Eureka!

Book: John Dean; Parchment: iStockphoto.com; Ghosted text: courtesy the Archimedes Palimpsest Project

Reviel Netz could hardly believe his eyes when he opened his e-mail one day in 1998. For a couple of years, the young classicist had been pounding away on an ambitious new project: the first comprehensive English translation of works by the ancient Greek scientist Archimedes of Syracuse. It was a challenging business. While some of the medieval source documents Netz needed were easily accessible in European archives, others were much harder to find. The oldest and most tantalizing manuscript was a parchment Byzantine volume dating from A.D. 975. Besides being the unique source for two of Archimedes's most intriguing treatises, The Method and Stomachion, it was the only one containing On Floating Bodies in the original Greek. Unfortunately, the text had been scrubbed and written over in the year 1229. Moreover, the manuscript was locked away in a French basement. No scholar had examined it for decades.

Netz, who specializes in the history and philosophy of science, was particularly keen to have a peek at the Greek manuscript's mathematical diagrams, to see whether they matched others he'd seen in Latin translations. At one point, he even tried contacting the secretive French owners himself, to no avail. Now, shortly before joining the Stanford faculty as an assistant professor of classics and philosophy, Netz started up his computer to find an astonishing e-mail—one that would impact his career forever. The precious Greek Archimedes manuscript had resurfaced. And it was about to go on the auction block at Christie's.

In a new book, The Archimedes Codex (Da Capo Press), Netz and his co-author, Baltimore curator William Noel, trace the incredible journey of that worm-eaten volume, from its genesis in a Constantinople monastery, through exile in the Holy Land and war-torn Europe, to its 21st-century resurrection in the United States. Along the way, it was recycled into a prayer book, defaced and neglected. Yet thanks to some remarkable international academic teamwork—by Netz and other scholars, manuscript conservators, computer imaging specialists and modern mathematicians—long-lost ideas of Archimedes are coming to light again.

According to Netz, newfound passages in the codex show that Archimedes had ideas about “actual infinity” that were far ahead of his time. Other parts suggest that Archimedes was the first person to think seriously about combinatorics, the mathematical study of combinations. In fact, Netz argues that if you add Archimedes' mathematical insights to his better-known contributions in physics and engineering—such as his law of buoyancy or the Archimedean screw—the proof is evident. “Archimedes,” he writes, “is the most important scientist who ever lived.”

Given Archimedes' long shadow, it is surprising how little scholars really know about the man. He was born in the Sicilian port city of Syracuse, a hub for commerce, science and art, sometime around 287 B.C. He was the son of an otherwise unknown astronomer named Phidias, and probably related to the king of Syracuse, Hiero II. Apparently he spent part of his youth studying overseas at Ptolemy's great forum for ancient scholars, the Museum of Alexandria in Egypt.

No one knows if Archimedes ever married or had children. He did leave behind rolled papyrus letters to his fellow scho-lars that were copied and treasured in libraries throughout the ancient world. His best-known treatise, On Floating Bodies, proclaimed the famous Archimedes Principle, which states that a body immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid. (Supposedly Archimedes had this splash of insight while stepping into his bathtub—though scholars doubt that the level-headed scientist actually shouted “Eureka!” over the discovery, running naked into the street.)

During his lifetime, Archimedes was renowned for his ingenious mechanical devices, including deadly accurate catapults, compound pulleys and his famous screw for moving water uphill. He calculated a nearly precise value for pi, showed how to determine the areas and volumes of geometric figures, and demonstrated novel ways to figure out their centers of gravity. His “method of exhaustion” for determining an area bound by a curve anticipated integral calculus, the basis of modern science. Sadly, though, when the great cities of the Roman Empire were sacked between the 3rd and 6th centuries, many of Archimedes' works were burned or lost forever.

The 10 Archimedes treatises that did make it intact through the Dark Ages gradually were converted from rolls into bound parchment manuscripts known as codices. Among the most dedicated preservationists were the Byzantine scribes of Constantinople. Writing in minuscule (lowercase) Greek with sharpened reed pens, they churned out three well-known Archimedes volumes in the 9th and 10th centuries. The first two, known as Codex A and Codex B, eventually were lost—though not before being translated into Latin manuscripts that inspired the great thinkers of the scientific revolution, including Galileo and Newton.

