the scholia I. Nos. 40 and 49 appear with the name of Psellus attached.

Schol. Vind. are not found without the admixture of foreign elements in any of our three sources. In 1 there are only very few such in the first hand. In q there are several new scholia in the first hand, for the most part due to the copyist himself. The collection of scholia on Book x. in q (Heiberg's q) is also in the first hand; it is not original, and it may perhaps be due to Psellus (Maglb. has some definitions of Book x. with a heading "scholia of ... Michael Psellus on the definitions of Euclid's 10th Element" and Schol. x. No. 9), whose name must have been attached to it in the common source of Maglb. and q; to a great extent it consists of extracts from Schol. Vind. taken from the same source as Vl. The scholia q' (in an ancient hand in q), confined to Book II., partly belong to Schol. Vind. and partly correspond to b1 (Bologna Ms.). qa and qb are in one hand (Theodorus Antiochita), the nearest to the first hand of q; they are doubtless due to an early possessor of the MS. of whom we know nothing more.

Va has, besides Schol. Vind., a number of scholia which also appear in other MSS., one in BFb, some others in P, and some in v (Codex Vat. 1038, 13th c.); these scholia were taken from a source in which many abbreviations were used, as they were often misunderstood by Va. Other scholia in Va which are not found in the older sourcessome appearing in Va alone-are also not original, as is proved by mistakes or corruptions which they contain; some others may be due to the copyist himself.

Vb seldom has scholia common with the other older sources; for the most part they either appear in Vb alone or only in the later sources as v or F2 (later scholia in F), some being original, others not. In Book X. Vb has three series of numerical examples, (1) with Greek numerals, (2) alternatives added later, also mostly with Greek numerals, (3) with Arabic numerals. The last class were probably the work of the copyist himself. As Vb belongs to the same time as the MS. (12th c.), these examples give an idea of the facility with which calculations were made in Byzantium at that time. They show too that the Greek method of writing numbers still preponderated in the IIth c., but that the use of the Arabic numerals (in the East-Arabian form) was thoroughly established in the 12th c.

Of collections in other hands in V distinguished by Heiberg (see preface to Vol. v.), V1 has very few scholia which are found in other sources, the greater part being original; V, V3 are the work of the copyist himself; V are so in part only, and contain several scholia from Schol. Vat. and other sources. V and V are later than 13th -14th c., since they are not found in f (cod. Laurent. XXVIII, 6) which was copied from V and contains, besides Va Vb, the greater part of V1 and VI. No. 20 of V2 (in the text).

In P there are, besides P3 (a quite late hand, probably one of the old Scriptores Graeci at the Vatican), two late hands (P), one of which has some new and independent scholia, while the other has

added the greater part of Schol. Vind., partly in the margin and partly on pieces of leaves stitched on.

Our sources for Schol. Vat. also contain other elements. In P there were introduced a certain number of extracts from Proclus, to supplement Schol. Vat. to Book I.; they are all written with a different ink from that used for the oldest part of the MS., and the text is inferior. There are additions in the other sources of Schol. Vat. (F and B) which point to a common source for FB and which are nearly all found in other MSS., and, in particular, in Schol. Vind., which also used the same source; that they are not assignable to Schol. Vat. results only from their not being found in Vat. Of other additions in F, some are peculiar to F and some common to it and b; but they are not original. F (scholia in a later hand in F) contains three original scholia; the rest come from V. B contains, besides scholia common to it and F, b or other sources, several scholia which seem to have been put together by Arethas, who wrote at least a part of them with his own hand.

Heiberg has satisfied himself, by a closer study of b, that the scholia which he denotes by b, B and b' are by one hand; they are mostly to be found in other sources as well, though some are original. By the same hand (Theodorus Cabasilas, 15th c.) are also the scholia denoted by b2, B2, b3 and B3. These scholia come in great part from Schol. Vind., and in making these extracts Theodorus probably used one of our sources, 1, mistakes in which often correspond to those of Theodorus. To one scholium is attached the name of Demetrius (who must be Demetrius Cydonius, a friend of Nicolaus Cabasilas, 14th c.); but it could not have been written by him, since it appears in B and Schol. Vind. Nor are all the scholia which bear the name of Theodorus due to Theodorus himself, though some are so.

As B3 (a late hand in B) contains several of the original scholia of b2, B3 must have used b itself as his source, and, as all the scholia in B3 are in b, the latter is also the source of the scholia in B3 which are found in other MSS. B and b were therefore, in the 15th c., in the hands of the same person; this explains, too, the fact that b in a late hand has some scholia which can only come from B. We arrive then at the conclusion that Theodorus Cabasilas, in the 15th c., owned both the MSS. B and b, and that he transferred to B scholia which he had before written in b, either independently or after other sources, and inversely transferred some scholia from B to b. Further, B2 are earlier than Theodorus Cabasilas, who certainly himself wrote B' as well as b2 and b3.

An author's name is also attached to the scholia VI. No. 6 and X. No. 223, which are attributed to Maximus Planudes (end of 13th c.) along with scholia on I. 31, X. 14 and X. 18 found in 1 in a quite late hand and published on pp. 46, 47 of Heiberg's dissertation. These seem to have been taken from lectures of Planudes on the Elements by a pupil who used 1 as his copy.

There are also in 1 two other Byzantine scholia, written by a late hand, and bearing the names Ioannes and Pediasimus respectively;

these must in like manner have been written by a pupil after lectures of Ioannes Pediasimus (first half of 14th c.), and this pupil must also have used 1.

