A Newly-Discovered Partial Hebrew Version of al-Khwārizmī's Algebra

Tony Levy A Newly-Discovered Partial Hebrew Version of al-Khwârizmï's Algebra The purpose of this brief note is to announce the discovery of a short Hebrew text that I have identified as an adaptation of the famous Algebra by Muhammad ibn Müsä al-Khwarizmï (ninth century). This discovery is significant because no one has ever suspected that this fundamental text of algebra had ever been translated into Hebrew.1 In the course of my study of Hebrew texts related to Abraham Ibn Ezra's Sefer ha-mispar, I examined a manuscript in Geneva, Biblioth èque Publique et Universitaire (BPU), MS heb. 10 (hereafter: G), which carries the title: "The [Book of] Number by the Wise Abraham 1 It had also been thought that no algebraic text was translated into Hebrew before Mordecai Finzi (Mantua, fifteenth century) prepared the Hebrew version of Abu Kämil's Algebra (ninth-tenth century). But this is no longer true. In my article, "L'algèbre arabe au Moyen Age: le témoignage des textes hébraïques (XIP-XVI' siècles)" (forthcoming in Arabie Sciences and Philosophy), I have studied a hitherto unknown Hebrew text on arithmetic, which includes an important chapter on algebra. It was completed in Sicily by the Spanish astronomer Isaac al-Ahdab at the end of the fourteenth century. I have shown that this Hebrew work is an annotated translation of a well-known Arabic book, A Compendium of the Arithmetical Rules by Ibn alBann ä3 (1256-1321). Aleph 2 (2002)225 Ibn Ezra and Aristotle's Physical Problems." The title is wrong, however, and in fact the manuscript contains no work by Ibn Ezra.2 Instead, most of it consists of an anonymous arithmetical text (fol. 2a38a ), followed by an anonymous commentary on Ibn Ezra's Sefer hamispar (fol. 39a-64b). Upon examination, it becomes clear that the first text falls into two unequal parts: an "introduction" (fol. 2a-5b), followed by the bulk of the text.3 It is on this "introduction" that I will focus below. I was particularly intrigued by a passage (fols. 3b:15-4b:2) that begins as follows: All numbers are divided between three notions, namely: the root, the square, and the number that is neither root nor square. It is possible to obtain (Heb.: le-hosi3am) [any of these three notions] from one another. [Thus,] when you say "a square equals five of its roots" you can know that the root is 5 and its square is 25.4 This passage called to my mind a passage from al-Khwârizmï's famous Algebra. It is in fact an adaptation of a passage of the Arabic text, which opens with the following sentence: "I found that the numbers that are required in calculating by al-jabr wa-l muqàbala are of three kinds, namely, roots, squares [amwàl, lit. wealth, possessions], and the simple number [cadad mufrad], [i.e.] that which is related neither to root nor to square." The passage in G ends just before the geometrical demonstrations for each of the three "compound" equations.5 What is the relationship between the Hebrew text and the Algebra? Having systematically compared the two, I conclude that the author of the Hebrew text was following in the footsteps of al-Khwarizmï, his rendition being now closer to and now more distant from the original. Let us consider a sample. The passage in the Hebrew text that immediately follows the previous quotation presents the six canonical 226 To ny Levy equations along with the corresponding methods for solving them and numerical examples. In modern notation, they are as follows: I ax2=bx: x2=5x (x=5, x2=25); V3X2= 4x (x=12, x2=144) IIax2=c: 5x2=80 (x=4, x2=16); V2x2=18 (x=6, x2=36) IIIbx=c: 9x=81 (x=9, x2=81) The old catalogue (J. Sénebier, Catalogue raisonné des manuscrits conservés dans la bibliothèque de la Ville et République de Genève [Geneva, 1779], 23-24) does not even indicate Ibn Ezra's name. The notes by Joseph Prijs and the complementary notes by D. Goldschmidt (not printed) correctly indicate...