Mathematics in Egypt: Mathematical Leather Roll (2014 Update)

Henry Rhind purchased a 10″ × 17″ leather roll (Egyptian Mathematical Leather Roll, EMLR) and a mathematical papyrus (Rhind Mathematical Papyrus, RMP) in 1858 on the streets of Cairo, Egypt. Both texts date to the Middle Kingdom (2050 BCE to 1550 BCE). Upon Henry Rhind’s unexpected death, both texts were gifted to the British Museum in 1864 (Seyf & Hall, 1927). The papyrus was published in 1879 in Germany, and reported as translated in the USA (Chace, 1927). In 1927 the EMLR was chemically softened, unrolled, and analyzed (Seyf & Hall, 1927).

The Middle Kingdom hieratic script was written right to left. There were 26 rational numbers listed in a right column, followed by a series of equivalent unit fractions. There were ten binary rational numbers: 1/2, 1/4 (twice), 1/8 (thrice), 1/16 (twice), 1/32, and 1/64. There were seven other even rational numbers: 1/6 (twice—but wrong once), 1/10, 1/12, 1/14, 1/20, and 1/30. There were also nine odd rational numbers: 2/3, 1/3 (twice), 1/5, 1/7, 1/9, 1/11, 1/13, and 1/15.

British Museum examiners found no description of how or why the equivalent unit fraction series were recorded (Gillings, 1972). There was a trivial scribal error associated with the 1/15 unit fraction series. The scribe mistakenly listed 1/6. A serious scribal error was associated with the 1/13 line, a problem that the examiners did not resolve. Seyf and Hall (1927) naively reported in The British Museum Quarterly the chemical analysis was more interesting than the Egyptian mathematical leather roll's apparent singular focus on additive unit fraction relationships.

Nearby Babylonian scribes used an infinite series base 60 system 1,000 years before finite Egyptian numeration statements appeared. Babylonians gained numerical accuracy by minimizing round-offs to 1/3,600 for two terms, and 1/216,000 for. Babylonian inverse prime number tables used for dividing fractions by fractions rounded off and degraded an intended high accuracy (Campbell-Kelly, et al, 2003).

The Egyptian binary fractions were restatements of and improvements to the Old Kingdom Horus-Eye infinite series system. The Egyptian system asked a “decimal fraction” numeration question: How can a Horus-Eye representation for the number one (1)—and all other numbers—that rounded off a seventh 1/64 term be represented by an exact unit fraction series?

Properties of infinite and finite Egyptian arithmetic were similar to those contained in our modern decimal system (Ore, 1948). Modern researchers, since 1927, minimize the EMLR's significance as a teaching document by overlooking non-additive aspects of the student test paper. Horus-Eye round off errors were implicitly corrected in the EMLR. But did the EMLR demonstrate that unit fractions were always intended to be recorded as exact unit fraction series?

Probing the intent of the text, EMLR examiners suggested that the 26 lines only contained simple additive information. Five modern arithmetic categories freshly parsed the EMLR's 26 unit fraction series in 2002. The first four additive categories were understood in 1927. However, a non-additive fifth category, an algebraic identity, included single and double least common multiples (LCM) implicitly scaled unit fractions to smaller unit fraction series (Gardner, 2002).

Today it is clear that the fifth category was not intended to be an algebraic identity. The student scribe had scaled 1/8 by two single LCMs, 3/3 and 5/5, as alternatives. The complex fifth method also scaled 1/8 by a pair of LCMs 25/25 and 6/6, a two-part alternative. The three alternative 1/8 series did not include shorthand notes. A modern translation of the scribal data therefore adds intermediate steps in blue.

1. 1/8 (3/3) = 3/24 = (2 + 1)/24 = 1/12 + 1/24

2. 1/8(5/5) = 5/40 = (4 + 1)/40 = 1/10 + 1/40

3. 1/8(25/25) = 25/200 = (8 + 17)/200 = 1/25 + 17/200

7/200(6/6) = 102/1200 = (80 + 16 + 6)/1200

1/25 + 1/15 + 1/75 + 1/200

As two confirmations, the RMP and Akhmim Wooden Tablet (AWT), the later housed in a Cairo, Egypt museum, scaled a hekat (a volume unit) by double LCMs to record sub-units. Hekat quotients scaled a hekat unity (64/64) when multiplied by 1/n. The main remainder was scaled by LCM 5/5. The 1/320th of a hekat unit was called ro (Vymazalova, 2002; Gardner, 2006). The RMP and AWT scribal shorthand was subtle. A modern translation of the longhand reports:

1. (64/64) (1/3) = 21/64 + (1/192)(5/5) = (16 + 4 + 1)/64 + 5/3(1/320) =

1/4 + 1/16 + 1/64 + (1 + 2/3) ro

2. (64/64)(1/10) = 6/64 + 1/640(5/5) = (4 + 2)/64 + 20/10(1/320) =

1/16 + 1/32 + 2 ro

Specific common EMLR and RMP rational number conversion methods had been searched out during the 20th century. As a scribal division issue, historians early on named a scribal "pick a number" guess "false position” (Eves, 1961). But were “false position” guesses explicitly or implicitly present in the EMLR, RMP, and/or the AWT as scribal division guesses?

Given that the EMLR recorded no intermediate calculations, no explicit prove of “false position” division guessing was found. Concerning the RMP and AWT explicit scribal division of a fraction by a second fraction inverted and multiplied the second fraction was found, without any use of a “false position” division guess (Gardner, 2006, M. Gardner, 2008).

In summary, the EMLR’s 26 lines of text responded to the same infinite series question, find an exact unit fraction series for a given unit fraction. Concerning details, the EMLR student scaled Horus Eye (binary) and other unit fractions by single and double LCMs m/m to record exact finite series. EMLR data reported concise and awkward finite series without “false position” in ways that were impossible in the older Egyptian and Babylonia infinite series systems.

References

Campbell-Kelly, M., Croarken, M., Flood, R., & E. Robson, E. (Eds.). (2003). The history of mathematical tables from Sumer to spreadsheets. Oxford: Oxford University Press.

Eves, H. (1961). An introduction to the history of mathematics. New York: Holt, Rinehart & Winston.

Gardner, M. (2002). The Egyptian Mathematical Leather Roll, attested short term and long term. In I. Grattan-Guiness, & B. C. Yadav (Eds.), History of the mathematical sciences (pp. 119–134). New Dehli: Hindustan Book Agency.

Gardner, M. (2006). The Arithmetic used to solve an ancient Horus-Eye problem. Ganita Bharati: Bulletin for the Indian Society for the History of Mathematics, David Pingree . Volume, 28, 157–173.

M. Gardner. (2008, July 21). Breaking the RMP 2/n table code. (Web log comment. Retrieved

From http://remprectotable.blogspot.com/)

Gillings, R. J. (1972). “The Egyptian Mathematical Leather Roll”, Mathematics in the Time of Pharaohs, 89-104.

Ore, O. (1948). Number Theory and its History. New York: McGraw Hill.

Seyf A., & Hall, H. H. (1927). Laboratory Notes: Egyptian Mathematical Leather Roll of the Seventeenth Century BC. British Museum Quarterly 2, 56–57.

Vymazalova, H. (2002). The Wooden Tablets from Cairo: The Use of the Grain Unit HK3T in ancient Egypt." Archiv Orientalni, 70(1), 27–42.