The third volume met an entirely different fate. (Download a pdf of the Path of the Palimpsest.) Sometime around 1229, an Eastern Orthodox priest named Ioannes Myronas needed parchment for a new prayer book. Codex C, he decided, would do nicely. Unbinding the precious manuscript, he scrubbed the pages with natural acid and a pumice stone—fortunately not too hard. Then he cut the folios in half, rotated them 90 degrees, folded them and wrote his prayers at a right angle to the faint traces of the earlier handwriting. (Such an overwritten manuscript is known as a palimpsest.) For centuries the stout little prayer book was used at the St. Sabas Monastery near Jerusalem. Eventually someone took it back to Constantinople and placed it in the library at the Metochion (embassy church) of the Jerusalem Holy Sepulchre.

By the beginning of the 20th century, scholars were pretty sure they had all the material on Archimedes they were going to find. Then, a Danish paleographer named Johan Ludwig Heiberg read an intriguing catalogue entry about an odd-looking prayer book in Istanbul that also contained mathematics text. In 1906, he went there to see the palimpsest for himself. Sure enough, barely visible beneath the priest's handwriting were the ancient words of Archimedes. Squinting through a magnifying glass under natural light, Heiberg transcribed what he could and photographed pages for later study. Then he put the volume back on the shelf, figuring it would stay there.

It didn't. During the chaos that enveloped Turkey after World War I, a Parisian book dealer somehow acquired the palimpsest and attempted to sell it. There were no takers, though—the volume was too ugly; the asking price too high. As Nazi tanks rolled through the streets of Paris, the dealer became even more anxious to sell. Although the evidence is sketchy, Noel speculates that the Jewish bookseller badly needed money for safe passage. He knew that the Nazis were collecting illuminated manuscripts, not worm-eaten math books. So, in an act of desperation, he dressed up the medieval prayer book with forged icons of the four evangelists, Matthew, Mark, Luke and John. In the process, four pages of Archimedes text were obliterated.

Apparently a sympathetic member of the French Resistance paid the dealer for the defaced palimpsest and hid it in his basement, where it sat moldering for decades. In the 1970s, its Gallic owners began casting about for a potential buyer—reassuring scholars around the world that at least the volume had survived. Stanford briefly contemplated purchasing it in the 1980s. But those negotiations, and others, eventually fell through.

When the palimpsest finally did come back on the market, at the 1998 Christie's auction, friends of Archimedes held their breath. Who would be the highest bidder? Would it be the Greek Ministry of Culture, trying to buy back a piece of its national patrimony? Would it be a university or museum friendly to outside researchers? Would the manuscript go back into private hands, never to be seen again?

Fortunately, the wealthy individual who made the winning $2 million bid—a man Netz and Noel identify only as “Mr. B”—was as interested in unlocking the secrets of the palimpsest as any academic. Several months after the auction, Mr. B paid a visit to the Walters Art Museum in Baltimore, where the British-born Noel is curator of manuscripts and rare books. As they walked to a nearby restaurant for lunch, the nervous young museum staffer congratulated Mr. B on his acquisition, noting that “it was extremely generous of him to even consider putting his great new treasure on deposit at the Walters.” Mr. B replied that he had already left the fragile volume on deposit—in a bag on Noel's desk.

“I swallowed hard,” Noel recalls vividly in The Archimedes Codex. “As the museum registrar would have been quick to point out, this did not conform to standard museum protocols for the transportation and documentation of objects worth several millions of dollars. I went with the flow. 'Great,' I told him, 'and what a good job it was that I had locked the door of my office on the way out.'”

