Al-Hassâr's Kitâb al-Bayân and the Transmission of the Hindu-Arabic Numerals

This article was a talk given at the 7th Maghrebi Colloque of the History of Arabic Mathematics held from 30 May to 1 June 2002 in Marrakech, Morocco. It presents a new manuscript of the mathematical work Kitâb al-Bayân by the Moroccan mathematician of the 12th centrury Al-Hassâr, together with related remarks on the transmission of the Hindu-Arabic numerals to the medieval West.

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The library of the University of Pennsylvania, Philadelphia, PA, has published online a provisional catalogue of about 400 manuscripts of scientific interest in Latin, Arabic and other languages from the private collection of Lawrence J. Schoenberg, Longboat Key, Florida. In 2001 I had the opportunity to inspect a number of astronomical [1] and mathematical manuscripts of the collection in Philadelphia – where a selected number of them had been transferred for an exhibition – and in Mr. Schoenberg's residence.

Here I give a short description of MS LJS 293 which caught my special attention, because in the internet catalogue its author, Abû Bakr, was declared to be the so far unidentified Alabuchri, author of a geometrical treatise known only in a Latin translation by Gerard of Cremona [2]. The contents of the Arabic manuscript, however, proves to be purely arithmetical and is, therefore, different from the Latin treatise (see below) [3].

MS LJS 293 comprises 87 folios (the fly-leaf is not counted). There are no folio numbers in the manuscript itself. I here follow the pagination added to the pages of the text in the library's scanned reproduction [4]. On folio 2r there appears the title and the author's name (written by the same hand which wrote the whole manuscript): Kitâb al-bayân wa-l-tadhkâr [this word is partly obliterated, but the reading is clear, because the title is repeated in the author's preface on folio 2v] fî san‘at ‘amal al-ghubâr, ta'lîf al-shaykh al-ajall Abû Bakr ibn Muhammad ibn ‘Ayyâsh al-Hassâr. Under the title there is an owner's name, ‘Alî ibn Tha‘lab al-Sâ‘âtî al-Baghdâdî. Deeper down on the page there is the notice, in Persian ductus: min kutub Muhammad ibn Muhammad ibn al-Hasan al-Tûsî ghafara Allâhu lahu wa-li-wâlidayhi ("Belonging to the books of M. b. M. b. al-H. al-Tûsî"). It is of some historical interest to know that the manuscript once belonged to the library of that famous Muslim scholar [5]. At the bottom of the page a more recent Eastern Arabic hand, [6] which has also added several more notes in the course of the text, has put down, for identification, in an upper line, the Western Arabic forms of the nine numerals and, in a lower line, their Eastern Arabic correspondences.

Figure 1: Folio 2r of al-Hassâr's Kitâb al-Bayân bearing at the bottom both the Western and the Eastern Arabic forms of the nine numerals (Source).

The work begins on folio 2v with the author's preface and a list of the headings of ten chapters forming Part I (both the chapters and the Part are similarly called bâb). The text itself begins on 3v.

The manuscript is written in a clear naskhi script, mostly not dotted. On 87r, there is the copyist's colophon. The copy is dated Safar 590 (equivalent January-February 1194); it was written in the famous Nizâmîya madrasa in Baghdad by Muhammad ibn ‘Abdallâh ibn al-M-h-l al-Baghdâdî al-Hâsib. The numerals (in ch. 2, fol. 5r, and through the whole manuscript) are Eastern Arabic numerals. On f. 5r the annotator has again added the corresponding Western Arabic forms.

The Kitâb al-Bayân and its author, Abû Bakr al-Hassâr, are known since more than a century [7]. Up to now, four manuscripts of the text had been located: Gotha, Pertsch 1489 (see below); Rabat, BG [8] Q 917; al-Zâwiya al-Hamzawîya (Morocco); and Damascus, Zahiriya ‘âmm 9760 [9]. The Gotha manuscript is in clear naskhî, written in 836 H / 1432. Contrary to the Schoenberg manuscript, where all numerals have been converted into Eastern Arabic numerals, the Gotha manuscript has retained the numerals in their Western Arabic form. The Damascus manuscript is dated 1003 H / 1594-95 and consists of 116 folios; more details are not known [10]. The Rabat manuscript is in Maghrebi script and, of course, has Western Arabic numerals; it is not dated (judging from the script, one would ascribe it to a period around or after 1500) [11]. As it seems, it contains only the ten chapters of Part I [12]. Regarding the manuscript in the Zâwiya al-Hamzawîya we have no further information.

