Book Review The Works of Archimedes: Translation and Commentary Volume 1: The Two Books On the Sphere and the Cylinder Reviewed by Alexander Jones The Works of Archimedes: Translation and He was the sub- Commentary. Volume 1: The Two Books On the ject of a biogra- Sphere and the Cylinder phy—now lost Edited and translated by Reviel Netz alas!—and stories Cambridge University Press, 2004 about him are told Hardcover x + 376 pp., $130.00 by ancient histo- ISBN 0-521-66160-9 rians and other writers who gen- Ancient Greek mathematics is associated in erally took little most people’s minds with two names: Euclid and interest in scien- Archimedes. The lasting fame of these men does tific matters. The not rest on the same basis. We remember Euclid as stories of Archi- the author of a famous book, the Elements, which medes’ inven- for more than two millennia served as the funda- tions; his solution mental introduction to ruler-and-compass geome- of the “crown try and number theory. About Euclid the man we problem”; the ma- know practically nothing, except that he lived be- chines by which, fore about 200 B.C. and may have worked in Alexan- as an old man, he dria. He wrote works on more advanced mathe- defended his native city, Syracuse, from the be- matics than the Elements, but none of these have sieging Roman fleet in 212 B.C.; and his death—still survived, though we have several fairly basic books doing geometry—at the hands of a Roman soldier on mathematical sciences (optics, astronomy, har- when Syracuse at last fell have never lost their ap- monic theory) under his name. All his writings dive peal. Archimedes became paradoxically emblem- straight into the mathematics with no introduc- atic of two stereotypes of the mathematician: a tions. There are hardly even any unreliable anec- man who could harness reasoning to the seemingly dotes about Euclid. superhuman performance of practical tasks, yet Archimedes, by contrast, is not just an author whose preoccupation with abstract problems could to us but a personality. He was famous in his time, make him fatally oblivious to his surroundings. not only among mathematicians and intellectuals. Alongside the public Archimedes of the stories, we also have the private Archimedes of the writ- Alexander Jones is professor of classics at the University ings. In the manner of his time, Archimedes wrote of Toronto. His email address is alexander.jones@ his mathematics in the form of substantial books utoronto.ca. built up of theorems that cumulatively lead to the 520 NOTICES OF THE AMS VOLUME 52, NUMBER 5 proof of a series of major results. Copies of these parchment sheets bound like modern books, and books were sent, in the form of papyrus rolls, to they could hold the equivalent of many of the old other mathematicians to be read, copied, and ap- rolls. The parchment codices were exceedingly preciated. The third century B.C., when Archimedes costly, both because of the materials (good parch- lived, was the heyday of the Greek mathematical ment was always expensive) and because of the cal- sciences. Mathematicians were scattered about the ligraphy, to say nothing of special skills such as Greek-speaking world, but there was a particular copying geometrical diagrams. Nevertheless, at concentration of them, as of other intellectuals, least three codex collections of Archimedes’ works in Alexandria. Archimedes sent his books from were made, each containing a different selection. Syracuse to three Alexandrian mathematicians: Yet it seems as if no one in the Byzantine Em- Eratosthenes, remembered for his measurement of pire read them. Archimedes’ name continues to the Earth and his scientifically based world map; crop up in Byzantine literature, but he is the Conon, whose identification of a new constella- Archimedes of the old anecdotes, not the mathe- tion in honor of Berenice, the queen of Egypt, was matical writer. No further copies of the Archimedes immortalized by the poets Callimachus and Cat- codices were made. In fact, by about A.D. 1300 all ullus; and one Dositheus. three codices were in situations where Byzantine There was a perverse, teasing streak in Archimedes’ scholars could no longer read them. Two of them relations with his Alexandrian correspondents. At a had somehow made their way to western Europe, time when, reflecting the cosmopolitan nature of where Greek learning was still rather scarce. Per- Greek culture after the conquests of Alexander the haps they were part of a royal gift, like the manu- Great, writers had embraced a standard form of Greek script of Ptolemy’s Almagest that the Byzantine in place of the many local