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The Egyptian Mathematical Leather Roll (EMLR) dates to the 1850 BC to 2000 BCE period of the Egyptian Middle Kingdom. The leather scroll has been housed in the British Museum from 1864 to the present. The leather roll was not softened and unrolled until 1927. Scholars initially had correctly reported the the text's practical arithmetical relationships, but overlooked theoretical considerations.
The text is an important historical document, recording 26 lines of hieratic era data. The data defines one scribal method that converted 1/p and 1/pq to non-optimal Egyptian fraction series. Early 1930's scholars had considered the EMLR data as likely being decoding door to understand RMP 2/n table construction methods. Today, scholars understand the scribal use of theoretical p, and q, prime numbers, as well as the EMLR student method that wrote 22 unit fractions, 1/8 three times, 1/7 two times, and 1/16 two times, in
total writing out 26 different Egyptian fraction series.
Concerning details, the text reports two methods to 1/p, and 1/pq. The first multiplied 1/p or 1/pq by the multiple 2, 3, 4, 5, 6, 7, and 25. Each conversion obtained not-so-elegant Egyptian fraction series.
The EMLR student used multiple and identity methods to compute non-optimal Egyptian fraction series. For example, 1/8 was converted by three multiples: 3, 5, 25 obtaining non-optimal Egyptian fraction series by:
 ,
 , and
 .
A identity method defining 1= (1/2 + 1/3 + 1/6) to convert 1/7, 1/9, 1/11 and 1/15 had been improperly considered:
 ,
 ,
 , and
 .
More likely, 1/7, 1/9, 1/11 and 1/15, were converted by the EMLR student using a multiple of 6, considering:
and so forth for 1/9, 1/11 and 1/15.
An ancient scribal error converted 1/13 to (1/28 + 1/49 + 1/96), a sum equaling 3/49 rather than 3/39. The error may show that a multiple 3 had been improperly applied.
Overall, the EMLR reports 22 unit fractions, prime and composite denominators, converted to Egyptian fractions by using seven multiples: 2, 3, 4, 5, 6, 7, and 25. A detailed discussion of the text is the found on Egyptian Mathematical Leather Roll.
It should be noted that advanced Egyptian scribes converted 2/n, n/p, and n/pq to short, and concise unit fraction series. In other words, advanced scribes generally converted small and large rational numbers to optimal Egyptian fraction series. Hence, Egyptian fraction classes introduced students to a non-optimal version of the Egyptian fraction system by learning a method that was closely related to a RMP 2/n tables method.
In summary, the EMLR was a leather roll containing 26 Egyptian fraction series. The text likely was produced by a student scribe interested in learning 2/n table construction methods. The EMLR likely used a single multiple method. Multiples were introduced by applying 2, 3, 4, 5, 6, 7 and 25, to a small number of binary and prime denominator rational numbers by creating non-optimal unit fraction series. An error in the EMLR incorrectly converted 1/13, showing that the student may not have been ready to graduate.
- 1
- Milo Gardner, The Egyptian Mathematical Leather Roll Attested Short Term and Long Term, History of Mathematical Sciences, Hindustan Book Company, 2002.
- 2
- Milo Gardner, "Mathematical Roll of Egypt", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, 2005
- 3
- Richard J Gillings, The Egyptian Mathematical Leather Roll, Australian Journal of Science 24 pgs 339-344, 1962.
- 4
- Richard J Gillings, The Egyptian Mathematical Leather Roll, Archive for History of Exact Sciences pgs 291-298, 1974
- 5
- Richard J Gillings, The Egyptian Mathematical Leather Roll Line 8, how did the Scribe do it?, Historia Mathematica pgs 456-457, 1981.
- 6
- S.R.K Glanville, "Mathematical Leather Roll in the British Museum", Journal of Egyptian Archaeology pgs 232-8, 1927
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