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November 23, 2004
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image: The Walters Art Museum
Scientists are decoding an ancient mathematics book that contains 2000-year-old ideas that could have changed the course of history. Written by the Greek mathematician Archimedes, who was born about 287 B.C., the book contains concepts that others wouldn't discover for centuries.

As shown on PBS’s NOVA, the ancient book is not exactly in pristine shape. “The manuscript was heavily damaged by mold,” Abigail Quandt, senior conservator of manuscripts and rare books at the Walters Art Museum told NOVA. “The parchment is perforated where the fungi have actually gone through and digested the collagen, and it means that the Archimedes text is just totally missing in these areas.”

But the larger problem with the book is that Archimedes’s text was hidden some 800 years ago by a medieval monk who ran out of paper for his prayer book. The monk took pages out of the Archimedes book, turned them sideways, washed away the ink, and wrote over it. Still, there is a faint trace of the Archimedes text, and some 900 years later, a team of scientists is teasing the two texts apart.

"We are trying to take advantage of the very slight differences in color of the two inks, of the inks from the Archimedes text and the later ink,” explained Roger Easton, associate professor of optical sciences at Rochester Institute of Technology. “So we did that by taking images at a wide variety of wavelengths."

image: The Walters Art Museum
Pages were scanned into a computer and then looked at in blue and red light. Then the computer processed those together and came up with a "false color" image in which the two different texts appeared as different colors that were easier to distinguish. "I was amazed by the fact that now for the first time I can look at pages that look hopeless with the naked eye and begin to use them as text from which you just read,” said Reviel Netz, assistant professor of classics at Stanford University. “We are able to recover the original text of Archimedes where it appears to have been lost. I think we will be able to read everything there for the first time.”

The invention of calculus, the mathematical basis for much of 21st century technology, is generally thought to have taken place near the end of the 17th century. But now it looks like Archimedes was well on his way to that crucial discovery. One of the treatises in the book was called On the Method of Mechanical Theorems and it took into account a difficult concept that is central to calculus: infinity.

"In these informal proofs Archimedes thought of solid bodies as consisting of infinitely many slices. For example, thinking of a salami as consisting of infinitely many circular slices of zero thickness," explains Chris Rorres, a former mathematics professor and Archimedes scholar. "This leads to logical problems since then each of the infinitely many slices has zero volume and the total volume of the body involves multiplying infinity by zero in some sense. Archimedes understood these logical difficulties with infinity."

"We always knew that Archimedes was making a step in the direction leading to modern calculus," said Netz. "What we have found right now is that, in a sense, Archimedes was already there. He already did develop a special tool with which you can sum up infinitely many objects and measure a volume."

While these concepts are ones we have known for quite some time now, if the mathematicians who followed Archimedes had had his Method to work with, the resulting progress of science and technology would likely have been affected. "If the book had been available 100 years before the development of the calculus, then things would have got going sooner,” said Alexander Jones, a classics professor at the University of Toronto.

Rorres agrees: “If the mathematicians and scientists of the Renaissance had been aware of these discoveries of Archimedes, this could have had a tremendous impact on the development of mathematics. We could have been on Mars today. We could have accomplished all of the things that people are predicting for a century from now. You would basically be raising the tide by increasing the knowledge of mathematics several hundred years ago."

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