Archimedes was the most creative, the most powerful, and in many ways the most interesting of the mathematicians of the Ancient World. This is the only available English translation of his work. Ergo, every library needs a copy of this book. Anyone interested in the work of Archimedes will want it too, though they may well be scared away by the price.

I can hear the objections already. "Wait a minute! What about Heath's translation of Archimedes? That's in the MAA's Basic Library List already, and since it is a Dover book, I can even afford a copy!"

Well, Heath's edition is useful, and it has served the English-speaking world well. But consider the first proposition in *On Sphere and Cylinder.* This is what Heath gives us:

*If a polygon be cricumscribed about a circle, the perimeter of the circumscribed polygon is greater than the perimeter of the circle.*

Let any two adjacent sides, meeting in A, touch the circle at P, Q respectively.

Then [*Assumptions,* 2]

PA + AQ > (arc PQ)

A similar inequality holds for each angle of the polygon; and, by addition, the required result follows.

And here is Netz's translation:

If a polygon is circumscribed around a circle, the perimeter of the circumscribed polygon is greater than the perimeter of the circle.

For let a polygon — the one set down — be circumscribed around a circle. I say that the perimeter of the polygon is greater than the perimeter of the circle.

For since BAΛ taken together is greater than the circumference BΛ through its containing the circumference while having the same limits, similarly ΔΓ, ΓB taken together are ΔB as well; and ΛK, KΘ taken together than ΛΘ; and ZHΘ taken together than ZΘ; and once more, ΔE, EZ taken together than ΔZ; therefore the whole perimeter of the polygon is greater than the circumference of the circle.

In sum, Heath tells us what (he thinks) Archimedes *meant*, but feels free to modernize notation and shorten the text. Netz gives us what Archimedes *wrote.*

Does it matter? Well, it depends what we are trying to do. If we are interested, for example, in how Archimedes dealt with generality, it seems very significant that he worked with a specific polygon (a pentagon, in fact), enumerating its sides one by one!

Proposition 1 is probably the easiest one in this book; two things should then be noted. First, the difference between Netz's literal translation and Heath's paraphrase gets much bigger as the complexity of the arguments increases. Second, Netz is considerably harder to read, parse, and absorb.

I think it's worth the effort. Reading Archimedes in Netz's translation, one feels much more clearly how different Greek mathematics is from modern mathematics. Rather than "a fellow of another college", Archimedes is revealed as an inhabitant of Ancient Syracuse working within the Ancient Greek mathematical tradition. We can understand and admire him, but we also understand how different he is from us.

In addition, Netz gives us useful extras. He discusses the diagrams as they appear in the textual tradition, noting in particular their variation. (The diagram in Heath is nothing like the diagram in the manuscripts, it seems.) He also gives us a translation of Eutocius' commentary on these two books, which provides insight on how Archimedes was read and understood (or not) a few centuries later. Finally, Netz's notes are interesting and different, focusing less on the mathematics and more on Archimedes' thought processes, mode of expression, and goals.

What is missing? Well, the most obvious thing is that this is only the first volume of Netz's translation. We'll have to wait for the rest. In addition, Heath's edition is prefaced by a long introduction discussing Archimedes' life and work. I hope Netz will undertake that eventually, perhaps after he finishes the translation itself.

Netz gives us only the English text. I would have liked (especially given the price) to have the Greek too. Netz says that he is mostly using Heiberg's text as published by Teubner (when he deviates from that, he tells us). Unfortunately, that edition is not very easy to obtain. Perhaps once the complete translation is done we can ask Cambridge to produce a version with the Greek text and facing translation, perhaps without all the notes, for weird folks like me.

For anyone seriously interested in Archimedes and in Greek mathematics, this is the edition to have. Have your library buy them one by one, and the financial pain will be less. And keep your eyes open for the other volumes.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College, editor of MAA Reviews, and crazy about books. It has taken him three years to write this review, in part because he wanted to read the book so carefully.