J. Sesiano, Books IV to VII of Diophantus' Arithmetica in the Arabic Translation Attributed to Qusta ibn Luqa. New York/Heidelberg/Berlin (Springer-Verlag). 1982. xii + 502 pp. Sources in the History of Mathematics and Physical Sciences, Vol. 3.
Reviewed by Jan P. Hogendijk . The review appeared in Historia Mathematica 12 (1985) pp. 82-85.
Only six of the thirteen books of the Arithmetica of Diophantus (ca. A.D. 250) are extant in Greek. The remaining books were believed to be lost, until the recent discovery of a medieval Arabic translation of four of the remaining books in a manuscript in the Shrine Library in Meshed in Iran (see the catalogue [Gulchin-i Ma'ani 1971-1972, pp. 235-236]. The manuscript was discovered in 1968 by F. Sezgin). The work under review is devoted to these four Arabic books; it contains an introduction, an English translation, a mathematical commentary, an edited Arabic text, and indexes. The four Arabic books have been the subject of previous publications by Dr. Rashed; see the appendix to this review for more details and a discussion of the possibility of dependency. Dr. Sesiano viewed the four Arabic books not as an entity in itself but as part of the Arithmetica as a whole, and he therefore based his work on a thorough study of the Greek books as well. His book contains many conclusions relevant to the Greek part of the Arithmetica, and enlightening (textual and other) comparisons between the Greek and the Arabic.
Chapter 1 of the Introduction begins with a discussion of Diophantus' authorship of the four Arabic books, their placement, and purpose. The Arabic books are in fact Books IV-VII; they immediately follow the Greek Books I-III, but precede the last three Greek Books (hitherto supposed to be IV-VI, and henceforth called "IV"-"VI"). Diophantus wrote Books IV-VII in order to train students in the application of the methods of Books II and III to new types of indeterminate equations (involving cubes and higher powers). Dr. Sesiano then lists the references to the Arithmetica in the Islamic literature, and continues with a well-written summary of the history of the Arithmetica in Byzantium.
Chapter 2 is about the Arabic manuscript, interpolations, and linguistic matters. Of special interest are the sections on fractions and powers (pp. 39-46).
Chapter 3 contains much more than is suggested by its title, "Tentative Reconstruction of the History of the Arithmetica." Dr. Sesiano begins with a discussion of the formal Greek division of (the solution of) a problem in (1) protasis, (2) diorismos, (3) ekthesis, (4) analysis, (5) synthesis, and (6) symperasma or final statement. The most important difference between the Greek and Arabic Arithmetica is the fact that the Greek contains only parts (l)-(4), whereas the Arabic contains all six parts. Dr. Sesiano then investigates interpolations and additions to the Greek and Arabic texts, errors in the Arabic text (this is the most important part of the discussion), and the quality of the translation. His final conclusions are that the Greek text of Books I-III and "IV"-"VI" is, aside from unsystematic interpolations, the same as the original written by Diophantus; thus an earlier hypothesis of Tannery has to be modified (pp. 74-75). The situation with regard to the Arabic text is more complicated: part of the Arithmetica (at least Books I-VII) plus early additions were rewritten by a commentator (Hypatia?), who added the syntheses; the final statements were added by a Greek scholiast, and the resulting version was translated into Arabic. Unfortunately we do not (yet?) have a manuscript of the Arabic translation of Books I-III. However, two quotations by Samaw'al ibn Yahya (ca. 1180) from the lost Arabic translation of Book I at least partially support Dr. Sesiano's hypothesis. The Introduction ends with conjectures on the missing three books of the Arithmetica.
The translation (part 2) and the mathematical commentary (part 3) are both very good. The edition of the Arabic text (part 4) is done with the utmost care; the emendations are all very carefully constructed. The reviewer would like to add only one suggestion: in line 439 Dr. Sesiano emends the nonsensical al-thani to al- ta‘atti. The word ta'attin does not occur elsewhere in the four Arabic books, and I would prefer to read al-masa'il (the manuscript is probably hopelessly corrupted). The book ends with an excellent Arabic index (pp. 435-460), a conspectus of the problems in the Arithmetica, a bibliography and a general index in six parts.
The Arabic books of the Arithmetica have been the subject of three earlier publications by Dr. R. Rashed: an edition of the Arabic text and a commentary in Arabic (mainly consisting of an algebraical transcription of the problems and solutions) published in [Rashed 1975b], and a French version of this commentary published in two articles  (introduction and Book IV) and [1975a] (Books V- VII). The book under review does not contain any reference to the work of Dr. Rashed on Diophantus, and some scholars have accused Dr. Sesiano of having plagiarized the work of Dr. Rashed. It is the purpose of this appendix to investigate whether there is any truth in these accusations. We note that the manuscript of the Arithmetica is very legible, and that the concise French summaries in [Rashed 1974, 1975a] have hardly any bearing on the interpretation of the details of the Arabic text. We will therefore concentrate on two questions:
The key to the first question will be provided by Dr. Sesiano's unpublished dissertation , which contains among other things a critical edition of the Arabic text of the Arithmetica (this edition is the basis of the edition in the work under review). First it is necessary to settle the chronology of [Sesiano 1975]. In , Sesiano refers to [Rashed 1974, 1975a] but not to [Rashed 1975b], which suggests that the latter publication was not available to him before he finished the dissertation. Sesiano's dissertation was accepted by the Department of History of Mathematics at Brown University on May 12, 1975 (see [Sesiano 1975, p. ii]). Dr. Rashed's preface in [1975b, p. 6] is dated "Paris, Dec. 10, 1974," so it is hardly conceivable that his edition was available (from Cairo) before Dr. Sesiano finished his thesis. The reviewer has inspected the complimentary copy of [Rashed 1975bl which was sent to Professor G. J. Toomer, who supervised Dr. Sesiano's thesis: this copy was dated August 10, 1975, by Dr. Rashed. We must conclude that [Sesiano 1975] is independent of [Rashed 1975bl.
