Moscow Mathematical Papyrus

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Moscow Mathematical Papyrus

The Moscow Mathematical Papyrus is also called the Golenischev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenišcev. It later entered the collection of the Pushkin State Museum of Fine Arts in Moscow, where it remains today. Based on the palaeography

Palaeography

Palaeography, pal?ography , or paleography is the study of ancient handwriting, and the practice of deciphering and reading historical manuscripts....
of the hieratic

Hieratic

Hieratic is a cursive writing system used in Pharaoh Ancient Egypt that developed alongside the Egyptian hieroglyphs system, to which it is intimately related....
text, it probably dates to the Eleventh dynasty of Egypt

Eleventh dynasty of Egypt

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Encyclopedia

Palaeography

Palaeography, pal?ography , or paleography is the study of ancient handwriting, and the practice of deciphering and reading historical manuscripts....
of the hieratic

Hieratic

Eleventh dynasty of Egypt

The Eleventh dynasty of ancient Egypt was one group of rulers, whose earlier members are grouped with the four preceding dynasties to form the First Intermediate Period, while the later members are considered part of the Middle Kingdom 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 the Soviet

Soviet Union

The Union of Soviet Socialist Republics was a Constitution of the Soviet Union socialist state that existed in Eurasia from 1922 to 1991.The name is a translation of the , romanization of Russian Soyuz Sovetskikh Sotsialisticheskikh Respublik, abbreviated ????, SSSR....
Orientalist Vasily Vasilievich Struve

Vasily Vasilievich Struve

Vasily Vasilievich Struve was a Soviet orientalist from the Friedrich_Georg_Wilhelm_von_Struve#Family, the founder of the Soviet scientific school of researchers on the Ancient East history....
in 1930. It is one of the two well-known Mathematical Papyri along with the Rhind Mathematical Papyrus

Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scotland antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum....
. The Moscow Mathematical Papyrus is older than the Rhind Mathematical Papyrus, while the latter is the larger of the two.

Problem 10: Surface area of a Quonset hut roof

The 10th problem of the Moscow Mathematical Papyrus asks for a calculation of the surface area of a Quonset type roof hut. This is one of the first examples of the estimation of a curvilinear area.

Problem 14: Volume of frustum of square pyramid

The 14th problem of the Moscow Mathematical Papyrus is the most difficult problem. It calculates the volume of a frustum

Frustum

A frustum is the portion of a solid?normally a Cone or pyramid ?which lies between two parallel planes cutting the solid. The term is commonly used in computer graphics to describe the 3d area which is visible on the screen ....
of a pyramid or cone. There are no known examples of a volume calculation of a complete pyramid or cone. Similarly, in Mesopotamia

Mesopotamia

Mesopotamia is the area of the Tigris-Euphrates river system, along the Tigris and Euphrates rivers, largely corresponding to modern Iraq, as well as some parts of northeastern Syria, some parts of southeastern Turkey, and some parts of the Khuzestan Province of southwestern Iran....
interest seems to have been in finding the volumes of frusta rather than complete pyramids or cones. The Babylonian mathematical tablet BM 85194, for example, sets out the calculation for the volume of a trapezium-sectioned fortification wall.

Problem 14 states that a pyramid has been 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 text of the example runs like this: "If you are told: a truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top: You are to square the 4; result 16. You are to double 4; result 8. You are to square this 2; result 4. You are to add the 16 and the 8 and the 4; result 28. You are to take 1/3 of 6; result 2. You are to take 28 twice; result 56. See, it is of 56. You will find (it) right"

This describes the correct calculation: which indicates that the Egyptians knew the correct formula for obtaining the volume

Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
of a truncated pyramid:

We do not know how the Egyptians arrived at the formula for the volume of a frustum

Frustum

A frustum is the portion of a solid?normally a Cone or pyramid ?which lies between two parallel planes cutting the solid. The term is commonly used in computer graphics to describe the 3d area which is visible on the screen ....
. The Babylonians had taken the incorrect approach of averaging the area of base and top and multiplying by height.

Touraeff, the first commentator, strangely saw Problem 14 as describing the more general formula for the volume of any frustum, a formula that was not derived for another 3000 years. He was not alone in this view.

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

Moscow Mathematical Papyrus

Problem 10: Surface area of a Quonset hut roof

Problem 14: Volume of frustum of square pyramid

See also

Full Text of the Moscow Mathematical Papyrus

Other references

Footnotes