- Archimedes Palimpsest
The Archimedes Palimpsest is a
palimpsest onparchment in the form of acodex which originally was a copy of an otherwise unknown work of the ancientmathematician ,physicist , andengineer Archimedes of Syracuse and other authors. Archimedes lived in thethird century BC , but the copy was made in thetenth century AD by an anonymous scribe. In thetwelfth century the codex was unbound and washed, in order that the parchment leaves could be folded in half and reused for a Christianliturgical 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 using digital processing of images produced byultraviolet ,X-ray , and visible light. [http://www.smithsonianmag.com/science-nature/archimedes.html] [http://www.archimedespalimpsest.org/digitalproduct1.html]In 1906 it was briefly inspected in
Constantinople and was published, from photographs, by the Danish philologist Johan Ludvig Heiberg; shortly thereafter Archimedes' Greek text was translated into English byThomas 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 , and a commentary onAristotle 's Categories byAlexander of Aphrodisias . [http://news.bbc.co.uk/1/hi/technology/6591221.stm]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 his method of double contradiction: assuming that the first is bigger than the second leads to a contradiction, as does the assumption that the first be smaller than the second; so the two must be equal. These proofs, still considered to be rigorous and correct, used what we might now consider secondary-school
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".Essentially, the method consists in dividing the two areas or volumes in infinitely many stripes of
infinitesimal width, and "weighing" the stripes of the first figure against those of the second, evaluated in terms of a finiteEgyptian fraction series. He considered this method as a usefulheuristic but always made sure to prove the results found in this manner using the rigorous arithmetic methods mentioned above.He was able to solve problems that would now be treated by
integral calculus , which was formally invented in the seventeenth century byIsaac Newton and Gottfried Leibniz, working independently. Among those problems were that of calculating thecenter of gravity of a solid hemisphere, the center of gravity of afrustum of a circularparaboloid , and the area of a region bounded by aparabola and one of itssecant lines. Contrary to exaggerations found in some twentieth century calculus textbookswho, he did not use anything likeRiemann sum s, either in the work embodied in this palimpsest or in any of his other works. (For explicit details of the method used, seeArchimedes' use of infinitesimals .)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. Moderncombinatorics 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 ofcombinatorics in Greek antiquity.Modern history
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 ofMar 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. JudgeKimba 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 theinformation technology field.The palimpsest is now at the
Walters Art Museum in Baltimore, where conservation continues (as it had suffered considerably frommold ).A team of imaging scientists from the
Rochester Institute of Technology and Johns Hopkins University has used computer processing of digital images from various spectral bands, including ultraviolet and visible light, to reveal more of Archimedes' text. Dr. Reviel Netz [http://www.archimedespalimpsest.org/scholarship_netz2.html] ofStanford University has been trying to fill 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 theStanford Linear Accelerator Center in Menlo Park, California, were used to begin deciphering the parts of the 174-page text that have not yet been revealed. The production of x-rayfluorescence 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." [http://news-service.stanford.edu/news/2005/may25/archimedes-052505.html]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 toAlexander of Aphrodisias . Doctor William Noel, the curator of manuscripts at theWalters Art Museum , 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 byHypereides , anAthenian politician from the fourth century BC, which has also been found within the palimpsest. [http://news.bbc.co.uk/1/hi/technology/6591221.stm] 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. [http://www.uni-koeln.de/phil-fak/ifa/zpe/indices/inhaltsverzeichnis_165.pdf]References
* Reviel Netz and William Noel, " [http://www.orionbooks.co.uk/HB-25714/The-Archimedes-Codex.htm The Archimedes Codex] ", Weidenfeld & Nicolson, 2007
* Dijksterhuis, E.J.,"Archimedes", Princeton U. Press, 1987, pages 129- 133. copyright 1938, ISBN 0-691-08421, 0-691-02400-6External links
* [http://www.archimedespalimpsest.org/ The Archimedes Palimpsest Project Web Page]
* [http://www.archimedespalimpsest.org/digitalproduct1.html Digital Palimpsest on the Web]
* [http://www.thewalters.org/archimedes/frame.html The Archimedes Palimpsest web pages at the Walters Art Museum]
* [http://www.pbs.org/wgbh/nova/archimedes/palimpsest.html The Nova Program outlined]
* [http://www.pbs.org/wgbh/nova/teachers/programs/3010_archimed.html The Nova Program teacher's version]
* "The Method": [http://books.google.com/books?id=suYGAAAAYAAJ English translation]
* [http://dftuz.unizar.es/~rivero/research/isisletter.htm Did Isaac Barrow read it?]
* [http://news-service.stanford.edu/news/2005/may25/archimedes-052505.html May 2005 Stanford Report: Heather Rock Woods, "Archimedes manuscript yields secrets under X-ray gaze"] May 19, 2005
* [http://www.law.washington.edu/courses/andrews/A503C_WiSp06/Documents/Greek_Orthodox_Patriarchate_of_Jerusalem_v.pdf The Greek Orthodox Patriarchate of Jerusalem v. Christies’s Inc., 1999 U.S. Dist. LEXIS 13257 (S.D. N.Y. 1999) ]
Wikimedia Foundation. 2010.