Note

See also the Executive Summary and Outline for Chapter VII: Euclid in Arabia.

[Folio 0089 | p. 75]

CHAPTER VII — EUCLID IN ARABIA.

WE are told by Ḥāji Khalfa[1] that the Caliph al-Manṣūr (754–775) sent a mission to the Byzantine Emperor as the result of which he obtained from him a copy of Euclid among other Greek books, and again that the Caliph al-Ma’mūn (813–833) obtained manuscripts of Euclid, among others, from the Byzantines. The version of the Elements by al-Ḥajjāj b. Yūsuf b. Maṭar is, if not the very first, at least one of the first books translated from the Greek into Arabic[2]. According to the Fihrist[3] it was translated by al-Ḥajjāj twice; the first translation was known as “Hārūnī” (“for Hārūn”), the second bore the name “Ma’mūnī” (“for al-Ma’mūn”) and was the more trustworthy. Six Books of the second of these versions survive in a Leiden MS. (Codex Leidensis 399, 1) now in part published by Besthorn and Heiberg[4]. In the preface to this MS. it is stated that, in the reign of Hārūn ar-Rashīd (786–809), al-Ḥajjāj was commanded by Yaḥyā b. Khālid b. Barmak to translate the book into Arabic. Then, when al-Ma’mūn became Caliph, as he was devoted to learning, al-Ḥajjāj saw that he would secure the favour of al-Ma’mūn “if he illustrated and expounded this book and reduced it to smaller dimensions. He accordingly left out the superfluities, filled up the gaps, corrected or removed the errors, until he had gone through the book and reduced it, when corrected and explained, to smaller dimensions, as in this copy, but without altering the substance, for the use of men endowed with ability and devoted to learning, the earlier edition being left in the hands of readers.”

The Fihrist goes on to say that the work was next translated by Isḥāq b. Ḥunain, and that this translation was improved by Thābit b. Qurra. This Abū Ya’qūb Isḥāq b. Ḥunain b. Isḥāq al-‘Ibādī (d. 910) was the son of the most famous of Arabic translators, Ḥunain b. Isḥāq al-‘Ibādī (809–873), a Christian and physician to the Caliph al-Mutawakkil (847–861). There seems to be no doubt that Isḥāq, who must have known Greek as well as his father, made his translation direct from the Greek. The revision must apparently have been the subject of an arrangement between Isḥāq and Thābit, as the latter died in 901 or nine years before Isḥāq. Thābit undoubtedly consulted Greek MSS. for the purposes of his revision. This is expressly stated in a marginal note to a Hebrew version of the Elements, made from Isḥāq’s, attributed to one of two scholars belonging to the same family, viz. either to Moses b. Tibbon (about 1244–1274) or to Jakob b. Machir (who died soon after 1306)[5]. Moreover Thābit observes, on the proposition which he gives as IX. 31, that he had not found this proposition and the one before it in the Greek but only in the Arabic; from which statement Klamroth draws two conclusions, (1) that the Arabs had already begun to interest themselves in the authenticity of the text and (2) that Thābit did not alter the numbers of the propositions in Isḥāq’s translation[6]. The Fihrist also says that Yuḥannā al-Qass (i.e. “the Priest”) had seen in the Greek copy in his possession the proposition in Book I. which Thābit took credit for, and that this was confirmed by Naẓīf, the physician, to whom Yuḥannā had shown it. This proposition may have been wanting in Isḥāq, and Thābit may have added it, but without claiming it as his own discovery[7]. As a fact, I. 45 is missing in the translation by al-Ḥajjāj.

[Folio 0090 | p. 76]

The original version of Isḥāq without the improvements by Thābit has probably not survived any more than the first of the two versions by al-Ḥajjāj; the divergences between the MSS. are apparently due to the voluntary or involuntary changes of copyists, the former class varying according to the degree of mathematical knowledge possessed by the copyists and the extent to which they were influenced by considerations of practical utility for teaching purposes[8]. Two MSS. of the Isḥāq-Thābit version exist in the Bodleian Library (No. 279 belonging to the year 1238, and No. 280 written in 1260–1)[9]; Books I.—XIII. are in the Isḥāq-Thābit version, the non-Euclidean Books XIV., XV. in the translation of Qusṭā b. Lūqā al-Ba’labakkī (d. about 912). The first of these MSS. (No. 279) is that (O) used by Klamroth for the purpose of his paper on the Arabian Euclid. The other MS. used by Klamroth is (K) Kjøbenhavn LXXXI, undated but probably of the 13th c., containing Books V.—XV., Books V.—X. being in the Isḥāq-Thābit version, Books XI.—XIII. purporting to be in al-Ḥajjāj’s translation, and Books XIV., XV. in the version of Qusṭā b. Lūqā. In not a few propositions K and O show not the slightest difference, and, even where the proofs show considerable differences, they are generally such that, by a careful comparison, it is possible to reconstruct the common archetype, so that it is fairly clear that we have in these cases, not two recensions of one translation, but arbitrarily altered and shortened copies of one and the same recension[10]. The Bodleian MS. No. 280 contains a preface, translated by Nicoll, which cannot be by Thābit himself because it mentions Avicenna (980–1037) and other later authors. The MS. was written at Marāġa in the year 1260–1 and has in the margin readings and emendations from the edition of Naṣīraddīn aṭ-Ṭūsī (shortly to be mentioned) who was living at Marāġa at the time. Is it possible that aṭ-Ṭūsī himself is the author of the preface[11]? Be this as it may, the preface is interesting because it throws light on the liberties which the Arabians allowed themselves to take with the text. After the observation that the book (in spite of the labours of many editors) is not free from errors, obscurities, redundancies, omissions etc., and is without certain definitions necessary for the proofs, it goes on to say that the man has not yet been found who could make it perfect, and next proceeds to explain (1) that Avicenna “cut out postulates and many definitions” and attempted to clear up difficult and obscure passages, (2) that Abū’l Wafā al-Būzjānī (939–997) “introduced unnecessary additions and left out many things of great importance and entirely necessary,” inasmuch as he was too long in various places in Book VI. and too short in Book X. where he left out entirely the proofs of the apotomae, while he made an unsuccessful attempt to emend XII. 14, (3) that Abū Ja’far al-Khāzin (d. between 961 and 971) arranged the postulates excellently but “disturbed the number and order of the propositions, reduced several propositions to one” etc. Next the preface describes the editor’s own claims[12] and then ends with the sentences, “But we have kept to the order of the books and propositions in the work itself (i.e. Euclid’s) except in the twelfth and thirteenth books. For we have dealt in Book XIII. with the (solid) bodies and in Book XII. with the surfaces by themselves.”

