Definitions

# Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus (RMP) (also designated as: 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. It dates to around 1650 B.C. The British Museum, where the papyrus is now kept, acquired it in 1864 along with the Egyptian Mathematical Leather Roll, also owned by Henry Rhind; there are a few small fragments held by the Brooklyn Museum in New York. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus. The Rhind Papyrus is larger than the Moscow Mathematical Papyrus, while the latter is older than the former.

The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt and is the best example of Egyptian mathematics. It was copied by the scribe Ahmes (i.e., Ahmose; Ahmes is an older transcription favoured by historians of mathematics), from a now-lost text from the reign of king Amenemhat III (12th dynasty). Written in the hieratic script, this Egyptian manuscript is 33 cm tall and over 5 meters long, and began to be transliterated and mathematically translated in the late 19th century. In 2008, the mathematical translation aspect is incomplete in several respects. The document is dated to Year 33 of the Hyksos king Apophis and also contains a separate later Year 11 on its verso likely from his successor, Khamudi.

In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving “Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secrets”.

## Mathematical problems

The papyrus begins with the RMP 2/n table and follows with 84 problems, written on both sides. Taking up roughly one third of the manuscript is the RMP 2/n table which expresses 2 divided by the odd numbers from 5 to 101 in terms only of unit fractions. Many scholars have claimed to have fully decoded the 2/n table, a task that, in certain subtle respects, is officially incomplete. There appears to have been two vulgar fraction methods used to create the table, one to convert 2/p and another to convert 2/pq vulgar fractions to Egyptian fractions. In 2008 it has been shown that one LCM method was used for both. An early reporting of the $frac\left\{2\right\}\left\{p\right\}$ method was noted by F. Hultsch in 1895, and confirmed by E.M. Bruins in 1945. Today it is called the H-B method. The $frac\left\{2\right\}\left\{pq\right\}$ method consisted of three methods, with two based on the Egyptian Mathematical Leather Roll and the third factoring $frac\left\{2\right\}\left\{95\right\}$ into $frac\left\{1\right\}\left\{5\right\} times frac\left\{2\right\}\left\{19\right\}$, with $frac\left\{2\right\}\left\{19\right\}$ converted by the H-B Method. By applying Occam's Razor, the singular 2/p and 2/pq LCM method solved the 2/95 problem by selecting a multiple 12, as Ahmes had used to convert 2/19, by writing out:

$frac\left\{2\right\}\left\{95\right\}$*$frac\left\{12\right\}\left\{12\right\}$ = $frac\left\{24\right\}\left\{1140\right\}$ = $frac\left\{19 + 3 + 2\right\}\left\{1140\right\}$ = $frac\left\{1\right\}\left\{60\right\}$ + $frac\left\{1\right\}\left\{380\right\}$ + $frac\left\{1\right\}\left\{570\right\}$

Ahmes often listed the additive (19 + 3 + 2) numerator of his shorthand info, omitting other data, thereby confusing math historians for 100 years (with several suggesting that 2/n table conversions had only been additive based).

The RMP's 84 problems begin with six division-by-10 problems, the central subject of the Reisner Papyrus. There are 15 problems dealing with addition, and 18 algebra problems. There are 15 algebra problems of the same type. They ask the reader to find x and a fraction of x such that the sum of x and its fraction equals a given integer. Problem #24 is the easiest, and asks the reader to solve this equation, x + 1/7x = 19. Ahmes, the author of the RMP, worked the problem this way:

(8/7)x = 19, or x = 133/8 = 16 + 5/8,

with 133/8 being the initial vulgar fraction find 16 as the quotient and 5/8 as the remainder term. Ahmes converted 5/8 to an Egyptian fraction series by (4 + 1)/8 = 1/2 + 1/8, making his final quotient plus remainder based answer x = 16 + 1/2 + 1/8.

Each of the RMP's other 14 algebra problems produced increasingly difficult vulgar fractions. Yet, all were easily converted to an optimal (short and small last term) Egyptian fraction series.

Two arithmetical progressions (A.P.) were solved, one being RMP 64. The method of solution followed the method defined in the Kahun Papyrus. The problem solved sharing 10 hekats of barley, between 10 men, by a difference of 1/8th of a hekat finding 1 7/16 as the largest term.

The second A.P. was RMP 40, the problem divided 100 loaves of bread between five men such that the smallest two shares (12 1/2) were 1/7 of the largest three shares' sum (87 1/2). The problem asked Ahmes to find the shares for each man, which he did without finding the difference (9 1/6) or the largest term (38 1/3). All five shares 38 1/3, 29 1/6, 20, 10 2/3 1/6, and 1 1/3) were calculated by first finding the five terms from a proportional A.P. that summed to 60. The median and the smallest term, x1, were used to find the differential and each term. Ahmes then multiplied each term by 1 2/3 to obtain the sum to 100 A.P. terms. In reproducing the problem in modern algebra, Ahmes also found the sum of the first two terms by solving x + 7x = 60.

