Anyway, if either NY Times or LA Times ever wanted to hire me to write for them I'd be SO there, even with my contempt for the LA Times. That might just be because it's LA.
NY TIMES ARTICLE
In Archimedes' Puzzle, a New Eureka Moment
By GINA KOLATA
Published: December 14, 2003
Twenty-two hundred years ago, the great Greek mathematician Archimedes wrote a treatise called the Stomachion. Unlike his other writings, it soon fell into obscurity. Little of it survived, and no one knew what to make of it.
But now a historian of mathematics at Stanford, sifting through ancient parchment overwritten by monks and nearly ruined by mold, appears to have solved the mystery of what the treatise was about. In the process, he has opened a surprising new window on the work of the genius best remembered (perhaps apocryphally) for his cry of "Eureka!" when he discovered a clever way to determine whether a king's crown was pure gold.
The Stomachion, concludes the historian, Dr. Reviel Netz, was far ahead of its time: a treatise on combinatorics, a field that did not come into its own until the rise of computer science.
The goal of combinatorics is to determine how many ways a given problem can be solved. And finding the number of ways that the problem posed in the Stomachion (pronounced sto-MOCK-yon) can be solved is so difficult that when Dr. Netz asked a team of four combinatorics experts to do it, it took them six weeks.
While Dr. Netz acknowledges that his findings cannot be proved with absolute certainty, he has presented the work to other scholars, and they say they agree with his interpretation.
On a recent snowy Sunday morning at Princeton University, three dozen academics gathered to hear Dr. Netz speak, and then congratulated him, saying his arguments made sense. "I'm convinced," said Dr. Stephen Menn, a McGill University historian of ancient mathematics, in an interview at the end of the two-hour session.
Among all of Archimedes' works, the Stomachion has attracted the least attention, ignored or dismissed as unimportant or unintelligible. Only a tiny fragment of the introduction survived, and as far as anyone could tell, it seemed to be about an ancient children's puzzle — also known as the Stomachion — that involved putting strips of paper together in different ways to make different shapes. It made no sense for a man of Archimedes' stature to care about such a game.
As a result, Dr. Netz said, "people said, `We don't know what it is about.' "
In fact, he has concluded, the prevailing wisdom was based on a misinterpretation. Archimedes was not trying to piece together strips of paper into different shapes; he was trying to see how many ways the 14 irregular strips could be put together to make a square.
The answer — 17,152 — required a careful and systematic counting of all possibilities. "It was hard," said Dr. Persi Diaconis, a Stanford statistician who worked on it along with a colleague, Dr. Susan Holmes, who is also his wife, and a second husband-and-wife team of combinatorial mathematicians, Dr. Ronald Graham and Dr. Fan Chung from the University of California, San Diego.
Independently, a computer scientist, Dr. William H. Cutler at Chicago Rawhide, a manufacturer of oil seals in Elgin, Ill., wrote a program that confirmed that the mathematicians' answer was correct.
Perhaps as remarkable as the discovery that Archimedes knew combinatorics is the story of a manuscript that dates to 975, written in Greek on parchment. It is one of three sets of copies of Archimedes' works that were available in the Middle Ages. (The others are lost, and neither contained the Stomachion.)
"For Archimedes, as for all others from antiquity, we don't have the original works," Dr. Netz said. "What we have are copies of copies of copies."
Investigators evaluate copies by asking whether they agree on the text they have in common, and by looking for unique passages, which lend them particular interest. By those measures, the manuscript was invaluable. But it was nearly lost.
In the 13th century, Dr. Netz explained, Christian monks, needing vellum for a prayer book, ripped the manuscript apart, washed it, folded its pages in half and covered it with religious text. After centuries of use, the prayer book — known as a palimpsest, because it contains text that is written over — ended up in a monastery in Constantinople.
Johan Ludvig Heiberg, a Danish scholar, found it in 1906, in the library of the Church of the Holy Sepulcher in Istanbul. He noticed faint tracings of mathematics under the prayers. Using a magnifying glass, he transcribed what he could and photographed about two-thirds of the pages. Then the document disappeared, lost along with other precious manuscripts in the strife between the Greeks and the Turks.