[Folio 0091 | p. 77]

After Thābit the Fihrist mentions Abū ‘Uthmān ad-Dimashqī as having translated some Books of the Elements including Book X. (It is Abū ‘Uthmān’s translation of Pappus’ commentary on Book X. which Woepcke discovered at Paris.) The Fihrist adds also that “Naẓīf the physician told me that he had seen the tenth Book of Euclid in Greek, that it had 40 propositions more than the version in common circulation which had 109 propositions, and that he had determined to translate it into Arabic.”

But the third form of the Arabian Euclid actually accessible to us is the edition of Abū Ja’far Muḥ. b. Muḥ. b. al-Ḥasan Naṣīraddīn aṭ-Ṭūsī (whom we shall call aṭ-Ṭūsī for short), born at Ṭūs (in Khurāsān) in 1201 (d. 1274). This edition appeared in two forms, a larger and a smaller. The larger is said to survive in Florence only (Pal. 272 and 313, the latter MS. containing only six Books); this was published at Rome in 1594, and, remarkably enough, some copies of this edition are to be found with 12 and some with 13 Books, some with a Latin title and some without[13]. But the book was printed in Arabic, so that Kästner remarks that he will say as much about it as can be said about a book which one cannot read[14]. The shorter form, which however, in most MSS., is in 15 Books, survives at Berlin, Munich, Oxford, British Museum (974, 1334[15], 1335), Paris (2465, 2466), India Office, and Constantinople; it was printed at Constantinople in 1801, and the first six Books at Calcutta in 1824[16].

[Folio 0092 | p. 78]

Aṭ-Ṭūsī’s work is however not a translation of Euclid’s text, but a re-written Euclid based on the older Arabic translations. In this respect it seems to be like the Latin version of the Elements by Campanus (Campano), which was first published by Erhard Ratdolt at Venice in 1482 (the first printed edition of Euclid)[17]. Campanus (13th c.) was a mathematician, and it is likely enough that he allowed himself the same liberty as aṭ-Ṭūsī in reproducing Euclid. Whatever may be the relation between Campanus’ version and that of Athelhard of Bath (about 1120), and whether, as Curtze thinks[18], they both used one and the same Latin version of 10th—11th c., or whether Campanus used Athelhard’s version in the same way as aṭ-Ṭūsī used those of his predecessors[19], it is certain that both versions came from an Arabian source, as is evident from the occurrence of Arabic words in them[20]. Campanus’ version is not of much service for the purpose of forming a judgment on the relative authenticity of the Greek and Arabian tradition; but it sometimes preserves traces of the purer source, as when it omits Theon’s addition to VI. 33[21]. A curious circumstance is that, while Campanus’ version agrees with aṭ-Ṭūsī’s in the number of the propositions in all the genuine Euclidean Books except V. and IX., it agrees with Athelhard’s in having 34 propositions in Book V. (as against 25 in other versions), which confirms the view that the two are not independent, and also leads, as Klamroth says, to this dilemma: either the additions to Book V. are Athelhard’s own, or he used an Arabian Euclid which is not known to us[22]. Heiberg also notes that Campanus’ Books XIV., XV. show a certain agreement with the preface to the Thābit-Isḥāq version, in which the author claims to have (1) given a method of inscribing spheres in the five regular solids, (2) carried further the solution of the problem how to inscribe any one of the solids in any other and (3) noted the cases where this could not be done[23].

[Folio 0093 | p. 79]

With a view to arriving at what may be called a common measure of the Arabian tradition, it is necessary to compare, in the first place, the numbers of propositions in the various Books. Ḥāji Khalfa says that al-Ḥajjāj’s translation contained 468 propositions, and Thābit’s 478; this is stated on the authority of aṭ-Ṭūsī, whose own edition contained 468[24]. The fact that Thābit’s version had 478 propositions is confirmed by an index in the Bodleian MS. 279 (called O by Klamroth). A register at the beginning of the Codex Leidensis 399, 1 which gives Isḥāq’s numbers (although the translation is that of al-Ḥajjāj) apparently makes the total 479 propositions (the number in Book XIV. being apparently 11, instead of the 10 of O)[25]. I subjoin a table of relative numbers taken from Klamroth, to which I have added the corresponding numbers in August’s and Heiberg’s editions of the Greek text.

Books	The Arabian Euclid			The Greek Euclid
	Isḥāq	aṭ-Ṭūsī	Campanus	Gregory	August	Heiberg
I	48	48	48	48	48	48
II	14	14	14	14	14	14
III	36	36	36	37	37	37
IV	16	16	16	16	16	16
V	25	25	34	25	25	25
VI	33	32	32	33	33	33
VII	39	39	39	41	41	39
VIII	27	25	25	27	27	27
IX	38	36	39	36	36	36
X	109	107	107	117	116	115
XI	41	41	41	40	40	39
XII	15	15	15	18	18	18
XIII	21	18	18	18	18	18
	462	452	464	470	469	465
[XIV	10	10	18	7	—	?
XV	6	6	13	10	—	?]
	478	468	495	487	—	?]

The numbers in the case of Heiberg include all propositions which he has printed in the text; they include therefore XIII. 6 and III. 12 now to be regarded as spurious, and X. 112—115 which he brackets as doubtful. He does not number the propositions in Books XIV., XV., but I conclude that the numbers in P reach at least 9 in XIV., and 9 in XV.

[Folio 0094 | p. 80]

The Fihrist confirms the number 109 for Book X., from which Klamroth concludes that Isḥāq’s version was considered as by far the most authoritative.

