Archimedes Palimpsest
The Archimedes Palimpsest is a
Following the
All images and transcriptions are now freely available on the web at the Archimedes Digital Palimpsest under the
History
Early
Archimedes lived in the 3rd century BC and wrote his proofs as letters in
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 in a school founded by the mathematician, engineer, and former Greek Orthodox archbishop of Thessaloniki,
This medieval Byzantine manuscript then traveled from Constantinople to
Modern
The Biblical scholar
The manuscript was still in the
Sometime between 1923 and 1930 the palimpsest was acquired by Marie Louis Sirieix, a "businessman and traveler to the Orient who lived in Paris."
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
Imaging and digitization
At the
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.[16]
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 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.[17]
Sometime after 1938, a forger placed four
In April 2007, it was announced that a new text had been found in the palimpsest, a commentary on Aristotle's Categories running to some 9 000 words. Most of this text was recovered in early 2009 by applying principal component analysis to the three color bands (red, green, and blue) of fluorescent light generated by ultraviolet illumination. 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
The transcriptions of the book were digitally encoded using the
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
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
- A commentary on Aristotle's Categories by Porphyry (or by Alexander of Aphrodisias)[23]
- Other works
Source:[1]
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.[citation needed]
The method that Archimedes describes was based upon his investigations of
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
When rigorously proving theorems, Archimedes often used what are now called Riemann sums.[dubious ] 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 determined 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.
Stomachion
In Heiberg's time, much attention was paid to Archimedes' brilliant use of
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.
See also
Notes
- ^ a b c Morelle, Rebecca (2007-04-26). "Text Reveals More Ancient Secrets". BBC News. Archived from the original on 19 February 2009. Retrieved 2009-03-31.
- ^ "Editions of Archimedes' Work". Brown University Library. Archived from the original on 8 August 2007. Retrieved 2007-07-23.
- ^ Reviel Netz, William Noel and Nigel Wilson. The Archimedes Palimpsest, Vol. 1. Catalogue and Commentary, Cambridge University press, 2011.
- ISBN 9781602397064. Archivedfrom the original on 2023-04-22. Retrieved 2020-10-29.
- ISBN 9780838906095. Archivedfrom the original on 2023-04-22. Retrieved 2020-10-29.
- ^ Bergmann, Uwe. "X-Ray Fluorescence Imaging of the Archimedes Palimpsest: A Technical Summary" (PDF). Archived (PDF) from the original on 2017-05-18. Retrieved 2013-09-29.
- ^ a b c d The Archimedes Palimpsest Project. "The History of the Archimedes Manuscript". Archived from the original on 2018-10-01. Retrieved 2015-03-16.
- ^ a b c d e Schulz, Matthias (June 22, 2007). "Revolutionary? Authentic? Stolen? The Story of the Archimedes Manuscript". Der Spiegel. Archived from the original on April 2, 2015. Retrieved March 16, 2015.
- ^ ISBN 9780802719799. Archivedfrom the original on 2023-04-22. Retrieved 2013-09-29.
- Free Cultural Works
- ^ "Reading Between the Lines, Smithsonian Magazine". Archived from the original on 2008-01-19. Retrieved 2009-03-31.
- ^ "archimedespalimpsest". Archived from the original on 21 February 2009.
This data is released for use under a Creative Commons license, with attribution
- ^ "Editions of Archimedes' Work". Brown University Library. Archived from the original on 8 August 2007. Retrieved 2007-07-23.
- ^ "NOVA – Official Website – Inside the Archimedes Palimpsest". PBS. Archived from the original on 2017-08-15. Retrieved 2017-08-24.
- ^ "Archimedes Palimpsest – Press Release". Archived from the original on 2015-09-24. Retrieved 2015-03-16.
- University of Pennsylvania Libraries. Archivedfrom the original on January 19, 2022. Retrieved January 4, 2022.
- ^ Netz, Reviel; Noel, William; Wilson, Nigel (eds.). The Archimedes Palimpsest, Vol. 1. Catalogue and Commentary; Vol. 2. Images and Transcriptions (2011 ed.). Cambridge University Press.
- ^ Rock Woods, Heather (May 19, 2005). "Placed under X-ray gaze, Archimedes manuscript yields secrets lost to time". Archived from the original on May 17, 2008. Retrieved February 8, 2016.
- ^ Carey, C.; et al. "Fragments of Hyperides' Against Diondas from the Archimedes Palimpsest" (PDF). Inhaltsverzeichnis. Zeitschrift für Papyrologie und Epigraphik. 165: 1–19. Archived (PDF) from the original on 2011-11-18. Retrieved 2009-10-11.
- ^ Porter, Dot (October 29, 2008). "The Digital Archimedes Palimpsest Released". stoa.org. The Stoa Consortium for Electronic Publication in the Humanities. Archived from the original on 2013-12-30. Retrieved 2013-12-29.
- ^ Archimedes Palimpsest. Archived from the original on 2023-04-22. Retrieved 2009-03-31 – via Google Books.
- ^ Aron, Jacob (February 11, 2015). "Glassed-in DNA makes the ultimate time capsule". New Scientist. Archived from the original on July 11, 2015. Retrieved January 4, 2022.
- ^ R. Chiaradonna, M. Rashed, D. Sedley and N. Tchernetska, A rediscovered Categories commentary, Oxford Studies in Ancient Philosophy 44:129–194 (2013); Porphyry is the preferred attribution see pp. 134, 137.
Additional sources
- Dijksterhuis, E.J. Dijksterhuis (1987). Archimedes. Princeton, NJ: ISBN 0-691-08421-1.
- Reviel Netz and William Noel. The Archimedes Codex, Weidenfeld & Nicolson, 2007
- The Nova Program outlined
- The Nova Program teacher's version
- The Method: English translation (Heiberg's 1909 transcription)
- Did Isaac Barrow read it?
- 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)
External links
- The Digital Archimedes Palimpsest (official web site)