dialects, Archimedes per- emperor Manuel Comnenus gave to the Norman sisted in writing his mathematics in the provincial king William I of Sicily about 1160. Whatever the Doric dialect of his native city. In the letters that he story, by 1300 these manuscripts had become part prefaced to his books, he speaks in less-than- of the small collection of Greek manuscripts in flattering terms of the mathematicians to whom he the papal library (one of them was in pretty bad con- is sending them, and on one occasion he sent them a dition). After the papacy was moved to Avignon in list of theorems without proofs, including two false 1309, the papal manuscripts seem to have been ones that were laid as a trap to catch anyone claim- dispersed, and only one of the Archimedes manu- ing priority of discovery. In spite of his best efforts scripts eventually resurfaced in the fifteenth to get their backs up, Archimedes’ contemporaries century. Greek humanism was now in full flower evidently thought well of his work, and later people in Italy, and good copies were made of this survivor not only preserved them but wrote commentaries on before it again vanished, this time for good. Ap- some of them, in some instances also stripping the parently the other manuscript had never been texts of their dialect for easier reading. We can still copied, though a painstaking Latin translation of read the commentaries of a very late Alexandrian the contents of both manuscripts had been made philosopher-mathematician, Eutocius, who lived at by the Dominican scholar William of Moerbeke in the time of the emperor Justinian, in the sixth cen- 1269, and this Latin version has survived. tury of our era. The third codex met a different fate: around It was about Eutocius’ time that Constantinople 1300 it was dismantled, its parchment leaves were began supplanting Alexandria as the focus of Greek cut in two, to some extent cleaned, and rewritten learning—the Islamic conquest of Egypt was just with a Greek prayer book. In this new guise, as a century away—and the selection of ancient Greek a palimpsest (recycled manuscript) it passed literature that was to survive through the Middle through one or more monastic libraries. By the Ages was to a large extent determined by which end of the nineteenth century it was in the library books were brought to Constantinople. There was of a monastery in Constantinople, and a scholar a very high risk of loss, especially in the seventh writing a catalogue of Greek manuscripts in that and eighth centuries, when classical learning, es- city noticed and copied a bit of the partially pecially in the sciences, was at a low ebb, but some- erased text. The Danish classicist Johan Ludvig how a dozen or so papyrus rolls of works of Heiberg, who had recently published an edition Archimedes were still around and in copyable of Archimedes’ works based on the manuscripts condition by the time intellectual conditions then known, recognized that the lines copied from were improving, about A.D. 800. (A few works of the palimpsest were part of Archimedes’ On the Archimedes, and apparently some others falsely at- Sphere and the Cylinder, and he made haste to tributed to him, were translated into Arabic about get access to the manuscript. Heiberg succeeded this time.) During the next two hundred years or in transcribing a substantial part of the faded so, a large number of manuscripts of ancient philo- Archimedean texts written crossways underneath sophical, scientific, and mathematical works were the prayers; they turned out to include two works produced. These were codices, manuscripts of that were otherwise entirely unknown and a third MAY 2005 NOTICES OF THE AMS 521 Uncovering New Views on Archimedes In 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 misguided project,” he recalled. “In scholarly terms it would be incomplete because I would not have access to the palimpsest.” That Netz forged ahead anyway proved to be fortuitous. When the long-lost Archimedes palimpsest resurfaced in 1998, his work on translating the first volume, On the Sphere and Cylinder, was ideal preparation for working on the palimpsest. “I was incredibly lucky,” he says. The program of preserving, imaging, and transcribing the Archimedes palimpsest is under way at the Walters Art Museum in Baltimore. Netz, now at Stanford University, and classics scholar Nigel Wilson 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 uncover additional information. Transcription of The Method is about one-third fin- ished. The palimpsest is the only extant source for The Method, and Netz lost no time in setting about studying it.