Egyptian Mathematical Leather Roll

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. 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 (Gillings 1981: 456-457). 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.

The British Museum Quarterly (1927) naively reported the chemical analysis to be more interesting than the document's additive contents. One minimalist reported that the Horus-Eye binary fraction system was superior to the Egyptian fraction notation.

The Middle Kingdom Egyptian fraction conversions of binary fractions corrected a Eye of Horus numeration error. The older Horus-Eye arithmetic had employed an infinite series numeration system that rounded-off to 6-term binary fraction series, throwing away 1/64 units. Horus-Eye fractions are related to modern decimals, with both numeration systems rounding off, (Ore 1944: 331-325). Note that the Horus-Eye definition of one (1): 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + … dropped off the last term 1/64th, (Gillings 1972: 210). Modern decimals' round-off rules are closely related.

Within the EMLR red auxiliary numbers (multiples) converted 26 1/p and 1/pq unit fractions to non-optimal Egyptian fraction series. In total 22 unique unit fractions were converted by eight multiples (2, 3, 4, 5, 6, 7, 10, and 25). A detailed analysis of the 26 conversions are linked below. Egyptian fractions represented a solution to the Eye of Horus round-off problem by always converting rational numbers to exact unit fraction series, frequently using multiples as the first of three-steps. The RMP 2/n table converted 51 rational numbers by 14 multiples.

Summary: Middle Kingdom Egyptian arithmetic methods were written in non-optimal and optimal unit fraction series. Early researchers minimized the EMLR’s significance. The EMLR, the Rhind Mathematical Papyrus and the RMP 2/n table demonstrate that one method converted all rational numbers to exact unit fraction series during the Middle Kingdom. That is, the EMLR and RMP 2/n table should be seen as one document. The EMLR used 8 multiples to convert 22 rational numbers, introducing students to 14 multiples used to convert 51 rational numbers in the RMP 2/n table.

Chronology

The following chronology shows several milestones that marked the recent progress toward reporting a clearer understanding of the EMLR's contents.

1895 – Hultsch suggested that all RMP 2/p series were coded by an algebraic identity, using a parameter A (Hultsch 1895).

1927 – Glanville prematurely concluded that EMLR arithmetic was purely additive (Glanville 1927).

1929 – Vogel reported the EMLR to be more important, though it contains only 25 unit fraction series (Vogel 1929)

1950 – Bruins independently confirmed Hultsch’s RMP 2/p analysis (Bruins 1950)

1972 – Gillings found solutions to an easier problem, the 2/pq series of the RMP (Gillings 1972: 95-96).

1982 – Knorr identified the RMP fractions 2/35, 2/91 and 2/95 as exceptions to the 2/pq problem (Knorr 1982).

ources

REFERENCES

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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.

--- “Platon et la table égyptienne 2/n”. Janus 46, Amsterdam, (1957): 253–263.

--- “Egyptian Arithmetic.” Janus 68, Amsterdam, (1981): 33–52.

--- “Reducible and Trivial Decompositions Concerning Egyptian Arithmetics”. Janus 68, Amsterdam, (1981): 281–297.

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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.

--- " 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.

--- “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.

--- “The Recto of the RMP and the EMLR”, Historia Mathematica, Toronto 6 (1979), 442-447.

--- “The Egyptian Mathematical Leather Role–Line 8. How Did the Scribe Do it?”,(Historia Mathematica1981), 456–457.

Glanville, S.R.K. "The Mathematical Leather Roll in the British Museum” Journal of Egyptian Archaeology 13, London (1927): 232–8

Griffith, Francis Llewelyn. The Petrie Papyri. Hieratic Papyri from Kahun and Gurob (Principally of the Middle Kingdom), Vol. 1, 2, Bernard Quaritch, London, 1898.

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links

*http://emlr.blogspot.com Egyptian Mathematical Leather Roll
*http://planetmath.org/encyclopedia/EgyptianMathematicalLeatherRoll2.html Planetmath
*http://planetmath.org/encyclopedia/EgyptianFraction2.html Planetmath
*http://en.wikipedia.org/wiki/RMP_2/n_table Wikipedia RMP 2/n Table
*http://rmprectotable.blogspot.com/RMP Breaking the RMP 2/n Table Code
*http://egyptianmath.blogspot.com History of Egyptian fractions
*http://translate.google.com/translate?hl=en&sl=fr&u=http://fr.wikipedia.org/wiki/Sylvia_Couchoud&sa=X&oi=translate&resnum=10&ct=result&prev=/search%3Fq%3Dcouchoud,%2Bsylvia%26hl%3Den%26rlz%3D1B2GGFB_enUS216US216