In the text of O, Book IV. consists of 17 propositions and Book XIV. of 12, differing in this respect from its own table of contents; IV. 15, 16 in O are really two proofs of the same proposition.

In al-Ḥajjāj’s version Book I. consists of 47 propositions only, I. 45 being omitted. It has also one proposition fewer in Book III., the Heronic proposition III. 12 being no doubt omitted.

In speaking of particular propositions, I shall use Heiberg’s numbering, except where otherwise stated.

The difference of 10 propositions between Thābit-Isḥāq and aṭ-Ṭūsī is accounted for thus:

1. The three propositions VI. 12 and X. 28, 29 which both Isḥāq and the Greek text have are omitted in aṭ-Ṭūsī.

2. Isḥāq divides each of the propositions XIII. 1—3 into two, making six instead of three in aṭ-Ṭūsī and in the Greek.

3. Isḥāq has four propositions (numbered by him VIII. 24, 25, IX. 30, 31) which are neither in the Greek Euclid nor in aṭ-Ṭūsī.

Apart from the above differences al-Ḥajjāj (so far as we know), Isḥāq and aṭ-Ṭūsī agree; but their Euclid shows many differences from our Greek text. These differences we will classify as follows[26].

1. Propositions.

The Arabian Euclid omits VII. 20, 22 of Gregory’s and August’s editions (Heiberg, App. to Vol. II. pp. 428—32); VIII. 16, 17; X. 7, 8, 13, 16, 24, 112, 113, 114, besides a lemma vulgo X. 13, the proposition X. 117 of Gregory’s edition, and the scholium at the end of the Book (see for these Heiberg’s Appendix to Vol. III. pp. 382, 408—416); XI. 38 in Gregory and August (Heiberg, App. to Vol. IV. p. 354); XII. 6, 13, 14; (also all but the first third of Book XV.).

The Arabian Euclid makes III. 11, 12 into one proposition, and divides some propositions (X. 31, 32; XI. 31, 34; XIII. 1—3) into two each.

The order is also changed in the Arabic to the following extent. V. 12, 13 are interchanged and the order in Books VI., VII., IX.—XIII. is:

6. 1—8, 13, 11, 12, 9, 10, 14—17, 19, 20, 18, 21, 22, 24, 26, 23, 25, 27—30, 32, 31, 33.

7. 1—20, 22, 21, 23—28, 31, 32, 29, 30, 33—39.

9. 1—13, 20, 14—19, 21—25, 27, 26, 28—36, with two new propositions coming before prop. 30.

10. 1—6, 9—12, 15, 14, 17—23, 26—28, 25, 29—30, 31, 32, 33—111, 115.

11. 1—30, 31, 32, 34, 33, 35—39.

12. 1—5, 7, 9, 8, 10, 12, 11, 15, 16—18.

13. 1—3, 5, 4, 6, 7, 12, 9, 10, 8, 11, 13, 15, 14, 16—18.

[Folio 0095 | p. 81]

2. Definitions.

The Arabic omits the following definitions: IV. Deff. 3—7, VII. Def. 9 (or 10), XI. Deff. 5—7, 15, 17, 23, 25—28; but it has the spurious definitions VI. Deff. 2, 5, and those of proportion and ordered proportion in Book V. (Deff. 8, 19 August), and wrongly interchanges V. Deff. 11, 12 and also VI. Deff. 3, 4.

The order of the definitions is also different in Book VII. where, after Def. 11, the order is 12, 14, 13, 15, 16, 19, 20, 17, 18, 21, 22, 23, and in Book XI. where the order is 1, 2, 3, 4, 8, 10, 9, 13, 14, 16, 12, 21, 22, 18, 19, 20, 11, 24.

3. Lemmas and porisms.

All are omitted in the Arabic except the porisms to VI. 8, VIII. 2, X. 3; but there are slight additions here and there, not found in the Greek, e.g. in VIII. 14, 15 (in K).

4. Alternative proofs.

These are all omitted in the Arabic, except that in X. 105, 106 they are substituted for the genuine proofs; but one or two alternative proofs are peculiar to the Arabic (VI. 32 and VIII. 4, 6).

The analyses and syntheses to XIII. 1—5 are also omitted in the Arabic.

Klamroth is inclined, on a consideration of all these differences, to give preference to the Arabian tradition over the Greek (1) “on historical grounds,” subject to the proviso that no Greek MS. as ancient as the 8th c. is found to contradict his conclusions, which are based generally (2) on the improbability that the Arabs would have omitted so much if they had found it in their Greek MSS., it being clear from the Fihrist that the Arabs had already shown an anxiety for a pure text, and that the old translators were subjected in this matter to the check of public criticism. Against the “historical grounds,” Heiberg is able to bring a considerable amount of evidence[27]. First of all there is the British Museum palimpsest (L) of the 7th or the beginning of the 8th c. This has fragments of propositions in Book X. which are omitted in the Arabic; the numbering of one proposition, which agrees with the numbering in other Greek MS., is not comprehensible on the assumption that eight preceding propositions were omitted in it, as they are in the Arabic; and lastly, the readings in L are tolerably like those of our MSS., and surprisingly like those of B. It is also to be noted that, although P dates from the 10th c. only, it contains, according to all appearance, an ante-Theonine recension.

Moreover there is positive evidence against certain omissions by the Arabians. Aṭ-Ṭūsī omits VI. 12, but it is scarcely possible that, if Eutocius had not had it, he would have quoted VI. 23 by that number[28]. This quotation of VI. 23 by Eutocius also tells against Isḥāq who has the proposition as VI. 25. Again, Simplicius quotes VI. 10 by that number, whereas it is VI. 13 in Isḥāq; and Pappus quotes, by number, XIII. 2 (Isḥāq 3, 4), XIII. 4 (Isḥāq 8), XIII. 16 (Isḥāq 19).

[Folio 0096 | p. 82]

On the other hand the contraction of III. 11, 12 into one proposition in the Arabic tells in favour of the Arabic.

Further, the omission of certain porisms in the Arabic cannot be supported; for Pappus quotes the porism to XIII. 17[29], Proclus those to II. 4, III. 1, VII. 2[30], and Simplicius that to IV. 15.