The RMP continues with 5 hekat division problems from the Akhmim Wooden Tablet, 15 problems similar to ones from the Moscow Mathematical Papyrus, 23 problems from practical weights and measures, especially the hekat, and three problems from recreational diversion subjects, the last the famous multiple of 7 riddle, written in the Medieval era as, "Going to St. Ives".

The Rhind Mathematical Papyrus also contains the following problem related to trigonometry:

The solution to the problem is given as the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face.

## Mathematical knowledge

Upon closer inspection, modern-day mathematical analyses of Ahmes' problem-solving strategies reveal a basic awareness of composite and prime numbers; arithmetic, geometric and harmonic means; a simplistic understanding of the Sieve of Eratosthenes, and perfect numbers.

The papyrus also demonstrates knowledge of solving first order linear equations and summing arithmetic and geometric series.

The papyrus calculates π as $\left(8/9\right)^2*4 simeq 3.1605$ (a margin of error of less than 1%). Two viable theories offering some insight into a possible motivation for such an accurate derivation have been proposed:

1. African crafts demonstrating snake curves and sets of equidistant concentric circles, and
2. Boardgame resembling Mancala, found in the Mortuary Temple of Seti I (both boardgames utilize small and large circles; see Mancala).

Other problems in the Rhind papyrus demonstrate knowledge of arithmetic progressions, algebra and geometry.

The papyrus also demonstrates knowledge of weights and measures, business, and recreational diversions.

## Influence of the RMP

The Egyptian use of arithmetic proportions in the Rhind Papyrus, problems 40 and 64, and the Kahun Papyrus, are briefly discussed by Gillings. In particular the use of the Remen, which has two values, is reflected in the foot which has two values, (the second being the nibw or ell which is two feet), and the cubit which has two values. Doubling is also seen in the subdivisions such as fingers and palms. Since doubling seems to have been the basis of most of the unit fraction calculations, which it was not (multiples were) up to and including the calculations of circles with dimensions given in khet (see Ancient Egyptian units of measurement), looking at how the remen and seked were used provided many insights to Greek and Roman geometers and architects. The actual and proposed readings/decodings of the RMP and Kahun 2/n tables is required to be fairly interjected.

In the Rhind Papyrus we first encounter the remen which is defined as the proportion of the diagonal of a rectangle to its sides when its other sides are whole units. Yet, a singular arithmetic proportion formula reported in the RMP and Kahun Papyrus offer an additional example beyond the remen's diagonal of a square, with its sides a cubit. We also find problems using the seked or unit rise to run proportion. Typical of the Classical orders of the Greeks and Romans, it was built upon the canon of proportions derived from the inscription grids of the Egyptians.

This document is one of the main sources of our knowledge of Egyptian mathematics.

## References

• Borbola J. Kiralykörök /the Hungarian reading and solving of the Rhind-papyrus/
• Borbola J. Olvassunk együtt magyarul /Hungarian reading and solving of the Moskow Mathematic Papyrus/
• *

• O'Connor and Robertson, 2000. Mathematics in Egyptian Papyri
• Williams, Scott W. Mathematicians of the African Diaspora, containing a page on Egyptian Mathematics Papyri
• Imhausen, A., Ägyptische Algorithmen. Eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten, Wiesbaden 2003.
• Gardner, Milo, Egyptian math (blog), "An Ancient Egyptian Problem and its Innovative Arithmetic Solution", Ganita Bharati, 2006, Vol 28, Bulletin of the Indian Society for the History of Mathematics, MD Publications, New Delhi, pp 157-173.
• http://rmprectotable.blogspot.com/, Breaking the RMP 2/n table code
• Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. ISBN 0-87169-232-5
• Allen, Don. April 2001. The Ahmes Papyrus and Summary of Egyptian Mathematics
• Chace, Arnold Buffum. 1927-1929. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations. Classics in Mathematics Education 8. 2 vols. Oberlin: Mathematical Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979). ISBN 0-87353-133-7
• Peet, Thomas Eric. 1923. The Rhind Mathematical Papyrus, British Museum 10057 and 10058. London: The University Press of Liverpool limited and Hodder & Stoughton limited
• Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0-7141-0944-4
• Truman State University, Math and Computer Science Division. Mathematics and the Liberal Arts: The Rhind/Ahmes Papyrus
• Egyptian Mathematical Leather Roll

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