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==See also== |
==See also== |
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* http://rmp36.blogspot.com/ |
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**[[Liber Abaci]] |
**[[Liber Abaci]] |
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Revision as of 18:08, 27 August 2010
Egyptian Mathematical Leather Roll (EMLR) | |
---|---|
British Museum in London | |
Date | ca 1650 BCE |
Place of origin | Thebes |
Language(s) | Hieratic |
Size | Length: 10 inches (25 cm) Width: 17 inches (43 cm) |
The Egyptian Mathematical Leather Roll (also referred to as EMLR) was a 10" x 17" leather roll purchased by Alexander Henry Rhind in 1858. It was sent to the British Museum in 1864, along with the Rhind Mathematical Papyrus, but the former was not chemically softened and unrolled until 1927 (Scott, Hall 1927).
The writing consists of Middle Kingdom hieratic characters written right to left. Scholars date the EMLR to the 17th Century BCE. [1]
Mathematical Content
The EMLR is an aid for computing fractions. The roll consists of 26 sums of unit fractions which equal another unit fraction. The sums appear in two colums, and are followed by two more columns which contain the exact same sums. [2]
Column 1 | Column 2 | Column 3 | Column 4 |
---|---|---|---|
There are 26 rational numbers listed. Each rational number is followed by its equivalent Egyptian fraction series. There were ten Eye of Horus numbers: 1/2, 1/4 (twice), 1/8 (thrice), 1/16 (twice), 1/32, 1/64 converted to Egyptian fractions. There were seven other even rational numbers converted to Egyptian fractions: 1/6 (twice–but wrong once), 1/10, 1/12, 1/14, 1/20 and 1/30. Finally, there were nine odd rational numbers converted to Egyptian fractions: 2/3, 1/3 (twice), 1/5, 1/7, 1/9, 1/11, 1/13 and 1/15, training patterns for scribal students to learn the RMP 2/n table method.
The British Museum examiners found no introduction or description to how or why the equivalent unit fraction series were computed [3]. Equivalent unit fraction series are associated with fractions 1/3, 1/4, 1/8 and 1/16. There was a trivial error associated with the final 1/15 unit fraction series. The 1/15 series was listed as equal to 1/6. Another serious error was associated with 1/13, an issue that the 1927 examiners did not attempt to resolve.
It has been noted that there are groups of unit fraction decompositions which are very similar. For instance lines 5 and 6 easily combine into the equation . It is easy to derive lines 11, 13, 24, 20, 21, 19, 23, 22, 25 and 26 by dividing this equation by 3, 4, 5, 6, 7, 8, 10, 15, 16 and 32 respectively. Several other algorithms have been proposed in the literature, but we do not know which methods were actually used by the scribe.[4]
Since 2001 it has been clear that LCM m scaled EMLR, RMP and all the hieratic texts. For example in RMP 37 Ahmes scaled 1/4 by 72 considering 72/288 writing out an awkward unit fraction series. RMP 37 showed that Ahmes used the awkward EMLR-type conversion methods learned as a young student. Clearly the EMLR was a student test document. The test answer sheet reported conversions of 1/p and 1/pq rational numbers to unit fraction series in ways that the student was later introduced to the advanced 2/n table LCM m conversion methods. [citation needed]
Chronology
The following chronology shows several milestones that marked the recent progress toward reporting a clearer understanding of the EMLR's contents, related to the RMP 2/n table.
- 1895 - Hultsch suggested that all RMP 2/p series were coded by aliquot parts. [5]
- 1927 - Glanville concluded that EMLR arithmetic was purely additive. [6]
- 1929 - Vogel reported the EMLR to be more important (than the RMP), though it contains only 25 unit fraction series. [7]
- 1950 - Bruins independently confirms Hultsch’s RMP 2/p analysis (Bruins 1950)
- 1972 - Gillings found solutions to an easier RMP problem, the 2/pq series (Gillings 1972: 95-96).
