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 Ishaq and Thabit, as the latter died in 901 or nine years before Ishaq. Thabit 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 Ishaq'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)1. Moreover Thabit 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 Thabit did not alter the numbers of the propositions in Ishaq's translation. 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 Thabit took credit for, and that this was confirmed by Nazif, the physician, to whom Yuḥannā had shown it. This proposition may have been wanting in Ishaq, and Thābit may have added it, but without claiming it as his own discovery. As a fact, I. 45 is missing in the translation by al-Hajjāj.

The original version of Ishaq without the improvements by Thabit has probably not survived any more than the first of the two versions by al-Hajjāj; 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. Two MSS. of the Ishaq-Thabit version exist in the Bodleian Library (No. 279 belonging to the year 1238, and No. 280 written in 1260-1); Books I-XIII. are in the Ishaq-Thabit version, the non-Euclidean Books XIV., XV. in the translation of Qusṭā b. Lūqā al-Ba‍labakki (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 Ishaq-Thabit version, Books XI.-XIII. purporting to be in al-Hajjāj'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

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

Klamroth, p. 279.

Klamroth, p. 306.

8 Steinschneider, p. 88.

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

་

shortened copies of one and the same recension'. The Bodleian MS. No. 280 contains a preface, translated by Nicoll, which cannot be by Thabit himself because it mentions Avicenna (980–1037) and other later authors. The MS. was written at Marāga in the year 1260-1 and has in the margin readings and emendations from the edition of Naşiraddin at-Tusi (shortly to be mentioned) who was living at Marāġa at the time. Is it possible that at-Ṭūsi himself is the author of the preface?? 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 Wafa 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 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."

After Thabit the Fihrist mentions Abū 'Uthman ad-Dimashqi as having translated some Books of the Elements including Book X. (It is Abū 'Uthman's translation of Pappus' commentary on Book X. which Woepcke discovered at Paris.) The Fihrist adds also that "Nazif 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-Hasan Naṣiraddin at-Tūsi (whom we shall call at-Tusi for short), born at Tūs (in Khurasan) 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

Klamroth, pp. 306-8.

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

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

this edition are to be found with 12 and some with 13 Books, some with a Latin title and some without'. 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. The shorter form, which however, in most MSS., is in 15 Books, survives at Berlin, Munich, Oxford, British Museum (974, 1334, 1335), Paris (2465, 2466), India Office, and Constantinople; it was printed at Constantinople in 1801, and the first six Books at Calcutta in 1824.

At-Tusi'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'). Campanus (13th c.) was a mathematician, and it is likely enough that he allowed himself the same liberty as at-Tūsi in reproducing Euclid. Whatever may be the relation between Campanus' version and that of Athelhard of Bath (about 1120), and whether, as Curtze thinks, 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 at-Tūsi used those of his predecessors', it is certain that both versions came from an Arabian source, as is evident from the occurrence of Arabic words in them. 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. A curious circumstance is that, while Campanus' version agrees with at-Tūsi'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 us1o. Heiberg also notes that Campanus' Books XIV., XV. show a certain agreement with the preface to the Thabit-Ishaq 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

1 Suter, Die Mathematiker und Astronomen der Euclidis elementorum geometricorum libri tredecim. Tusini nunc primum arabice impressi. Romae in licentia superiorum.

Araber, p. 151. The Latin title is Ex traditione doctissimi Nasiridini typographia Medicea MDXCIV. Cum

2 Kästner, Geschichte der Mathematik, 1. p. 367. Suter has a note that this Ms. is very old, having been copied from the original in the author's lifetime.

4 Suter, p. 151.

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

• Sonderabdruck des Jahresberichtes über die Fortschritte der klassischen Alterthumswissenschfat vom Okt. 1879-1882, Berlin, 1884.

7 Klamroth, p. 271.

8 Curtze, op. cit. p. 20; Heiberg, Euklid-Studien, p. 178. • Heiberg's Euclid, vol. v. p. ci.

10 Klamroth, pp. 273—+

to inscribe any one of the solids in any other and (3) noted the cases where this could not be done'.

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. Haji Khalfa says that al-Hajjaj's translation contained 468 propositions, and Thābit's 478; this is stated on the authority of at-Tūsi, whose own edition. contained 468. The fact that Thabit'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, I which gives Ishaq's numbers (although the translation is that of al-Hajjaj) apparently makes the total 479 propositions (the number in Book XIV. being apparently 11, instead of the 10 of O3). 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.

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.

1 Heiberg, Zeitschrift fur Math. u. Physik, XXIX., hist.-litt. Abtheilung, p. 21. Klamroth, p. 274; Steinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abth. p. 98.

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

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-Hajjaj'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 Thabit-Isḥāq and at-Tūsi is accounted for thus:

(1) The three propositions VI. 12 and X. 28, 29 which both Ishaq and the Greek text have are omitted in at-Tūsi.

(2) Ishaq divides each of the propositions XIII. 1-3 into two, making six instead of three in at-Tūsi and in the Greek.

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

Apart from the above differences al-Hajjāj (so far as we know), Ishaq and at-Tusī agree; but their Euclid shows many differences from our Greek text. These differences we will classify as follows1.

I. 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:

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

VII. I-20, 22, 21, 23-28, 31, 32, 29, 30, 33-39.

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

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

XI. I–30, 31, 32, 34, 33, 35–39.

XII. I-5, 7, 9, 8, 10, 12, 11, 15, 16—18.

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

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