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Stanford Report, November 6, 2002

Scholars decode ancient text, shake up pre-calculus history


Reviel Netz, an assistant professor of classics, might not have actually shouted "Eureka!" on a visit last year to the Walters Art Museum in Baltimore, but that's what he was thinking.

A scholar of Greek mathematics, Netz was hanging out with one of his colleagues and frequent collaborators, Professor Ken Saito of the Osaka Prefecture University in Japan, when they flew together to Baltimore in January 2001 to look at a recently rediscovered codex of Archimedes treatises.

"It was basically just tourism," Netz recalled.

On a lark they examined a theretofore unread section of The Method of Mechanical Theorems, which is the book's biggest claim to fame; no other copy of the work is known to exist. What they discovered made their jaws drop.

A section from The Archimedes Palimpsest, which classics Professor Reviel Netz stumbled on during a visit to the Walters Art Museum in Baltimore. Closer examination showed the Greeks understood the concept of infinity. ROCHESTER INSTITUTE OF TECHNOLOGY, WALTERS ART MUSEUM, JOHNS HOPKINS UNIVERSITY


The Archimedes Palimpsest, as the book is called, is in terrible shape. (A palimpsest is a manuscript that has been written on more than once; in this case, a 13th-century Greek prayer book overlays the 10th-century script of the treatises.) The pages have been battered, gouged, scorched by fire and blotched by fungus. Without the use of computer technology, they would be mostly unreadable.

But when the palimpsest caught the attention of the great Danish philologist Johan Ludvig Heiberg in 1906, the underlying script was much more legible. At that time, the volume was in a library collection in Constantinople -- present-day Istanbul -- and, until Heiberg went to examine it, nobody seems to have realized its importance; the book contained the ancient Greek mathematician's previously unknown treatise on The Method of Mechanical Theorems.

Heiberg's find made the front page of the New York Times on July 16, 1907, under the headline "Big Literary Find in Constantinople." But then, suddenly, the palimpsest went missing again (probably stolen, Netz believes), reappearing -- unbeknownst to pretty much the entire world -- in the 1930s in a private collection in Paris. Only when Christie's Auction House sold it to another private collector in 1998 did scholars realize the book still survived. The Times published another story: "Archimedes Text Sold for $2 million," reads the headline on Oct. 30 of that year.

The buyer permitted the text to be displayed as part of an exhibition titled Eureka! The Archimedes Palimpsest at the Walters Art Museum, where it is presently being scanned and conserved.

Infinity and beyond

The ancient Greeks developed mathematics into a theoretical discipline. But conventional wisdom has always held that they disliked dealing with infinity because it's a messy concept. "Infinities give rise to all sorts of slippery problems," Netz explained.

In the 17th century, however, mathematics underwent a fundamental shift, thanks mostly to the efforts of England's Sir Isaac Newton and Germany's Gottfried Wilhelm Leibniz, who built upon Galileo's work and who are credited with inventing calculus.

In the 19th century, mathematicians rebuilt the calculus to create a rigorous and precise "science of infinity," Netz explained.

For the past 100 years -- that is, since Heiberg first looked at the palimpsest -- scholars have known that Archimedes toyed heuristically with concepts of infinity. But what Netz and Saito found in the palimpsest was that Archimedes actually had dealt with infinitely large sets in a mathematical proof.

"It has always been thought that modern mathematicians were the first to be able to handle infinitely large sets, and that this was something the Greek mathematicians never attempted to do," Netz writes in an essay on Archimedes published in the Nov. 1 issue of Science magazine. "But in the palimpsest we found Archimedes doing just that. He compared two infinitely large sets and stated that they have an equal number of members. No other extant source for Greek mathematics has that."

The proof in question is too complex to explain here, but suffice it to say that its shakes up the historical view of pre-calculus to its very foundations. And this is something Netz particularly enjoys.

"As an undergraduate, I was told the given fact that a feature of Greek thinking in general is the abhorrence of infinity," he said. "We tend to think that cultures are monoliths, and I really like this example showing that cultures are not monoliths."

In other words, it is impossible to pigeonhole ancient Greek thought or, for that matter, the intellectual culture of any civilization. People like to simplify forces that shape history, and this can lead to conceptually crude, underdeveloped ideas -- such as that the Victorians were sexually repressed.

Archimedes, who lived in the third century B.C., is probably best known as the protagonist in the apocryphal story about water displacement -- if it were a Hollywood film, it would be titled Bath, Interrupted -- who jumps out of the tub and shouts "Eureka!" As Netz writes in the Science essay, this is a "trivial observation." Yet its pithy simplicity overshadows the mathematician's more important contributions.

"The bath anecdote does not give us the true measure of the man," Netz writes. "In On Floating Bodies, Archimedes made the following, astonishingly subtle deduction: In a stable body of liquid, each column of equal volume must have equal weight; otherwise, liquid would flow from the heavier to the lighter. ... Now, that's something to cry 'eureka' about."

Reviel Netz