Mr. B left more than the palimpsest at the Walters. He also supplied generous funding for its conservation and study, placing Noel in charge of the whole operation. Working with tiny scissors and tweezers in the Walters' basement laboratory, senior conservator Abigail Quandt and her team began taking the volume apart—necessary if scholars were to read any of the Archimedes text that was in the book's gutter. It was a delicate operation: folios were afflicted by mold, water damage, wood glue, candle wax, tears, burns and holes; those with forgeries on them also bore deliberate distressing and nicks, as well as Blu-tack adhesive and rust from a paper clip. On average, a page was liberated from the volume every 15 days; the entire process took almost four years.

A second task was to highlight the faint handwriting, to make it easier to read. Ultraviolet lamps, standard tools of the paleographer's trade, were somewhat effective at illuminating the hidden text on the parchment. But what the Walters team really wanted was help digitizing the pages, so they'd be easy to retrieve, magnify and share with other scholars online. They particularly needed a way to sort out which of the crisscrossed lines of handwriting were 10th-century Archimedes text, and which were the markings of a 13th-century priest.

Rising to the challenge, three scientists from the Rochester Institute of Technology, Xerox and Johns Hopkins University collaborated on an innovative multispectral imaging system—one that made the Archimedes text pop out on the computer screen in red-tinted pixels, while the prayer book text stayed black. During the course of five years, they imaged the pages, digitally stitched the images together and made them available electronically to scholars across the globe. The pages defaced by the World War II forger posed another technological hurdle. That problem eventually was solved by scientists at the Stanford Synchrotron Radiation Laboratory. (See sidebar.)

The ultimate challenge was deciphering and editing the Archimedes texts so they could be published. Around the world, classicists jostled to take on the job. One obvious choice was Nigel Wilson of Lincoln College, Oxford, who had catalogued the manuscript for the Christie's auction. Wilson was renowned for his talents at deciphering and analyzing ancient scripts, skills that would be vital to the project's success.

Noel also wanted someone on the team who could understand the Archimedes text and diagrams from a mathematical perspective. A professor at Cambridge University recommended Netz, an Israeli classicist who had earned his phd there in 1995 before heading to mit for postdoctoral work. Netz had served in the Israeli army. He wrote award-winning poetry, in Hebrew. His doctoral thesis had focused on the thought processes of ancient Greek mathematicians. And as it happened, he was translating Archimedes into English for Cambridge University Press.

Netz, of course, was more than eager to sign on to the palimpsest project. Still, when he first laid eyes on the homely little volume cradled in the basement lab at the Walters, he couldn't help feeling anxious. “It was definitely in the most horrible shape,” he recalls now, sitting in the living room of his Stanford condominium, which he shares with his wife, Maya Arad, and two young children. Moreover, he says, “I was quite young and inexperienced [at paleography]. To be quite honest, my worry was whether I would be up to the challenge.”

Buoyed by Heiberg's old transcripts and photos, Wilson's expertise and the enhanced computer images, Netz gradually gained confidence in his ability to decipher the manuscript. He was particularly enthralled by its delicate mathematical diagrams. As Netz explains in The Archimedes Codex, Greek mathematicians didn't communicate their ideas with numerals and shorthand symbols; they illustrated their reasoning with prose and accompanying diagrams. Taken with the drawings descended from the other codices, the palimpsest's drawings offered vital clues to how Archimedes' mind worked.

Other clues lay hidden within the Greek text. Early in 2001, Netz and Ken Saito, a scholar from the University of Osaka, flew to Baltimore to examine some folios containing passages from Archimedes' little-known treatise on mechanical theorems, The Method. In it, Archimedes describes how to calculate the volume of a cylindrical cut. Puzzling over a nearly illegible passage, Netz and Saito suddenly noticed four letters they hadn't before: epsilon . . . gamma . . . epsilon . . . theta, or eget. Apparently the letters had something to do with megethos, the Greek word for magnitude.

Archimedes was not afraid to crunch big numbers. In his treatise The Sand-Reckoner, he devised a method for counting how many grains of sand the universe theoretically could contain (8 x1063 in scientific notation). But modern mathematicians draw a clear line between calculating with potential infinity—summing numbers “as big as you wish”—and calculating with actual infinity. Was this faint passage from the codex evidence that Archimedes had laid a stronger foundation for the calculus than previously established?

