Perhaps the ultimate acknowledgement that a work of classical antiquity is truly impossible to read is the provision of a facing translation in the Teubner text. In the case of Archimedes, probably the most famous of ancient mathematicians, the distinction is certainly well deserved, and there is considerable courage involved in any attempt to translate this difficult, elliptical, and interpolated (not to mention highly technical) set of writings.

The book under review, a translation of the two books *On the Sphere and the Cylinder*, is an example of particular courage well applied: it is only the first volume of a multi-volume translation project intended to cover all the works of Archimedes included in the standard Greek edition (Teubner, ed. J. L. Heiberg, 2nd edn 1910-15). The volume includes not only a translation but also extensive commentary, as well as a translation of the important ancient commentary by Eutocius of Ascalon, notes on that commentary, and a critical edition of the diagrams that accompany both texts in the manuscript tradition. The work is of high quality and will undoubtedly remain an important one for years to come — though perhaps less because of the translation itself than because of the accompanying material.

The translation itself is probably the best ever done in terms of faithfulness to the text and to Archimedes’ own way of thinking. It is based not only upon the best available Greek text but also upon re-examination of manuscripts, including a palimpsest that had been lost for almost all of the twentieth century. Netz has re-thought many of Heiberg’s editorial decisions and discusses his thoughts at length in the commentary (not only in places where he questions Heiberg’s choices, but also often where he agrees or is unsure), so that at times one is almost tempted to treat the translation as a critical edition in its own right.

This temptation is deliberate on the part of the author, whose stated goal is to produce “a reliable translation that may serve as basis for scholarly comment” (p. 3). It may be doubted whether true scholarly comment can ever be based on anything other than the original text, but if such use is possible for any translation, it is possible for this one.

The corollary of the translation’s fidelity to the original, however, is that it is almost completely unreadable. Of course, geometrical proofs are not considered compelling reading by non-mathematicians at the best of times, but most people with a reasonable background (say, that provided in a year of high school geometry) can get through most of the works of Archimedes as presented in a number of other translations, which use modern mathematical notation. No amount of training in modern mathematics, however, will suffice to get a reader through this translation, which presents Greek geometry as it really was: a very different way of thinking from our mathematics.

The choice to produce so difficult a translation was a very conscious one. Netz comments that “the purpose of a scholarly translation as I understand it is to remove all barriers having to do with the foreign language itself, leaving all other barriers intact” (p. 3). Moreover, his interests ^{1} lie precisely in the differences between ancient mathematical thought and our own, and these issues are extensively explored in the commentary, so that at times the translation seems to be almost filling the role of a vehicle on which to hang the commentary. And the commentary is not only readable, but positively fascinating. It discusses exactly how Archimedes constructed his argument, which portions of it are probably not his own but later interpolations, how the community of mathematicians functioned in the time of Archimedes, how Archimedes’ arguments move from specific cases to general principles (or fail to do so), how the text interacts with the diagrams, and many equally intriguing topics. Almost the only thing the commentary does not do, in fact, is to explain in terms intelligible to a reader trained in modern mathematical notation what is being said in the corresponding section of the text.

The commentary is not presented in the line-by-line or lemma-by-lemma format usual for classical scholarship, but in large chunks interspersed with the translation (two chunks, one on textual matters and one interpretive, after each theorem) and, for smaller points, in the form of footnotes to the translation. This format allows the commentary to be much more readable and discursive than a normal commentary; there is nothing constrained or compact about it, and matters of editorial choice that would normally be swept under the carpet are made gloriously explicit. In general this unconstrained quality will probably be welcome, both to classicists, who will appreciate asides like “in Greek mathematics, you cannot step into the same diagram twice” (p. 137) and self-aware qualifications like “Needless to say, had Heiberg bracketed Steps 26-43 I would probably have found something nice to say about them” (pp. 196-7, following an argument against the authenticity of those steps), and to non-classicists, who will I think be particularly pleased to find textual problems discussed in normal prose rather than in the code of an apparatus criticus. At the same time, the expansiveness seems inseparable from some speculative tendencies that may be less universally applauded. A not insignificant proportion of the comments contain suggestions about questions such as Archimedes’ feelings which, while interesting, are so unsusceptible of demonstration as to be normally excluded from the domain of scholarship.

