which it might have been expected to lead. In this he differs from Tannery, who says that, as Serenus' treatise on the sections of cones and cylinders was added to the mutilated Conics of Apollonius consisting of four Books only, in order to make up a convenient volume, so the tract on Polygonal Numbers was added to the remains of the Arithmetica, though forming no part of the larger work1. Thus Tannery would seem to deny the genuineness of the whole tract on Polygonal Numbers, though in his text he only signalises the portion beginning with the enunciation of the problem "Given a number, to find in how many ways it can be a polygonal number" as a "vain attempt by a commentator" to solve this problem. Hultsch, on the other hand, thinks we may conclude that Diophantus really solved the problem. He points out moreover that the beginning of the tract is like the beginning of Book I. of the Arithmetica in containing definitions and preliminary propositions. Then came the difficult problem quoted, the discussion of which breaks off in our text after a few pages; and to this it would be easy to tack on a great variety of other problems. Again, says Hultsch, the supplementary propositions added by Bachet may serve to give an approximate idea of the difficulty of the problems which were probably treated in Books VII... and the following. And between these and the bold combination of a triangular and a square number in the Cattle-Problem stretches, as Tannery says, a wide domain which was certainly not unknown to Diophantus, but was his hunting-ground for the most various problems. Whether Diophantus dealt with plane numbers, and with other figured numbers, such as prisms and tetrahedra, is uncertain.

The name of Diophantus was used, as were the names of Euclid, Archimedes and Heron in their turn, for the purpose of palming off the compilations of much later authors. Tannery prints in his edition three fragments under the head of "Diophantus Pseudepigraphus." The first, which is not "from the Arithmetic of Diophantus" as its heading states, is worth notice as containing some particulars of one of "two methods of finding the square root of any square number"; we are told to begin by writing the number "according to the arrangement of the Indian method," i.e. according to the Indian numerical notation which reached us through the Arabs. The fragment is taken from a Paris MS. * Dioph. 11. p. 3, 3-14.

:

(Supplem. gr. 387), where it follows a work with the title ̓Αρχή τῆς μεγάλης καὶ Ἰνδικῆς ψηφιφορίας (ί.ε. ψηφοφορίας), written in 1252 and raided about half a century later by Maximus Planudes. The second fragment1 is the work edited by C. Henry in 1879 as Opusculum de multiplicatione et divisione sexagesimalibus Diophanto vel Pappo attribuendum. The third, beginning with Διοφάντου ἐπιπεδομετρικά, is a compilation made in the Byzantine period out of late reproductions of the γεωμετρούμενα and στερεομετρούμενα of Heron. The second and third fragments, like the first, have nothing to do with Diophantus.

CHAPTER II

THE MSS. OF AND WRITERS ON DIOPHANTUS

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 edition1. 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, II. pp. xxii-xxxiv.

• 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:

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

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.

1. 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 ἀρξάμενος, 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 ἀρξάμενος 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