Before these scholia were edited by Heiberg, very few of them had been published in the original Greek. The Basel editio princeps has a few (V. No. I, VI. Nos. 3, 4 and some in Book X.) which are taken, some from the Paris MS. (Paris. Gr. 2343) used by Grynaeus, others probably from the Venice MS. (Marc. 301) also used by him; one published by Heiberg, not in his edition of Euclid but in his paper on the scholia, may also be from Venet. 301, but appears also in Paris. Gr. 2342. The scholia in the Basel edition passed into the Oxford edition in the text, and were also given by August in the Appendix to his Vol. II.

Several specimens of the two series of scholia (Vat. and Vind.) were published by C. Wachsmuth (Rhein. Mus. XVIII. p. 132 sqq.) and by Knoche (Untersuchungen über die neu aufgefundenen Scholien des Proklus, Herford, 1865).

The scholia published in Latin were much more numerous. G. Valla (De expetendis et fugiendis rebus, 1501) reproduced apparently some 200 of the scholia included in Heiberg's edition. Several of these he obtained from two Modena MSS. which at one time were in his possession (Mutin. III B, 4 and II E, 9, both of the 15th c.); but he must have used another source as well, containing extracts from other series of scholia, notably Schol. Vind. with which he has some 87 scholia in common. He has also several that are new.

Commandinus included in his translation under the title "Scholia antiqua" the greater part of the Schol. Vat. which he certainly obtained from a MS. of the class of Vat. 192; on the whole he adhered closely to the Greek text. Besides these scholia Commandinus has the scholia and lemmas which he found in the Basel editio princeps, and also three other scholia not belonging to Schol. Vat., as well as one new scholium (to XII. 13) not included in Heiberg's edition, which are distinguished by different type and were doubtless taken from the Greek MS. used by him along with the Basel edition.

In Conrad Dasypodius' Lexicon mathematicum published in 1573 there is (on fol. 42-44) "Graecum scholion in definitiones Euclidis libri quinti elementorum appendicis loco propter pagellas vacantes annexum." This contains four scholia, and part of two others, published in Heiberg's edition, with some variations of readings, and with some new matter added (for which see pp. 64-6 of Heiberg's pamphlet). The source of these scholia is revealed to us by another work of Dasypodius, Isaaci Monachi Scholia in Euclidis elementorum geometriae sex priores libros per C. Dasypodium in latinum sermonem translata et in lucem edita (1579). This work contains, besides excerpts from Proclus on Book 1. (in part closely related to Schol. Vind.), some 30 scholia included in Heiberg's edition, several new scholia, and the above-mentioned scholia to the definitions of Book V. published in Greek in 1573. After the scholia follow "Isaaci Monachi

(two

prolegomena in Euclidis Elementorum geometriae libros definitions of geometry) and "Varia miscellanea ad geometriae cognitionem necessaria ab Isaaco Monacho collecta" (mostly the same as pp. 252, 24-272, 27 in the Variae Collectiones included in Hultsch's Heron); lastly, a note of Dasypodius to the reader says that these scholia were taken "ex clarissimi viri Joannis Sambuci antiquo codice manu propria Isaaci Monachi scripto." Isaak Monachus is doubtless Isaak Argyrus, 14th c.; and Dasypodius used a MS. in which, besides the passage in Hultsch's Variae Collectiones, were a number of scholia marked in the margin with the name of Isaak (cf. those in b under the name of Theodorus Cabasilas). Whether the new scholia are original cannot be decided until they are published in Greek; but it is not improbable that they are at all events independent arrangements of older scholia. All but five of the others, and all but one of the Greek scholia to Book V., are taken from Schol. Vat.; three of the excepted ones are from Schol. Vind., and the other three seem to come from F (where some words of them are illegible, but can be supplied by means of Mut. III B, 4, which has these three scholia and generally shows a certain likeness to Isaak's scholia).

Dasypodius also published in 1564 the arithmetical commentary of Barlaam the monk (14th c.) on Eucl. Book II., which finds a place in Appendix IV. to the Scholia in Heiberg's edition.

CHAPTER VII.

EUCLID IN ARABIA.

WE are told by Ḥāji Khalfa1 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-Hajjāj b. Yūsuf b. Maṭar is, if not the very first, at least one of the first books translated from the Greek into Arabic". According to the Fihrist it was translated by al-Hajjāj twice; the first translation was known as "Hārūni" ("for Hārūn"), the second bore the name "Ma'muni" ("for al-Ma'mun") and was the more trustworthy. Six Books of the second of these versions survive in a Leiden MS. (Codex Leidensis 399, 1) which is being published by Besthorn and Heiberg. In the preface to this MS. it is stated that, in the reign of Harūn ar-Rashid (786-809), al-Hajjāj was commanded by Yaḥyā b. Khalid b. Barmak to translate the book into Arabic. Then, when al-Ma'mun became Caliph, as he was devoted to learning, al-Hajjāj saw that he would secure the favour of al-Ma'mun "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 Ishaq b. Hunain, and that this translation was improved by Thabit b. Qurra. This Abū Ya'qub Isḥāq b. Hunain b. Ishaq al-'Ibadi (d. 910) was the son of the most famous of Arabic translators, Hunain b. Ishaq al-Ibādi (809-873), a Christian and physician to the Caliph alMutawakkil (847-861). There seems to be no doubt that Ishaq, who

1 Lexicon bibliogr. et encyclop. ed. Flügel, 111. pp. 91, 92.

2 Klamroth, Zeitschrift der Deutschen Morgenländischen Gesellschaft, xxxv. p. 303.
Fihrist (tr. Suter), p. 16.

Codex Leidensis 399, 1. Euclidis Elementa ex interpretatione al-Hadschdschadschii cum commentariis al-Narizii, Hauniae, part 1. i. 1893, part 1. ii. 1897, part II. i. 1905.