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FOR full details of the various MSS. and of their mutual relations, reference should be made to the prefaces to the first and second volumes of Tannery's edition'. Tannery's account needs only to be supplemented by a description given by Gollob' of another MS. supposed by Tannery to be non-existent, but actually rediscovered in the Library of the University of Cracow (Nr 544). Only the shortest possible summary of the essential facts will be given here.

After the loss of Egypt the work of Diophantus long remained almost unknown among the Byzantines; perhaps one copy only survived (of the Hypatian recension), which was seen by Michael Psellus and possibly by the scholiast to Iamblichus, but of which no trace can be found after the capture of Constantinople in 1204. From this one copy (denoted by the letter a in Tannery's table of the MSS.) another MS. (a) was copied in the 8th or 9th century; this again is lost, but is the true archetype of our MSS. The copyist apparently intended to omit all scholia, but, the distinction between text and scholia being sometimes difficult to draw, he included a good deal which should have been left out. For example, Hypatia, and perhaps scholiasts after her, seem to have added some alternative solutions and a number of new problems; some of these latter, such as II. 1-7, 17, 18, were admitted into the text as genuine.

The MSS. fall into two main classes, the ante-Planudes class, as we may call it, and the Planudean. The most ancient and the best of all is Matritensis 48 (Tannery's A), which was written in the 13th century and belongs to the first class; it is evidently a most faithful copy of the lost archetype (a). Maximus Planudes wrote a systematic commentary on Books I. and II., and his scholia,

1 Dioph. 1. pp. iii-v, 11. pp. xxii-xxxiv.

2 Eduard Gollob, "Ein wiedergefundener Diophantuscodex" in Zeitschrift für Math. u. Physik, XLIV. (1899), hist.-litt. Abtheilung, pp. 137-140.

which are edited by Tannery for the first time, are preserved in the oldest representative which we possess of the Planudean class, namely, Marcianus 308 (Tannery's B1), itself apparently copied from an archetype of the 14th century now lost, with the exception of ten leaves which survive in Ambrosianus Et 157 sup.

Tannery shows the relation of the MSS. in the following diagram:

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Auria's recension made up out of MSS. 2, 3, 15 above and Xylander's translation:

25. Parisinus 2380=D.

26. Ambrosianus E 5 sup.

27. MS. (Patavinus) of Broscius (Brozek) now at Cracow.
Lost MS. of Cardinal du Perron.


The addition of a few notes as regards the most important and interesting of the MSS., in the order of their numbers in Tannery's arrangement, will now sufficiently complete the story.

I. The best and most ancient MS., that of Madrid (Tannery's A), was unfortunately spoiled at a late date by corrections made, especially in the first two Books, from some MS. of the Planudean class, in such a way that the original reading is sometimes entirely erased or made quite illegible. In these cases recourse must be had to the Vatican MS. 191.

2. The MS. Vaticanus graecus 191 was copied from A before it had suffered the general alteration by means of a MS. of the other class, though not before various other corrections had been made in different hands not easily distinguished; thus sometimes has readings which Tannery found to have arisen from some correction in A. A appears to have been at Rome for a considerable period at the time when was copied; for the librarian who wrote the old table of contents1 at the beginning of V inserted in the margin in one place the word apkáuevos, which had been omitted, direct from the original (A).

3. Vat. gr. 304 was copied from V, not from A; Tannery inferred this mainly from a collation of the scholia, and he notes that the word ap§áμevos above mentioned is here brought into the text by the erasure of some letters. This MS. 304, being very clearly written, was used thenceforward to make copies from. The next five MSS. do not appear to have had any older source.

4. The MS. Parisinus 2379 (Tannery's C) was that used by Bachet for his edition. It was written by one Ioannes Hydruntinus after 1545, and has the peculiarity that the first two Books were copied from the MS. Vat. gr. 200 (a MS. of the Planudean class), evidently in order to include the commentary of Planudes, while the MS. Vat. gr. 304 belonging to the pre-Planudes class was followed in the remaining Books, no doubt because it was considered superior. Thus the class of which C is the chief representative is a sort of mixed class.


5, 6. Parisinus 2378 P, and Neapolitanus III C 17, were copied by Angelus Vergetius. In the latter Vergetius puts the

1 The MS. was made up of various MSS. before separated. The old table of contents has Διοφάντου ἀριθμητική· ἁρμονικὰ διάφορα. The ἁρμονικά include the Introduction to Harmony by Cleonides, but without any author's name. This fact sufficiently explains the error of Ramus in saying, Schola mathematica, Bk 1. p. 35, "Scripserat et Diophantus harmonica."

2 Dioph. 1. p. 2, 5-6.

numbers A, B, г, A, E, Z, H at the top of the pages (as we put headlines) corresponding to the different Books, implying that he regarded the tract on Polygonal Numbers as Book VII.