It reappeared in the 1970's, in the hands of a French family that had bought it in Istanbul in the early 20's and held it for five decades before trying to sell it. They had trouble finding a buyer, however, in part because there was some question of whether they legally owned it. But also, the manuscript looked terrible. It had been ravaged by mold in the years the family kept it, and it was ragged and ugly.
In 1998, an anonymous billionaire bought it for $2 million and lent it to the Walters Art Museum in Baltimore, where it still resides.
"I should emphasize how incredibly uncommon the situation is," Dr. Netz said.
With the manuscript in hand, a small group of scholars set out to reconstruct the original Greek text. It was not easy. "You look with the naked eye and you see nothing, absolutely nothing," Dr. Netz said.
Ultraviolet light revealed faint traces of writing, but it included both the prayers and the mathematics. "The major problem is the combination of the fact that many characters are hidden with the fact that many are so faint that they are invisible," Dr. Netz said. Then there are the gaps where the pages were ripped or eaten away by mold.
Computer imaging helped. Dr. Roger Easton of the Rochester Institute of Technology, Dr. Keith Knox of the Boeing Corporation and Dr. William Christens-Barry of Johns Hopkins University managed to write programs to pick out writing from the "noise" around it, and in many places the Greek letters fairly pop off the computer screen.
"The product of the software is incredible," Dr. Netz said. But it too has limitations, especially near the tattered edges of the pages. To reconstruct the writings, Dr. Netz and Dr. Nigel Wilson, a classics professor at Oxford University, are using every tool available: ultraviolet light, the computer images, Mr. Heiberg's photographs and their own intimate knowledge of ancient Greek texts. Still, in some areas, "the text is likely to remain a conjecture," Dr. Netz said.
It was chance that led Dr. Netz to his first insight into the nature of the Stomachion. Last August, he says, just as he was about to start transcribing one of the manuscript pages, he got a gift in the mail, a blue cut-glass model of a Stomachion puzzle. It was made by a retired businessman from California who found Dr. Netz on the Internet as a renowned Archimedes scholar.
Looking at the model, Dr. Netz realized that a diagram on the page he was transcribing was actually a rearrangement of the pieces of the Stomachion puzzle. Suddenly, he understood what Archimedes was getting at.
The diagram involved 14 pieces, and the word "multitude" seemed to be associated with it. Mr. Heiberg and those who followed him thought this meant that you could get many figures by rearranging the pieces.
"This is part of the reason people didn't see what it was about," Dr. Netz said. But the old interpretation seemed trivial, hardly worth Archimedes' time.
As he examined the manuscript pages, piecing together their text, he realized that what Archimedes was really asking seemed to be, "How many ways can you put the pieces together to make a square?" That question, Dr. Netz said, "has mathematical meaning."
"People assumed there wasn't any combinatorics in antiquity," he went on. "So it didn't trigger the observation when Archimedes says there are many arrangements and he will calculate them. But that's what Archimedes did; his introductions are always to the point."
But did Archimedes solve the problem? "I am sure he solved it or he would not have stated it," Dr. Netz said. "I do not know if he solved it correctly."
As for the name, derived from the Greek word for stomach, mathematicians are uncertain. But Dr. Diaconis has a hunch.
"It comes from `stomach turner,' " he said. "If you get involved with it, that's what happens."
LA TIMES ARTICLE
13th-Century Text Hides Words of Archimedes
By Jia-Rui Chong, Times Staff Writer
December 26, 2006
The book cost $2 million at auction, but large sections are unreadable.Some of its 348 pages are torn or missing and others are covered with sprawling purple patches of mildew. Sooty edges and water stains indicate a close escape from a fire.
"This manuscript is, by far, the worst of any manuscript I've ever seen," said William Noel, curator of manuscripts for the Walters Art Museum in Baltimore, where it now resides. "It's a book that is on its last legs.
"The sheepskin parchment originally contained a 10th century Greek text, which was erased by a 13th century scribe who replaced it with prayers. Seven hundred years later, a forger painted gilded pictures of the Evangelists on top of the faded words. Underneath it all, however, is an exceptional treasure — the oldest surviving copy of works by the ancient Greek mathematician and engineer Archimedes of Syracuse, who lived in the 3rd century BC.
About 80% of the text had been transcribed and translated in the 1910s after it was rediscovered in an Istanbul monastery, but since then much of it became unreadable again because of deterioration. Fully deciphering its mysteries has had to wait for advanced technologies, some of which had never been applied to ancient manuscripts.
