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It should be in the public domain (obviously), so I'd thought I could find the English text on the web somewhere. Apparently not?

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    There is: http://www.archive.org/details/diophantusofalex00heatiala for *Diophantus of Alexandria; a study in the history of Greek algebra* by Sir Thomas L Heath (1910).2011-08-24
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    Look at [this text by Norbert Schappacher](http://www-irma.u-strasbg.fr/~schappa/NSch/Publications_files/Dioph.pdf) for some interesting history.2011-08-24
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    @Didier, that comment could be an answer.2011-08-24
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    The fact that Diophantus' writings are in the public domain does not automatically mean that a translation into English would be in the public domain.2011-08-24
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    @Didier, that's exactly what I was looking for. Thanks!2011-08-24
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    @Theo: I included your comment in my answer since I fully concur with your suggestion to look at this paper by Norbert. Tell me if this is a problem...2011-08-24
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    @Didier: Thanks for notifying me. As you probably suspected, I have no problem with that whatsoever, on the contrary!2011-08-24
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    If you want a physical copy, some print-on-demand companies offer copies of the Heath book (e.g. on amazon) for not too much money. There is a Dover edition too: ISBN 1443730238.2011-08-24
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    In Steven Hawking's "God Made the Integers" he has books II, III, and V, in English.2014-12-26
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    A [consolidated answer](https://math.stackexchange.com/a/2698252/34365) with history and sources for 10 of the 13 books of Diophantus Arithmetica is [below](https://math.stackexchange.com/a/2698252/34365)2018-03-19

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Context

Diophantus' Arithmetica consists of 13 books written in Greek in ~270 CE (the dates vary by ~500 years from 70AD to ~500AD). The original Greek text is lost to us. The earliest surviving texts are from copies made in Arabic and transmitted through the Islamic world, before returning to Western Europe in the 1600s through translations into Greek and Latin.

In 1621 CE, Bachet published in Latin the standard Diophantus which Fermat read and annotated with his observations. This makes available 6 of the 13 books. (see Translations below for Download)

In 1968, an Arabic text was discovered in Iran containing Books 4-7 of the Arithmetica.

The modern view of the Arithmetica is therefore that we have available 10 of the 13 original books: Books 1-3 from the Greek/Latin text, Books 4-7 from the Arabic text, Books "4-6" from the remaining books 8-13 but without knowing which is which, and 3 books still lost.

Translations to English

The English translation of the Bachet text is by Thomas Heath (1910), available freely here: https://archive.org/details/diophantusofalex00heatiala It has Books 1-3 & "4-6".

The English translation (1982) by Jacques Sesiasno of Books 4-7 can be found here: https://www.amazon.com/dp/0387906908/

Historical References

The history of Diophantus's Arithmetica here: www-irma.u-strasbg.fr/~schappa/NSch/Publications_files/1998cBis_Dioph.pdf

A review of Sesiano's translation, with its history, is here: http://www.jphogendijk.nl/reviews/sesiano.html

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There is one such translation (freely available), included in the book Diophantus of Alexandria; a study in the history of Greek algebra by Sir Thomas L. Heath (1910).

For some interesting history, user @t.b. recommended (and I fully concur) to look at the paper Diophantus of Alexandria: a text and its history (2005) by Norbert Schappacher (this paper is freely available as well).

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This link says there exists a book, but the cost seems a bit high.

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    This is a translation into English by Sesiano of an Arabic translation of what may be some of the "lost" books of *Arithmetica*. (The manuscript was found not that many years ago in a shrine library.) The text is probably substantially modified from the original. Any decent university library will have it. Yes, the cost is high, but the profit margin is undoubtedly much less than on a routine calculus book.2011-08-24