Moscow and Rhind Mathematical Papyri

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The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts and perhaps our best indication of what ancient Egyptian mathematics might have been like near 2000 BC. They are both written on papyrus.

In addition to these two historical texts, there is other evidence demonstrating an ancient Egyptian knowledge of basic mathematics and even surveying as early as 3000 BC. [1] (http://ag.arizona.edu/ABE/People/Faculty_Homepages/Cuellos_Homepage/Thoughts/ibe7.htm) See also Timeline of mathematics.

Contents

The Moscow Mathematical Papyrus

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Mpap.JPG
14th problem of the Moscow Mathematical Papyrus (Struve 1930)

The Moscow Mathematical Papyrus is also called the Golenischev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Semenovič Goleniščev. It later entered the collection of the Pushkin State Museum of Fine Arts (http://www.museum.ru/gmii/) in Moscow, where it remains today. Based on the palaeography of the hieratic text, it probably dates to the Eleventh dynasty of Egypt. Approximately 18 feet long and varying between 1 1/2 and 3 inches wide, its format was divided into 25 problems with solutions by Vasilij Struve in 1930. The mathematics, however, is illegible in some spots and erroneous in others. Nevertheless, one problem in particular, the 14th, has received some heightened interest among present-day historians.

The 14th problem of the Moscow Mathematical Papyrus is the most difficult problem. It calculates the volume of a frustum. The problem states that a pyramid has been divided (or truncated) in such a way that the top area is a square of length 2 units, the bottom a square of length 4 units, and the height 6 units, as shown.

Missing image
Mfrus3.GIF


The volume is found to be 56 units, which is correct. The calculation shows that the Egyptians knew the general formula for the volume of a frustum, as displayed on the bottom of the picture. This is remarkable since the Greeks at the time of Euclid did not solve this problem. We do not know how the Egyptians arrived at the formula for the volume of a frustum.

The solution to the 14th problem has been used to argue that the Egyptians had more technical knowledge than usually assumed. This would help to explain how they could build the Great Pyramid of Giza, since it is not clear how it could be constructed, in any way, in a reasonable time frame without mechanical help.

The Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus (i.e. 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 1865; there are a few small fragments held by the Brooklyn Museum in New York.

The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt. 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.

Besides describing how to obtain an approximation of <math>\pi<math> accurate to within less than one per cent, it also describes one of the earliest attempts at squaring the circle. Although it might be an overstatement to suggest that the papyrus represents a rudimentary attempt at analytical geometry, Ahmes did make use of a kind of an analogue of the cotangent.

Furthermore, quoting Mathpages.com,

... the 2/n table of the Rhind Papyrus, which dates from more than a thousand years before Pythagoras, seems to show an awareness of prime and composite numbers, a crude version of the 'Sieve of Eratosthenes,' a knowledge of the arithmetic, geometric, and harmonic means, and of the 'perfectness' of the number 6. This all seems to suggest a greater number-theoretic sophistication than is generally credited to the ancient Egyptians. (The Rhind Papyrus 2/N Table (http://mathpages.com/home/rhind.htm))

The Rhind papyrus also shows that the ancient Egyptians knew how to solve first order linear equations (O'Connor and Robinson, 2000). Arithmetic and geometric series also appear in the papyrus (Williams, [2] (http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_algebra.html#areithmetic%20series)). As a footnote, the Berlin Papyrus (circa 1800 BC) shows that the ancient Egyptians could solve a second-order algebraic equation (Williams, [3] (http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin)).

Related articles

References

General

Moscow Mathematical Papyrus

  • Allen, Don. April 2001. The Moscow Papyrus (http://www.math.tamu.edu/~don.allen/history/egypt/node4.html) and Summary of Egyptian Mathematics (http://www.math.tamu.edu/~don.allen/history/egypt/node5.html).
  • 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 0871692325
  • Mathpages.com. The Prismoidal Formula (http://www.mathpages.com/home/kmath189/kmath189.htm).
  • Struve, Vasilij Vasil'evič, and Boris Aleksandrovič Turaev. 1930. Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau. Quellen und Studien zur Geschichte der Mathematik; Abteilung A: Quellen 1. Berlin: J. Springer
  • Truman State University, Math and Computer Science Division. Mathematics and the Liberal Arts: Ancient Egypt (http://math.truman.edu/~thammond/history/AncientEgypt.html) and The Moscow Mathematical Papyrus (http://math.truman.edu/~thammond/history/MoscowPapyrus.html).
  • Zahrt, Kim R. W. Thoughts on Ancient Egyptian Mathematics (http://www.iusb.edu/~journal/2000/zahrt.html).

Rhind Mathematical Papyrus

  • Allen, Don. April 2001. The Ahmes Papyrus (http://www.math.tamu.edu/~don.allen/history/egypt/node3.html) and Summary of Egyptian Mathematics (http://www.math.tamu.edu/~don.allen/history/egypt/node5.html).
  • 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 0873531337
  • 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 0714109444
  • Truman State University, Math and Computer Science Division. Mathematics and the Liberal Arts: The Rhind/Ahmes Papyrus (http://math.truman.edu/~thammond/history/RhindPapyrus.html).it:Papiro di Rhind

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