The unusual cast of detectives includes not only the imaging specialists who helped photograph the Dead Sea Scrolls, but also a Stanford University physicist who studies trace metals in spinach with a particle accelerator. Together, they have been carrying out one of the most remarkable "salvage jobs" in the history of codicology, the study of ancient manuscripts.
Archimedes, it turns out, is only one secret of the text.
Among the mathematicians of antiquity, Archimedes was one of the greatest and most cunning.He was one of the earliest to devise ways to calculate the area beneath curves and was the first to prove that a circle's circumference and diameter are related by the constant pi. He developed the Archimedes Screw to lift water and invented deadly devices, such as the Claw of Archimedes, which was designed to grapple enemy warships.
Archimedes died in 212 BC, when Syracuse was sacked by the Romans. Legend holds that he was drawing figures in the sand. "Don't disturb my circles," he supposedly told the soldier who killed him.
Knowledge of Archimedes' work is derived from three books.Codex A, transcribed around the 9th century, contained seven major treatises in Greek. Codex B, created around the same time, had at least one additional work by Archimedes and survived only in Latin translation. Codex C has been an enigma.
It was originally copied down in 10th century Constantinople, now known as Istanbul. Three centuries later, the manuscript was in Palestine. By then, it was no longer a precious vestige of ancient learning but an obscure text that could be put to better use as a prayer book. A scribe began by unbinding the pages. He washed them with citrus juice or milk and sanded them with a pumice stone. He cut the sheets in half, turned them 90 degrees and stitched the new book down the middle.The scribe wrote prayers over the blank pages. Codex C had become a "palimpsest" — a recycled book.
The book eventually was brought back to Constantinople, where it sat until the 1890s, when a Greek scholar wrote down a fragment of erased text that he was able to read. That fragment was brought to the attention of Danish philologist Johan Ludvig Heiberg in 1906, then the foremost authority on Archimedes.
Armed with a magnifying glass, he translated everything he could read, publishing his work in 1910. The palimpsest disappeared amid the chaos of World War I, only resurfacing in 1998, when a French family named Guersan offered it for auction at Christie's in New York.
An anonymous book collector paid $2 million and deposited it at the Walters Art Museum for conservation. Mold had attacked much of the manuscript, and four forged paintings of the Evangelists made in the 20th century covered some of its most important pages."That was our worst nightmare," said Abigail Quandt, senior conservator of rare books and manuscripts at the Walters Art Museum.
Roger L. Easton Jr., a 56-year-old imaging specialist at the Rochester Institute of Technology, had just come off his success revealing hidden text in the Dead Sea Scrolls. Christie's had commissioned him to make ultraviolet images of the palimpsest for the auction catalog, and now he offered his help to the museum.
Easton and his colleagues began their work in 2000. They tinkered with different methods for capturing the image with the ultraviolet light, which makes the parchment glow more whitish.They then merged those images with another set taken under a tungsten light, which enhanced the reddish hue of the Archimedes text.
The resulting "pseudocolor" image made it easier to distinguish the black prayer book writing from the burnt sienna words of Archimedes. Using this painstaking method, Easton and his team took two years to uncover another 15% of the text.They were stymied in penetrating the rest.
Two more years passed before Stanford physicist Uwe Bergmann, 43, read a magazine article about the Archimedes palimpsest that mentioned it had originally been written with iron gall ink.
One of Bergmann's projects at Stanford was investigating the process of photosynthesis in plants by using the synchrotron X-rays to image small clusters of manganese atoms in spinach. "Why not find traces of iron in an ancient book?" he asked.
Bergmann sent an e-mail to the Walters Art Museum, and the museum agreed to a test. Bergmann set up the palimpsest experiment at the Stanford Synchrotron Radiation Laboratory. Spread over an area the size of a football field, the synchrotron is part of the Stanford Linear Accelerator Center, a Department of Energy facility set in the foothills of Menlo Park.
The synchrotron hurls electrons at near light speed, forcing them to give off X-rays as they veer around bends. That X-ray beam is channeled away into the laboratories. Bergmann figured the powerful and precise beam could be used to make iron molecules fluoresce, thus allowing him with a sensitive-enough detector to pick up even the faintest traces of ink. Bergmann first had to determine the exposure time. Too much time and the powerful synchrotron X-ray could damage the parchment. Then, they adjusted the intensity of the beam, which could be so strong that it blinded the detectors that picked up the glow from the iron gall ink.