Lastly, some propositions omitted in the Arabic are required in later propositions. Thus X. 13 is used in X. 18, 22, 23, 26 etc.; X. 17 is wanted in X. 18, 26, 36; XII. 6, 13 are required for XII. 11 and XII. 15 respectively.

It must also be remembered that some of the things which were properly omitted by the Arabians are omitted or marked as doubtful in Greek MSS. also, especially in P, and others are rightly suspected for other reasons (e.g. a number of alternative proofs, lemmas, and porisms, as well as the analyses and syntheses of XIII. 1—5). On the other hand, the Arabic has certain interpolations peculiar to our inferior MSS. (cf. the definition VI. Def. 2 and those of proportion and ordered proportion).

Heiberg comes to the general conclusion that, not only is the Arabic tradition not to be preferred offhand to that of the Greek MSS., but it must be regarded as inferior in authority. It is a question how far the differences shown in the Arabic are due to the use of Greek MSS. differing from those which have been most used as the basis of our text, and how far to the arbitrary changes made by the Arabians themselves. Changes of order and arbitrary omissions could not surprise us, in view of the preface above quoted from the Oxford MS. of Thābit-Isḥāq, with its allusion to the many important and necessary things left out by Abū ‘l Wafā and to the author’s own rearrangement of Books XII., XIII. But there is evidence of differences due to the use by the Arabs of other Greek MSS. Heiberg[31] is able to show considerable resemblances between the Arabic text and the Bologna MS. b in that part of the MS. where it diverges so remarkably from our other MSS. (see the short description of it above, p. 49); in illustration he gives a comparison of the proofs of XII. 7 in b and in the Arabic respectively, and points to the omission in both of the proposition given in Gregory’s edition as XI. 38, and to a remarkable agreement between them as regards the order of the propositions of Book XII. As above stated, the remarkable divergence of b only affects Books XI. (at end) and XII.; and Book XIII. in b shows none of the transpositions and other peculiarities of the Arabic. There are many differences between b and the Arabic, especially in the definitions of Book XI., as well as in Book XIII. It is therefore a question whether the Arabians made arbitrary changes, or the Arabic form is the more ancient, and b has been altered through contact with other MSS. Heiberg points out that the Arabians must be alone responsible for their definition of a prism, which only covers a prism with a triangular base. This could not have been Euclid’s own, for the word prism already has the wider meaning in Archimedes, and Euclid himself speaks of prisms with parallelograms and polygons as bases (XI. 39; XII. 10). Moreover, a Greek would not have been likely to leave out the definitions of the “Platonic” regular solids.

[Folio 0097 | p. 83]

Heiberg considers that the Arabian translator had before him a MS. which was related to b, but diverged still further from the rest of our MSS. He does not think that there is evidence of the existence of a redaction of Books I.—X. similar to that of Books XI., XII. in b; for Klamroth observes that it is the Books on solid geometry (XI.—XIII.) which are more remarkable than the others for omissions and shorter proofs, and it is a noteworthy coincidence that it is just in these Books that we have a divergent text in b.

An advantage in the Arabic version is the omission of VII. Def. 10, although, as Iamblichus had it, it may have been deliberately omitted by the Arabic translator. Another advantage is the omission of the analyses and syntheses of XIII. 1—5; but again these may have been omitted purposely, as were evidently a number of porisms which are really necessary.

One or two remarks may be added about the Arabic versions as compared with one another. Al-Ḥajjāj’s object seems to have been less to give a faithful reflection of the original than to write a useful and convenient mathematical text-book. One characteristic of it is the careful references to earlier propositions when their results are used. Such specific quotations of earlier propositions are rare in Euclid; but in al-Ḥajjāj we find not only such phrases as “by prop. so and so,” “which was proved” or “which we showed how to do in prop. so and so,” but also still longer phrases. Sometimes he repeats a construction, as in I. 44 where, instead of constructing “the parallelogram BEFG equal to the triangle C in the angle EBG which is equal to the angle D” and placing it in a certain position, he produces AB to G, making BG equal to half DE (the base of the triangle CDE in his figure), and on GB so constructs the parallelogram BHKG by I. 42 that it is equal to the triangle CDE, and its angle GBH is equal to the given angle.

Secondly, al-Ḥajjāj, in the arithmetical books, in the theory of proportion, in the applications of the Pythagorean I. 47, and generally where possible, illustrates the proofs by numerical examples. It is true, observes Klamroth, that these examples are not apparently separated from the commentary of an-Nairīzī, and might not therefore have been due to al-Ḥajjāj himself; but the marginal notes to the Hebrew translation in Municn MS. 36 show that these additions were in the copy of al-Ḥajjāj used by the translator, for they expressly give these proofs in numbers as variants taken from al-Ḥajjāj[32].

These characteristics, together with al-Ḥajjāj’s freer formulation of the propositions and expansion of the proofs, constitute an intelligible reason why Isḥāq should have undertaken a fresh translation from the Greek. Klamroth calls Isḥāq’s version a model of a good translation of a mathematical text; the introductory and transitional phrases are stereotyped and few in number, the technical terms are simply and consistently rendered, and the less formal expressions connect themselves as closely with the Greek as is consistent with intelligibility and the character of the Arabic language. Only in isolated cases does the formulation of definitions and enunciations differ to any considerable extent from the original. In general, his object seems to have been to get rid of difficulties and unevennesses in the Greek text by neat devices, while at the same time giving a faithful reproduction of it[33].

[Folio 0098 | p. 84]

There are curious points of contact between the versions of al-Ḥajjāj and Thābit-Isḥāq. For example, the definitions and enunciations of propositions are often word for word the same. Presumably this is owing to the fact that Isḥāq found these definitions and enunciations already established in the schools in his time, where they would no doubt be learnt by heart, and refrained from translating them afresh, merely adopting the older version with some changes[34]. Secondly, there is remarkable agreement between the Arabic versions as regards the figures, which show considerable variations from the figures of the Greek text, especially as regards the letters; this is also probably to be explained in the same way, all the later translators having most likely borrowed al-Ḥajjāj’s adaptation of the Greek figures[35]. Lastly, it is remarkable that the version of Books XI.—XIII. in the Kjøbenhavn MS. (K), purporting to be by al-Ḥajjāj, is almost exactly the same as the Thābit-Isḥāq version of the same Books in O. Klamroth conjectures that Isḥāq may not have translated the Books on solid geometry at all, and that Thābit took them from al-Ḥajjāj, only making some changes in order to fit them to the translation of Isḥāq[36].

