History of Mathematics Project | Moscow Mathematical Papyrus

History of
Mathematics

Project

around 1850 BCE

Moscow Mathematical Papyrus

Ancient Egyptian mathematical papyrus

The Moscow papyrus is one of the most famous ancient Egyptian mathematical texts. It was most likely authored in the 13th dynasty and originally dated to the 12th dynasty of ancient Egypt, dating it to around 1850 BCE.

The Moscow papyrus was purchased by Egyptologist Vladimir Golenishchev in Thebes in 1892 or 1893. The papyrus contains 25 problems in no particular order, and its contents provide less detail than present in the Rhind papyrus. The papyrus is best known for its geometry problems, including the surface area and volume of a frustum and the surface area of a hemisphere or half-cylinder, while its remaining problems are more prosaic.

Artifact dimensions

5.5 m × 5.7 cm (varies between 3.8 and 7.6 cm wide)

Original artifact location

Thebes (historical name), al-Wadi al-Jadid, Egypt (current name)

Current artifact location

Pushkin State Museum of Fine Arts, Moscow

Catalog number

E4676

Timeline

Interactive Content

Computational Explanation

The Rhind papyrus and the Moscow papyrus are both written in hieratic, the ancient Egyptian writing of abridged hieroglyphics. While the former is the larger of the two, the latter is the older. The Moscow papyrus measures 5 meters long and 8 centimeters wide and contains 25 mathematical problems. The problems follow no particular order, and the solutions of the problems provide much less detail than those in the Rhind papyrus. Information is available only about some of its sections since there are some parts unreadable. The following is a summary of the problems that are readable.

Aha Problems (1, 19 and 25)

Problems 1, 19 and 25 of the Moscow papyrus are Aha problems that involve finding unknown quantities where the sum of the quantity and parts of it are given. For instance, problem 19 asks for the calculation of a quantity taken 1 ½ times and then added to 4 to make 10. In other words:

In[]:=

Reducex+

x+410,x

Out[]=

x4

Ship's Part Problems (2 and 3)

Problems 2 and 3 are ship's part problems. One of the problems calculates the length of a ship's rudder and the other computes the length of a ship's mast given that it is 1/3 + 1/5 of the length of a cedar log originally 30 cubits long.

Baku Problems (11 and 23)

Problems 11 and 23 are Baku problems that calculate the output of workers. For example, problem 23 finds the output of a shoemaker cutting and decorating sandals.

Beer-Making Problems

The remaining non-geometric problems are all about measuring the strength of the beer made from a certain amount of grain, known to them as pefsu:

pefsu

numberofloavesofbread(orjugsofbeer)

numberofhekatsofgrain

Geometry Problems (6, 7, 10, 17 and 18)

Seven out of the 25 problems are geometry problems and range from computing the area of triangles or rectangles (4, 6, 7, 17 and 18) to finding the surface area of a hemisphere (problem 10) and finding the volume of a square pyramidal frustum.

Problem 10: Area of a Hemispherical Basket

The 10th problem of the Moscow papyrus seems to asks for a calculation of the surface area of a hemisphere.

The text of problem 10 states:

Example of calculating a basket.
You are given a basket with a mouth of 4 1/2. What is its surface?
Take 1/9 of 9 (since) the basket is half an egg-shell. You get 1.
Calculate the remainder which is 8. Calculate 1/9 of 8. You get 2/3 + 1/6 + 1/18.
Find the remainder of this 8 after subtracting 2/3 + 1/6 + 1/18. You get 7 + 1/9.
Multiply 7 + 1/9 by 4 + 1/2. You get 32.
Behold this is its area. You have found it correctly.

In[]:=

(4+1/2)(7+1/9)

Out[]=

The solution amounts to computing the area of a hemisphere (

2π

), by approximating π as 256/81 ≈ 3.16049:

In[]:=

Reduce2pi

(4+1/2)

32,pi

Out[]=

pi

256

In[]:=

N

256



Out[]=

3.16049

Using the actual value of pi

π≈3.1416

(which was not known to the ancient Egyptians), the area of that basket is:

In[]:=

SphericalPlot3D1,{θ,π/2,π},{ϕ,0,2π},

label



Out[]=

Problem 14: Volume of a Square Pyramidal Frustum

One of the most important objects exhibited in the Pushkin State Museum of Fine Arts is problem 14 of the Moscow mathematical papyrus, which is the only section of the papyrus on display. This portion of the papyrus clearly contains a diagram of a trapezoid that represents a truncated pyramid:

The problem states that a pyramid has been truncated in such a way that the top area is a square of length 2, the bottom a square of length 4 and the height 6:

The volume is found to be 56 cubic units, which is correct:

In[]:=

frustum=Hexahedron[{(*topsquare2x2*){-1,-1,6},{1,-1,6},{1,1,6},{-1,1,6},(*bottomsquare4x4*){-2,-2,0},{2,-2,0},{2,2,0},{-2,2,0}}];Volume[frustum]

Out[]=

The text of the example states:

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.

The solution to the problem indicates that the Egyptians knew the correct formula for obtaining the volume of a truncated pyramid:

In[]:=

volume[a_,b_,h_]:=

+ab+

)

… where

and

are the side lengths of the top and base of the truncated pyramid and

its height. For the frustum in question:

In[]:=

volume[2,4,6]

Out[]=

Researchers have speculated how the Egyptians might have arrived at the formula for the volume of a frustum, but the derivation of this formula is not given in the papyrus (Gillings 1964). Finally, it is interesting to note that computing the exact surface area of that frustum was something that the ancient Egyptians would not have been able to do exactly since it involves knowing about irrational numbers, and in particular the square root of 37:

In[]:=

SurfaceArea[frustum]

Out[]=

4(5+3

)

In[]:=

N[%]

Out[]=

92.9932

Other Resources

Additional Reading

Clagett, M. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society, Vol. 232. Philadelphia: American Philosophical Society, 1999.
Gillings, R. J. "The Volume of a Truncated Pyramid in Ancient Egyptian Papyri." Mathematics Teacher, Vol. 57, pp. 552–555, 1964.
Gillings, R. J. Mathematics in the Time of the Pharoahs. New York: Dover. pp. 246–247, 1972.
Imhausen, A. Ägyptische Algorithmen. Eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten. Wiesbaden, Germany: 2003.
Imhausen, A. "Egyptian Mathematics." Ch. 1 in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Ed. V. J. Katz). Princeton, NJ: Princeton University Press, pp. 7–56, 2007.
Imhausen, A. "Moscow Mathematical Papyrus. " § 9.1.6 in Mathematics in Ancient Egypt. A Contextual History. Princeton, NJ: Princeton University Press, p. 69, 2016.
Rossi, C. Architecture and Mathematics in Ancient Egypt. Cambridge, England: Cambridge University Press, p. 218, 2007.
Struve, V. V. and Turaev, B. Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau. Quellen und Studien zur Geschichte der Mathematik; Abteilung A: Quellen 1. Berlin: J. Springer, 1930.

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Image Credits

Carlos Dorce: The Mathematical Tourist