Egyptian Mathematical Leather Roll

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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.

One review includes the Middle Kingdom Egyptian fraction conversions of binary fractions corrected a Eye of Horus numeration error. The Old Kingdom Horus-Eye arithmetic 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. The Middle Kingdom correction wrote exact finite series thereby eliminating binary round off errors.

Preceding the RMP 2/n table by 200 years the EMLR used red auxiliary numbers scaled by LCM (multiples) converted 26 1/p and 1/pq unit fractions to non-optimal Egyptian fraction series using a method that Ahmes described in RMP 36. The EMLR LCM scaled 1/p and 1/pq by Egyptian multiplication and division methods that allowed additive red auxiliary numerators to define final unit fraction answers. In total 22 unique unit fractions were converted by eight multiples (2, 3, 4, 5, 6, 7, 10, and 25), written as 2/2, 3/3, 4/4, 5/5, 6/6, 10/10 and 25/25, Egyptian fractions represented a solution to the Eye of Horus round-off problem by converting any rational number to an exact unit fraction series by selecting LCMs. The RMP 2/n table converted 51 rational numbers by selecting 14 optimized LCMs.

Summary: Middle Kingdom Egyptian arithmetic was written in non-optimal and optimal unit fraction series correcting Old Kingdom round-off errors. Early 1900s researchers minimized the EMLR’s significance. The EMLR, the Kahun Papyrus (KP) 2/n table and Rhind Mathematical Papyrus, and the RMP 2/n table demonstrate that red auxiliary numbers assisted in converting rational numbers to exact unit fraction series. The EMLR and 2/n tables can be seen using one LCM method. The EMLR, and 8 non-optimal LCMs, defined a method also used KP 2/n table and RMP 2/n table.

[edit] 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 (Hultsch 1895).

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

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

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 (Knorr 1982).

2002 – Gardner identifies five abstract EMLR patterns

2008 – Gardner identifies one EMLR pattern, based on RMP red auxiliary numbers LCMs

[edit] Sources

REFERENCES

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--- " Mathematical Roll of Egypt", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, Nov. 2005.

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--- “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.

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Vogel, Kurt. “Erweitert die Lederolle unserer Kenntniss ägyptischer Mathematik Archiv fur Geschichte der Mathematik, V 2, Julius Schuster, Berlin (1929): 386-407

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