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.

The Moscow Mathematical Papyrus

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. (Note that the mathematics used to solve the 14th problem is both legible and correct!)

Recent interest in the 14th problem of the Moscow Mathematical Papyrus mainly revolves around the Great Pyramid of Giza, because modern calculations show that the Great Pyramid could not have been constructed, in any way, in a reasonable time frame without mechanical help. (See Great Pyramid of Giza: Talk.) The Moscow Mathematical Papyrus helps to demonstrate this further by showing that the ancient Egyptians did have some of the technical knowledge necessary for such an achievement (perhaps kept secret from the remainder of the Egyptian population as well as the world and known only to priests/royalty). The 14th 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.

Finding the volume of the frustrum represented (by the shaded region) in the Moscow papyrus is complicated and relies on using the general formula shown above. What is surprising is that this general formula for finding the volume of a frustrum may be mathematically derived using only the methods of Calculus. Common knowledge before has held that the origins of calculus only go so far back as the ancient Greeks. (See History of calculus.) Therefore, present day revelations challenge these assumptions. The Moscow Mathematical Papyrus also contains another calculus-like question, pertaining to calculating the surface area of a curvilinear area. (See references below.)

See also the interesting discussion about the Great Pyramid of Giza and the golden ratio at Egyptian Mathematics (http://www-history.mcs.st-andrews.ac.uk/HistTopics/Egyptian_mathematics.html) (bottom of web page). Note that in a new book recently published, modern day Egyptologist and architect Corinna Rossi (Architecture and Mathematics in Ancient Egypt, Cambridge University Press, 2004, pp. 23-56) presents fascinating and exorbitant evidence indicating ancient Egyptian knowledge of the golden ratio (1.618...) as demonstrated by a modern and comprehensive architectural analysis of ancient Egyptian structures. For more details on present day Revisionistic arguments similar to what is mentioned in the discussion above, see origins of chess and Alphabet: History and diffusion.

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, aquired 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 π accurate to within less than one per cent, it also describes one of the earliest attempts at squaring the circle and in the process provides persuasive evidence against the theory that the Egyptians deliberately built their pyramids to enshrine the value of π in the proportions. Even though it would be a strong overstatement to suggest that the papyrus represents even rudimentary attempts 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. See Mathematics in Egyptian Papyri (http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Egyptian_papyri.html). See also some very informative articles posted in the references below.

Related Articles

• Papyrus Harris I
• Rollin Papyrus

References

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).
• Bryant University Community Web Site. The History of Geometry in Egypt (http://web.bryant.edu/~history/h453proj/spring_99/geometry/Egyptian.htm).
• 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
• Bryant University Community Web Site. The History of Geometry in Egypt (http://web.bryant.edu/~history/h453proj/spring_99/geometry/Egyptian.htm).
• 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).