Archimedes Palimpsest

The Archimedes Palimpsest is a palimpsest on parchment in the form of a codex. It originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse and other authors, which was overwritten with a religious text. Archimedes lived in the third century BC, but the copy of his work was made in the tenth century AD by an anonymous scribe. In the twelfth 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. 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. The erasure was incomplete, and Archimedes' work is now readable after scientific and scholarly work from 1998 to 2008 using digital processing of images produced by ultraviolet, infrared, and visible light, and X-ray. [1][2]

In 1906 it was briefly inspected in Constantinople by the Danish philologist Johan Ludvig Heiberg. With the aid of black and white photographs he arranged to have taken, he published a transcription of the Archimedes' text. Shortly thereafter Archimedes' Greek text was translated into English by Thomas Heath. Before that it was not widely known among mathematicians, physicists, or historians. It contains

"Equilibrium of Planes"
"Spiral Lines"
"Measurement of a Circle"
"Sphere and Cylinder"
"On Floating Bodies" (only known copy in Greek)
"The Method of Mechanical Theorems" (only known copy)
"Stomachion" (only known copy)

The palimpsest also contains speeches by the fourth century BC politician Hypereides, a commentary on Aristotle's Categories by Alexander of Aphrodisias, and other works. [3]

Mathematical content

The most remarkable of the above works is The Method, of which the palimpsest contains the only known copy.

In his other works, Archimedes often proves the equality of two areas or volumes with Eudoxes' method of exhaustion, an ancient Greek counterpart of the modern method of limits. Since the Greeks were aware that some numbers were irrational, their notion of a real number was a quantity Q which is approximated by two sequences, one providing an upper bound and the other a lower bound. If you find two sequences U and L, with U always bigger than Q, and L always smaller than Q, and if the two sequences eventually came closer together than any prespecified amount, then Q is found, or exhausted, by U and L.

Archimedes would use exhaustion to prove his theorems. This involved approximating the figure whose area he wanted to compute into sections of known area, which provide upper and lower bounds for the area of the figure. He would then prove that the two bounds become equal when the subdivision becomes arbitrarily fine. These proofs, still considered to be rigorous and correct, used geometry with rare brilliance. Later writers often criticized Archimedes for not explaining how he arrived at his results in the first place. This explanation is contained in The Method.

The method that Archimedes' describes was based upon his investigations of physics, on the center of mass and the law of the lever. He would compare the area or volume of a figure for which he knew the total mass and the location of the center of mass with the area or volume of another figure he didn't know anything about. He would divide both figures into infinitely many slices of infinitesimal width, and he would balance each slice of one figure against a corresponding slice of the second figure on a lever. The essential point is that the two figures are oriented differently, so that the corresponding slices are at different distances from the fulcrum, and the condition that the slices balance is not the same as the condition that they are equal.

Once he shows that each slice of one figure balances each slice of the other figure, he concludes that the two figures balance each other. But the center of mass of one figure is known, and the the total mass can be placed at this center and it would still balance. The second figure has an unknown mass, but the position of its center of mass might be restricted to lie at a certain distance from the fulcrum by a geometrical argument, by symmetry. The condition that the two figures balance now allows him to calculate the total mass of the other figure. He considered this method as a useful heuristic but always made sure to prove the results he found using exhaustion, since the method did not provide upper and lower bounds.

Using this method, Archimedes was able to solve several problems that would now be treated by integral calculus, which was given its modern form in the seventeenth 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. (For explicit details, see Archimedes' use of infinitesimals.)

When rigorously proving theorems, Archimedes often used what are now called Riemann sums. In On the Sphere and Cylinder, he gives upper and lower bounds for the surface area of a sphere by cutting the sphere into sections of equal width. He then bounds the area of each section by the area of an inscribed and circumscribed cone, which he proves have a larger and smaller area correspondingly. He adds the areas of the cones, which is a type of Riemann sum for the area of the sphere considered as a surface of revolution.

