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 Aritkmetz'ca 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.