The Schoenberg manuscript is at present the fifth known represen¬tative of the text and, as it seems, the oldest. In 1901, H. Suter has given a detailed description of the book's contents based on the Gotha manuscript [13]. The work has also been translated into Hebrew, in 1271, by Moses ben Tibbon [14]. It is now also known that al-Hassâr has edited two versions of his Arithmetic, a shorter one, the Kitâb al-Bayân, perhaps for the use of students, and an extensive version, the Kitâb al-Kâmil [15]. The life time of al-Hassâr has so far been roughly ascribed to the 12th century CE. The Schoenberg manuscript, copied in 1194 CE in Baghdad, now gives a firm terminus ante quem for his activity. The Kitâb al-Bayân must have been written, in the Maghreb, sufficiently earlier so that it could reach Baghdad, in the Arabic East, to be copied there in 1194.

The contents of MS LJS 293 can be easily compared to, and identified with, the detailed description of MS Gotha given by Suter.

The Preface and the ten chapters (bâb) of Part I (al-bâb al-awwal) correspond to Suter, pp. 13-23 (3v-25r). Then follows a section on the multiplication of fractions, in 72 bâbs, 25r-66r (= Suter, pp. 23-28). Hereafter follow more, unnumbered, chapters (bâb) on the treatment of fractions, up to 87r (Suter, pp. 23-35). Here ends MS LJS 293. In the colophon it is said: tamma al-juz' al-awwal [now called juz'] min al-bayân … yalîhi fî 1-thânî bâb qismat al-kasr ("Finished is the first Part [juz'] of the Bayân …; there follows in the second [sc. Part] the chapter [bâb] of the division of the fraction" (sic). But this remaining part of the work (about a quarter of what is in MS Gotha – in Suter reaching until p. 39) is absent from MS LJS 293, which ends here.

Figure 2: Colophon of Kitâb al-Bayân (folio 87r). (Source).

I take [16] the opportunity of this talk presented in a gathering of Maghrebi colleagues to proceed to the second part of my conference in Arabic, and you will see the reason of that in a moment. The introduction of the Indian numerical system with the nine numerals and the zero from India to the Arabic Orient occurred in the 8th century; this is well known and widely recognized and needs not to be presented in detail here [17]. Also, it is well known that this knowledge – like most of the sciences that developed in the Arabic Orient – was transmitted later to the Islamic West (that is, the countries of North Africa and al-Andalus), where it was diffused and developed. The Europeans adopted these nine numerals in Spain and after a while they adopted also the whole Indian calculus from the Arabs of al-Andalus [18]. Among the proofs attesting the transfer of the Indian calculus to the West is the testimony of Abû Sahl Dunas b. Tamîm in Qayrawân in 955-56 in one of his works where he said:

"The Indians imagined nine signs for the unities. I talked about that sufficiently in a book that I wrote on the Indian calculus known under the title Hisâb al-ghubâr, that is calculus of dust [19]."

I mention also another early proof, viz. the occurrence of the nine digits in two Latin manuscripts, one of them dating from 976 CE and the other from 992 CE. These manuscripts are now preserved in El Escorial Library near Madrid. In these two documents, the nine numerals are not part of the original text, but were inserted at some later point, before or when copying the manuscript of 976 [20].

Figure 3: Eastern and Western Arabic numerals in folio 5r of al-Hassâr's Kitâb al-Bayân (Source).

Also, it is well known that the forms of the nine numerals in the Maghreb as we know them differ in various respects from the ones spread in the East. This difference is not an isolated phenomenon, since the Maghreb differs from the Mashriq (Orient) in various other respects, such as the style of writing as such (for instance, the way of writing the qâf or "q" and the or "f"), and the order of letters in the alphabet and the order of letters and their numerical values in the abjad system. I did not come across any convincing argument explaining the reasons of these differences and the way in which they were introduced. Naturally, we must assume that these letters came from the Mashriq as an integral part of the Indian numerical system, but how could the nine digits take their different forms in the Maghreb? This is a disturbing question, and the best way to solve it would be to find early traces of the writing of the nine digits in the Maghreb. Unfortunately, we still lack such evidence, and this is the reason why I address these words to the Maghrebi colleagues.

