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Solari. MDCXXI. Bachet's Greek text is based, as he tells us, upon a MS. which he calls "codex Regius," now in the Bibliothèque Nationale at Paris (Parisinus 2379); this MS. is his sole authority, except that Jacobus Sirmondus had part of a Vatican MS. (Vat. gr. 304) transcribed for him. He professes to have produced a good Greek text, having spent incalculable labour upon its emendation, to have inserted in brackets all additions which he made to it, and to have given notice of all corrections, except those of an obvious or trifling nature; a few passages he has left asterisked, in cases where correction could not be safely ventured upon. He is careful to tell us what previous works relating to the subject he had been able to consult. First he mentions Xylander (he spells the name as Xilander throughout), who had translated the whole of Diophantus, and commented upon him throughout, "except that he scarcely touched a considerable part of the fifth book, the whole of the sixth and the treatise on multangular numbers, and even the rest of his work was not very successful, as he himself admits that he did not thoroughly understand a number of points." Then he speaks of Bombelli (as already mentioned) and of the Zetetica of Vieta (in which the author treats in his own way a large number of Diophantus' problems: Bachet thinks that he so treated them because he despaired of restoring the book completely). Neither Bombelli nor Vieta (says Bachet) made any attempt to demonstrate the difficult porisms and abstruse theorems in numbers which Diophantus assumes as known in many places, or sufficiently explained the causes of his operations and artifices. All these omissions on the part of his predecessors he thinks he has supplied in his notes to the various problems and in the three books of "Porisms" which he prefixed to the work'. As regards his Latin translation, he says that he gives us Diophantus in Latin from the version of Xylander most carefully corrected, in which he would have us know that he has done two things in particular, first,

Cramoisy, via Jacobaea, sub Ciconiis." The copy (from the Library of Trinity College, Cambridge) which I used in preparing my first edition has the former words; a copy in the Library of the Athenaeum Club has the latter.

On the nature of some of Bachet's proofs Nicholas Saunderson (formerly Lucasian Professor) remarks in Elements of Algebra, 1740, àpropos of Dioph. 111. 15: "M. Bachet indeed in the 16th and 17th props. of his second book of Porisms has given us demonstrations, such as they are, of the theorems in the problem: but in the first place he demonstrates but one single case of those theorems, and in the next place the demonstrations he gives are only synthetical, and so abominably perplexed withal, that in each demonstration he makes use of all the letters in the alphabet except I and O, singly to represent the quantities he has there occasion for."

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corrected what was wrong and filled the numerous lacunae, secondly, explained more clearly what Xylander had given in obscure or ambiguous language; "I confess however," he says, "that this made so much change necessary, that it is almost fairer to attribute the translation to me than to Xilander. But if anyone prefers to consider it as his, because I have held fast, tooth and nail, to his words when they do not misrepresent Diophantus, I have no objection." Such sentences as these, which are no rarity in Bachet's book, are certainly not calculated to increase our respect for the author. According to Montucla", "the historian of the French Academy tells us" that Bachet worked at this edition during the course of a quartan fever, and that he himself said that, disheartened as he was by the difficulty of the work, he would never have completed it, had it not been for the stubbornness which his malady generated in him.

As the first edition of the Greek text of Diophantus, this work, in spite of any imperfections we may find in it, does its author all honour.

The same edition was reprinted and published with the addition of Fermat's notes in 1670: Diophanti Alexandrini Arithmeticorum libri sex, et de numeris multangulis liber unus. Cum commentariis C. G. Bacheti V.C. et obseruationibus D. P. de Fermat Senatoris Tolosani. Accessit Doctrinae Analyticae inuentum nouum, collectum ex variis eiusdem D. de Fermat Epistolis. Tolosae, Excudebat Bernardus Bosc, è Regione Collegii Societatis Jesu. MDCLXX. This edition was not published by Fermat himself, but by his son after his death. S. Fermat tells us in the preface that this publication of Fermat's notes to Diophantus was part of an attempt to collect together from his letters and elsewhere his contributions to mathematics. The "Doctrinae Analyticae Inuentum nouum" is a collection made by Jacobus de Billy'

1 Deinde Latinum damus tibi Diophantum ex Xilandri versione accuratissimè castigata, in qua duo potissimum nos praestitisse scias velim, nam et deprauata correximus, hiantesque passim lacunas repleuimus: et quae subobscurè, vel ambiguè fuerat interpretatus Xilander, dilucidius exposuimus; fateor tamen, inde tantam inductam esse mutationem, vt propemodum aequius sit versionem istam nobis quàm Xilandro tribuere. Si quis autem potius ad eum pertinere contendat, quòd eius verba, quatenus Diophanto fraudi non erant, mordicus retinuimus, per me licet." 2 1.323.

