book review: the works of archimedes: translation and

520 NOTICES OF THE AMS VOLUME 52, NUMBER 5

Book Review

The Works of Archimedes:Translation and CommentaryVolume 1: The Two Books On theSphere and the CylinderReviewed by Alexander Jones

The Works of Archimedes: Translation andCommentary. Volume 1: The Two Books On theSphere and the CylinderEdited and translated by Reviel NetzCambridge University Press, 2004Hardcover x + 376 pp., $130.00ISBN 0-521-66160-9

Ancient Greek mathematics is associated in most people’s minds with two names: Euclid andArchimedes. The lasting fame of these men doesnot rest on the same basis. We remember Euclid asthe author of a famous book, the Elements, whichfor more than two millennia served as the funda-mental introduction to ruler-and-compass geome-try and number theory. About Euclid the man weknow practically nothing, except that he lived be-fore about 200 B.C. and may have worked in Alexan-dria. He wrote works on more advanced mathe-matics than the Elements, but none of these havesurvived, though we have several fairly basic bookson mathematical sciences (optics, astronomy, har-monic theory) under his name. All his writings divestraight into the mathematics with no introduc-tions. There are hardly even any unreliable anec-dotes about Euclid.

Archimedes, by contrast, is not just an authorto us but a personality. He was famous in his time,not only among mathematicians and intellectuals.

He was the sub-ject of a biogra-phy—now lostalas!—and storiesabout him are toldby ancient histo-rians and otherwriters who gen-erally took littleinterest in scien-tific matters. Thestories of Archi-medes’ inven-tions; his solutionof the “crownproblem”; the ma-chines by which,as an old man, he

defended his native city, Syracuse, from the be-sieging Roman fleet in 212 B.C.; and his death—stilldoing geometry—at the hands of a Roman soldierwhen Syracuse at last fell have never lost their ap-peal. Archimedes became paradoxically emblem-atic of two stereotypes of the mathematician: aman who could harness reasoning to the seeminglysuperhuman performance of practical tasks, yetwhose preoccupation with abstract problems couldmake him fatally oblivious to his surroundings.

Alongside the public Archimedes of the stories,we also have the private Archimedes of the writ-ings. In the manner of his time, Archimedes wrotehis mathematics in the form of substantial booksbuilt up of theorems that cumulatively lead to the

Alexander Jones is professor of classics at the Universityof Toronto. His email address is [email protected].

MAY 2005 NOTICES OF THE AMS 521

proof of a series of major results. Copies of thesebooks were sent, in the form of papyrus rolls, toother mathematicians to be read, copied, and ap-preciated. The third century B.C., when Archimedeslived, was the heyday of the Greek mathematicalsciences. Mathematicians were scattered about theGreek-speaking world, but there was a particularconcentration of them, as of other intellectuals, in Alexandria. Archimedes sent his books from Syracuse to three Alexandrian mathematicians: Eratosthenes, remembered for his measurement ofthe Earth and his scientifically based world map;Conon, whose identification of a new constella-tion in honor of Berenice, the queen of Egypt, wasimmortalized by the poets Callimachus and Cat-ullus; and one Dositheus.

There was a perverse, teasing streak in Archimedes’relations with his Alexandrian correspondents. At atime when, reflecting the cosmopolitan nature ofGreek culture after the conquests of Alexander theGreat, writers had embraced a standard form of Greekin place of the many local dialects, Archimedes per-sisted in writing his mathematics in the provincialDoric dialect of his native city. In the letters that heprefaced to his books, he speaks in less-than-flattering terms of the mathematicians to whom heis sending them, and on one occasion he sent them alist of theorems without proofs, including two falseones that were laid as a trap to catch anyone claim-ing priority of discovery. In spite of his best effortsto get their backs up, Archimedes’ contemporariesevidently thought well of his work, and later peoplenot only preserved them but wrote commentaries onsome of them, in some instances also stripping thetexts of their dialect for easier reading. We can stillread the commentaries of a very late Alexandrianphilosopher-mathematician, Eutocius, who lived atthe time of the emperor Justinian, in the sixth cen-tury of our era.

