Hidden text reveals Archimedes' genius

FROM ancient Syracuse, through the medieval Holy Land to Istanbul and, finally, California, it has been a long journey for a musty old prayer book. But what is written on it makes the journey worthwhile. "This is Archimedes' brain on parchment," says William Noel, curator of ancient manuscripts at the Walters Art Museum in Baltimore, Maryland. Hidden beneath the lines of ancient prayers and layers of dirt, candle wax and mould lies the oldest written account of the thoughts of the great mathematician.

This invaluable artifact is a classic example of a palimpsest: a manuscript in which the original text has been scraped off and overwritten. It was discovered more than a century ago, but only in the past eight years have scholars uncovered its secrets. Using advanced imaging techniques, they have peered behind the 13th-century prayers inscribed on its surface to reveal the text and diagrams making up seven of Archimedes' treatises. They include the only known copies of The Method of Mechanical Theorems, On Floating Bodies and fragments of The Stomachion in their original Greek.

As the investigation drew to a close in August, the impact of these discoveries became clear. What one of the experts described as "a very drab and dirty object" sheds fresh light on how Archimedes developed proofs and theorems, and shows that he may have employed and understood the concept of infinity more rigorously than previously thought. It also suggests that Archimedes discovered the field of mathematics called combinatorics, an important technique in modern computing. These are remarkable discoveries, yet it is only through a chain of chance events that the text was discovered at all.

The story begins in ancient Greece. Little is known about Archimedes' life other than that he was born in Syracuse, Sicily, around 287 BC, educated at Alexandria in Egypt and was the son of an astronomer. He is probably most famous for devising a way of calculating an object's density. King Hiero asked Archimedes to see if a crown was made of solid gold or, as he suspected, a mix of cheaper metals. Legend has it that Archimedes' moment of inspiration occurred in the bath. He realised that by dividing his weight by the volume of water his body displaced, he could to calculate its average density. The same would work for any object, Hiero's crown included. In his excitement at solving the problem he is said to have jumped out of the bath shouting "Eureka!"

Archimedes wrote his mathematical treatises on scrolls. Though the originals are all lost, copies had been made onto papyrus and parchment. Today only three books containing Archimedes' texts remain: codices A, B and C. Of these, the first two are medieval Latin translations, held in the Vatican library. It is now known that the third, codex C, was written on parchment in Constantinople - the modern-day Turkish city of Istanbul - around AD 1000. It is the only one containing The Method and also contains a fragment of The Stomachion. Somehow, it wound up in the monastery of St Sabas near Jerusalem, where in 1229 a Christian monk unceremoniously pulled the manuscript apart, scraped the pages clean, rotated them by 90 degrees, folded them in two and wrote an orthodox prayer book called the Euchologion over it.

The prayer book lost several leaves through heavy use, but remained otherwise intact and eventually found its way to the Church of the Holy Sepulchre in Istanbul. There it lay until, in 1906, Johan Ludvig Heiberg, a professor of the history of mathematics from the University of Copenhagen in Denmark, studied the manuscript and realised the significance of the mathematical text faintly visible in the margins and beneath the prayers. He identified it as containing The Method, The Stomachion and On Floating Bodies alongside further works by Archimedes and other unidentified texts.

A few months later, the manuscript went missing. It resurfaced briefly when a French family living in Istanbul announced they had bought it, and there it remained, untouched for several decades. Descendants tried unsuccessfully to sell it to public institutions in Paris and London in the early 1990s, and finally, in 1998, offered the manuscript on open auction at Christie's in New York.

It sold for $2 million to an anonymous millionaire known as "Mr B". Fortunately, he turned out to be both enlightened and generous. He responded to an email from Noel asking to display the palimpsest at the Walters Art Museum. That simple request kicked off a new chapter in the saga: the Archimedes Palimpsest Project. It brought together an international team of conservators, mathematicians, imaging experts and physicists to unlock the secrets hidden within the prayer book. Mr B funded the work, spending almost as much as he had paid for the manuscript. Many involved worked for free out of the conviction that Archimedes deserved to be heard from the grave.

Among the eager scholars lining up to examine the palimpsest, one man had a head start. Nigel Wilson, a classics scholar and retired tutor at Lincoln College, Oxford, UK, had been asked to examine and describe the palimpsest for Christie's catalogue, and almost 30 years earlier he had identified a fragment from a single palimpsest folio - then held at the University of Cambridge - as containing text by Archimedes. That folio turned out to be one of the manuscript's missing pages.

Abigail Quandt, a senior conservator of rare books and manuscripts at the Walters Art Museum, took on the painstaking job of conserving and disbinding the manuscript. Wilson quickly became part of the project. "I realised at once that if you could apply even an ultraviolet lamp to the manuscript, you'd be able to read a great deal more than was read in 1907", when Heiberg was first transcribing the palimpsest.