Now published in Euvres de Fermat by P. Tannery and C. Henry, Vol. 1. (1891), . pp. 289-342 (the Latin original), and Vol. 111. (1896), pp. 241-274 (French translation).

Now published in Œuvres de Fermal, 111. 323-398 (French translation). De Billy had already published in 1660 a book under the title Diophantus geometra sive opus contextum ex arithmetica et geometria.

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from various letters which Fermat sent to him at different times. The notes upon Diophantus' problems, which his son hopes will prove of value very much more than commensurate with their bulk, were (he says) collected from the margin of his copy of Diophantus. From their brevity they were obviously intended for the benefit of experts', or even perhaps solely for Fermat's own, he being a man who preferred the pleasure which he had in the work itself to any reputation which it might bring him. Fermat never cared to publish his investigations, but was always perfectly ready, as we see from his letters, to acquaint his friends and contemporaries with his results. Of the notes themselves this is not the place to speak in detail. This edition of Diophantus is rendered valuable only by the additions in it due to Fermat; for the rest it is a mere reprint of that of 1621. So far as the Greek text is concerned, it is very much inferior to the first edition. There is a far greater number of misprints, omissions of words, confusions of numerals; and, most serious of all, the brackets which Bachet inserted in the edition of 1621 to mark the insertion of words in the text are in this later edition altogether omitted. These imperfections have been already noticed by Nesselmann'. Thus the reprinted edition of 1670 is untrustworthy as regards the text.

In 1535 Simon Stevin published a French version of the first four books of Diophantus'. It was based on Xylander and was a free reproduction, not a translation, Stevin himself observing that the MS. used by Xylander was so full of mistakes that the text of

1 Lectori Beneuolo, p. iii: "Doctis tantum quibus pauca sufficiunt, harum obseruationum auctor scribebat, vel potius ipse sibi scribens, his studiis exerceri malebat quam gloriari; adeo autem ille ab omni ostentatione alienus erat, vt nec lucubrationes suas typis mandari curauerit, et suorum quandoque responsorum autographa nullo seruato exemplari petentibus vltrò miserit; norunt scilicet plerique celeberrimorum huius saeculi Geometrarum, quam libenter ille et quantâ humanitate, sua iis inuenta patefecerit."

"Was dieser Abdruck an äusserer Eleganz gewonnen hat (denn die Bachet'sche Ausgabe ist mit äusserst unangenehmen, namentlich Griechischen Lettern gedruckt), das hat sie an innerm Werthe in Bezug auf den Text verloren. Sie ist nicht bloss voller Druckfehler in einzelnen Worten und Zeichen (z. B. durchgehends statt, 900) sondern auch ganze Zeilen sind ausgelassen oder doppelt gedruckt (z. B. III. 12 eine Zeile doppelt, IV. 25 eine doppelt und gleich hinterher eine ausgelassen, IV. 52 eine doppelt, v. 11 eine ausgelassen, desgleichen v. 14, 25, 33, VI. 8, 13 und so weiter), die Zahlen verstümmelt, was aber das Aergste ist, die Bachet'schen kritischen Zeichen sind fast überall, die Klammer durchgängig weggefallen, so dass diese Ausgabe als Text des Diophant völlig unbrauchbar geworden ist," p. 283.

* Included in L'Arithmetique de Simon Stevin de Bruges...A Leyde, De l'Imprimerie de Christophle Plantin, CIɔ. 1Ɔ . LXXXV.

Diophantus could not be given word for word'. Albert Girard added the fifth and sixth books to the four, and this complete version appeared in 1625.

In 1810 was published an excellent translation (with additions) of the fragment upon Polygonal Numbers by Poselger: Diophantus von Alexandrien über die Polygonal-Zahlen. Uebersetzt mit Zusätzen von F. Th. Poselger. Leipzig, 1810.

