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 leather roll'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 may have been corrections to the Horus-Eye, or Eye of Horus, numeration system. The 500 to 1,000 year older Horus-Eye arithmetic employed an infinite series numeration system that rounded-off its fractions to 6-term binary fraction series, throwing away 1/64 units. Horus-Eye fractions are indirectly 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 in closely related ways.

Summary: Middle Kingdom Egyptian arithmetic methods were written in hard to read unit fraction series. Early researchers minimized the EMLR’s significance. The EMLR, the Rhind Mathematical Papyrus and the RMP 2/n table demonstrate a multiple method that converts rational numberss to exact unit fraction series. That is, considering the RMP 2/n table, and the EMLR as one document, Middle Kingdom Egyptian fractions were generally converted to Egyptian fractions by a single multiple method.

[edit] 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).

1990s – A multiple method was used in the EMLR to find 1/p and 1/pq unit fraction series. The multiple was improved in the RMP, written as (p + 1). The improved method was used to convert 21 of 24 Ahmes’ 2/(pq) series. Ahmes’ multiple method has been reported as an algebraic identity: 2/(pq) = 1/A x A/(pq), with A = (p + 1). However, the multiple form, written as (p + 1), is seen in a simplest form. For example, 2/21: (3+ 1)/(3+ 1)x 2/21 = 8/84 = (6 + 2)/84 = 1/14 + 1/42, as Ahmes' shorthand suggests.

There are five categories (a–e) that summarize the rational numbers converted in the EMLR’s 26 unit fraction series. Three are identities (a, b, c) and one (d) is a possible remainder. The first four categories are best seen as multiples of the initial rational number. The first four categories have been improperly reported as additive arithmetic since 1927. The multiple aspects of the EMLR conversion method(s) were not formally explored and published until 2002.

Detailing the fifth method (e), it was first reported as an algebraic identity in 2002. In 2006 it was reported as a simple multiple. In either case, the EMLR student used a set of multiples to convert 26 rational number. The multiples were 2, 3, 4, 5, 6, 7 and 25.

To analyze the five EMLR categories, for decoding purposes, the following information is offered.

a. Four rational numbers used the identity 1 = 1/2 + 1/2 was written as 1/n = 1/(2n) + 1/(2n). As a multiple of 2, or 1 = 2/2 = (1 + 1)/2 = 1/2 + 1/2.

b. Ten rational numbers use the identity 1/2 = 1/3 + 1/6 were written as 1/(2p) = 1/p x (1/3 + 1/6). As multiple of 3, or 1/2p = 1/2p x 3/3 = (2 + 1)/6p = 1/3p + 1/6p.

c. Four rational numbers use the identity 1 = 1/2 + 1/3 + 1/6 were written as 1/p = 1/p x (1/2 + 1/3 + 1/6) = 1/2p + 1/3p + 1/6p.

d. Three rationals used a remainder 1/p-1/(p + 1) = 1/p x (p + 1) were written as 1/p = 1/(p +1) + 1/p x (p + 1). Several multiples may have also used this method, thus subtraction was not an EMLR requirement.

e. Five rational numbers may have used an advanced algebraic identity method 1/(pq) = 1/A x A/(pq), or a simple multiple method, setting the multiple to the variable (p + 1).

For example, the EMLR student set used the multiple (1) 25/25, or (2) p = 1, q = 8, A = 25, such that:

1. 1/8 x 25/25 = 25/200 = 1/5 x 25/40 = 1/5 x 5/8, with 5/8 = 1/5 + 1/3 + 1/15 + 1/40

2. 1/8 = 1/25 x 25/8 = 1/5 x 25/40 = 1/5 x 5/8, with 5/8 = 1/5 + 1/3 + 1/15 + 1/40

ans: 1/8 = 1/25 + 1/15 + 1/75 + 1/200

The EMLR data proposed seven partitioning values (A = 2, 3, 4, 5, 6, 7, and 25). The partitioning A values were multiples of 2, 3, 4, 5, 6, 7, and 25, raising unit fractions to solvable Egyptian fraction series. That is, EMLR unit fraction were multiplied by 2/2, 3/3, 4/4, 5/5, 6/6, 7/7, or 25/25, creating non-optimal Egyptian fraction series. The EMLR data reports a scribal training method. The EMLR's methods can be better understood by introducing the modern idea of Occam's Razor, that the simplest method is likely the historical method. In either case, A values, or multiples, the EMLR was a student scribes' introduction to a generalized Rhind Mathematical Papyrus 2/nth table method.

In other words, the EMLR student converted 1/p and 1/pq by raising fractions to a multiple, thereby allowing non-optimal unit fraction series to be written. Following the EMLR technique, the student was preparing to generally convert RMP 2/p and 2/pq to optimal Egyptian fraction series, as needed in weights and measures and other practical applications.

It should be noted that an EMLR 1/13th error may be historically resolved by the student using either methods c, d, e, or a closely related multiple method. That is, the 1/13 conversion error may have been related to a failed attempt to apply a method that the EMLR instructor requested the student to solve. The EMLR student may have been unprepared to convert 1/13 to an Egyptian fraction series without studying optimal 2/nth table conversion methods.

In summary, the EMLR conversion methods have proven to be a student’s introduction to a generalized RMP 2/n table's conversion method. The EMLR method converted six classes of vulgar fractions to non-optimal Egyptian fraction series by using multiples. Taken as a whole, the EMLR non-optimal Egyptian fraction series defined an aspect of scribal division. The improved RMP conversion method employed a factoring process.

The EMLR'S place in arithmetic history has been difficult to reach for a number of considerations. The above chronology, read in the context of Occam's Razor, provides a list of relevant considerations. Seen in basic terms, the EMLR was of one three elementary texts available to represent 2000 BCE to 1650 BCE arithmetic. The second text was the Reisner Papyrus. The Reisner scribe divided work output of 10 workers in terms of integer quotients and Egyptian fraction remainders. The third document was the Ebers Papyrus, a medical text. The medical texts defined hekat (volume) sub-units. The hekat and its sub-units were partitioned by divisors n into quotients and exact Egyptian fraction remainders. The new quotient and Egyptian fraction remainder method eliminated an Old Kingdom round off error that had been present in the Eye of Horus system. The new Egyptian fraction remainder method had been created the need for EMLR and RMP 2/n table multiples.

[edit] Sources

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