Ibn Fallūs’, in Erdem, iv, 11 (May 1988), 467–83.Google Scholar
31 The terms ‘difference of how’ (tafäḍul fī ‘l-kaif) and ‘difference of how much’ (tafāḍul fī ‘l-kaif) are used to describe a geometrical and arithmetical progression respectively.
32 MS No. 226, p. 18 f.; Bughya (ed. Sawīsī), 28Google Scholar f; this fascinating problem is connected to a legendary Indian ruler who could not keep his promise. A wise man had solved a riddle posed to him at court. But beforehand he had asked the ruler for a reward: the quantity of rice which the doubling of one grain of rice as many times as there are chess fields would yield. The ruler had to realize that this inconspicious demand would swallow up his entire kingdom.
33 Ibn Ghāzī restricts the validity of formula (2) correctly to all cases where n is an ‘even-even’ ( = even) number. Apart from that, he explicitly states that (2) and (3) may have an optional q. The relevant passage runs:
This is the second type of ‘the difference of how’, i.e. of a geometrical proportion (nisba handasīya). Its numbers follow each other proportionately, not (necessarily) in the ratio of ½. This is meant by ‘halving’. For the sake of practice, I have chosen the following example:
Let four successive numbers be in the ratio of ⅓, in the shape of 1, 3, 9, 27. If we want to sum them up, then we multiply the smallest, i.e. 1, by the difference between 1 and 27. The difference is 26 and the result will be 26. Divide this by the difference between 3 and 1, which is 2. You will get 13. This is the sum of all that is before the 27. Add it to the 27, the largest number. This is 40, the sum of all numbers.
35 The canonical ounce, i.e. ūqīya, of Arabia had 40 dirham and weighed 125 grammes; see Hinz, Walther, Islamische Masse und Gewichte (Leiden, 1955), 35.Google Scholar
36 MS No. 226, p. 23.
43 MS No. 37, p. 8/12 f.
44 MS No. 1789, p. 11/–6 f.
46 MS No. 37, p. 108/6 f.
47 The Arabic text (ibid., 109) runs as follows: wa-lau marra rākib ḥimar bi-unās, fa-sallama ‘alaihim;’ wa-qāla: al-salām ‘alaikum yd mi'at rajul wa-rajul!’ fa-qāla aḥaduhum: ‘lau anna ma ‘anā mithlunā wa-mithl niṣfinā wa-rub’ inā wa-inta wa-ḥimāruka la-kunnā mi'ata wa-wāḥid.’
In the oldest known version of this problem, Metrodoros ‘the Egyptian’ (first half of the sixth century) counts dancing girls instead of bedouins; see Hunger, Herbert and Vogel, Kurt, Ein byzantinisches Rechenbuch des 15. Jahrhunderts. 100 Aufgaben aus dem Codex Vindobonensis Phil. Gr. 65. Text, Übersetzung und Kommentar (Wien, 1963), 39–40.Google Scholar
48 MS No. 695, p. 68/–5 f.
51 MS No. 101/1, p. 15/–1 f.
54 MS No. 101/1, p. 47 ff.
55 See Saidan, A., The arithmetic of al-Uqlīdisī, The story of Hindu-Arabic arithmetic as told in Kitāb Fuṣūl fī al-Ḥisāb al-Hindī by Abū al-Ḥasan Aḥmed ibn Ibrāhīm al-Uqlīdisī. Translated and annotated by Saidan, A. S. (Dordrecht, 1978), p. 35Google Scholar, n. 1.6.
56 p. 2.