The possibility had been suggested before, by Netz's predecesso at Stanford, Wilbur Knorr. Regarded as one of the world's most distinguished historians of ancient mathematics before his death in 1997 at age 51, Knorr had argued that ancient Greek mathematicians had sophisticated notions about infinity that far surpassed the musings of their philosopher friends.

The more Netz read in The Method, the more he was convinced that Knorr's intuition was right. “The Greeks could very well envisage actual infinity. They could even operate with it,” he concludes in The Archimedes Codex. “For various reasons, in most contexts they preferred to avoid it. But this avoidance was a conscious decision, not some kind of reflection of shortcoming on the Greeks' part. They were ahead of the infinity game.”

Another game that interested Archimedes was stomachion, from the Greek word for stomachache. Players started out with 14 polygon-shaped puzzle pieces made out of ivory. Perhaps Greek children amused themselves by making animal shapes out of the pieces. The real challenge, though, was to fit them into a square.

Near the end of the palimpsest, Archimedes describes this ancient puzzle and starts to make a proposition about it. Then, maddeningly, the text stops. Apparently the priest who made the palimpsest either threw the rest of the treatise away or used its parchment for something else.

While fiddling with a model of the stomachion sent to him by an Archimedes admirer, Netz had his own eureka moment: there was more than one way to fit the pieces into a square. Way more than one, maybe. Netz hypothesized that Archimedes must have figured out a way to calculate stomachion's possible combinations; otherwise he wouldn't have bothered writing about it. But how did he come up with the answer, and what was it?

Netz took the puzzle to a pair of Stanford colleagues, mathematician Persi Diaconis and statistician Susan Holmes, who teamed up with mathematicians from UC-San Diego. Working independently, a Chicago-area computer scientist tackled the problem by writing a program. Several weeks later, Netz had his answer: there were 17,152 different ways of arranging the pieces into a square. The computer scientist arrived at the number first, but the mathematicians demonstrated that Archimedes could have done so with pen and papyrus.

For Netz, the complex solution suggested a fascinating possibility. In Stomachion, he asserts, Archimedes was thinking about much more than a child's game. Centuries before Fermat and Pascal took a deep interest in the subject, Archimedes was laying the foundation for combinatorics—the modern science of sets and their possible combinations—which in turn gave rise to the science of probability. It was a field that wouldn't flower fully until the age of computers.

While scholars value the Archimedes palimpsest chiefly for its mathematical treatises, the volume also contains 30 intriguing folios recycled from other ancient manuscripts. Among them: an early Christian commentary on the soul; two lost speeches by Hyperides, a founding father of ancient Greek democracy; and an early commentary on Aristotle's Categories, probably written by Alexander of Aphrodisias in the 2nd or 3rd century A.D.

As his transcription of the Archimedes text nears completion, Netz is plowing into the barely legible Aristotle commentary to see if it might yield clues to the origins of Greek logic. He's also busy cultivating the next generation of Archimedes scholars. During his Stanford courses on ancient Greek, Netz pulls up images of the palimpsest on a large computer screen, to give his budding paleographers practice at deciphering ancient texts. Last year, Randy Folse, '09, made a small but significant contribution to the palimpsest project, when he noticed a punctuation mark in The Method's title that Netz had overlooked. In essence, this changed the title from Archimedes' Method of Mechanical Theorems to Method; or, Archimedes on Mechanical Theorems.

As Netz reminds his students, there's still plenty about Archimedes that scholars don't know. The Arabic translations made of his works beginning in the 9th century are largely unstudied. Technology experts are still at work, testing new approaches to reading the most camouflaged portions of Codex C. And there may be more palimpsests out there, tucked in the recesses of Greek Orthodox monasteries throughout the Middle East. “In general, people don't recognize the extent to which ancient science is an undeveloped field,” Netz observes ruefully. “They have the sense that surely everything has been worked over for a couple of centuries; that there's nothing new to find.” In truth, he says, “There's a world to be found.”


THERESA JOHNSTON, '83, is a Palo Alto writer and frequent contributor to Stanford