From the facts (1) that aṭ-Ṭūsī’s edition had the same number of propositions (468) as al-Ḥajjāj’s version, while Thābit-Isḥāq’s had 478, and (2) that aṭ-Ṭūsī has the same careful references to earlier propositions, Klamroth concludes that aṭ-Ṭūsī deliberately preferred al-Ḥajjāj’s version to that of Isḥāq[37]. Heiberg, however, points out (1) that aṭ-Ṭūsī left out VI. 12 which, if we may judge by Klamroth’s silence, al-Ḥajjāj had, and (2) al-Ḥajjāj’s version had one proposition less in Books I. and III. than aṭ-Ṭūsī has. Besides, in a passage quoted by Ḥāji Khalfa[38] from aṭ-Ṭūsī, the latter says that “he separated the things which, in the approved editions, were taken from the archetype from the things which had been added thereto,” indicating that he had compiled his edition from both the earlier translations[39].

There were a large number of Arabian commentaries on, or reproductions of, the Elements or portions thereof, which will be found fully noticed by Steinschneider[40]. I shall mention here the commentators etc. referred to in the Fihrist, with a few others.

[Folio 0099 | p. 85]

1. Abū ‘l ‘Abbās al-Faḍl b. Ḥātim an-Nairīzī (born at Nairīz, died about 922) has already been mentioned[41]. His commentary survives, as regards Books I.—VI., in the Codex Leidensis 399, 1, now edited, as to four Books, by Besthorn and Heiberg, and as regards Books I.—X. in the Latin translation made by Gherard of Cremona in the 12th c. and now published by Curtze from a Cracow MS.[42] Its importance lies mainly in the quotations from Heron and Simplicius.

2. Aḥmad b. ‘Umar al-Karābīsī (date uncertain, probably 9th—10th c.), “who was among the most distinguished geometers and arithmeticians[43].”

3. Al-‘Abbās b. Sa’īd al-Jauharī (fl. 830) was one of the astronomical observers under al-Ma’mūn, but devoted himself mostly to geometry. He wrote a commentary to the whole of the Elements, from the beginning to the end; also the “Book of the propositions which he added to the first book of Euclid[44].”

4. Muḥ. b. ‘Īsā Abū ‘Abdallāh al-Māhānī (d. between 874 and 884) wrote, according to the Fihrist, (1) a commentary on Eucl. Book V., (2) “On proportion,” (3) “On the 26 propositions of the first Book of Euclid which are proved without reductio ad absurdum[45].” The work “On proportion” survives and is probably identical with, or part of, the commentary on Book V.[46] He also wrote, what is not mentioned by the Fihrist, a commentary on Eucl. Book X., a fragment of which survives in a Paris MS.[47]

5. Abū Ja’far al-Khāzin (i.e. “the treasurer” or “librarian”), one of the first mathematicians and astronomers of his time, was born in Khurāsān and died between the years 961 and 971. The Fihrist speaks of him as having written a commentary on the whole of the Elements[48], but only the commentary on the beginning of Book X. survives (in Leiden, Berlin and Paris); therefore either the notes on the rest of the Books have perished, or the Fihrist is in error[49]. The latter would seem more probable, for, at the end of his commentary, al-Khāzin remarks that the rest had already been commented on by Sulaimān b. ‘Uṣma (Leiden MS.)[50] or ‘Oqba (Suter), to be mentioned below. Al-Khāzin’s method is criticised unfavourably in the preface to the Oxford MS. quoted by Nicoll (see p. 77 above).

6. Abū ‘l Wafā al-Būzjānī (940—997), one of the greatest Arabian mathematicians, wrote a commentary on the Elements, but

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did not complete it[51]. His method is also unfavourably regarded in the same preface to the Oxford MS. 280. According to Ḥāji Khalfa, he also wrote a book on geometrical constructions, in thirteen chapters. Apparently a book answering to this description was compiled by a gifted pupil from lectures by Abū ‘l Wafā, and a Paris MS. (Anc. fonds 169) contains a Persian translation of this work, not that of Abū ‘l Wafā himself. An analysis of the work was given by Woepcke[52], and some particulars will be found in Cantor[53]. Abū ‘l Wafā also wrote a commentary on Diophantus, as well as a separate “book of proofs to the propositions which Diophantus used in his book and to what he (Abū ‘l Wafā) employed in his commentary[54].”

7. Ibn Rāhawaihi al-Arjānī also commented on Eucl. Book X.[55]

8. ‘Alī b. Aḥmad Abū ‘l-Qāsim al-Anṭākī (d. 987) wrote a commentary on the whole book[56]; part of it seems to survive (from the 5th Book onwards) at Oxford (Catal. MSS. orient. II. 281)[57].

9. Sind b. ‘Alī Abū ‘ṭ-Ṭaiyib was a Jew who went over to Islam in the time of al-Ma’mūn, and was received among his astronomical observers, whose head he became[58] (about 830); he died after 864. He wrote a commentary on the whole of the Elements; “Abū ‘Alī saw nine books of it, and a part of the tenth[59].” His book “On the Apotomae and the Medials,” mentioned by the Fihrist, may be the same as, or part of, his commentary on Book X.

10. Abū Yūsuf Ya’qūb b. Muḥ. ar-Rāzī “wrote a commentary on Book X., and that an excellent one, at the instance of Ibn al-‘Amīd[60].”

