Archimedes Palimpsest

History

Early

Archimedes lived in the 3rd century BC and wrote his proofs as letters in Doric Greek addressed to contemporaries, including scholars at the Great Library of Alexandria. These letters were first compiled into a comprehensive text by Isidorus of Miletus, the architect of the Hagia Sophia patriarchal church, sometime around AD 530 in the then Byzantine Greek capital city of Constantinople.

A copy of Isidorus' edition of Archimedes was made around AD 950 by an anonymous scribe, again in the Byzantine Empire, in a period during which the study of Archimedes flourished in Constantinople, especially at the Magnaura, a school founded by the mathematician, engineer, and former Greek Orthodox archbishop of Thessaloniki, Leo the Geometer, a cousin of the Patriarch.

This medieval Byzantine manuscript then traveled from Constantinople to Jerusalem, likely sometime after the Crusader sack of Byzantine Constantinople in 1204. There, in 1229, the Archimedes codex was unbound, scraped and washed, along with at least six other partial parchment manuscripts, including one with works of Hypereides. Their leaves were folded in half, rebound and reused for a Christian liturgical text of 177 later numbered leaves, of which 174 are extant (each older folded leaf became two leaves of the liturgical book).

The manuscript includes a colophon which was deciphered in 2002 and found to contain the date April 13, 1229, which is likely when the prayerbook was finished. This was notably only a few months after the implementation of the Treaty of Jaffa, which returned Jerusalem to Christian control, bringing the Sixth Crusade to its end. The volatile political situation in the region may have resulted in a shortage of parchment, possibly providing one reason the scribe may have recycled the earlier manuscript.

The palimpsest remained near Jerusalem through at least the 16th century at the isolated Greek Orthodox monastery of Mar Saba. At some point before 1840 the palimpsest was brought back by the Greek Orthodox Patriarchate of Jerusalem to its library (the Metochion of the Holy Sepulcher) in Constantinople.

Modern

The Biblical scholar Constantin von Tischendorf visited Constantinople in the 1840s, and, intrigued by the Greek mathematics visible on the palimpsest he found in a Greek Orthodox library, removed a leaf of it (which is now in the Cambridge University Library). In 1899, the Greek scholar Papadopoulos-Kerameus produced a catalog of the library's manuscripts and included a transcription of several lines of the partially visible underlying text. Upon seeing these lines Johan Heiberg, the world's authority on Archimedes, realized that the work was by Archimedes. When Heiberg studied the palimpsest in Constantinople in 1906, he confirmed that the palimpsest included works by Archimedes thought to have been lost. Heiberg was permitted by the Greek Orthodox Church to take careful photographs of the palimpsest's pages, and from these he produced transcriptions, published between 1910 and 1915, in a complete works of Archimedes. Shortly thereafter Archimedes' Greek text was translated into English by Thomas Heath. Before that it was not widely known among mathematicians, physicists or historians.

The manuscript was still in the Greek Orthodox Patriarchate of Jerusalem's library (the Metochion of the Holy Sepulchre) in Constantinople in 1920. Shortly thereafter, during a turbulent period for the Greek community in Turkey that saw a Turkish victory in the Greco-Turkish War (1919–22) along with the Greek genocide and the forced population exchange between Greece and Turkey, the palimpsest disappeared from the Greek church's library in Istanbul.

Sometime between 1923 and 1930, the palimpsest was acquired by Marie Louis Sirieix, a "businessman and traveler to the Orient who lived in Paris." Though Sirieix claimed to have bought the manuscript from a monk, who would not in any case have had the authority to sell it, Sirieix had no receipt or documentation for a sale of the valuable manuscript. Stored secretly for years by Sirieix in his cellar, the palimpsest suffered damage from water and mold. In addition, after its disappearance from the Greek Orthodox Patriarchate's library, a forger added copies of medieval evangelical portraits in gold leaf onto four pages in the book in order to increase its sales value, further damaging the text. These forged gold leaf portraits nearly obliterated the text underneath them, and x-ray fluorescence imaging at the Stanford Linear Accelerator Center would later be required to reveal it.

Sirieix died in 1956, and, in 1970, his daughter began attempting quietly to sell the valuable manuscript. Unable to sell it privately, in 1998, she finally turned to Christie's to sell it in a public auction, risking an ownership dispute. The ownership of the palimpsest was immediately contested in federal court in New York, where the auction was to take place, in the case of the Greek Orthodox Patriarchate of Jerusalem v. Christie's, Inc. The Greek church contended that the palimpsest had been stolen from its library in Constantinople in the 1920s, during a period of extreme persecution. 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 American buyer. The lawyer who represented the anonymous buyer stated that the buyer was "a private American" who worked in "the high-tech industry", but was not Bill Gates.