The other MSS. of the first class call for no notice, and we pass to the Planudean class.

9. Tannery, as he tells us, congratulated himself upon finding in Ambrosianus Et 157 sup. ten pages of the archetype of the class, and eagerly sought for new readings. So far, however, as he was able to carry his collation, he found no difference from the principal representative of the class (B1) next to be mentioned.

10. The MS. Marcianus 308 (= B1) of the 15th century formerly belonged to Cardinal Bessarion, and was seen by Regiomontanus at Venice in 1464. It contains the recension by Planudes with his commentary.

II. It seems certain that the Wolfenbüttel MS. Guelferbytanus Gudianus I (15th c.) was that which Xylander used for his translation; Tannery shows that, if this was not the MS. lent to Xylander by Andreas Dudicius Sbardellatus, that MS. must have been lost, and there is no evidence in support of the latter hypothesis. It is not possible to say whether the Wolfenbüttel MS. was copied from Marcianus 308 (B1) or from the complete MS. of which Ambrosianus Et 157 sup. preserves the ten leaves.

12. Palatinus gr. 391 (end of 16th c.) has notes in German in the margin which show that it was intended to print from it; it was written either by Xylander himself or for him. It is this MS. of which Claudius Salmasius (Claude de Saumaise, 1588-1653) told Bachet that it contained nothing more than the six Books, with the tract on Polygonal Numbers.

13. Reginensis 128 was copied at the end of the 16th century from the Wolfenbüttel MS.

14, 15. Ambrosianus A 91 sup. and Vaticanus gr. 200 both come from B1; as they agree in omitting V. 28 of Diophantus, one was copied from the other, probably the latter from the former. They were both copied by the same copyist for Mendoza in 1545. Vat. gr. 200 has headings which make eight Books; according to Tannery the first Book is numbered a', the fourth 8; before V. 20 (in Bachet's numbering)—should this be IV. 20?-is the heading Διοφάντου ε", before the fifth Book Διοφάντου 5, before the sixth Διοφάντου ζον, and before the tract on Polygonal Numbers Acopávтov nor; this wrong division occurs in the next three MSS.

H. D.


(16, 17, 18 in the diagram), all of which seem to be copied from Vat. 200.

The MSS. numbered 20, 21, 22, 23 in the diagram are of the hybrid class derived from Parisinus 2379 (C). Scorialensis -I-15 and Scorialensis R-II-3, the latter copied from the former, have the first Book divided into two (cf. p. 5 above), and so make seven Books of the Arithmetica and an eighth Book of the Polygonal Numbers.

27. The Cracow MS. has the same division into Books as the MSS. last mentioned. According to Gollob, the collation of this MS., so far as it was carried in 1899, showed that it agrees in the main with A (the best MS.), B, (Marcianus 308) and C (Parisinus 2379); but, as it contains passages not found in the two latter, it cannot have been copied from either of them.

25. Parisinus 2380 appears to be the copy of Auria's Diophantus mentioned by Schulz as having been in the library of Carl von Montchall and bearing the title "Diophanti libri sex, cum scholiis graecis Maximi Planudae, atque liber de numeris polygonis, collati cum Vaticanis codicibus, et latine versi a Josepho Auria1."

The first commentator on Diophantus of whom we hear is Hypatia, the daughter of Theon of Alexandria; she was murdered by Christian fanatics in 415 A.D. According to Suidas she wrote commentaries on Diophantus, on the Astronomical Canon (sc. of Ptolemy) and on the Conics of Apollonius. Tannery suggests that the remarks of Michael Psellus (11th c.) at the beginning of his letter about Diophantus, Anatolius, and the Egyptian method of arithmetical reckoning were taken bodily from some MS. of Diophantus containing an ancient and systematic commentary; and he believes this commentary to have been that of Hypatia. I have already mentioned the attractive hypothesis of Tannery that Hypatia's commentary extended only to our six Books, and that this accounts for the loss of the rest.

Georgius Pachymeres (1240 to about 1310) wrote in Greek a paraphrase of at least a portion of Diophantus. Sections 25-44 of

1 Schulz, Diophantus, pref. xliii.

2 Suidas s.v. Υπατία: ἔγραψεν ὑπόμνημα εἰς Διόφαντον, < εἰς> τὸν ἀστρονομικὸν κανόνα, εἰς τὰ κωνικὰ ̓Απολλωνίου ὑπόμνημα. So Tannery reads, following the best MSS. ; he gives ample reasons for rejecting Kuster's conjecture εἰς Διοφάντου τὸν ἀστρονομικὸν κανόνα, viz. (r) that the order of words would have been τὸν Διοφάντου ἀστρονομικὸν κανόνα, (2) that there is nothing connecting Diophantus with astronomy, while Suidas mentions, s.v. Θέων, a commentary εἰς τὸν Πτολεμαίου πρόχειρον κανόνα.

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