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The **Archimedes Palimpsest** is a palimpsest (ancient overwritten manuscript) on parchment in the form of a codex (hand-written bound book, as opposed to a scroll). It originally was a 10th century copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes (*c.* 287 BC–*c.* 212 BC) of Syracuse and other authors, which was overwritten with a religious text. The manuscript currently belongs to an American private collector.

## Contents |

Archimedes lived in the 3rd century BC, but the copy of his work was made in the 10th century AD by an anonymous scribe. In the 12th century the original Archimedes codex was unbound, scraped and washed, along with at least six other parchment manuscripts, including one with works of Hypereides. The parchment leaves had been folded in half and reused for a Christian liturgical text 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, visible and raking light, and X-ray.^{[1]}^{[2]}

In 1906 it was briefly inspected in Istanbul 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 T. L. Heath. Before that it was not widely known among mathematicians, physicists or historians. It contains:

- "On the Equilibrium of Planes"
- "Spiral Lines"
- "Measurement of a Circle"
- "On the 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 4th century BC politician Hypereides, a commentary on Aristotle's *Categories* by Alexander of Aphrodisias, and other works.^{[3]}

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 divided both figures into infinitely many slices of infinitesimal width, and balanced 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 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. 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 did not know about differentiation, so he could not calculate any integrals other than those that 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 that 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 Mnatsakanian 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, that is, to put the pieces back in 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). A reconciliation of the Suter board with this Codex board was published by Richard Dixon Oldham, FRS, in *Nature* in March, 1926, sparking a Stomachion 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 a 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.

The Biblical scholar Constantin von Tischendorf visited Constantinople in the 1840s, and, intrigued by the Greek mathematics visible on the palimpsest, brought home a page of it. (This page is now in the Cambridge University Library.) It was Johan Heiberg who realized, when he studied the palimpsest in Constantinople in 1906, that the text was of Archimedes, and included works otherwise lost. Heiberg took photographs, from which he produced transcriptions, published between 1910 and 1915 in a complete works of Archimedes. It is not known how the palimpsest subsequently wound up in France.^{[4]}

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* v. *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. 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.^{[5]} (The German magazine *Der Spiegel* reported that the buyer is probably Jeff Bezos.)^{[5]}

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 625 nm; Infrared: 700, 735, and 870 nm; and Raking Light: 910 and 470 nm.^{[6]} 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^{[7]} of Stanford University and Nigel Wilson have produced a diplomatic transcription of the text, filling in gaps in Heiberg's account with these images. All images are currently hosted on the website.^{[8]}

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 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."^{[9]}

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.^{[3]} 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.^{[10]}

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 were 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 a Creative Commons License, and processed images of the palimpsest in original page order were posted as a Google Book.^{[11]} In late 2011 it was the subject of the Walters Art Museum exhibit "Lost and Found: The Secrets of Archimedes".

**^**"Reading Between the Lines, Smithsonian Magazine". http://www.smithsonianmag.com/science-nature/archimedes.html. Retrieved 2009-03-31.**^**"The Archimedes Palimpsest Project". Archived from the original on 21 February 2009. http://www.archimedespalimpsest.org/digitalproduct1.html. Retrieved 2009-03-31.- ^
^{a}^{b}Morelle, Rebecca (2007-04-26). ""Text Reveals More Ancient Secrets", BBC News". Archived from the original on 19 February 2009. http://news.bbc.co.uk/1/hi/technology/6591221.stm. Retrieved 2009-03-31. **^**"History of the Archimedes Manuscript". http://www.archimedespalimpsest.org/palimpsest_history1.html. Retrieved 2009-03-31.- ^
^{a}^{b}Hisrhfield, Alan.*Eureka Man*, Walker & Co, NY, 2009; p. 187. **^**"File Naming Conventions". http://archimedespalimpsest.net/Documents/Internal/FileNamingConventions.txt. Retrieved 2009-03-31.**^**"The Scholarship of the Palimpsest". Archived from the original on 15 May 2009. http://www.archimedespalimpsest.org/scholarship_netz2.html. Retrieved 2009-03-31.**^**Archimedespalimpsest.net**^**"Placed under X-ray gaze, Archimedes manuscript yields secrets lost to time". http://news-service.stanford.edu/news/2005/may25/archimedes-052505.html. Retrieved 2009-03-31.**^**Carey, C. et al., "Fragments of Hyperides’**Against Diondas**from the Archimedes Palimpsest", "Inhaltsverzeichnis",*Zeitschrift für Papyrologie und Epigraphik*, vol. 165, pp. 1-19. Retrieved 2009-10-11.**^***Archimedes Palimpsest*. http://books.google.com/books?id=_zX8OG3QoF4C&printsec=frontcover&cad=0. Retrieved 2009-03-31.

- 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 0-691-08421-1, ISBN 0-691-02400-6 (paperback)

- The Archimedes Palimpsest Project Web Page
- Digital Palimpsest on the Web
- The Nova Program outlined
- The Nova Program teacher's version
*The Method*: English translation (Heiberg's 1909 transcription)- Did Isaac Barrow read it?
- May 2005 Stanford Report: Heather Rock Woods, "Archimedes manuscript yields secrets under X-ray gaze" May 19, 2005
- Will Noel: Restoring The Archimedes Palimpsest (YouTube), Ignite (O'Reilly), August 2009
- The Greek Orthodox Patriarchate of Jerusalem v. Christies’s Inc., 1999 U.S. Dist. LEXIS 13257 (S.D. N.Y. 1999) (via Archive.org)
- Eureka! 1,000-year-old text by Greek maths genius Archimedes goes on display
*Daily Mail*, October 18, 2011.

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