Imaging and digitization

The anonymous new owner loaned the book to the Walters Art Museum in Baltimore, where the palimpsest was the subject of an extensive imaging study from 1999 to 2008, and conservation (as it had suffered considerably from mold while in Sirieix's cellar). This was directed by Dr. William 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.

The target audiences for the digitisation are Greek scholars, math historians, people building applications, libraries, archives, and scientists interested in the production of the images.

A team of imaging scientists including Dr. Roger L. Easton, Jr. from the Rochester Institute of Technology, Dr. William A. Christens-Barry from Equipoise Imaging, and Dr. Keith Knox (then with Boeing LTS, now retired from the USAF Research Laboratory) used computer processing of digital images from various spectral bands, including ultraviolet, visible, and infrared wavelengths to reveal most of the underlying text, including that 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 625 nm; Infrared: 700, 735, and 870 nm; and Raking Light: 910 and 470 nm. The team digitally processed these images to reveal more of the underlying text with pseudocolor. They also digitized the original Heiberg images. Dr. Reviel Netz 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, a forger placed four Byzantine-style religious images in the manuscript in an effort to increase its sales 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 Bergmann and Bob Morton to begin deciphering the parts of the 174-page text that had 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.

Contents

Works contained within

On the Equilibrium of Planes

On Spirals

Measurement of a Circle

On the Sphere and Cylinder

On Floating Bodies

The Method of Mechanical Theorems

Ostomachion

Speeches by the 4th-century BC politician Hypereides (Against Diondas and Against Timandros)

A commentary on Aristotle's Categories by Porphyry (or by Alexander of Aphrodisias)

Fragments of a tenth-century Menaion

Fragments of a hagiographical text, Life of St. Pantoleon

Two other unidentified fragments

The Method of Mechanical Theorems

The palimpsest contains the only known copy of The Method of Mechanical Theorems.

In his other works, Archimedes often proves the equality of two areas or volumes with Eudoxus' 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 approximated by two sequences, one providing an upper bound and the other a lower bound. If one finds two sequences U and L, and U is 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 used 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 then proved 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 compared the area or volume of a figure of which he knew the total mass and center of mass with the area or volume of another figure he did not know anything about. He viewed plane figures as made out of infinitely many lines as in the later method of indivisibles, and balanced each line, or 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 the figures 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 total mass can be placed at this center and it still balances. 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.

Stomachion

In Heiberg's time, much attention was paid to Archimedes' brilliant use of indivisibles to solve problems about areas, volumes, and centers of gravity. Less attention was given to the Ostomachion, 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, that is, to put the pieces back into their box. No pieces have been identified as such; the rules for placement, such as whether pieces are allowed to be turned over, are not known; and there is doubt about the board.

The board illustrated here, as also by Netz, is one proposed by Heinrich Suter in translating an unpointed Arabic text in which twice and equals are easily confused; Suter makes at least a typographical error at the crucial point, equating the lengths of a side and diagonal, in which case the board cannot be a rectangle. But, as the diagonals of a square intersect at right angles, the presence of right triangles makes the first proposition of Archimedes' Ostomachion immediate. Rather, the first proposition sets up a board consisting of two squares side by side (as in tangram). A reconciliation of the Suter board with this Codex board was published by Richard Dixon Oldham, FRS, in Nature in March, 1926, sparking an Ostomachion craze that year.

Modern combinatorics reveals that the number of ways to place the pieces of the Suter board to reform their square, allowing them to be turned over, is 17,152; the number is considerably smaller – 64 – if pieces are not allowed to be turned over. The sharpness of some angles in the Suter board makes fabrication difficult, while play could be awkward if pieces with sharp points are turned over. For the Codex board (again as with Tangram) there are three ways to pack the pieces: as two unit squares side by side; as two unit squares one on top of the other; and as a single square of side the square root of two. But the key to these packings is forming isosceles right triangles, just as Socrates gets the slave boy to consider in Plato's Meno – Socrates was arguing for knowledge by recollection, and here pattern recognition and memory seem more pertinent than a count of solutions. The Codex board can be found as an extension of Socrates' argument in a seven-by-seven-square grid, suggesting an iterative construction of the side-diameter numbers that give rational approximations to the square root of two.

The fragmentary state of the palimpsest leaves much in doubt. But it would certainly add to the mystery had Archimedes used the Suter board in preference to the Codex board. However, if Netz is right, this may have been the most sophisticated work in the field of combinatorics in Greek antiquity. Either Archimedes used the Suter board, the pieces of which were allowed to be turned over, or the statistics of the Suter board are irrelevant.