But there are two essential differences between Archimedes' method and 19th century methods:

Archimedes didn't know about differentiation, so he couldn't calculate any integrals other than those which came from center of mass considerations, by symmetry. While he had a notion of linearity, to find the volume of a sphere he had to balance two figures at the same time, he never figured out how to change variables or integrate by parts.
When calculating approximating sums, he imposed the further constraint that the sums provide rigorous upper and lower bounds. This was required because the greeks lacked algebraic methods which could establish that error terms in an approximation are small.

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 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.

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. Reviel Netz of Stanford University has argued that Archimedes discussed the number of ways to solve the puzzle. Modern combinatorics leads to the result that this number is 17,152. Due to the fragmentary state of the palimpsest it is unknown whether or not Archimedes came to the same result. This may have been the most sophisticated work in the field of combinatorics in Greek antiquity.

Modern history

A typical page from the Archimedes Palimpsest. The text of the prayer book is seen from top to bottom, the original Archimedes manuscript is seen as fainter text below it running from left to right

The page from the palimpsest after imaging, the original Archimedes text is now seen clearly

From the 1920s, the manuscript lay unknown in the Paris apartment of a collector 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 buyer who worked in the information technology field.

At the Walters Art Museum in Baltimore, the palimpsest was the subject of an extensive imaging study from 1999 to 2008, and conservation (as it had suffered considerably from mold). This was directed by Dr. Will Noel, curator of manuscripts at the Walters Art Museum, and managed by Michael B. Toth of R.B. Toth Associates, with Dr. Abigail Quandt performing the conservation of the manuscript.

A team of imaging scientists including Dr. Roger Easton from the Rochester Institute of Technology, Dr. Bill Christens-Barry from Equipoise Imaging, and Dr. Keith Knox with Boeing LTS used computer processing of digital images from various spectral bands, including ultraviolet and visible light, to reveal most of the underlying text, including of Archimedes. After imaging and digitally processing the entire palimpsest in three spectral bands prior to 2006, in 2007 they reimaged the entire palimpsest in 12 spectral bands, plus raking light: UV: 365 nanometers; Visible Light: 445, 470, 505, 530, 570, 617, and 625nm; Infrared: 700, 735, and 870nm; and Raking Light: 910 and 470nm.[4] The team digitally processed these images to reveal more of the underlying text with pseudocolor. Dr. Reviel Netz [5] of Stanford University and Nigel Wilson have produced a diplomatic transcription of the text, filling in gaps in Heiberg's account with these images.

Sometime after 1938, one owner of the manuscript forged four Byzantine-style religious images in the manuscript in an effort to increase its value. It appeared that these had rendered the underlying text forever illegible. However, in May 2005, highly-focused X-rays produced at the Stanford Linear Accelerator Center in Menlo Park, California, were used by Drs. Uwe Bergman and Bob Morton 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 near 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." [6]

In April 2007 it was announced that a new text had been found in the palimpsest, which was a commentary on the work of Aristotle attributed to Alexander of Aphrodisias. Dr. Will Noel said in an interview: "You start thinking striking one palimpsest is gold, and striking two is utterly astonishing. But then something even more extraordinary happened." This referred to the previous discovery of a text by Hypereides, an Athenian politician from the fourth century BC, which has also been found within the palimpsest.[7] It is from his speech Against Diondas, and was published in 2008 in the German scholarly magazine Zeitschrift für Papyrologie und Epigraphik, vol. 165, becoming the first new text from the palimpsest to be published in a scholarly journal.[8]

The transcriptions of the book were digitally encoded using the Text Encoding Initiative guidelines, and metadata for the images and transcriptions included identification and cataloging information based on Dublin Core Metadata Elements. The metadata and data was managed by Dr. Doug Emery of Emery IT.

On October 29, 2008, (the tenth anniversary of the purchase of the palimpsest at auction) all data, including images and transcriptions, were hosted on the Digital Palimpsest Web Page for free use under Creative Commons License, and processed images of the palimpsest in original page order were posted as a Google Book.[9]

References

Reviel Netz and William Noel, The Archimedes Codex, Weidenfeld & Nicolson, 2007
Dijksterhuis, E.J.,"Archimedes", Princeton U. Press, 1987, pages 129- 133. copyright 1938, ISBN 0-691-08421, 0-691-02400-6

External links