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| {{Infobox manuscript
| | 43 years old Supply and Distribution Manager Alphonse from Newmarket, has many hobbies and interests including musical instruments, ganhando dinheiro na internet and aerobics. Of late has made vacation to Churches of Peace in Jawor and Swidnica.<br><br>Feel free to visit my website: [http://comoficarrico.comoganhardinheiro101.com como ficar rico] |
| <!----------Name---------->
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| | name = Egyptian Mathematical Leather Roll (EMLR)
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| | location = [[British Museum]] in London
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| <!----------Image---------->
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| <!----------General---------->
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| | Also known as =
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| | Type =
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| | Date = ca 1650 BCE
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| | Place of origin = [[Thebes, Egypt|Thebes]]
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| | Language(s) = [[Hieratic]]
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| | Scribe(s) =
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| | Illuminated by =
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| <!----------Form and content---------->
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| | Material =
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| | Size = Length: {{convert|10|in|cm}}<br>Width: {{convert|17|in|cm}}
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| The '''Egyptian Mathematical Leather Roll (EMLR)''' was a 10 × 17 in (25 × 43 cm) 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).
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| The writing consists of [[Middle Kingdom of Egypt|Middle Kingdom]] [[hieratic]] characters written right to left. Scholars date the EMLR to the 17th century BCE.<ref name="Clagett">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, pp. 17–18, 25, 37–38, 255–257</ref>
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| ==Mathematical content==
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| This leather roll is an aid for computing [[Egyptian fraction]]s. It contains 26 sums of unit fractions which equal another unit fraction. The sums appear in two columns, and are followed by two more columns which contain exactly the same sums.<ref name="Imhausen">Annette Imhausen, in The Mathematics of Egypt, Mesopotamia, China, India, and Islam:
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| A Sourcebook, Edited by Victor J. Katz, 2007, pp. 21–22</ref>
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| {| class="wikitable" border="1" cellpadding="5" align="center" | style="margin: 1em auto 1em auto; width: 75%"
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| |+'''The Egyptian Mathematical Leather Roll<ref name="Imhausen"/>'''
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| !|Column 1 !! |Column 2!!align="center" |Column 3 !! |Column 4
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| | <math>\frac{1}{10} + \frac{1}{40} = \frac{1}{8}</math> || <math>\frac{1}{30} + \frac{1}{45} + \frac{1}{90}= \frac{1}{15}</math> || <math>\frac{1}{10} + \frac{1}{40} = \frac{1}{8}</math> || <math>\frac{1}{18} + \frac{1}{36} = \frac{1}{12}</math>
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| | <math>\frac{1}{5} + \frac{1}{20} = \frac{1}{4}</math> || <math>\frac{1}{24} + \frac{1}{48} = \frac{1}{16}</math> || <math>\frac{1}{5} + \frac{1}{20} = \frac{1}{4}</math> || <math>\frac{1}{21} + \frac{1}{42} = \frac{1}{14}</math>
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| | <math>\frac{1}{4} + \frac{1}{12} = \frac{1}{3}</math> || <math>\frac{1}{18} + \frac{1}{36} = \frac{1}{12}</math> || <math>\frac{1}{4} + \frac{1}{12} = \frac{1}{3}</math> || <math>\frac{1}{45} + \frac{1}{90} = \frac{1}{30}</math>
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| | <math>\frac{1}{10} + \frac{1}{10} = \frac{1}{5}</math> || <math>\frac{1}{21} + \frac{1}{42} = \frac{1}{14}</math> || <math>\frac{1}{10} + \frac{1}{10} = \frac{1}{5}</math> || <math>\frac{1}{30} + \frac{1}{60} = \frac{1}{20}</math>
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| | <math>\frac{1}{6} + \frac{1}{6} = \frac{1}{3}</math> || <math>\frac{1}{45} + \frac{1}{90} = \frac{1}{30}</math> || <math>\frac{1}{6} + \frac{1}{6} = \frac{1}{3}</math> || <math>\frac{1}{15} + \frac{1}{30} = \frac{1}{10}</math>
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| | <math>\frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{1}{2}</math> || <math>\frac{1}{30} + \frac{1}{60} = \frac{1}{20}</math> || <math>\frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{1}{2}</math> || <math>\frac{1}{48} + \frac{1}{96} = \frac{1}{32}</math>
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| | <math>\frac{1}{3} + \frac{1}{3} = \frac{2}{3}</math> || <math>\frac{1}{15} + \frac{1}{30} = \frac{1}{10}</math> || <math>\frac{1}{3} + \frac{1}{3} = \frac{2}{3}</math> || <math>\frac{1}{96} + \frac{1}{192} = \frac{1}{64}</math>
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| | <math>\frac{1}{25} + \frac{1}{15} + \frac{1}{75} + \frac{1}{200} = \frac{1}{8}</math> || <math>\frac{1}{48} + \frac{1}{96} = \frac{1}{32}</math> || <math>\frac{1}{25} + \frac{1}{15} + \frac{1}{75} + \frac{1}{200} = \frac{1}{8}</math> ||