It was about Eutocius’ time that Constantinoplebegan supplanting Alexandria as the focus of Greeklearning—the Islamic conquest of Egypt was justa century away—and the selection of ancient Greekliterature that was to survive through the MiddleAges was to a large extent determined by whichbooks were brought to Constantinople. There wasa very high risk of loss, especially in the seventhand eighth centuries, when classical learning, es-pecially in the sciences, was at a low ebb, but some-how a dozen or so papyrus rolls of works ofArchimedes were still around and in copyable condition by the time intellectual conditions were improving, about A.D. 800. (A few works ofArchimedes, and apparently some others falsely at-tributed to him, were translated into Arabic aboutthis time.) During the next two hundred years orso, a large number of manuscripts of ancient philo-sophical, scientific, and mathematical works wereproduced. These were codices, manuscripts of

parchment sheets bound like modern books, andthey could hold the equivalent of many of the oldrolls. The parchment codices were exceedinglycostly, both because of the materials (good parch-ment was always expensive) and because of the cal-ligraphy, to say nothing of special skills such ascopying geometrical diagrams. Nevertheless, atleast three codex collections of Archimedes’ workswere made, each containing a different selection.

Yet it seems as if no one in the Byzantine Em-pire read them. Archimedes’ name continues tocrop up in Byzantine literature, but he is theArchimedes of the old anecdotes, not the mathe-matical writer. No further copies of the Archimedescodices were made. In fact, by about A.D. 1300 allthree codices were in situations where Byzantinescholars could no longer read them. Two of themhad somehow made their way to western Europe,where Greek learning was still rather scarce. Per-haps they were part of a royal gift, like the manu-script of Ptolemy’s Almagest that the Byzantine emperor Manuel Comnenus gave to the Normanking William I of Sicily about 1160. Whatever thestory, by 1300 these manuscripts had become partof the small collection of Greek manuscripts inthe papal library (one of them was in pretty bad con-dition). After the papacy was moved to Avignon in1309, the papal manuscripts seem to have been dispersed, and only one of the Archimedes manu-scripts eventually resurfaced in the fifteenth century. Greek humanism was now in full flowerin Italy, and good copies were made of this survivorbefore it again vanished, this time for good. Ap-parently the other manuscript had never beencopied, though a painstaking Latin translation ofthe contents of both manuscripts had been madeby the Dominican scholar William of Moerbeke in1269, and this Latin version has survived.

The third codex met a different fate: around1300 it was dismantled, its parchment leaves werecut in two, to some extent cleaned, and rewrittenwith a Greek prayer book. In this new guise, as a palimpsest (recycled manuscript) it passed through one or more monastic libraries. By theend of the nineteenth century it was in the library of a monastery in Constantinople, and a scholar writing a catalogue of Greek manuscripts in thatcity noticed and copied a bit of the partially erased text. The Danish classicist Johan Ludvig Heiberg, who had recently published an edition of Archimedes’ works based on the manuscripts then known, recognized that the lines copied fromthe palimpsest were part of Archimedes’ On theSphere and the Cylinder, and he made haste to get access to the manuscript. Heiberg succeededin transcribing a substantial part of the fadedArchimedean texts written crossways underneaththe prayers; they turned out to include two worksthat were otherwise entirely unknown and a third


Uncovering New Viewson ArchimedesIn 1996 when as a postdoc Reviel Netz launched his project of translating the works of Archimedes,his historian colleagues were not encouraging. “They said that obviously it would be a misguidedproject,” he recalled. “In scholarly terms it would be incomplete because I would not have accessto the palimpsest.” That Netz forged ahead anyway proved to be fortuitous. When the long-lostArchimedes palimpsest resurfaced in 1998, his work on translating the first volume, On the Sphereand Cylinder, was ideal preparation for working on the palimpsest. “I was incredibly lucky,” hesays.

The program of preserving, imaging, and transcribing the Archimedes palimpsest is under wayat the Walters Art Museum in Baltimore. Netz, now at Stanford University, and classics scholar NigelWilson of Oxford University are leading an international team analyzing the content of the palimpsest.The museum’s work on Sphere and Cylinder has been completed. The imaging of The Method is

also finished, and the Stanford Linear Accelerator is now helping to uncoveradditional information. Transcription of The Method is about one-third fin-ished.