The existing examples of the writing of the nine numerals that we know of date all from the end of the 13th century and afterwards, and I don't know of any trace in any manuscript, coin, or inscription that goes back to an earlier date. On the other hand, the Latin manuscripts in which we find these digits are numerous, starting in 976 CE and through the 11th century and afterwards. These Latin sources use the nine numerals in their Maghrebi form. Undoubtedly they derived them from Arabic sources.

Of course, we know of Maghrebi mathematicians active before 1300 CE, like Ibn al-Yâsamîn and Abû Bakr al-Hassâr, but unfortunately the manuscripts of their works date of later times, posterior to the date of their composition, such as the manuscript of Ibn al-Yâsamîn [21] or most of the manuscripts of al-Hassâr. When we recently found an older manuscript of al-Bayân – which I described in the first part of this conference – copied in Baghdad in 1194 CE, we found that the copyist changed the forms of the nine numerals into the forms used in the Orient at that time. In my opinion, we cannot consider later manuscript copies of Maghrebi mathematicians who lived and worked before 1300 CE as proofs for their way of writing of the nine numerals, since it is natural that the copyists used the forms spread in their times and in their regions, while they did not make use of the ancient original forms.

For all these reasons, I would like to take the opportunity of this conference to address to the Maghrebi colleagues the urgent demand to search for written forms of Maghrebi numerals in the manuscripts preserved in the numerous libraries in the Maghreb. These manuscripts need not necessarily to be mathematical manuscripts. The numerals might come and be used in any text and at any occasion, for example in the tables of contents of books and in writing the dates, etc. We urgently need specimens of the nine numerals dating from the 350 years that followed the introduction of the Indian numerical system into the countries of the Maghreb, namely from the middle of the 10th until approximately the end of the 13th century. After that date, we have plenty of evidence, but before this period it is still completely absent. You are, dear colleagues, the ones who would be able to examine the very numerous manuscripts that exist in the Maghrebi libraries, most of which were never described nor studied before and have thus remained unnoticed in the world of scholarship.

Bibliography and further reading

Abattouy, Mohammed, "Al- Hassâr" (2006). URL is here.

Aballagh, Mohammed & Djebbar, Ahmed, "Découverte d'un écrit mathématique d'al-Hassâr (XIIe s.) : le Livre I du Kâmil." Historia Mathematica vol. 14 (1987), pp. 147-158.

Hassar, al-, Abû Bakr, Kitâb al-Bayân. Online publication is here.

Djebbar, Ahmed, "On mathematical activities in North Africa since the 9th century". Amuchma-Newsletter 15 (1995).

Kunitzsch, Paul, "A Hitherto Unknown Arabic Manuscript of the Almagest, in Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften vol. 14 (2001), 31-37.

Kunitzsch, Paul, "A New Manuscript of Abû Bakr al-Hassâr's Kitâb al-Bayân", in Suhayl (Barcelona), vol. 3 (2003): pp. 187-192.

Kunitzsch, Paul, "The Transmission of Hindu-Arabic Numerals Reconsidered", in The Enterprise of Science in Islam. New Perspectives, edited by J. P. Hogendijk ans A. I. Sabra, Cambridge (Mass.): The MIT Press, 2003, pp.4-21.

Lamrabet, Driss, Introduction à l'histoire des mathématiques maghrébines. Rabat: Dâr al-ma?ârif, 1994.

H. Suter, "Das Rechenbuch des Abu Zakarija el-Hassar", in Bibliotheca Mathematica, 3. Folge, 2. Band, 1901, 12-40.


[1] MS LJS 268 is an Arabic manuscript of the Almagest, copied in 1381 AD in Saragossa. See the description by P. Kunitzsch, "A Hitherto Unknown Arabic Manuscript of the Almagest", in Zeitschrift fur Geschichte der Arabisch-Islamischen Wissenschaften vol. 14 (2001), pp. 31-37.

[2] Liber mensurationum ... Ababuchri; cf. F. Sezgin, Geschichte des arabischen Schrifttums V, Leiden, 1974, pp. 389-391; R. Lemay, article "Gerard of Cremona", in Dictionary of Scientific Biography, vol. XV [Suppl. I] , New York, 1978, pp. 173 ff., esp. p. 188 (n. 82). The treatise has been edited by H.L.L. Busard, "Le Liber mensurationum d'Abu Bekr", in Journal des Savants, 1968, pp. 65-124.