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| | <math>\frac{1}{50} + \frac{1}{30} + \frac{1}{150} + \frac{1}{400} = \frac{1}{16}</math> || <math>\frac{1}{96} + \frac{1}{192} = \frac{1}{64}</math> || <math>\frac{1}{50} + \frac{1}{30} + \frac{1}{150} + \frac{1}{400} = \frac{1}{16}</math> ||
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| | <math>\frac{1}{25} + \frac{1}{50} + \frac{1}{150} = \frac{1}{6}</math> || || <math>\frac{1}{25} + \frac{1}{50} + \frac{1}{150} = \frac{1}{6}</math> ||
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| | <math>\frac{1}{9} + \frac{1}{18} = \frac{1}{6}</math> || || <math>\frac{1}{9} + \frac{1}{18} = \frac{1}{6}</math> ||
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| | <math>\frac{1}{7} + \frac{1}{14} + \frac{1}{28} = \frac{1}{4}</math> || || <math>\frac{1}{7} + \frac{1}{14} + \frac{1}{28} = \frac{1}{4}</math> ||
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| | <math>\frac{1}{12} + \frac{1}{24} = \frac{1}{8}</math> || || <math>\frac{1}{12} + \frac{1}{24} = \frac{1}{8}</math> ||
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| | <math>\frac{1}{14} + \frac{1}{21} + \frac{1}{42} = \frac{1}{7}</math> || || <math>\frac{1}{14} + \frac{1}{21} + \frac{1}{42} = \frac{1}{7}</math> ||
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| | <math>\frac{1}{18} + \frac{1}{27} + \frac{1}{54} = \frac{1}{9}</math> || || <math>\frac{1}{18} + \frac{1}{27} + \frac{1}{54} = \frac{1}{9}</math> ||
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| | <math>\frac{1}{22} + \frac{1}{33} + \frac{1}{66} = \frac{1}{11}</math> || || <math>\frac{1}{22} + \frac{1}{33} + \frac{1}{66} = \frac{1}{11}</math> ||
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| | <math>\frac{1}{28} + \frac{1}{49} + \frac{1}{196} = \frac{1}{13}</math> || || <math>\frac{1}{28} + \frac{1}{49} + \frac{1}{196} = \frac{1}{13}</math> ||
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| | || || <math>\frac{1}{30} + \frac{1}{45} + \frac{1}{90} = \frac{1}{15}</math> ||
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| | || || <math>\frac{1}{24} + \frac{1}{48} = \frac{1}{16}</math> ||
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| |}
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| Of the 26 rational numbers listed, ten are [[Eye of Horus]] numbers: 1/2, 1/4 (twice), 1/8 (thrice), 1/16 (twice), 1/32, 1/64 converted to Egyptian fractions. There are seven other even rational numbers converted to Egyptian fractions: 1/6 (listed 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. | |
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| The British Museum examiners found no introduction or description to how or why the equivalent unit fraction series were computed.<ref>Gillings, Richard J. “The Egyptian Mathematical Leather Role–Line 8. How Did the Scribe Do it?” (Historia Mathematica 1981), 456–457.</ref> 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.
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| ==Modern analysis==
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| The original mathematical texts never explain where the procedures and formulas came from. This holds true for the EMLR as well. Scholars have attempted to deduce what techniques the ancient Egyptians may have used to construct both the unit fraction tables of the EMLR and the 2 /n tables known from the [[Rhind Mathematical Papyrus]] and the [[Lahun Mathematical Papyri]]. Both types of tables were used to aid in computations dealing with fractions, and for the conversion of measuring units.<ref name="Imhausen"/>
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| It has been noted that there are groups of unit fraction decompositions in the EMLR which are very similar. For instance lines 5 and 6 easily combine into the equation 1/3 + 1/6 = 1/2. 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.<ref>Gillings, Richard J., Mathematics in the Time of the Pharaohs, Dover Publications, 1982 reprint (1972) ISBN 0-486-24315-X</ref>
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| Some of the problems would lend themselves to a solution via an algorithm which involves multiplying both the numerator and the denominator by the same term and then further reducing the resulting equation:
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| : <math>\frac{1}{pq} = \frac{1}{N}\times\frac{N}{pq} </math>
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| This method leads to a solution for the fraction 1/8 as appears in the EMLR when using N=25 (using modern mathematical notation):
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| : <math>1/8 = 1/25 \times 25/8 = 1/5 \times 25/40 = 1/5 \times (3/5 + 1/40) </math>
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| : <math>= 1/5 \times (1/5 + 2/5 + 1/40) = 1/5 \times (1/5 + 1/3 + 1/15 + 1/40) = 1/25 + 1/15 + 1/75 + 1/200</math> <ref name="MG">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.</ref>
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| ==Modern conclusions==
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| The EMLR has been considered a student scribe test document since 1927, the year that the text was unrolled at the British Museum. The scribe practiced conversions of rational numbers 1/p and 1/pq to alternative unit fraction series. Reading available Middle Kingdom math records, [[RMP 2/n table]] being one, modern students of Egyptian arithmetic may see that trained scribes improved conversions of 2/n and n/p to concise unit fraction series by applying algorithmic and non-algorithmic methods.
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| ==Chronology==
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| 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.
