Moscow and Rhind Mathematical Papyri

The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts discovered. They are both written on papyrus.

The Moscow Mathematical Papyrus

The Moscow Mathematical Papyrus is also called the Golenischev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Goleniščev. It later entered the collection of the Pushkin State Museum of Fine Arts 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 Vasily 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.

The volume is found to be 56 cubic 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. We do not know how the Egyptians arrived at the formula for the volume of a frustum.

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.

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. Other topics covered include what we today recognise as algebra, geometry and trigonometry. 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”.

References

General

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 0871692325

Moscow Mathematical Papyrus

Allen, Don. April 2001. The Moscow Papyrus and Summary of Egyptian Mathematics.
Mathpages.com. The Prismoidal Formula.
Struve, Vasilij Vasil'evič, and Boris 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 and The Moscow Mathematical Papyrus.
Zahrt, Kim R. W. Thoughts on Ancient Egyptian Mathematics.

Rhind Mathematical Papyrus

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 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.

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The Moscow Mathematical Papyrus

The Rhind Mathematical Papyrus

See also

References