The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity’s Greatest Scientist , Reviel Netz and William Noel , Da Capo Press, Philadelphia, 2007. $27.50 (313 pp.). ISBN 978-0-306-81580-5
On the afternoon of 29 October 1998, Christie’s auction house in New York sold off a scrappy, late Byzantine prayer book for $2 million. A very wealthy, anonymous American won the prize. It was a big event in the small world of Greek mathematics, for underneath the prayers were rubbed-out works of Archimedes (287–212 BCE). Prior to the sale, only Johan Ludvig Heiberg had ever studied the undertext. Heiberg had spent his 1906 summer vacation in Constantinople photographing and transcribing it. The palimpsest contained the only known Greek text of On Floating Bodies, a work whose Latin translation was important for the emergence of early modern science; a scrap of the Stomachion; and the Method, a masterpiece unlike any other extant work of ancient Greek mathematics.
Sometime in the 1920s the prayer book disappeared. Quiet attempts were made to sell it in the 1930s and again from the 1960s on, with potential buyers including Yale and Stanford universities and the J. Paul Getty Museum. The text today is not what Heiberg saw. He was not allowed to undo the binding to examine each page; glue, mold, forged illuminations, and missing folios now inhibit the modern reader. Newer technologies were needed to reveal the undertext. The result is that although Heiberg saw things that we may never see, researchers can now see much that he couldn’t.
Reviel Netz and William Noel’s The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity’s Greatest Scientist consists of two personal introductions to the palimpsest, its history, the work in reading it, and its mathematics—especially in recent discoveries (see the article by Netz in Physics Today, Physics Today 0031-9228 53 6 2000 32 https://doi.org/10.1063/1.1306365 June 2000, page 32 ). Noel is curator of manuscripts at the Walters Art Museum in Baltimore, Maryland, where the owner deposited the palimpsest, and is the director of the Archimedes Palimpsest Project. Netz, professor of classics and philosophy at Stanford University and one of the most creative historians of mathematics of our time, has worked extensively on the palimpsest. In alternating chapters with Netz, Noel narrates with some verve his own story in learning about the palimpsest; he surveys both the document’s history from the origins of the undertext to its sojourn at SLAC at 2006 for advanced imaging and the discovery of works by other ancient Greek authors.
Two interesting discoveries in the mathematics of the text have been made. In the Method, Archimedes uses indivisibles (a misnomer) to treat an n-dimensional figure as being composed of n–1-dimensions, such as a triangle of parallel lines or a cylinder of rectangles. He combines that method with the principle of the balance; however, proposition 14 of the Method works only with indivisibles. Archimedes uses a lemma he proves elsewhere that is appropriate to finite-sized sets and saw no problem in generalizing to the infinite: He calculated with infinities. Did he, as Netz claims, have a notion of equinumerous sets as having a one-to-one correspondence?
Initially, the purpose of the Stomachion, or “Bellyache,” the translation plausibly suggested by Netz, was unknown, except that it involves a game with 11 triangles, two quadrilaterals, and one irregular pentagon that together can form a square. Building on recent work of Fabio Acerbi showing that Hipparchus (c. 180–125 BCE) had engaged in sophisticated combinatorics and reading more of the text than Heiberg could, Netz reasonably proposes that the Stomachion also engages in combinatorics: How many ways can 14 figures form a square? Does this mean that Archimedes and Hipparchus studied combinatorics, or did they just work two different counting problems that happened to be difficult combinatoric ones?
Much of The Archimedes Codex is delightful. The story of the palimpsest is exciting, and few can explain difficult issues in Greek mathematics with the simplicity and elegance that Netz achieves. However, I cannot recommend the book without deep reservations. Both authors indulge everywhere in extravagance and occasionally even in melodrama. Some hyping, including the book’s title, is harmless journalism, if not to my taste, but a discussion of the many harmful instances would exceed the space of a short review. My brief examples center on two issues: the noble goal of explaining to a general audience the importance of studying the language of texts and Netz’s adaptation of the already extravagant quip of 20th-century philosopher and mathematician Alfred North Whitehead on Plato, that the history of modern science is a footnote to Archimedes.
After disparaging all reports of Archimedes’ life in ancient sources, Netz proposes his own implausible, linguistic recovery of Archimedes’ life—that Archimedes’ father had philosophical interests in choosing his son’s name. For the record, “Archimedes” means something like “Chief Counselor,” not Netz’s proposed “the Mind of the Principle” (page 36).
Archimedes certainly is the central and greatest figure in ancient Greek mathematics and in its revival, which was crucial for the emergence of 16th-and 17th-century mechanics. However, if all you knew of Greek mathematics was from Netz and Noel’s book, you might incorrectly conclude that Archimedes discovered the method of exhaustion, conic sections, the notions of infinity and indivisibles, mathematical mechanics, and more. Netz also goes on to turn Archimedes into a scientific magician. In describing (very elegantly, I might add) how Archimedes finds the center of mass of a triangle in his On the Equilibrium of Planes, Netz says that Archimedes has told us “without even looking how the world must behave, where a triangle must balance,” and Netz calls it an “act of magic” (page 147). Maybe, but Archimedes does begin the treatise with a series of seven crucial, empirically grounded assumptions about centers of mass and balances, and he consciously employs an idealization that the triangle, a plane figure, has uniformly distributed weight.
Netz hypes Archimedes’ importance. Surely, many basic concepts in early modern science are absent from Archimedes—for example, experimentalism, algebra, probabilistic and genetic explanations, even mechanism. Archimedes was uninterested in biology, medicine, and the nature of matter. Netz’s claim, based on six lines of the Method, that Archimedes “foresaw a glimpse of Set Theory” (page 202) baffles me inasmuch as Archimedes’ argument about infinite sets generates paradoxes discovered in the 17th century and not resolved until the 19th century.
The Method and the Stomachion are important works, and Netz is doing some great things with them, and with the rest of the mathematical texts in the palimpsest. Yet barely a shred of evidence exists that the Method influenced anyone; only Hipparchus was perhaps influenced by the Stomachion, and even the evidence for that is circumstantial at best.
The Archimedes Codex is a fun read. When it succeeds, it does so very well. But I worry that readers will not know when it does not—when a conjecture is wild, an inconvenient manuscript omitted, the study of chronology twisted to serve an entertaining but false thesis, and so forth. The subject is so interesting and the authors so uniquely positioned to present it that I wish they had chosen a more restrained approach. The book is also a clever ploy to attract interest in finding the missing folios. And in that attempt, we can all wish the authors luck.
Henry Mendell is a professor of philosophy at California State University in Los Angeles. His interests are in ancient philosophy, particularly the history of philosophy that focuses on astronomy and mathematics.