After two years of refining their technique, Bergmann and his colleagues began the laborious process of imaging the palimpsest this summer.Each side of a page, mounted in frame that moved in front of the beam, took 12 hours to record. The machines processed the pages continuously for two weeks.
Beneath a moldy, torn painting of St. John emerged two layers of writing.On the edge of the first page, they saw a signature dated April 14, 1229: "By the hand of presbyter Ioannes Myronas."It was the name of the priest who had erased Archimedes. In an office near Memorial Church at Stanford, Reviel Netz flicked off the lights. Netz, a slight 38-year-old with dark hair, leaned close to the screen of his laptop.
Bergmann's X-ray work had produced a black-and-white picture of a page from "The Method of Mechanical Theorems," a text found only in the palimpsest. One phrase — "let them be arranged so they balance on point theta" — had already been translated by Heiberg, although he had had to guess about the word "on," which was unreadable.
Netz, a professor of classics, looked at the X-ray image and nodded. He smiled.The actual word was "around.""That's not trivial," he said, explaining that the change altered the meaning of Archimedes' calculations involving an object's center of gravity.
The X-ray image also revealed a section of "The Method" that had been hidden from Heiberg in the fold between pages. It contained part of a discussion on how to calculate the area inside a parabola using a new way of thinking about infinity, Netz said. It appeared to be an early attempt at calculus — nearly 2,000 years before Isaac Newton and Gottfried Wilhelm Leibniz invented the field.
The discoveries may seem small, but they are significant in the understanding of ancient mathematics, Netz said. One passage he studied several years ago involved the innumerable slices and lines that could be made from a triangular prism similar to a wedge of cheese. Netz said the passage, which was unreadable to Heiberg, showed that Archimedes was grappling with the concept of infinity long before other mathematicians.
For Netz, a specialist in ancient mathematics and cognitive history, the chance to decipher the palimpsest "is the fulfillment of an incredible dream," he said. One of his biggest breakthroughs involves a quirky part of the palimpsest called the "Stomachion," which literally means "Belly-Teaser.
"Stomachions were children's games in which 14 geometrical shapes were rearranged to create new shapes. Heiberg translated fragments of the manuscript but paid little attention to it, thinking it was just a game. Netz saw a deeper significance.
Archimedes asked a more restricted question in his "Stomachion": How many different ways could you combine the 14 triangles to make a square?
Netz believes the fragments address an area of mathematics known as combinatorics that scholars have only recently believed interested the Greeks. For all the high-tech efforts, there are still gaps remaining in the Archimedes text, perhaps 2%, Netz guessed. Among the jumbled fragments are clues that perhaps the deepest secrets are yet to be found.
A century ago, Heiberg copied down two lines that he couldn't identify. They began: "The youngest had been abroad for so long that the sisters wouldn't even know who was who."The passage was not Archimedes.
In 2002, scholars were able to cross-reference the quote. It came from "Against Timandros," written by a 4th century BC Athenian orator named Hyperides. Although Hyperides is little-known now, contemporaries frequently compared him to Demosthenes, an acknowledged master of oratory.
No complete versions exist of "Against Timandros," which Hyperides had written as part of a lawsuit over an inheritance, said Judson Herrman, a classicist at Allegheny College in Pennsylvania. Further study determined there were 20 pages of Hyperides in the palimpsest, including a previously unknown text called "Against Diondas."
The palimpsest, it turns out, took parchment from seven texts, including what are believed to be a commentary on Aristotle's "On the Soul" and a group of biographies of the saints, plus two still unidentified texts.The works are even more difficult to discern than the Archimedes because the ink is different and the pages more thoroughly scrubbed.
"I have been cursing all morning," Herrman said of his work on a few lines of Hyperides. The scientists aren't giving up. Easton's team recently began experimenting with precisely tuned light-emitting diodes, or LEDs, to illuminate the text.
The team also is using angled light to detect the outlines of letters etched in the parchment by the acid in the ink. The team made progress on a few pages, but it may take decades — or longer — before technologies are developed that can unveil all the texts."We'll probably leave something for future scientists to work on," Netz said.
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