# Archimedes Palimpsest

The Archimedes Palimpsest is a palimpsest on parchment in the form of a codex which originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse. Archimedes lived in the third century BC, but the copy was made in the 10th century by an anonymous scribe. In the 12th century the codex was unbound and washed, in order that the parchment leaves could be folded in half and reused for a Christian liturgical text. Fortunately, the erasure was incomplete, and Archimedes' work is still in large part legible today, using combinations of ultraviolet and visible light. It was a book of nearly 90 pages before being made a palimpsest of 177 pages; the older leaves folded so that each became two leaves of the liturgical book.

In 1906 it was briefly inspected in Constantinople and was published, from photographs, by the Danish philologist Johan Ludvig Heiberg (1854-1928); shortly thereafter it was translated into English by Thomas Heath. Before that it was not widely known among mathematicians, physicists, or historians.

Many statements on the web on the topic of the Archimedes Palimpsest are full of hyperbole. They can leave the erroneous impression that none of Archimedes' works are known except this one, or that only since the late 1990s, when modern techniques began to be used to fill in some lacunae, did anyone know the content of this palimpsest.

## What Archimedes did

Although the only mathematical tools at its author's disposal were what we might now consider secondary-school geometry, Archimedes used those methods with rare brilliance, explicitly using infinitesimals to solve problems that would now be treated by integral calculus, which was independently reinvented in the 17th century by Isaac Newton and Gottfried Leibniz. Among those problems were that of calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. Contrary to historically ignorant statements found in some 20th century calculus textbooks, he did not use anything like Riemann sums, either in the work embodied in this palimpsest or in any of his other works. For explicit details of the method used, see how Archimedes used infinitesimals.

A problem solved exclusively in the Method is the calculation of the volume of a cylindrical wedge, a result that reappears as theorem XVII (schema XIX) of Kepler's Stereometria.

Some pages of the Method remained unused by the author of the Palimpsest and thus they are still—probably forever—lost. Between them, an announced result concerned the volume of the intersection of two cylinders, a figure that Apostol and Mnatsakian have renamed n = 4 Archimedean globe (and the half of it, n = 4 Archimedean dome), whose volume relates to the n-polygonal pyramid. This is amusing because the collaboration on indivisibles between Galileo and Cavalieri—ranging between years 1626 to around 1635—has as a main argument the hull and pyramid of the n = ∞ dome. So in some sense it is true that the Method is only a theorem behind the modern infinitesimal theory {not clear!}.

Historian Reviel Netz of Stanford University, with technical assistance from several persons at the Rochester Institute of Technology, has been trying to fill in gaps in Heiberg's account. In Heiberg's time, much attention was paid to Archimedes' brilliant use of infinitesimals to solve problems about areas, volumes, and centers of gravity. Less attention was given to the Stomachion, a problem treated in the Palimpsest that appears to deal with a children's puzzle. Netz has shown that Archimedes found that the number of ways to solve the puzzle is 17,152. This is perhaps the most sophisticated work in the field of combinatorics in classical antiquity.

## The palimpsest's modern career

From the 1920s, the manuscript lay unknown in the Paris apartment of an amateur of manuscripts and his heirs. In 1998 the ownership of the palimpsest was disputed in federal court in New York in the case of the Greek Orthodox Patriarchate of Jerusalem versus Christie's, Inc. At some time in the distant past, the Archimedes manuscript had lain in the library of Mar Saba, near Jerusalem, a monastery bought by the Patriarchate in 1625. The plaintiff contended that the palimpsest had been stolen from one of its monasteries in the 1920s. Judge Kimba Wood decided in favor of Christie's Auction House on laches grounds, and the palimpsest was bought for \$2 million by an anonymous IT person.

The palimpsest is now on display at the Walters Art Museum in Baltimore, where work of conservation continues, (as it had suffered considerably from mould), and a more accurate edition of the manuscript, including its drawn geometrical figures, is expected.

Four pages that had been painted over with Byzantine-style religious images, which turned out to be 20th century forgeries intended to increase the value of the prayer book, rendered the underlying text of Archimedes forever illegible, it appeared. Then, in May 2005, highly focused X-rays produced at the Stanford Linear Accelerator Center in Menlo Park, California were used to begin deciphering the parts of the 174-page text that have not yet been revealed. The production of x-ray fluorescence was described by Keith Hodgson, director of SSRL. "Synchrotron light is created when electrons traveling the speed of light take a curved path around a storage ring—emitting electromagnetic light in X-ray through infrared wavelengths. The resulting light beam has characteristics that make it ideal for revealing the intricate architecture and utility of many kinds of matter—in this case, the previously hidden work of one of the founding fathers of all science." [1] (http://news-service.stanford.edu/news/2005/may25/archimedes-052505.html).

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