In order to answer Q1 we will study the emendations which the two editors made in the Arabic text. We will refer to the emendations by the emendation number in the apparatus in [Sesiano 1982], the work under review. Thus 405 will refer to emendation 1215(405) on page 335 (note that 1215 is the number of the line in which the emendation occurs). Let R, S, and D be the collections of emendations in [Rashed 1975], [Sesiano 1982] (the work under review). and [Sesiano 1975] (his Dissertation). We are interested in the intersection of the sets R and S, which consists of the emendations which are in R and in S. This set which is the disjoint union of the sets V (the emendations which belong to the three sets R, S and D) and W (the emendations which are in R and in S, but not in D). The emendations in W are in both [Rashed 1975] and [Sesiano 1982], but not in Sesiano's dissertation.
The reviewer has found that
V = (2, 15, 32, 38, 43, 51, 71, 76, 110, 123, 132, 138, 141, 143, 148, 151, 158, 174, 186, 190, 195, 202, 206, 210, 221, 223, 226, 230, 249, 251, 257, 258, 259, 260, 277, 279, 281, 286, 289, 295, 296, 305, 306, 314, 316, 317, 326, 327, 328, 333, 336, 338, 341, 342, 343, 345, 346, 356, 357, 358, 362, 365, 366, 368, 371, 374, 383, 385, 387, 388, 401, 402, 409, 410, 411, 413, 414, 427, 429, 431, 432, 445, 446, 447, 448, 461, 467, 471, 474, 475, 480, 484, 487, 496, 501, 513, 514, 515, 521, 522, 523, 526, 528, 534, 537, 538, 542, 545, 547, 550, 559, 561, 562, 564, 566, 575, 585, 586, 587, 593, 608, 619, 620, 624, 628, 630, 636, 639, 640, 641, 644, 650, 654, 655, 662, 670, 679, 683, 689, 698, 701, 702, 706, 707, 708, 712, 713, 718, 720, 725, 728, 729, 732, 733, 735, 736, 738, 740, 744, 746, 750, 752, 755, 756, 759, 767, 769, 779, 780, 783, 792, 794, 798, 802, 803, 805, 811, 814, 819, 831, 857, 861, 865, 867, 869, 879, 880, 884, 889, 890, 892, 893, 915, 929, 931, 936, 939, 945, 948, 965, 974);
W = (222, 287, 297, 468, 527).
Dr. Sesiano cannot possibly have adopted the emendations in V from Dr. Rashed, since they belong to D, which is independent of R. The emendations in W are hardly noteworthy. For example, No. 222 is the emendation of (61)^3 (the cube of 61) = 531, 441 (stated in the text) to (81)^3 = 531, 441, which must have escaped Sesiano in D (though he had already made exactly the same emendation No. 223 in D). In addition we should note that S contains a number of improvements of the text, not found in R (for example 75, 77, 85, 100-105, 124, 188, 193, 207, 244, 291, 331, 404, 416-419, 456, 649; compare also [Sesiano 1982, 21, footnote 2, and the margin of lines 3105, 3149, 3192]). This rules out any possibility of dependency.
Ad Q2. Dr. Sesiano's mathematical commentary in  is much more elaborate than that in [Rashed 1974, 1975a]. The same can be said of Dr. Sesiano's discussion of the relations between Book IV of the Arithmetica and the Fakhri of the l0th-century Arabic mathematician Al-Karaji (compare [Sesiano 1982, 10-11, 19-60, 180-214] and [Rashed 1974, 104-105]). Note that Dr. Sesiano discovered a new problem in the Fakhri, corresponding to Arithmetica IV:19 (see [Sesiano 1982, 194]). Finally, Dr. Sesiano's interpretation of the marginal gloss to the Fakhri (in [Sesiano 1982, 5] differs from the interpretation of it by Dr. Rashed in [Rashed 1974, 103]. Here again, plagiarism is out of the question.
I should like to make it very clear that this appendix was not written with the intention of belittling or criticizing Dr. Rashed's work on Diophantus. On the contrary, historians of mathematics should be grateful to Dr. Rashed for having made his work on the Arabic Diophantus available. My only purpose was to prove that the work of Dr. Sesiano is not in any way dependent on the work of Dr. Rashed on the Arabic Diophantus.
Gulchin-i Ma'ani, A. 1971-1972. Fihrist-i kutub-i khatti-yi kitabkhana-i Astan-i quds-i Ridawi. [In Persian] Meshed, 1350 H (1971-1972).
Rashed, R. 1974. Les travaux perdus de Diophante, I. Revue d'Histoire des Sciences 17, 97-122.
-- 1975a. Les travaux perdus de Diophante, II. Revue d'Histoire des Sciences 18, 3-30.
-- 1975b. Usul al-jabr li-Diyufantis. Tarjamat Qusta ibn Luqa. Haqqaqahu wa-qaddama lahu Rushdi Rashid. [In Arabic] Cairo, Al-Hay'a al-misriyya al-`amma li'l-kitab; al-turath al-`ilmi al-arabi No. I.
Sesiano, J. 1975. The Arabic text of Books IV to VII of Diophuntus "Arithmetica" in the Translation of Qusta ibn Luqa; Edited, with Translation and Commentary. Dissertation, Brown University, Providence, R.I. Available through University Microfilms International, Ann Arbor, Mich., No. DCJ 76-15709.
-- 1982. See the beginning of this review.