In 1822 Otto Schulz, professor in Berlin, published a very meritorious German translation with notes: Diophantus von Alexandria arithmetische Aufgaben nebst dessen Schrift über die Polygon-Zahlen. Aus dem Griechischen übersetzt und mit Anmerkungen begleitet von Otto Schulz, Professor am BerlinischCölnischen Gymnasium zum grauen Kloster. Berlin, 1822. In der Schlesingerschen Buch- und Musikhandlung. The work of Poselger just mentioned was with the consent of its author incorporated in Schulz's edition along with his own translation and notes upon the larger treatise, the Arithmetica. According to Nesselmann Schulz was not a mathematician by profession; he produced, however, a thoroughly useful edition, with notes chiefly upon the matter of Diophantus and not on the text (with the exception of a very few emendations): notes which, almost invariably correct, help much to understand the author. Schulz's translation is based upon the edition of Bachet's text published in 1670.

Another German translation was published by G. Wertheim in 1890: Die Arithmetik und die Schrift über Polygonalzahlen des Diophantus von Alexandria. Übersetzt und mit Anmerkungen begleitet von G. Wertheim (Teubner). Though it appeared before the issue of Tannery's definitive text, it is an excellent translation, the translator being thoroughly equipped for his task; it is valuable also as containing Fermat's notes, also translated into German, with a large number of other notes by the translator clucidating both Diophantus and Fermat, and generalising a number of the problems which, with very few exceptions, receive only particular solutions from Diophantus himself. Wertheim has also included 46 epigramproblems from the Greck anthology and the enunciation of the famous Cattle-Problem attributed to Archimedes.

1 See Bibliotheca Mathematica VII,, 1906-7, p. 59.

2 L'Arithmetique de Simon Stevin de Bruges, Reueuë, corrigee & augmentee de plusieurs traictez et annotation par Albert Girard Samielois Mathematicien. A Leide, de l'Imprimerie des Elzeviers CI5.15.cxxv. Reproduced in the edition of Les Euvres Mathematiques de Simon Stevin de Bruges. Par Albert Girard. Leyde, C1ɔ. 1ɔ . CXXXIV.

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No description is necessary of the latest edition, by Tannery, in which we at last have a definitive Greek text of Diophantus with the ancient commentaries, etc., Diophanti Alexandrini opera omnia cum Graecis commentariis. Edidit et Latine interpretatus est Paulus Tannery (Teubner). The first volume (1893) contains the text of Diophantus, the second (1895) the Pseudepigrapha, Testimonia veterum, Pachymeres' paraphrase, Planudes' commentary, various ancient scholia, etc., and 38 arithmetical epigrams in the original Greek with scholia. Any further edition will necessarily be based on Tannery, who has added all that is required in the shape of introductions, etc.

Lastly we hear of other works on Diophantus which, if they were ever written, are lost or remain unpublished. First, we find it asserted by Vossius (as some have understood him) that the Englishman John Pell wrote an unpublished Commentary upon Diophantus. John Pell (1611-1685) was at one time professor of mathematics at Amsterdam and gave lectures there on Diophantus, but what Vossius says about his commentary may well be only a recommendation to undertake a commentary, rather than a historical assertion of its completion. Secondly, Schulz states in his preface that he had lately found a note in Schmeisser's Orthodidaktik der Mathematik that Hofrath Kausler by command of the Russian Academy prepared an edition of Diophantus'. This seems however to be a misapprehension on the part of Schulz. Kausler is probably referring, not to a translation of Diophantus, but to his memoir of 1798 published in Nova Acta Acad. Petropol. XI. p. 125, which might easily be described as an Ausarbeitung of Diophantus' work.

I find a statement in the New American Cyclopaedia (New York, D. Appleton and Company), Vol. VI., that "a complete translation of his (Diophantus') works into English was made by the late Miss Abigail Lousada, but has not been published."

1 The whole passage of Schmeisser is: "Die mechanische, geistlose Behandlung der Algebra ist ins besondere von Herrn Hofrath Kausler stark gerligt worden. In der Vorrede zu seiner Ausgabe des Uflakerschen Exempelbuchs beginnt er so: 'Seit mehreren Jahren arbeitete ich für die Russisch-Kaiserliche Akademie der Wissenschaften Diophants unsterbliches Werk über die Arithmetik aus, und fand darin einen solchen Schatz von den feinsten, scharfsinnigsten algebraischen Auflösungen, dass mir die mechanische, geistlose Methode der neuen Algebra mit jedem Tage mehr ekelte u.s.w.'" (p. 33).

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