- 1982 - Knorr identifies RMP unit fractions 2/35, 2/91 and 2/95 as exceptions to the 2/pq problem.[8]
- 2002 - Gardner identifies five abstract EMLR patterns. [9]
- 2008 - Gardner identifies one EMLR pattern, based on RMP scaling LCMs and intermediate red auxiliary numbers
References
- ^ Clagett, Marshall. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society, 1999, pg 17-18, 25, 37-38, 255-257
- ^ a b Annette Imhausen, in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Edited by Victor J. Katz, 2007, pg 21-22
- ^ Gillings, Richard J. “The Egyptian Mathematical Leather Role–Line 8. How Did the Scribe Do it?” (Historia Mathematica 1981), 456–457.
- ^ Gillings, Richard J., Mathematics in the Time of the Pharaohs, Dover Publications, 1982 reprint (1972) ISBN 0-486-24315X
- ^ Hultsch, F, Die Elemente der Aegyptischen Theihungsrechmun 8, Ubersich uber die Lehre von den Zerlegangen, (1895):167-71
- ^ Glanville, S.R.K. "The Mathematical Leather Roll in the British Museum” Journal of Egyptian Archaeology 13, London (1927): 232–8
- ^ Vogel, Kurt. “Erweitert die Lederolle unserer Kenntniss ägyptischer Mathematik Archiv fur Geschichte der Mathematik, V 2, Julius Schuster, Berlin (1929): 386-407
- ^ Knorr, Wilbur R. “Techniques of Fractions in Ancient Egypt and Greece”. Historia Mathematica 9 Berlin, (1982): 133–171.
- ^ Gardner, Milo. “The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term” History of the Mathematical Sciences”, Ivor Grattan-Guinness, B.C. Yadav (eds), New Delhi, Hindustan Book Agency, 2002:119-134.
See also
Further Reading
- Boyer, Carl B. A History of Mathematics. New York: John Wiley, 1968.
- Brown, Kevin S. The Akhmin Papyrus 1995 --- Egyptian Unit Fractions 1995
- Bruckheimer, Maxim and Y. Salomon. “Some Comments on R. J. Gillings’ Analysis of the 2/n Table in the Rhind Papyrus.” Historia Mathematica 4 Berlin (1977): 445–452.
- Bruins, Evert M. Fontes matheseos: hoofdpunten van het prae-Griekse en Griekse wiskundig denken. Leiden, E. J. Brill, 1953.
- Bruins, Evert M. “Platon et la table égyptienne 2/n”. Janus 46, Amsterdam, (1957): 253–263.
- Bruins, Evert M. “Egyptian Arithmetic.” Janus 68, Amsterdam, (1981): 33–52.
- Bruins, Evert M. “Reducible and Trivial Decompositions Concerning Egyptian Arithmetics”. Janus 68, Amsterdam, (1981): 281–297.
- Burton, David M. History of Mathematics: An Introduction, Boston Wm. C. Brown, 2003.
- Chace, Arnold Buffum, et al. The Rhind Mathematical Papyrus, Oberlin, Mathematical Association of America, 1927.
- Collier, Mark and Steven Quirke (eds): Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical Oxford, Archaeopress, 2004.
- Cooke, Roger. The History of Mathematics. A Brief Course, New York, John Wiley & Sons, 1997.
- Couchoud, Sylvia. “Mathématiques égyptiennes”. Recherches sur les connaissances mathématiques de l’Egypte pharaonique., Paris, Le Léopard d’Or, 1993.
- Daressy, Georges. “Akhmim Wood Tablets”, Le Caire Imprimerie de l’Institut Francais d’Archeologie Orientale, 1901, 95–96.
- Eves, Howard, An Introduction to the History of Mathematics, New York, Holt, Rinehard & Winston, 1961
- Fowler, David H. The mathematics of Plato's Academy: a new reconstruction. New York, Clarendon Press, 1999.
- Gardiner, Alan H. “Egyptian Grammar being an Introduction to the Study of Hieroglyphs, Oxford, Oxford University Press, 1957.
- Gardner, Milo. "Mathematical Roll of Egypt", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, Nov. 2005.