11. The Fihrist next mentions al-Kindī (Abū Yūsuf Ya’qūb b. Isḥāq b. aṣ-Ṣabbāḥ al-Kindī, d. about 873), as the author (1) of a work “on the objects of Euclid’s book,” in which occurs the statement that the Elements were originally written by Apollonius, the carpenter (see above, p. 5 and note), (2) of a book “on the improvement of Euclid’s work,” and (3) of another “on the improvement of the 14th and 15th Books of Euclid.” “He was the most distinguished man of his time, and stood alone in the knowledge of the old sciences collectively; he was called ‘the philosopher of the Arabians’; his writings treat of the most different branches of knowledge, as logic, philosophy, geometry, calculation, arithmetic, music, astronomy and others[61].” Among the other geometrical works of al-Kindī mentioned by the Fihrist[62] are treatises on the closer investigation of the results of Archimedes concerning the measure of the diameter of a circle in terms of its circumference, on the construction of the figure of the two mean proportionals, on the approximate determination of the chords

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of the circle, on the approximate determination of the chord (side) of the nonagon, on the division of triangles and quadrilaterals and constructions for that purpose, on the manner of construction of a circle which is equal to the surface of a given cylinder, on the division of the circle, in three chapters etc.

12. The physician Naẓīf b. Yumn (or Yaman) al-Qass (“the priest”) is mentioned by the Fihrist as having seen a Greek copy of Eucl. Book X. which had 40 more propositions than that which was in general circulation (containing 109), and having determined to translate it into Arabic[63]. Fragments of such a translation exist at Paris, Nos. 18 and 34 of the MS. 2457 (952, 2 Suppl. Arab. in Woepcke’s tract); No. 18 contains “additions to some propositions of the 10th Book, existing in the Greek language[64].” Naẓīf must have died about 990[65].

13. Yūḥannā b. Yūsuf b. al-Ḥārith b. al-Biṭrīq al-Qass (d. about 980) lectured on the Elements and other geometrical books, made translations from the Greek, and wrote a tract on the “proof” of the case of two straight lines both meeting a third and making with it, on one side, two angles together less than two right angles[66]. Nothing of his appears to survive, except that a tract “on rational and irrational magnitudes,” No. 48 in the Paris MS. just mentioned, is attributed to him.

14. Abū Muḥ. al-Ḥasan b. ‘Ubaidallāh b. Sulaimān b. Wahb (d. 901) was a geometer of distinction, who wrote works under the two distinct titles “A commentary on the difficult parts of the work of Euclid” and “The Book on Proportion[67].” Suter thinks that another reading is possible in the case of the second title, and that it may refer to the Euclidean work “on the divisions (of figures)[68].”

15. Qusṭā b. Lūqā al-Ba’labakkī (d. about 912), a physician, philosopher, astronomer, mathematician and translator, wrote “on the difficult passages of Euclid’s book” and “on the solution of arithmetical problems from the third book of Euclid[69]”; also an “introduction to geometry,” in the form of question and answer[70].

16. Thābit b. Qurra (826—901), besides translating some parts of Archimedes and Books V.—VII. of the Conics of Apollonius, and revising Isḥāq’s translation of Euclid’s Elements, also revised the translation of the Data by the same Isḥāq and the book On divisions of figures translated by an anonymous writer. We are told also that he wrote the following works: (1) On the Premisses (Axioms, Postulates etc.) of Euclid, (2) On the Propositions of Euclid, (3) On the propositions and questions which arise when two straight lines are cut by a third (or on the “proof” of Euclid’s famous postulate). The last tract is extant in the MS. discovered by Woepcke (Paris 2457, 32°). He is also credited with “an excellent work” in the shape of an “Introduction to the Book of Euclid,” a treatise on

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Geometry dedicated to Ismā’īl b. Bulbul, a Compendium of Geometry, and a large number of other works for the titles of which reference may be made to Suter, who also gives particulars as to which are extant[71].

17. Abū Sa’īd Sinān b. Thābit b. Qurra, the son of the translator of Euclid, followed in his father’s footsteps as geometer, astronomer and physician. He wrote an “improvement of the book of …… on the Elements of Geometry, in which he made various additions to the original.” It is natural to conjecture that Euclid is the name missing in this description (by Ibn abī Uṣaibi’a); Casiri has the name Aqāṭon[72]. The latest editor of the Ta’rīkh al-Ḥukamā, however, makes the name to be Iflāṭon (= Plato), and he refers to the statement by the Fihrist and Ibn al-Qifṭī attributing to Plato a work on the Elements of Geometry translated by Qusṭā. It is just possible, therefore, that at the time of Qusṭā the Arabs were acquainted with a book on the Elements of Geometry translated from the Greek, which they attributed to Plato[73]. Sinān died in 943.

18. Abū Sahl Wījan (or Waijan) b. Rustam al-Kūhī (fl. 988), born at Kūh in Ṭabaristān, a distinguished geometer and astronomer, wrote, according to the Fihrist, a “Book of the Elements” after that of Euclid[74]; the 1st and 2nd Books survive at Cairo, and a part of the 3rd Book at Berlin (5922)[75]. He wrote also a number of other geometrical works: Additions to the 2nd Book of Archimedes on the Sphere and Cylinder (extant at Paris, at Leiden, and in the India Office), On the finding of the side of a heptagon in a circle (India Office and Cairo), On two mean proportionals (India Office), which last may be only a part of the Additions to Archimedes’ On the Sphere and Cylinder, etc.

19. Abū Naṣr Muḥ. b. Muḥ. b. Ṭarkhān b. Uzlaġ al-Fārābī (870—950) wrote a commentary on the difficulties of the introductory matter to Books I. and V.[76] This appears to survive in the Hebrew translation which is, with probability, attributed to Moses b. Tibbon[77].

20. Abū ‘Alī al-Ḥasan b. al-Ḥasan b. al-Haitham (about 965—1039), known by the name Ibn al-Haitham or Abū ‘Alī al-Baṣrī, was a man of great powers and knowledge, and no one of his time approached him in the field of mathematical science. He wrote several works on Euclid the titles of which, as translated by Woepcke from Usaibi’a, are as follows[78]:

1. Commentary and abridgment of the Elements. 2. Collection of the Elements of Geometry and Arithmetic, drawn from the treatises of Euclid and Apollonius. 3. Collection of the Elements of the Calculus deduced from the principles laid down by Euclid in his Elements.

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4. Treatise on “measure” after the manner of Euclid’s Elements.

