Review of:

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 [1974] (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:

- Q1. Did Dr. Sesiano in his edition of the Arabic text borrow from [Rashed 1975b]?
- Q2. Is the commentary in Sesiano's book dependent on [Rashed 1974, 1975a]?

The key to the first question will be provided by Dr. Sesiano's unpublished
dissertation [1975], 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
[1975], 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 [1982] 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.

REFERENCES

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.