Occasionally such inferences about Archimedes himself, or even other questions, are based on re-interpretations of manuscript evidence with which classicists may not be entirely comfortable. For example, when there are signs of haste or abbreviation in our versions of an ancient text, scholars studying other ancient technical works frequently believe that those works were originally fuller but have been abbreviated in transmission while Netz normally takes any evidence of haste as evidence of Archimedes’ own haste. Perhaps Archimedes’ works are different from those of other ancient writers, and Netz is correct in his tacit assumption that deliberate additions were far more likely than deliberate subtractions in the transmission of this text: but one would feel more comfortable if he argued this principle rather than assuming it. This is true especially on points such as the one where Netz suggests that a certain reading may be an error due to the presence in the archetype of a particular abbreviation for

One of the most original and most interesting contributions of Netz’s work is a critical edition of the diagrams that accompany Archimedes’ text. Netz claims, with a good deal of plausibility, that the diagrams go back to Archimedes himself, and he has carefully documented the various forms in which each appears in the different manuscripts. Many previous translations of Archimedes have simply redrawn the diagrams to make them adhere to the modern conventions of diagram construction, and even Heiberg was less scrupulous about the diagrams than about the Greek text. Netz is truly a pioneer in his interest in the ancient conventions of mathematical diagrams, and his reconstructions and commentary on them are particularly useful — not to mention fascinating.

The translation of Eutocius’ commentaries is another particularly valuable aspect of Netz’s work. The commentaries are extremely important, not only because they help us understand what Archimedes actually wrote and what he meant by it (for example, a significant amount of Archimedes’ text has been lost from the direct manuscript tradition and is preserved only in Eutocius’ version), but also because they reveal a great deal about how the ancients understood and used the works of Archimedes and because they preserve substantial amounts of the work of other ancient mathematicians whose writings are now otherwise lost. For this reason Eutocius’ commentaries are included in Heiberg’s edition of Archimedes, but they are omitted from most previous translations, thus depriving monolingual English speakers of an important aid to understanding Archimedes. The inclusion of the commentaries in this translation thus gives it a significant advantage over others. Netz also provides notes to Eutocius; though these are much less extensive than his commentary on Archimedes, they are also very valuable, especially given how little work has been done on Eutocius. A critical edition of Eutocius’ diagrams is similarly important.

This first volume of Netz’s translation contains a work of Archimedes that is already available in two other English versions: T. L. Heath, *The Works of Archimedes* (Cambridge 1897) and E. J. Dijksterhuis’ Dutch version translated into English by C. Dikshoorn as *Archimedes* (Copenhagen 1956). Both these versions are very free renditions — Dijksterhuis’ work in particular is a retelling rather than a translation — and convert all the mathematics into notation more intelligible to the modern reader; in addition, Heath’s version is based on an inferior Greek text. Thus Netz’s claim to be producing the first English translation of Archimedes (p. 2), while exaggerated to some extent, is not without some justification.

There are already, however, complete translations in several other commonly-read languages. Particularly notable are the French translation in the Budé text (by C. Mugler, 1970-2) and the Latin one in the Teubner (by Heiberg as above). Mugler’s version is a literal translation closely resembling Netz’s in many ways, while Heiberg’s sticks closely to the Greek in many places but converts the mathematics into modern notation when necessary for modern comprehension. Both these translations also include translations of Eutocius’ commentaries. Thus Netz’s translation fills a gap that, while significant, really existed only for those unable to read French, while his notes and diagrams meet a much wider need.

All five translations will continue to be useful, but for different purposes. Most readers need the help provided by the modern notation in Heath, Dijksterhuis, and Heiberg and will probably prefer those translations to that of Netz, though they may well want to use Netz’s commentary. Those who know some Greek and are trying to learn to read and understand Archimedes in the original will probably continue to use Heiberg’s version as their primary aid in this task, though again such readers may want to use Netz’s commentary. But readers who do not know Greek and who are curious to know not so much what Archimedes said but how he said it, how an ancient mathematician’s mind worked, and how ancient geometry really functioned are the ones who will really want to use Netz’s translation.

The book is on the whole well produced, and there are not many typographical errors.^{2} All symbols and conventions are explained with admirable precision and detail in the introduction, and unnecessary jargon is avoided. The English style, however, suffers from occasional lapses, some of which make particular passages unclear or difficult to read.^{3}

In general, this work is an important, interesting, and very welcome contribution to the field of ancient mathematics. Though it will not replace the earlier translations, it is a significant addition to the resources available for understanding Archimedes, and the high level of scholarship involved in its creation means that it will be a valuable resource even for those few scholars who do not “need” a translation to read Archimedes.

**Notes**

1. See for example Netz’s previous book, *The Shaping of Deduction in Greek Mathematics*, Cambridge 1999.

2. A few that could cause confusion are “circles” for “circles’ ” (p. 77), “it” for “is” (p. 88), and “nowhere” divided “now-here” (p. 107).

3. For example (none of these is from the translation itself): “It appears, that neither of the two.” (as a complete sentence, p. 94); “The introductory section is difficult to entangle, in that it moves from theme to theme, in a non-linear direction …” (p. 20); “Writing was crucial to Archimedes’ intellectual life who, living in Syracuse, seems …” (p. 13); “Perhaps his argument ran like: all the triangles have their two sides equal …” (p. 59).