5. Memoir on the solution of the difficulties in Book I. 6. Memoir for the solution of a doubt about Euclid, relative to Book V. 7. Memoir on the solution of a doubt about the stereometric portion. 8. Memoir on the solution of a doubt about Book XII. 9. Memoir on the division of the two magnitudes mentioned in X. 1 (the theorem of exhaustion). 10. Commentary on the definitions in the work of Euclid (where Steinschneider thinks that some more general expression should be substituted for “definitions”).

The last-named work (which Suter calls a commentary on the Postulates of Euclid) survives in an Oxford MS. (Catal. MSS. orient. I. 908) and in Algiers (1446, 1°).

A Leiden MS. (966) contains his Commentary “on the difficult places” up to Book V. We do not know whether in this commentary, which the author intended to form, with the commentary on the Musādarāt, a sort of complete commentary, he had collected the separate memoirs on certain doubts and difficult passages mentioned in the above list.

A commentary on Book V. and following Books found in a Bodleian MS. (Catal. II. p. 262) with the title “Commentary on Euclid and solution of his difficulties” is attributed to b. Haitham; this might be a continuation of the Leiden MS.

The memoir on X. 1 appears to survive at St Petersburg, MS. de l’Institut des langues orient. 192, 5° (Rosen, Catal. p. 125).

21. Ibn Sīnā, known as Avicenna (980—1037), wrote a Compendium of Euclid, preserved in a Leiden MS. No. 1445, and forming the geometrical portion of an encyclopaedic work embracing Logic, Mathematics, Physics and Metaphysics[79].

22. Aḥmad b. al-Ḥusain al-Ahwāzī al-Kātib wrote a commentary on Book X., a fragment of which (some 10 pages) is to be found at Leiden (970), Berlin (5923) and Paris (2467, 18°)[80].

23. Naṣīraddīn aṭ-Ṭūsī (1201—1274) who, as we have seen, brought out a Euclid in two forms, wrote:

1. A treatise on the postulates of Euclid (Paris, 2467, 5°). 2. A treatise on the 5th postulate, perhaps only a part of the foregoing (Berlin, 5942, Paris, 2467, 6°). 3. Principles of Geometry taken from Euclid, perhaps identical with No. 1 above (Florence, Pal. 298). 4. 105 problems out of the Elements (Cairo). He also edited the Data (Berlin, Florence, Oxford etc.)[81].

24. Muḥ. b. Ashraf Shamsaddīn as-Samarqandī (fl. 1276) wrote “Fundamental Propositions, being elucidations of 35 selected proposi-

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tions of the first Books of Euclid,” which are extant at Gotha (1496 and 1497), Oxford (Catal. I. 967, 2°), and Brit. Mus.[82]

25. Mūsā b. Muḥ. b. Maḥmūd, known as Qāḍīzāde ar-Rūmī (i.e. the son of the judge from Asia Minor), who died between 1436 and 1446, wrote a commentary on the “Fundamental Propositions” just mentioned, of which many MSS. are extant[83]. It contained biographical statements about Euclid alluded to above (p. 5, note).

26. Abū Dā’ūd Sulaimān b. ‘Uqba, a contemporary of al-Khāzin (see above, No. 5), wrote a commentary on the second half of Book X., which is, at least partly, extant at Leiden (974) under the title “On the binomials and apotomae found in the 10th Book of Euclid[84].”

27. The Codex Leidensis 399, 1 containing al-Ḥajjāj’s translation of Books I.—VI. is said to contain glosses to it by Sa’īd b. Mas’ūd b. al-Qass, apparently identical with Abū Naṣr Ġars al-Na’ma, son of the physician Mas’ūd b. al-Qass al-Baġdādī, who lived in the time of the last Caliph al-Musta’ṣim (d. 1258)[85].

28. Abū Muḥammad b. ‘Abdalbāqī al-Baġdādī al-Faraḍī (d. 1141, at the age of over 70 years) is stated in the Ta’rīkh al-Ḥukamā to have written an excellent commentary on Book X. of the Elements, in which he gave numerical examples of the propositions[86]. This is published in Curtze’s edition of an-Nairīzī where it occupies pages 252—386[87].

29. Yaḥyā b. Muḥ. b. ‘Abdān b. ‘Abdalwāḥid, known by the name of Ibn al-Lubūdī (1210—1268), wrote a Compendium of Euclid, and a short presentation of the postulates[88].

30. Abū ‘Abdallāh Muḥ. b. Mu’ādh al-Jayyānī wrote a commentary on Eucl. Book V. which survives at Algiers (1446, 3°)[89].

31. Abū Naṣr Manṣūr b. ‘Alī b. ‘Irāq wrote, at the instance of Muḥ. b. Aḥmad Abū ‘r-Raiḥān al-Bīrūnī (973—1048), a tract “on a doubtful (difficult) passage in Eucl. Book XIII.” (Berlin, 5925)[90].

[1]

Lexicon bibliogr. et encyclop. ed. Flügel, III. pp. 91, 92.

[2]

Klamroth, Zeitschrift der Deutschen Morgenländischen Gesellschaft, XXXV. p. 303.

[3]

Fihrist (tr. Suter), p. 16.

[4]

Codex Leidensis 399, 1. Euclidis Elementa ex interpretatione al-Hadschdschadii cum commentariis al-Narizii, Hauniae, part I. i. 1893, part I. ii. 1897, part II. i. 1900, part II. ii. 1905, part III. i. 1910.

[5]

Steinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abtheilung, pp. 85, 86, 99.

[6]

Klamroth, p. 279.

[7]

Steinschneider, p. 88.

[8]

Klamroth, p. 306.

[9]

These MSS. are described by Nicoll and Pusey, Catalogus cod. mss. orient. bibl. Bodleianae, pt. II. 1835 (pp. 257—262).

[10]

Klamroth, pp. 306—8.

[11]

Steinschneider, p. 98. Heiberg has quoted the whole of this preface in the Zeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. p. 16.

[12]

This seems to include a rearrangement of the contents of Books XIV., XV. added to the Elements.

[13]

Suter, Die Mathematiker und Astronomen der Araber, p. 151. The Latin title is Euclidis elementorum geometricorum libri tredecim. Ex traditione doctissimi Nasiridini Tusini nunc primum arabice impressi. Romae in typographia Medicea MDXCIV. Cum licentia superiorum.

