# Mathematical Papyruses

## Papyruses, Mathematical

extant mathematical works of ancient Egypt, dating from the period of the Middle Kingdom (c. 21st century to c. 18th century B.C.). The most famous are the Rhind Papyrus, now in the British Museum (London), and the Moscow Papyrus, now in the A. S. Pushkin Museum of Fine Arts (Moscow).

The Rhind Papyrus, named after its owner, the Egyptologist H. Rhind, was first studied and published in German by A. Eisenlohr; it is also called the Ahmes Papyrus, after its compiler, the scribe Ahmes (c. 2000 B.C.). The papyrus is a collection of solutions of 84 practical problems, involving operations with fractions, the calculation of the area of a rectangle, triangle, trapezoid, and circle (the area of a circle is equal to the square of eight-ninths of its diameter), and the calculation of the volume of a rectangular parallelepiped and cylinder. It also contains arithmetic problems involving proportional division and the calculation of the ratios of the amount of grain to the number of loaves of bread or jars of beer obtained from the grain. The solution of problem 79 leads to the calculation of the sum of a geometric progression. However, no general rules, much less theoretical generalizations, are given for the solution of these problems.

The Moscow Papyrus was studied by the Russian Egyptologists B. A. Turaev (1917) and V. V. Struve (1927) and published in its entirety in German in 1930. It contains the solutions of 25 problems, approximately of the same type as those in the Rhind Papyrus. Problems nos. 10 and 14 are of particular interest. The solution of problem 14 is based on an exact formula for the volume of a truncated pyramid with square base. In problem 10, the area of the lateral surface of a semicylinder with altitude equal to its diameter (or, possibly, the surface of a hemisphere) is calculated. This is the first example in the mathematical literature involving the determination of the area of a curved surface.

The mathematical papyruses shed light on the mathematical knowledge of ancient Egypt.

### REFERENCES

Bobynin, V. V. Matematika drevnykh egiptian. Moscow, 1882.
Vygodskii, M. Ia. Arifmetika i algebra vdrevnem mire, 2nd ed. Moscow, 1967.
Veselovskii, I. N. “Egipetskaia nauka i Gretsiia.” In Trudy In-ta istorii estestvoznaniia AN SSSR, vol. 2. Moscow, 1948.
Eisenlohr, A. Ein mathematisches Handbuch der alten Ágypter, vols. 1–2. Leipzig, 1877–91.
Peet, T. E. The Rhind Mathematical Papyrus. Liverpool, 1923.
Struve, W. W. Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moscau. Berlin, 1930.
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