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The Archimedes Palimpsest is a parchment codex palimpsest , originally a Greek copy of an otherwise unknown work of Archimedes of Syracuse and other authors. It was commissioned in the 10th-century Eastern Roman Byzantine Empire at a time when mathematical studies there were revived by the former Greek Orthodox archbishop of Thessaloniki, Leo the Geometer.

The text was later overwritten with a Christian religious work by 13th-century monks in Palestine, where the manuscript had been taken following the fall of Byzantine Constantinople in to the knights of the Fourth Crusade.

Concealed for over 70 years, a forger added pictures to some of its pages to increase its value. The text under these forged pictures, as well as additional text previously unreadable, has been revealed by scientific and scholarly work from to on images produced by ultraviolet , infrared , visible and raking light , and X-ray.

All images and scholarly transcriptions with metadata are now freely available on the web at the Archimedes Digital Palimpsest see External Links below , now hosted on OPenn [3] and other web sites for free use under the Creative Commons License CC BY.

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 in the then Byzantine Greek capital city of Constantinople.

A copy of Isidorus' edition of Archimedes was made around AD by an anonymous scribe, again in the Byzantine Empire, in a period during which the study of Archimedes flourished in Constantinople in a school founded by the mathematician, engineer, and former Greek Orthodox archbishop of Thessaloniki, Leo the Geometer , a cousin to the patriarch. This medieval Byzantine manuscript then traveled from Constantinople to Jerusalem , likely sometime after the Crusader sack of Byzantine Constantinople in Their leaves were folded in half, rebound and reused for a Christian liturgical text of later numbered leaves, of which are extant each older folded leaf became two leaves of the liturgical book.

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

The Biblical scholar Constantin von Tischendorf visited Constantinople in the s, and, intrigued by the Greek mathematics visible on the palimpsest he found in a Greek Orthodox library, brought home a leaf of it which is now in the Cambridge University Library. In 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. When Heiberg studied the palimpsest in Constantinople in , 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 and in a complete works of Archimedes.

Shortly thereafter Archimedes' Greek text was translated into English by T. Before that it was not widely known among mathematicians, physicists or historians. Sometime between and the palimpsest was acquired by Marie Louis Sirieix, a "businessman and traveler to the Orient who lived in Paris. Stored secretly for years by Sirieix in his cellar, the palimpsest suffered damage from water and mold.

In addition, after disappearing 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.

Sirieix died in , and in his daughter began attempting quietly to sell the valuable manuscript. Unable to sell it privately, in she finally turned to Christie's to sell it in a public auction, risking an ownership dispute. Christie's , Inc. The Greek church contended that the palimpsest had been stolen from its library in Constantinople in the s during a period of extreme persecution. Simon Finch, 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.

At the Walters Art Museum in Baltimore , the palimpsest was the subject of an extensive imaging study from to , and conservation as it had suffered considerably from mold while in Sirieix's cellar. This was directed by Dr. Toth of R. Toth Associates, with Dr. Abigail Quandt performing the conservation of the manuscript. A team of imaging scientists including Dr. Roger L. Easton, Jr. 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 of Archimedes.

The team digitally processed these images to reveal more of the underlying text with pseudocolor. They also digitized the original Heiberg images. 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 , 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. Uwe Bergman and Bob Morton to begin deciphering the parts of the 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. Most of this text was recovered in early by applying principal component analysis to the three color bands red, green, and blue of fluorescent light generated by ultraviolet illumination. 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. 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. On October 29, , 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 a Creative Commons License , [15] and processed images of the palimpsest in original page order were posted as a Google Book.

In , in an experiment into the preservation of digital data, Swiss scientists encoded text from the Archimedes Palimpsest into DNA. It contains: [6]. 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 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 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 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.

Using this method, Archimedes was able to solve several problems now 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. 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. Some pages of the Method remained unused by the author of the palimpsest and thus they are still lost. 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 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, 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' Stomachion immediate. Rather, the first proposition sets up a board consisting of two squares side by side as in Tangram. 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,; 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. From Wikipedia, the free encyclopedia.

Main article: The Method of Mechanical Theorems. Main article: Stomachion. Retrieved

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## The Archimedes Palimpsest

The thousand-year-old manuscript contains the earliest surviving writings by Archimedes, the Greek thinker who is regarded as the greatest mathematician of antiquity. In this interactive, see how sophisticated technology uncovers Archimedes' faded text and diagrams from beneath another Greek text that was written over it. Also, below, follow a time line that tells the fascinating story of the page volume's journey from its creation in Constantinople to the auction block at Christie's in New York. Follow the 1,year journey of an ancient document, and watch as modern technology makes the erased text reappear. Circa B. Before his death at Syracuse in B.

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## The Archimedes Palimpsest Project

The subject of this website is a manuscript of extraordinary importance to the history of science, the Archimedes Palimpsest. This thirteenth century prayer book contains erased texts that were written several centuries earlier still. These erased texts include two treatises by Archimedes that can be found nowhere else, The Method and Stomachion. The manuscript sold at auction to a private collector on the 29th October Since that date the manuscript has been the subject of conservation, imaging and scholarship, in order to better read the texts. The Archimedes Palimpsest project, as it is called, has shed new light on Archimedes and revealed new texts from the ancient world.