The palimpsest is the only extant source for The Method, and Netz lostno time in setting about studying it. He has formed some intriguing new in-sights. For example, in The Method Archimedes constructs in the course ofa proof a one-to-one correspondence between two infinite (in fact, un-countable) sets. “This was completely unknown,” Netz states. “The as-sumption was that Greek mathematicians never made this kind of claim” abouttwo infinite sets being the same size.

Another work unique to the palimpsest is The Stomachion. This mere half-page fragment has puzzled scholars. “There wasn’t an interpretation of it,”Netz says. “No one really ventured to say what it was.” He has now conjec-

tured that the purpose of The Stomachion was to treat a combinatorics problem, a surprising idea,for it was not thought that combinatorics existed in ancient mathematics.

In fact Netz is changing many of the standard views on Archimedes and his work. “The original pic-ture historians had of Archimedes is as a practical engineer,” Netz remarks. “I don’t find any evidencefor this.…He was strictly a mathematician.” Even in Archimedes’ works on physics Netz sees mathe-matics as the ultimate goal. For example, Netz believes that Archimedes invented statics as a way ofderiving results in geometry. The strategy was to use imaginary situations involving bodies in equi-librium to derive proportions that lead to measurements of geometric figures.

Netz sees in Archimedes’ work a personality that is “very playful, cunningly and even maliciously playful.” Hellenistic mathematicians produced works that juxtaposed different things in surprisingways and that set challenges and puzzles for readers. Netz says this style of presentation has paral-lels in the larger cultural tendencies of Alexandrian and Hellenistic society. His next planned book, Lu-dic Proof: Greek Mathematics and the Alexandrian Aesthetic, will examine the playful strands runningthrough Hellenistic mathematics.

“Archimedes, among the truly great, is relatively neglected,” Netz comments. “There is a Newton in-dustry and an Einstein industry, but there isn’t an Archimedes industry, and there ought to be one.” Hebelieves one barrier to the study of Archimedes has been the lack of a complete and faithful translationof his works into English. Netz’s translation is so faithful, he says, that one could use it for serious his-torical studies. It is not an easy read, but then it is not an easy thing to plunge into the mind of a writerfrom a completely different culture and time. The translation is “very Greek—it’s a Greek book,” Netzsays. He has not transcribed the mathematics into modern notation, preferring instead to let the readerpuzzle through the mathematics just as Archimedes’ contemporaries did. Says Netz, “If you are really interested in Archimedes, invest the extra effort to see what he did.”

—Allyn Jackson

Reviel Netz


As well as the intrinsic interest of Archimedes’mathematics in On the Sphere and the Cylinder,there are two “loose ends” that provoked Eutociusto collect for us some remarkable specimens ofGreek geometry that we would not otherwise knowabout. These are the solutions by several geome-ters of the problem of finding two mean propor-tionals between given magnitudes (i.e., given A andD, to find B and C such that A : B = B : C = C : D )and another problem mathematically equivalent tosolving a cubic equation. Both problems fall intothe class that Greek mathematicians sensed fromexperience, though they could not prove, to be insoluble using only the postulates of Euclid’s Elements, and the solutions that Eutocius preservesillustrate how they extended their “toolbox” by allowing certain mechanical constructions, inter-sections of conic sections, or special curves.

In due course Netz will follow this volume with twomore. Of the books awaiting translation, some are,like On the Sphere and the Cylinder, works of puremathematics, while others give a mathematical treat-ment of problems in statics such as centers of grav-ity of figures and conditions of stability of floatingsolids. Netz does not intend to include in his scopethe medieval Arabic tradition of Archimedes’ works.In fact, it is likely that none of the works that pass un-der Archimedes’ name in Arabic, aside from thosethat we also have in Greek, are authentic, althoughseveral contain interesting mathematics. An editionand translation collecting these would be a worth-while project.

