BookReview Netz WorksofArchimedes-1

7/28/2019 BookReview Netz WorksofArchimedes-1

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Bryn Mawr Classical Review 2004.07.14

Reviel Netz, The Works of Archimedes: Translated into English, togetherwith Eutocius' Commentaries, with Commentary, and Critical Edition ofthe Diagrams. Vol. 1: The Two Books On the Sphere and the Cylinder.Cambridge: Cambridge University Press, 2004. Pp. x, 375. ISBN 0-521-66160-9. $125.00.

Reviewed by Eleanor Dickey, Columbia University ([email protected])Word count: 1954 words

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

The book under review, a translation of the two books On the Sphere and theCylinder, is an example of particular courage well applied: it is only the first volumeof 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 asa translation of the important ancient commentary by Eutocius of Ascalon, notes onthat commentary, and a critical edition of the diagrams that accompany both texts inthe manuscript tradition. The work is of high quality and will undoubtedly remain animportant one for years to come -- though perhaps less because of the translation itselfthan because of the accompanying material.

The translation itself is probably the best ever done in terms of faithfulness to the textand to Archimedes' own way of thinking. It is based not only upon the best availableGreek 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 ofHeiberg'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 heagrees or is unsure), so that at times one is almost tempted to treat the translation as acritical 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 bedoubted whether true scholarly comment can ever be based on anything other than theoriginal 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 almostcompletely unreadable. Of course, geometrical proofs are not considered compellingreading by non-mathematicians at the best of times, but most people with a

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reasonable background (say, that provided in a year of high school geometry) can getthrough most of the works of Archimedes as presented in a number of othertranslations, which use modern mathematical notation. No amount of training inmodern 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. Netzcomments that "the purpose of a scholarly translation as I understand it is to removeall 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 ancientmathematical thought and our own, and these issues are extensively explored in thecommentary, so that at times the translation seems to be almost filling the role of avehicle on which to hang the commentary. And the commentary is not only readable,

but positively fascinating. It discusses exactly how Archimedes constructed hisargument, which portions of it are probably not his own but later interpolations, how

the community of mathematicians functioned in the time of Archimedes, howArchimedes' arguments move from specific cases to general principles (or fail to doso), 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 termsintelligible to a reader trained in modern mathematical notation what is being said inthe corresponding section of the text.

The commentary is not presented in the line-by-line or lemma-by-lemma formatusual for classical scholarship, but in large chunks interspersed with the translation(two chunks, one on textual matters and one interpretive, after each theorem) and, forsmaller 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 thatwould normally be swept under the carpet are made gloriously explicit. In generalthis unconstrained quality will probably be welcome, both to classicists, who willappreciate asides like "in Greek mathematics, you cannot step into the same diagramtwice" (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 aboutthem" (pp. 196-7, following an argument against the authenticity of those steps), andto non-classicists, who will I think be particularly pleased to find textual problemsdiscussed in normal prose rather than in the code of an apparatus criticus. At the sametime, the expansiveness seems inseparable from some speculative tendencies that

may be less universally applauded. A not insignificant proportion of the commentscontain suggestions about questions such as Archimedes' feelings which, whileinteresting, are so unsusceptible of demonstration as to be normally excluded fromthe domain of scholarship.

Occasionally such inferences about Archimedes himself, or even other questions, arebased on re-interpretations of manuscript evidence with which classicists may not beentirely comfortable. For example, when there are signs of haste or abbreviation inour versions of an ancient text, scholars studying other ancient technical worksfrequently believe that those works were originally fuller but have been abbreviatedin transmission while Netz normally takes any evidence of haste as evidence ofArchimedes' own haste. Perhaps Archimedes' works are different from those of otherancient writers, and Netz is correct in his tacit assumption that deliberate additionswere far more likely than deliberate subtractions in the transmission of this text: but




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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 certainreading may be an error due to the presence in the archetype of a particularabbreviation for and then attributes that abbreviation to Archimedes' own haste,without discussing whether the abbreviation in question (which he characterizes as"very common shorthand," referring the reader to a mention of its occurrence in theByzantine period) was ever used in ancient times (p. 182). Similar discomfort occurswhen Netz suggests that a sentence beginning in the manuscripts may havedeveloped from an original because a temporary misaccentuation of asinterrogative led to change to interrogative (p. 75) -- despite the fact that is far less likely than to follow either a relative or an interrogative pronoun.

One of the most original and most interesting contributions of Netz's work is a criticaledition of the diagrams that accompany Archimedes' text. Netz claims, with a gooddeal of plausibility, that the diagrams go back to Archimedes himself, and he hascarefully documented the various forms in which each appears in the differentmanuscripts. 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 mathematicaldiagrams, and his reconstructions and commentary on them are particularly useful --not to mention fascinating.

The translation of Eutocius' commentaries is another particularly valuable aspect ofNetz's work. The commentaries are extremely important, not only because they helpus 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 manuscripttradition 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 andbecause they preserve substantial amounts of the work of other ancientmathematicians whose writings are now otherwise lost. For this reason Eutocius'commentaries are included in Heiberg's edition of Archimedes, but they are omittedfrom most previous translations, thus depriving monolingual English speakers of animportant aid to understanding Archimedes. The inclusion of the commentaries inthis translation thus gives it a significant advantage over others. Netz also providesnotes to Eutocius; though these are much less extensive than his commentary onArchimedes, they are also very valuable, especially given how little work has beendone 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 alreadyavailable 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 asArchimedes (Copenhagen 1956). Both these versions are very freerenditions -- 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 tosome 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 Bud text (by C.Mugler, 1970-2) and the Latin one in the Teubner (by Heiberg as above). Mugler'sversion is a literal translation closely resembling Netz's in many ways, while




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Heiberg's sticks closely to the Greek in many places but converts the mathematicsinto modern notation when necessary for modern comprehension. Both thesetranslations also include translations of Eutocius' commentaries. Thus Netz'stranslation fills a gap that, while significant, really existed only for those unable toread French, while his notes and diagrams meet a much wider need.

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

The book is on the whole well produced, and there are not many typographicalerrors.2 All symbols and conventions are explained with admirable precision anddetail 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 tothe field of ancient mathematics. Though it will not replace the earlier translations, itis a significant addition to the resources available for understanding Archimedes, andthe 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 readArchimedes.

Notes:

1. See for example Netz's previous book, The Shaping of Deduction in GreekMathematics, 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 neitherof the two." (as a complete sentence, p. 94); "The introductory section is difficult toentangle, 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 twosides equal ..." (p. 59).

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