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). Alexander Henry Rhind (1833 – 1863) was a Scottish lawyer and Egyptologist. ... Year 1858 (MDCCCLVIII) was a common year starting on Friday (link will display the full calendar) of the Gregorian Calendar (or a common year starting on Wednesday of the 12-day slower Julian calendar). ... The British Museum in London, England is one of the worlds greatest museums of human history and culture. ... The Rhind Mathematical Papyrus ( papyrus British Museum 10057 and pBM 10058), is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. ...

The writing consists of Middle Kingdom hieratic characters are written right to left. There are 26 rational numbers listed. Each rational number are followed by its equivalent Egyptian fraction series. There are 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 are 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 are 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. The Middle Kingdom is: a old name for China a period in the History of Ancient Egypt, the Middle Kingdom of Egypt This is a disambiguation page — a navigational aid which lists pages that might otherwise share the same title. ... Development of hieratic script from hieroglyphs; after Champollion. ... An Egyptian fraction is the sum of distinct unit fractions, such as . ... The Eye of Horus The Eye of Horus (originally, The Eye of Ra) is an ancient Egyptian symbol of protection and Royal Power, from the deity Horus or Ra. ...

The British Museum examiners found no introduction or description to how or why the equivalent unit fraction series are computed (Gillings 1981: 456-457). Equivalent unit fraction series are associated with fractions 1/3, 1/4, 1/8 and 1/16. There is a trivial error associated with the final 1/15 unit fraction series. The 1/15 series is listed as equal to 1/6. Another serious error is 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.

The Middle Kingdom Egyptian fraction conversions of the older binary fractions may be corrections to the Horus-Eye, or Eye of Horus, numeration system. The 500-1,000 year older Horus-Eye arithmetic methods employed an infinite series numeration system rounded-off to 6-term binary fraction series. 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 + … drops off the last term 1/64th, (Gillings 1972: 210). Modern decimals' round-off rules vary based on the situation. Old Kingdom Horus-Eye round-off rules did not vary. The Eye of Horus The Eye of Horus (originally, The Eye of Ra) is an ancient Egyptian symbol of protection and Royal Power, from the deity Horus or Ra. ...

Because the Middle Kingdom Egyptian arithmetic methods were written in hard to read unit fraction series, researchers frequently minimized the EMLR’s significance. One minimalist reported that the Horus-Eye binary fraction system was superior to the Egyptian fraction notation. The EMLR, and Rhind Mathematical Papyrus may have demonstrated ways to convert intermediate rational numbers to exact unit fraction series, possible facts that are implied by the EMLR's contents. Considering the RMP, and the EMLR, as one document, Middle Kingdom Egyptian fraction system defined exact forms of division using methods that have not been reported in the older Horus-Eye numeration and arithmetic system. The Rhind Mathematical Papyrus ( papyrus British Museum 10057 and pBM 10058), is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. ...

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 may summarize the EMLR’s 26 unit fraction series. Three are identities (a, b, c) and one (d) is a possible remainder. The first four categories can been seen a multiples of the initial rational number. The first four categories have been understood in additive arithmetic terms since 1927. However, the ideas of multiples and other non-additive conversion methods were not explored in 1927.

Going on to the fifth method (e), it may have been an algebraic identity. Or, more likely, it may have been a simple multiple. In either case, the EMLR student used a set of methods that converted 26 rational number using multiples of 2, 3, 4, 5, 7 and 25, plus an identity 1 = 1/2 + 1/3 + 1/6.

To analyze each of the five EMLR categories, 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

Interestingly, the EMLR's four proposed A values (2, 3, 4, 5, 7 and 25) can best be seen as multiples of 2, 3, 4, 5, 7, and 25, thereby raising the initial unit fraction to a scaled vulgar fraction. The initial EMLR unit fraction can therefore be multiplied by 2/2, 3/3, 4/4, 5/5, 7/7, or 25/25, as needed, thereby creating easy to convert vulgar fractions. The EMLR vulgar fraction calculates non-optimal Egyptian fraction series. The EMLR multiple method is central in importance when considering Occam's Razor, the simplest method is most likely the historical method. In either case, A values, or multiples, the EMLR is fairly read as practicing Rhind Mathematical Papyrus 2/nth table conversion methods. For the House episode, see Occams Razor (House episode) Occams razor (sometimes spelled Ockhams razor) is a principle attributed to the 14th-century English logician and Franciscan friar William of Ockham. ... The Rhind Mathematical Papyrus ( papyrus British Museum 10057 and pBM 10058), is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. ...

That is, the EMLR student converted 1/p and 1/pq by raising the initial fraction to a range of multiples, and a larger vulgar fractions, thereby allowing non-optimal unit fraction series to be written. Later the student would have learned to convert RMP 2/p and 2/pq vulgar fractions to optimal Egyptian fraction series by using slightly modified EMLR methods.

It should be noted that the 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 was related to a failed attempt to apply a method that the EMLR instructor may have requested his student to solve. The EMLR student may have been unprepared to convert 1/13 to an Egyptian fraction series without studying the 2/nth table and its optimal Hultch-Bruins 2/p methods.

In summary, the EMLR has proven to be a student’s introduction to RMP 2/n table unit fraction conversion methods, methods that included six vulgar fractions created from multiples. Taken as a whole, the non-optimal vulgar fractions defined an aspect of scribal division. The improved and optimal RMP arithmetic method of division converted any rational number to an Egyptian fraction series after employing a factoring process. The RMP obtained elegant and exact unit fraction series by selecting the multiple (p + 1)/(p + 1), as cited in 21 0f 24 RMP 2/pq conversions. Another conversion method was used for all RMP 2/p conversions.

The EMLR'S actual place in history has been difficult to reach for a number of considerations. The above chronology, read in the context of Occam's Razor, provides an outline of the relevant considerations. Seen in its most basic terms, the EMLR was of one two elementary texts written during the 2000BC to 1650 BC period. The second basic text was the Reisner Papyrus. The Reisner outlines scribal division in terms of remainder arithmetic includes quotients and remainders. Reisner and EMLR vulgar and unity fraction remainders were converted to Egyptian fraction series using the EMLR and RMP multiple methods, and other methods defined in the RMP 2/nth table. THE REISNER PAPYRUS is one of the most basic of the hieratic mathematical texts. ...

Sources

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links

• http://emlr.blogspot.com Egyptian Mathematical Leather Roll
• http://egyptianmath.blogspot.com History of Egyptian fractions