Netz’s English Archimedes could hardly be moredifferent from Heath’s. To begin with, it is ruthlesslyliteral. Archimedes, like all Greek geometers, wrotehis mathematics in continuous prose, using wordsto represent concepts and relations, and letters ofthe Greek alphabet to name them. The vocabularyand sentence structures of Greek geometrical writ-ing were highly standardized and formulaic by thethird century B.C., a quality that gives the argumentssomething of the same clarity and freedom fromambiguity that notation provides in modern math-ematics. But a mathematical argument written outas prose (sometimes referred to as “rhetorical”mathematics) has two characteristics in contrast tonotation: the relation to the spoken word is muchmore immedate, so that one can read the argumentaloud correctly even if one does not understand themathematics at all, but it takes much more spaceon the page to write—and more time for the eyeto take in—each step of a proof. Mathematiciansare often more comfortable with translations ofearly mathematics that employ notations to com-press the argument, whereas nonmathematiciansinterested in the history of science may find therhetorical style more approachable, though in the case of Archimedes they soon find that the

that had hitherto only been available in Moerbeke’sLatin version. After a rather obscure history in thetwentieth century, the Archimedes palimpsest isnow in private hands and the subject of an ambi-tious project of conservation and research intendedto recover text that Heiberg was unable to read. Inthe meantime, Heiberg’s second edition, which isvery good, remains after a century the basis forstudying what Archimedes wrote.

Astonishingly, given Archimedes’ fame and the importance of the works, Archimedes’ books havenever really been translated into English. ThomasHeath, who made an excellent translation of Euclid’sElements, published The Works of Archimedes in1897 (before Heiberg’s rediscovery of the palimpsest),but except for the prefatory letters this was not atranslation so much as a mathematical paraphrasethat not only uses modern notation but liberally reorganizes parts of the proofs. E. J. Dijksterhuis’sArchimedes, first published in 1956, contains math-ematical paraphrases of the proofs that are closer tothe reasoning of the original than Heath’s, and hisbook, which has been reprinted by Princeton Uni-versity Press, is still the best general introduction to Archimedes’ thought. But neither Heath nor Dijksterhuis gives a complete restatement of all the proofs, and the “feel” of Archimedes’ writing isentirely lost.

Now Reviel Netz offers us the first installmentof a translation of all the works of Archimedesthat survive in Greek. The first volume contains asingle large work, On the Sphere and the Cylinder,together with the commentary on that work by Eutocius. On the Sphere and the Cylinder is the bookin which Archimedes proved several relations between the volumes and surface areas of spheres,segments of spheres, and cylinders, including ofcourse the proof that the volume of a sphere is two-thirds the volume of the smallest containing cylin-der; a diagram illustrating this relation is reputedto have been inscribed on Archimedes’ tomb. Thecentral proofs employ what are traditionally called“exhaustion methods”, which are rigorous limit arguments founded on inscribed or circumscribedsolids (such as the solids of rotation of polygons).Because exhaustion arguments are not easy totransfer from one geometrical figure to another, the proofs have a certain virtuosic character, verydifferent from the more readily generalizable approaches of European mathematicians to suchproblems leading up to the calculus. In anotherbook, The Method (to appear in a subsequent vol-ume), Archimedes presented a more heuristic andless rigorous strategy for demonstrating volumeand area relations by comparison of infinitesimalslices of figures. This work, however, was unknownto modern mathematicians until Heiberg discoveredit in the palimpsest.


The first of three views (seenext page) of a page from theArchimedes Palimpsest (fols.93v-92r). The photograph at

right is taken in regular light.

Photographs taken by theRochester Institute of

Technology and JohnsHopkins University. Copyright:

The owner of the ArchimedesPalimpsest.

easy flow of words expresses some very difficultmathematics.