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| * 1895 – Hultsch suggested that all RMP 2/p series were coded by aliquot parts.<ref>Hultsch, F, Die Elemente der Aegyptischen Theihungsrechmun 8, Ubersich uber die Lehre von den Zerlegangen, (1895):167–71</ref>
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| * 1927 – Glanville concluded that EMLR arithmetic was purely additive.<ref>Glanville, S.R.K. "The Mathematical Leather Roll in the British Museum” Journal of Egyptian Archaeology 13, London (1927): 232–8</ref>
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| * 1929 – Vogel reported the EMLR to be more important (than the RMP), though it contains only 25 unit fraction series.<ref>Vogel, Kurt. “Erweitert die Lederolle unserer Kenntniss ägyptischer Mathematik Archiv fur Geschichte der Mathematik, V 2, Julius Schuster, Berlin (1929): 386–407</ref>
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| * 1950 – Bruins independently confirms Hultsch’s RMP 2/''p'' analysis (Bruins 1950)
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| * 1972 – Gillings found solutions to an easier RMP problem, the 2/''pq'' series (Gillings 1972: 95–96).
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| * 1982 – Knorr identifies RMP unit fractions 2/35, 2/91 and 2/95 as exceptions to the 2/''pq'' problem.<ref>[[Wilbur Knorr|Knorr, Wilbur R.]] “Techniques of Fractions in Ancient Egypt and Greece”. Historia Mathematica 9 Berlin, (1982): 133–171.</ref>
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| * 2002 – Gardner identifies five abstract EMLR patterns.<ref name="MG"/>
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| ==See also==
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| Egyptian mathematical texts:
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| * [[Moscow Mathematical Papyrus]]
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| * [[Lahun Mathematical Papyri]]
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| * [[Berlin Papyrus 6619]]
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| * [[Akhmim Wooden Tablet]]
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| * [[Reisner Papyrus]]
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| * [[Rhind Mathematical Papyrus]]
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| Other:
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| *[[Liber Abaci]]
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| *[[:fr:Sylvia Couchoud|Sylvia Couchoud]] {{fr icon}}
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| ==References==
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| <references/>
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| ==Further reading==
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| *Brown, Kevin S. The Akhmin Papyrus 1995 – Egyptian Unit Fractions 1995
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| *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.
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| *Bruins, Evert M. “Platon et la table égyptienne 2/n”. Janus 46, Amsterdam, (1957): 253–263.
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| *Bruins, Evert M. “Egyptian Arithmetic.” Janus 68, Amsterdam, (1981): 33–52.
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| *Bruins, Evert M. “Reducible and Trivial Decompositions Concerning Egyptian Arithmetics”. Janus 68, Amsterdam, (1981): 281–297.
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| *Daressy, Georges. “Akhmim Wood Tablets”, Le Caire Imprimerie de l’Institut Francais d’Archeologie Orientale, 1901, 95–96.
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| *Gardner, Milo. "Mathematical Roll of Egypt", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, Nov. 2005.
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| *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.
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| *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.
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| *Gillings, Richard J. “The Recto of the RMP and the EMLR”, Historia Mathematica, Toronto 6 (1979), 442–447.
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| *Gillings, Richard J. “The Egyptian Mathematical Leather Role–Line 8. How Did the Scribe Do it?” (Historia Mathematica 1981), 456–457.
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| *[[Battiscombe Gunn|Gunn, Battiscombe George]]. Review of ”The Rhind Mathematical Papyrus” by T. E. Peet. The Journal of Egyptian Archaeology 12 London, (1926): 123–137.
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| *Imhausen, Annette. “Egyptian Mathematical Texts and their Contexts”, Science in Context, vol 16, Cambridge (UK), (2003): 367–389.
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| *Legon, John A.R. “A Kahun Mathematical Fragment”. Discussions in Egyptology, 24 Oxford, (1992).
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| *Lüneburg, H. “Zerlgung von Bruchen in Stammbruche” Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers, Wissenschaftsverlag, Mannheim, 1993. 81–85.
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| *Rees, C. S. “Egyptian Fractions”, Mathematical Chronicle 10, Auckland, (1981): 13–33.
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| *Roero, C. S. “Egyptian mathematics” Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences” I. Grattan-Guinness (ed), London, (1994): 30–45.
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| *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.
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| *Sylvester, J. J. “On a Point in the Theory of Vulgar Fractions”: American Journal Of Mathematics, 3 Baltimore (1880): 332–335, 388–389.
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| ==External links==
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| *[http://planetmath.org/encyclopedia/EgyptianMathematicalLeatherRoll2.html EMLR]
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| *[http://planetmath.org/encyclopedia/AhmesBirdFeedingRateMethod.html Theoretical (expected) economic control numbers]
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| *[http://planetmath.org/encyclopedia/RMP35To38PlusRMP66.html RMP 35–38 plus RMP 66]
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| [[Category:Egyptian fractions]]
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| [[Category:Ancient Egyptian literature]]
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| [[Category:Mathematics manuscripts]]
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| [[Category:Ancient Egyptian objects in the British Museum]]
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| [[Category:Egyptian mathematics]]
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