[14]

Kästner, Geschichte der Mathematik, I. p. 367.

[15]

Suter has a note that this MS. is very old, having been copied from the original in the author’s lifetime.

[16]

Suter, p. 151.

[17]

Described by Kästner, Geschichte der Mathematik, I. pp. 289—299, and by Weissenborn, Die Ubersetzungen des Euklid durch Campano und Zamberti, Halle a. S., 1882, pp. 1—7. See also infra, Chapter VIII, p. 97.

[18]

Sonderabdruck des Jahresberichtes über die Fortschritte der klassischen Alterthumswissenschaft vom Okt. 1879—1882, Berlin, 1884.

[19]

Klamroth, p. 271.

[20]

Curtze, op. cit. p. 20; Heiberg, Euklid-Studien, p. 178.

[21]

Heiberg’s Euclid, vol. V. p. ci.

[22]

Klamroth, pp. 273—4.

[23]

Heiberg, Zeitschrift für Math. u. Physik, XXIX., hist.-litt. Abtheilung, p. 21.

[24]

Klamroth, p. 274; Steinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abth. p. 98.

[25]

Besthorn-Heiberg read “11?” as the number, Klamroth had read it as 21 (p. 273).

[26]

See Klamroth, pp. 275—6, 280, 282—4, 314—15, 326; Heiberg, vol. v. pp. xcvi, xcvii.

[27]

Heiberg in Zeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. p. 3 sqq.

[28]

Apollonius, ed. Heiberg, vol. II. p. 218, 3—5.

[29]

Pappus, v. p. 436, 5.

[30]

Proclus, pp. 303—4.

[31]

Zeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. p. 6 sqq.

[32]

Klamroth, p. 310; Steinschneider, pp. 85—6.

[33]

Klamroth, p. 290, illustrates Isḥāq’s method by his way of distinguishing ἐφαρμόζειν (to be congruent with) and ἐφαρμόζεσθαι (to be applied to), the confusion of which by translators was animadverted on by Savile. Isḥāq avoided the confusion by using two entirely different words.

[34]

Klamroth, pp. 310—1.

[35]

ibid. p. 287.

[36]

ibid. pp. 304—5.

[37]

ibid. p. 274.

[38]

Ḥāji Khalfa, I. p. 383.

[39]

Heiberg, Zeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. pp. 2, 3.

[40]

Steinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abth. pp. 86 sqq.

[41]

Steinschneider, p. 86, Fihrist (tr. Suter), pp. 16, 67; Suter, Die Mathematiker und Astronomen der Araber (1900), p. 45.

[42]

Supplementum ad Euclidis opera omnia, ed. Heiberg and Menge, Leipzig, 1899.

[43]

Fihrist, pp. 16, 38; Steinschneider, p. 87; Suter, p. 65.

[44]

Fihrist, pp. 16, 25; Steinschneider, p. 88; Suter, p. 12.

[45]

Fihrist, pp. 16, 25, 58.

[46]

Suter, p. 26, note, quotes the Paris MS. 2467, 16° containing the work “on proportion” as the authority for this conjecture.

[47]

MS. 2457, 39° (cf. Woepcke in Mém. prés. à l’acad. des sciences, XIV., 1856, p. 669).

[48]

Fihrist, p. 17.

[49]

Suter, p. 58, note b.

[50]

Steinschneider, p. 89.

[51]

Fihrist, p. 17.

[52]

Woepcke, Journal Asiatique, Sér. v. T. v. pp. 218—256 and 309—359.

[53]

Gesch. d. Math. vol. I₃, pp. 743—6.

[54]

Fihrist, p. 39; Suter, p. 71.

[55]

Fihrist, p. 17; Suter, p. 17.

[56]

Fihrist, p. 17.

[57]

Suter, p. 64.

[58]

Fihrist, p. 17, 29; Suter, pp. 13, 14.

[59]

Fihrist, p. 17.

[60]

Fihrist, p. 17; Suter, p. 66.

[61]

Fihrist, p. 17, 10—15.

[62]

The mere catalogue of al-Kindī’s works on the various branches of science takes up four octavo pages (11—15) of Suter’s translation of the Fihrist.

[63]

Fihrist, pp. 16, 17.

[64]

Woepcke, Mém. prés. à l’acad. des sciences, XIV. pp. 666, 668.

[65]

Suter, p. 68.

[66]

Fihrist, p. 38; Suter, p. 60.

[67]

Fihrist, p. 26, and Suter’s note, p. 60.

[68]

Suter, p. 211, note 23.

[69]

Fihrist, p. 43.

[70]

Fihrist, p. 43; Suter, p. 41.

[71]

Suter, pp. 34—8.

[72]

Fihrist (ed. Suter), p. 59, note 132; Suter, p. 52, note b.

[73]

See Suter in Bibliotheca Mathematica, IV₃, 1903-4, pp. 296—7, review of Julius Lippert’s Ibn al-Qifṭī. Ta’rīch al-ḥukamā, Leipzig, 1903.

[74]

Fihrist, p. 40.

[75]

Suter, p. 75.

[76]

Suter, p. 55.

[77]

Steinschneider, p. 92.

[78]

Steinschneider, pp. 92—3.

[79]

Steinschneider, p. 92; Suter, p. 89.

[80]

Suter, p. 57.

[81]

Suter, pp. 150—1.

[82]

Suter, p. 157.

[83]

ibid. p. 175.

[84]

ibid. p. 56.

[85]

ibid. pp. 153—4, 227.

[86]

Gartz, p. 14; Steinschneider, pp. 94—5.

[87]

Suter in Bibliotheca Mathematica, IV₃, 1903, pp. 25, 295; Suter has also an article on its contents, Bibliotheca Mathematica, VII₃, 1906-7, pp. 234—251.

[88]

Steinschneider, p. 94; Suter, p. 146.

[89]

Suter, Nachträge und Berichtigungen, in Abhandlungen zur Gesch. der math. Wissenschaften, XIV., 1902, p. 170.

[90]

Suter, p. 81, and Nachträge, p. 172.