Netz does not hold that it is the translator’sbusiness to cater to the comfort of the reader; hewrites (p. 3) that “the purpose of a scholarly trans-lation as I understand it is to remove all barriershaving to do with the foreign language itself, leav-ing all other barriers intact.” And whatever onemay say of Netz’s Archimedes, it cannot in any waybe charged with looseness. He maintains a close and consistent correspondence between the Greekterminology and his English equivalents. The for-mulaic character of Greek geometrical prose makesit often possible to omit certain common wordswithout ambiguity, and the geometers took fulladvantage of this means of shortening their sen-tences: for example, the conventional Greek way ofsaying “the angle contained by AB and BC” wouldtranslate word-for-word as “the contained by ABC.”Netz carefully marks all the implied words by enclosing them in angle brackets, as in this case “the<angle> contained by ABC.” He also adheres morethan any other translator I know to the Greek wordorder, even when this goes against natural habitsof English. This is particularly apparent when ratios are being expressed and manipulated: for example, a sentence that might be represented bythe notation

Z : H = AC : CB = AD2 : DB2

is rendered: “it is: as CD to CB, that is Z to H, the<square> on AD to the <square> on DB.” Readinglong stretches of this is hard going, though one soongets used to the oddness, and the reader will prob-ably find that writing out each step symbolicallyas one works through the proofs makes them easier to follow and verify. It is a pity that Netz

has not provided such a step-by-step synopsis toaccompany the translation.

For Archimedes’ text Netz has relied on Heiberg’sedition. (In subsequent volumes he intends to takeaccount of new readings arising from study of thepalimpsest.) The figures are a different matter.Heiberg, like most scholars editing works of Greekmathematics, preferred to redraw the figures thataccompany each theorem according to the senseof the text rather than reproduce with necessarycorrections the drawings that appear in the man-uscripts. In many cases the resulting reconstructedfigures look quite different from the transmittedones, though they are usually mathematically equiv-alent. Netz has made a close study of the role offigures in Greek mathematics, and in his earlierbook, The Shaping of Deduction in Greek Mathe-matics (1999), he showed how the figures were intended as an integral and indispensible part ofthe proofs, not mere adjuncts to aid visualization.So here Netz has gone back to the manuscripts andproduced the first “edition” of the figures, report-ing all significant variations in the manuscript versions. Like his close translation, this brings thereader a step closer to seeing Archimedes’ mathe-matics as an ancient reader would have seen it.

Netz’s intention, which I have quoted, of “leav-ing all other barriers intact” sounds forbidding, butit applies only to the bare translation; taking thebook as a whole, he provides the reader with agreat deal of help in getting over those barriers. Thishelp takes several forms. There are frequent foot-notes clarifying and justifying unobvious steps inArchimedes’ arguments and cross-referencing themwith the relevant parts of Eutocius’ commentary,which appears later in the volume. After each the-orem Netz gives two sections of commentary: onediscussing textual matters (this will be of


particular interest to people comparing the trans-lation to Heiberg’s edition of the Greek), the otheroffering more general remarks. The general com-ments are not, for the most part, mathematical inscope, but discuss linguistic and stylistic aspectsof the text. For mathematical commentary, Netzrefers us to Eutocius and Dijksterhuis, who arecertainly worthy and sensible guides, though Dijk-sterhuis is difficult to use in close comparisonwith Netz’s translation, because the lettering ofthe diagrams is different, and Eutocius is writingfor a reader more conversant in the idioms of an-cient mathematics than most modern readers arelikely to be. A possible consequence of Netz’s de-cision not to provide his own mathematical

commentary is thatthere may be placeswhere he has notdetected that thetext as transmittedby the manuscriptsand edited byHeiberg is incorrect.I have noticed oneplace where thisseems to be thecase: in the fourththeorem of Book 2(p. 203, step 24) astatement about ra-tios in Heiberg’stext is both falseand different fromwhat Eutociusseems to have readin his copy of thesame passage.

In general thetranslation has been made consistently and withcare, and I have noticed very few misprints, exceptin the bibliography, where the typesetters seem tohave been uncharacteristically creative (and, oddly,Lewis Carroll appears disguised as Carol whereverhe turns up). The typography, layout, and drafts-manship of the diagrams are of a standard that onecan unfortunately no longer take for granted inscholarly books. A regrettable corollary is the highprice; individuals planning to own all three volumeswill need deep pockets.

The photograph at left istaken in ultraviolet light,showing some of theunderlying text ofArchimedes’ Sphere andCylinder.

The photograph below is aprocessed image, making theArchimedes undertext anddrawing appear in red.

book review: the works of archimedes: translation and

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