Archimedes Palimpsest
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The Archimedes Palimpsest is a palimpsest on parchment, in which a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse, who lived in the third century BC, was made in the 10th century. 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 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 known among either mathematicians or historians of mathematics.
Media hype
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. Among those problems were that of the center of gravity of a solid hemisphere, that of the center of gravity of a frustum of a circular paraboloid, and that of 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.
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 lawsuit
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. 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 sold for $2 million.
Now in a museum
The palimpsest is now on display at the Walters Art Museum in Baltimore, where work of conservation continues, and a more accurate edition of the manuscript, including its drawn geometrical figures, is expected.
