mr. Marmillaw 74,00 Math. 51 7438D THE PREFACE HE first edition of this book, which was the first English. Diophantus, appeared in 1885, and has long been out of print. Inquiries made for it at different times suggested to me that it was a pity that a treatise so unique and in many respects so attractive as the Arithmetica should once more have become practically inaccessible to the English reader. At the same time I could not but recognise that, after twenty-five years in which so much has been done for the history of mathematics, the book needed to be brought up to date. Some matters which in 1885 were still subject of controversy, such as the date of Diophantus, may be regarded as settled, and some points which then had to be laboured can now be dismissed more briefly. Practically the whole of the Introduction, except the chapters on the editions of Diophantus, his methods of solution, and the porisms and other assumptions found in his work, has been entirely rewritten and much shortened, while the chapters on the methods and on the porisms etc., have been made fuller than before. The new text of Tannery (Teubner 1893, 1895) has enabled a number of obscure passages, particularly in Books V and VI, to be cleared up and, as a basis for a reproduction of the whole work, is much superior to the text of Bachet. I have taken the opportunity to make my version of the actual treatise somewhat fuller and somewhat closer to the language of the original. In other respects also I thought I could improve upon a youthful work which was my first essay in the history of Greek mathematics. When writing it I was solely concerned to make Diophantus himself known to mathematicians, 81171 and I did not pay sufficient attention to Fermat's notes on the various problems. It is well known that it is in these notes that many of the great propositions discovered by Fermat in the theory of numbers are enshrined; but, although the notes are literally translated in Wertheim's edition, they do not seem to have appeared in English; moreover they need to be supplemented by passages from the correspondence of Fermat and from the Doctrinae analyticae Inventum Novum of Jacques de Billy. The histories of mathematics furnish only a very inadequate description of Fermat's work, and it seemed desirable to attempt to give as full an account of his theorems and problems in or connected with Diophantine analysis as it is possible to compile from the scattered material available in Tannery and Henry's edition of the Oeuvres de Fermat (1891-1896). So much of this material as could not be conveniently given in the notes to particular problems of Diophantus I have put together in the Supplement, which is thus intended to supply a missing chapter in the history of mathematics. Lastly, in order to make the book more complete, I thought it right to add some of the more remarkable solutions of difficult Diophantine problems given by Euler, for whom such problems had a great fascination; the last section of the Supplement is therefore devoted to these solutions. T. L. H. October, 1910. INTRODUCTION CHAPTER I DIOPHANTUS AND HIS WORKS THE divergences between writers on Diophantus used to begin, as Cossali said', with the last syllable of his flame. There is now, however, no longer any doubt that the name was Diophantos, not Diophantes'. The question of his date is more difficult. Abū'lfaraj, the Arabian historian, in his History of the Dynasties, places Diophantus under the Emperor Julian (A.D. 361-3), but without giving any authority; and it may be that the statement is due simply to a confusion of our Diophantus with a rhetorician of that name, mentioned in another article of Suidas, who lived in the time of Julian'. On the other hand, Rafael Bombelli in his Algebra, 1 Cossali, Origine, trasporto in Italia, primi progressi in essa dell' Algebra (Parma, 1797–9), I. p. 61: "Su la desinenza del nome comincia la diversità tra gli scrittori." 2 Greek authority is overwhelmingly in favour of Diophantos. The following is the evidence, which is collected in the second volume of Tannery's edition of Diophantus (henceforward to be quoted as "Dioph.," "Dioph. 11. p. 36" indicating page 36 of Vol. II., while "Dioph. 11. 20" will mean proposition 20 of Book 11.): Suidas s. v. 'Trarla (Dioph. 11. p. 36), Theon of Alexandria, on Ptolemy's Syntaxis Book 1. c. 9 (Dioph. 11. p. 35), Anthology, Epigram on Diophantus (Ep. XIV. 126; Dioph. 11. p. 60), Anonymi prolegomena in Introductionem arithmeticam Nicomachi (Dioph. 11. p. 73), Georgii Pachymerae paraphrasis (Dioph. 11. p. 122), Scholia of Maximus Planudes (Dioph. 11. pp. 148, 177, 178 etc.), Scholium on lamblichus In Nicomachi arithm. introd., ed. Pistelli, p. 127 (Dioph. 11. p. 72), a Scholium on Dioph. 11. 8 from the MS. “A” (Dioph. It. p. 26ο), which is otherwise amusing (Η ψυχή σου, Διόφαντε, εἴη μετὰ τοῦ Σατανᾶ ἕνεκα τῆς δυσκολίας τῶν τε ἄλλων σου θεωρημάτων καὶ δὴ καὶ τοῦ παρόντος θεωρή Maros, "Your soul to perdition, Diophantus, for the difficulty of your problems in general and of this one in particular "); John of Jerusalem (10th c.) alone (Vita Ioannis Damasceni XI. Dioph. 11. p. 36), if the reading of the MS. Parisinus 1559 is right, wrote, in the plural, ὡς Πυθαγόραι ἢ Διόφανται, where however Διόφανται is clearly a mistake for Διόφαντοι. 3 Λιβάνιος, σοφιστὴς ̓Αντιοχεύς, τῶν ἐπὶ Ἰουλιανοῦ τοῦ βασιλέως χρόνων, καὶ μέχρι Θεοδοσίου τοῦ πρεσβυτέρου· Φασγανίου πατρός, μαθητὴς Διοφάντου. H. D. I |