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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 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. Other area and volume problems from the Middle Kingdom are found in the Rhind Mathematical Papyrus.

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.

• Papyrus Harris I
• Rhind Mathematical Papyrus
• Rollin Papyrus

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