Rhind Mathematical Papyrus

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The Rhind Mathematical Papyrus (RMP, also designate 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. 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. ^[1]

The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt and 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 was first translated in the late 19th century.

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”. The papyrus has 84 problems with worked examples, written on both sides. Taking up roughly one third of the manuscript is a 2 / n table which expresses 2 divided by the odd numbers from 5 to 101 in terms only of unit fractions. There are two basic vulgar fraction methods used, one to convert 2/p and another to convert 2/pq vulgar fractions to Egyptian fractions. The 2/p method was noted by F. Hultsch in 1895, and confirmed by E.M. Bruins in 1945, commonly called the H-B method. The 2/pq method consists of three methods, with two based on the Egyptian Mathematical Leather Roll and the third factoring 2/95 into 1/5 x 2/19, with 2/19 converted by the H-B Method. Other topics covered include arithmetic progressions, algebra, geometry, weights and measures, business, and recreational diversions.

The RMP's 84 problems begin with 6 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".

[edit] See also

• Moscow Mathematical Papyrus
• Papyrus Harris I

[edit] References

1. ^ Great Soviet Encyclopedia, 3rd edition, entry on "Папирусы математические", available online here

• Rhind Papyrus. MathWorld–A Wolfram Web Resource.
• 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).
• 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.