- Gillings, Richard J. “The Egyptian Mathematical Leather Roll”. Australian Journal of Science 24 (1962): 339-344, Mathematics in the Time of the Pharaohs. Cambridge, Mass.: MIT Press, 1972. New York: Dover, reprint 1982.
- Gillings, Richard J. “The Recto of the Rhind Mathematical Papyrus: How Did the Ancient Egyptian Scribe Prepare It ?” Archive for History of Exact Sciences 12 (1974), 291–298.
- Gillings, Richard J. “The Recto of the RMP and the EMLR”, Historia Mathematica, Toronto 6 (1979), 442-447.
- Gillings, Richard J. “The Egyptian Mathematical Leather Role–Line 8. How Did the Scribe Do it?” (Historia Mathematica 1981), 456–457.
- Griffith, Francis Llewelyn. The Petrie Papyri. Hieratic Papyri from Kahun and Gurob (Principally of the Middle Kingdom), Vol. 1, 2, Bernard Quaritch, London, 1898.
- Gunn, Battiscombe George. Review of ”The Rhind Mathematical Papyrus” by T. E. Peet. The Journal of Egyptian Archaeology 12 London, (1926): 123–137.
- Imhausen, Annette. “Egyptian Mathematical Texts and their Contexts”, Science in Context, vol 16, Cambridge (UK), (2003): 367-389.
- Joseph, George Gheverghese. The Crest of the Peacock/the non-European Roots of Mathematics, Princeton, Princeton University Press, 2000
- Klee, Victor, and Wagon, Stan. Old and New Unsolved Problems in Plane Geometry and Number Theory, Mathematical Association of America, 1991.
- Legon, John A.R. “A Kahun Mathematical Fragment”. Discussions in Egyptology, 24 Oxford, (1992).
- Lüneburg, H. “Zerlgung von Bruchen in Stammbruche” Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers, Wissenschaftsverlag, Mannheim, 1993. 81–85.
- Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity (2 ed.). Dover Publications. ISBN 978-048622332-2.
- Ore, Oystein. Number Theory and its History, New York, McGraw-Hill, 1948
- Rees, C. S. “Egyptian Fractions”, Mathematical Chronicle 10 , Auckland, (1981): 13–33.
- Robins, Gay. and Charles Shute, The Rhind Mathematical Papyrus: an Ancient Egyptian Text" London, British Museum Press, 1987.
- Roero, C. S. “Egyptian mathematics” Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences” I. Grattan-Guinness (ed), London, (1994): 30–45.
- Sarton, George. Introduction to the History of Science, Vol I, New York, Williams & Son, 1927
- Scott, A. and Hall, H.R., “Laboratory Notes: Egyptian Mathematical Leather Roll of the Seventeenth Century BC”, British Museum Quarterly, Vol 2, London, (1927): 56.
- Sigler, Laurence E. (trans.) (2002). Fibonacci's Liber Abaci. Springer-Verlag. ISBN 0-387-95419-8.
- Sylvester, J. J. “On a Point in the Theory of Vulgar Fractions”: American Journal Of Mathematics, 3 Baltimore (1880): 332–335, 388–389.
- van der Waerden, Bartel Leendert. Science Awakening, New York, 1963
External links
- http://emlr.blogspot.com Egyptian Mathematical Leather Roll
- http://planetmath.org/encyclopedia/EgyptianMathematicalLeatherRoll2.html EMLR
- http://rmprectotable.blogspot.com/ Breaking the RMP 2/n Table Code
- http://rmp36.blogspot.com/ RMP 36 and the 2/n table
- http://planetmath.org/encyclopedia/AhmesBirdFeedingRateMethod.html theoretical (expected) economic control numbers
- fr:Sylvia Couchoud Template:Fr icon
- http://mathforum.org/kb/message.jspa?messageID=6579539&tstart=0 Math forum and two ways to calculate 2/7
- http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html New and Old Ahmes Papyrus classifications
- http://planetmath.org/encyclopedia/RMP35To38PlusRMP66.html RMP 35-38 plus RMP 66