The Heiberg translation was a constant reference point for the team. He had examined the palimpsest using only a magnifying glass, but it was in considerably better condition then than it was by the end of the century. At the time, there were 177 folios, of which three have since been lost. "In 1906, mildew had not yet begun to attack it," says Wilson. Luckily, Heiberg took several photographs of the manuscript, which were rediscovered at the Danish Royal Library in Copenhagen and digitally reproduced. They filled in some of the gaps where the parchment had been eaten away by mould.

Most of the text, however, was retrieved using multispectral imaging, a technique in which wavelengths of light not visible to the human eye are beamed at the parchment and the reflected light is captured and converted by computer into a visible image. Algorithms then enhanced selected parts of the text, revealing traces of ink that are too faint to see.

Absorbing thoughts
This work was led by three imaging specialists: Roger Easton, professor of imaging at the Rochester Institute of Technology, New York; Keith Knox from Boeing in Seattle, Washington; and William Christens-Barry of Equipoise Imaging in Ellicott City, Maryland. Between them they refined the imaging technique specifically for the palimpsest by combining two different wavelengths from the red and blue parts of the spectrum. The parchment reflected both red and blue light, making it appear almost white. The pigments in the prayer ink absorbed these wavelengths and appeared black, while Archimedes' text absorbed the blue and reflected the red, appearing as a legible red script.

Yet some parts of the text remained obscure. Physicist Uwe Bergmann at the Stanford Linear Accelerator Center in California read about the problems the imaging project was having in the German magazine Der Spiegel and thought he might be able to help. The ink in the script contained iron, and Bergmann realised that a technique called X-ray fluorescence might reveal it. X-ray fluorescence relies on the fact that when an X-ray photon strikes an iron atom it knocks out an electron, which is immediately replaced by another electron dropping in from a higher energy state to fill the gap. This releases a photon of light with a characteristic wavelength. "We record that flash, note the position where the X-ray beam struck the page and reconstruct this in a digital image."

Even then, the task was far from straightforward, as the inks used for the prayer book also contained iron. To make matters worse, several pages had been covered with illustrations containing metals such as gold, lead and zinc, probably painted in the 1930s by forgers attempting to increase the manuscript's value. Despite these problems, Bergmann and his team managed to decode 15 pages that had failed to yield their secrets to multi-spectral imaging analysis. Of special interest were the pages within The Method, in which there were hints that Archimedes discussed infinity. This was a huge surprise.

It has long been accepted wisdom among historians that the ancient Greeks did not use infinity. Historian Ken Saito of Osaka Prefecture University in Japan and Reviel Netz, a professor of classics at Stanford and an expert on pre-modern mathematics, now think otherwise. Netz edited the Archimedes text and believes that the great mathematician not only knew about infinity, but was calculating with it, using an early form of calculus. "We always knew about Archimedes' role in perfecting the Greek method of dealing with infinity in a roundabout way," but now there was evidence of Archimedes talking about infinity as a kind of number, Netz says - unique in Greek thought, as far as he can tell.

Netz also proposes an intriguing explanation for the Stomachion (see Diagram) - the name given to an ancient puzzle, or "tangram", in one of Archimedes' treatises. A tangram is a puzzle in which a square is divided into different geometric shapes and, like a jigsaw, must be put back together. In the treatise the square is divided into 11 triangles, two quadrilaterals and a pentagon. Many assumed that Archimedes simply included it as a challenging game.

But then Netz was given a tangram by someone who had read about his work. Its shapes were ordered differently to the way he'd expected, and that sparked his own eureka moment. He realised that Archimedes might have included the Stomachion to demonstrate multiple solutions to a problem. This suggests that the question Archimedes was tackling was: "how many ways are there to complete a square, given the 14 pieces of the puzzle?", Netz says. "This would be interesting as an example of a very early study of combinatorics - the study of the number of ways in which a given problem can be solved."

Several mathematicians raced to work out the number of unique solutions, but it was Bill Cutler, a mathematician and computer scientist based in Palatine, Illinois, who produced software that came up with an answer: 536. This number was finally confirmed on paper, using a method Netz believes Archimedes would have approved of (SCIAMVS, vol 5, p 67).

For Noel, one of the most striking discoveries was finding the name of the 13th-century scribe whose work caused the researchers so much difficulty. Noel and Netz dedicated their book about the project to him: Ioannes Myronas. For Wilson, though, there is still important unfinished business. "I can't identify the hand of the scribe who penned the Archimedes text itself," he confesses. "I'm still looking for him. He's a wanted man."