[3] Cf. also J.P. Hogendijk, "A Medieval Arabic Treatise on Mensuration by Qadî Abû Bakr", in Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften vol. 6 (1990), pp. 130-150, where comparison is made between this work and the Latin Liber mensurationum.

[4] The manuscript has been scanned and can now be read in internet on the homepage of the library of the University of Pennsylvania, Philadelphia. URL is here.

[5] The script of this owner's note is different both from the script of a collation notice allegedly in the hand of al-Tûsî, reproduced in M.T.M. Rizvi, Ahwâl wa-âthâr-i Khwâja Nasir al-Dîn Tûsî, Tehran 1354sh, p. 166. I am grateful to Prof. G. Saliba, New York, for providing me with a xerox of this notice from the book and from the script of al-Tûsî's Persian translation of al-Sufî's Book on the Constellations, contained in MS Istanbul, Aya Sofya 2595, dated 647 H/1250, which is sometimes assumed to be al-Tust's autograph (the text of the Istanbul MS has been published, in facsimile, in Tehran, 1348sh/1969). Which of the three could really be in al-Tûsî's hand remains uncertain.

[6] This is evident from the word qalam which is written with two dots over the q, in the Eastern Arabic style.

[7] For the history of its identification and the present state of its know¬ledge, see M. Aballagh and A. Djebbar, "Découverte d'un écrit mathématique d'al-Hassar (Xlle s.): Le livre I du Kamil", in Historia Mathematica vol. 14 (1987), pp. 147-158; D. Lamrabet, Introduction à l'histoire des mathématiques maghrébines, 1994, pp. 56-60 (no. 330).

[8] Now BNRM: Bibliothèque Nationale du Royaume du Maroc (National Library of the Kingdom of Morocco) (note added by the editor).

[9] Lamrabet, notice 7, p. 57 n. 23.

[10] See Al-Fihris al-‘amm li-makhtûtât Dâr al-Kutub al-Zâhiriya, Damascus 1407 H/1987, p. 485 (n° 9760).

[11] The published catalogue of manuscripts in the Bibliothèque Générale in Rabat has not yet reached the collection "Q". Dr. Sa'id Lamrabeti of the manuscript department of the library was kind enough to send me information on the manuscript and copies of the first eight and the last three pages of the text (letter of June 12, 2001). The manuscript was once in the possession of Ahmad ibn Muhammad ibn Nasîr al-Dar?î (1057-1129 H/ 1647-1717).

[12] Cf. Lamrabet, notice 7, pp. 57-59.

[13] H. Suter, "Das Rechenbuch des Abu Zakarija el-Hassar", in Bibliotheca Mathematica, 3. Folge, 2. Band, 1901, pp. 12-40.

[14] Cf. Suter, note 13, p. 12; Aballagh and Djebbar, op. cit., p. 148; Lamrabet, notice 7, p. 57.

[15] See, especially, Aballagh and Djebbar, op. cit.

[16] The rest of the article (except the footnotes) was written by the author in Arabic and translated into English by the editor.

[17] It is remarkable that the first mention of the Hindu calculus and the nine numerals in the Middle East occurred in a letter of Severus Sebokht, Archbishop of Qansarîn (south to Aleppo in Syria) in 662 CE, that is almost one century before this system was known in Arabic culture. See E. Reich, "Ein Brief des Severus Sebokt", in Sic itur ad astra... Festschrift... Paul Kunitzsch, edited by M. Folkerts and R. Lorch, Wiesbaden 2000, pp. 479-489.

[18] "The Transmission of Hindu-Arabic Numerals Reconsidered", in The Enterprise of Science in Islam. New Perspectives, edited by J. P. Hogendijk ans A. I. Sabra, Cambridge (Mass.): The MIT Press, pp. 4-21.

[19] See J. Reinaud, in an "Addition" to his Mémoire sur l'Inde, in Mémoires de l'Institut Impérial de France, Académie des Inscriptions et Belles-Lettres, vol. 18 (1855), p. 565.

[20] "Codex Vigilanus" and "Codex Emilianus", both containing the Etymologiae of Isidor of Seville.

[21] Rabat, National Library (Al-Khizana al-'âmma before), MS K 222. See Abû Fâris, "Dalîl jadid ‘alâ ‘urûbat al-arqâm al-musta'mala fî al-maghrib al-‘arabî", Al-Lisan al-‘arabî, vol. 10 (1392 H/1973), pp. 231-233; see the photographs of numerals on pp. 232-233.

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