Moscow Mathematical Papyrus Explained

Moscow Mathematical Papyrus
Location:Pushkin State Museum of Fine Arts in Moscow
Width:300px
Date:13th dynasty, Second Intermediate Period of Egypt
Place Of Origin:Thebes
Language(S):Hieratic
Size:Length: 18feet
Width: 1.5to

The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geometry, and algebra. Golenishchev bought the papyrus in 1892 or 1893 in Thebes. It later entered the collection of the Pushkin State Museum of Fine Arts in Moscow, where it remains today.

Based on the palaeography and orthography of the hieratic text, the text was most likely written down in the 13th Dynasty and based on older material probably dating to the Twelfth Dynasty of Egypt, roughly 1850 BC.[1] Approximately 5.5 m (18 ft) long and varying between 1.5and wide, its format was divided by the Soviet Orientalist Vasily Vasilievich Struve[2] in 1930[3] into 25 problems with solutions.

It is a well-known mathematical papyrus, usually referenced together with the Rhind Mathematical Papyrus. The Moscow Mathematical Papyrus is older than the Rhind Mathematical Papyrus, while the latter is the larger of the two.

Exercises contained in the Moscow Papyrus

The problems in the Moscow Papyrus follow no particular order, and the solutions of the problems provide much less detail than those in the Rhind Mathematical Papyrus. The papyrus is well known for some of its geometry problems. Problems 10 and 14 compute a surface area and the volume of a frustum respectively. The remaining problems are more common in nature.[1]

Ship's part problems

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.[1]

Aha problems

Aha problems involve finding unknown quantities (referred to as aha, "stack") if the sum of the quantity and part(s) of it are given. The Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1, 19, and 25 of the Moscow Papyrus are Aha problems. For instance, problem 19 asks one to calculate a quantity taken times and added to 4 to make 10.[1] In other words, in modern mathematical notation one is asked to solve

3
2

x+4=10

.

Pefsu problems

Most of the problems are pefsu problems (see: Egyptian algebra): 10 of the 25 problems. A pefsu measures the strength of the beer made from a hekat of grain

pefsu=

numberloavesofbread(orjugsofbeer)
numberofheqatsofgrain

A higher pefsu number means weaker bread or beer. The pefsu number is mentioned in many offering lists. For example, problem 8 translates as:

(1) Example of calculating 100 loaves of bread of pefsu 20

(2) If someone says to you: "You have 100 loaves of bread of pefsu 20

(3) to be exchanged for beer of pefsu 4

(4) like 1/2 1/4 malt-date beer"

(5) First calculate the grain required for the 100 loaves of the bread of pefsu 20

(6) The result is 5 heqat. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer

(7) The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain.

(8) Calculate 1/2 of 5 heqat, the result will be 2 1/2

(9) Take this 2 1/2 four times

(10) The result is 10. Then you say to him:

(11) "Behold! The beer quantity is found to be correct."[1]

Baku problems

Problems 11 and 23 are Baku problems. These calculate the output of workers. Problem 11 asks if someone brings in 100 logs measuring 5 by 5, then how many logs measuring 4 by 4 does this correspond to? Problem 23 finds the output of a shoemaker given that he has to cut and decorate sandals.[1]

Geometry problems

Seven of the twenty-five problems are geometry problems and range from computing areas of triangles, to finding the surface area of a hemisphere (problem 10) and finding the volume of a frustum (a truncated pyramid).[1]

Two geometry problems

Problem 10

The tenth problem of the Moscow Mathematical Papyrus asks for a calculation of the surface area of a hemisphere (Struve, Gillings) or possibly the area of a semi-cylinder (Peet). Below we assume that the problem refers to the area of a hemisphere.

The text of problem 10 runs like this: "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."[1] [4]

The solution amounts to computing the area as

Area=(((2 x diameter) x

8
9

) x

8
9

) x diameter=

128
81

(diameter)2

The formula calculates for the area of a hemisphere, where the scribe of the Moscow Papyrus used

256
81

3.16049

to approximate π.

Problem 14: Volume of frustum of square pyramid

The fourteenth problem of the Moscow Mathematical calculates the volume of a frustum.

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.[1]

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

V=

1
3

h(a2+ab+b2)

where a and b are the base and top side lengths of the truncated pyramid and h is the height. 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.[5]

Summary

Richard J. Gillings gave a cursory summary of the Papyrus' contents.[6] Numbers with overlines denote the unit fraction having that number as denominator, e.g.

\bar{4}=

1
4
; unit fractions were common objects of study in ancient Egyptian mathematics.
The contents of The Moscow Mathematical Papyrus! No.! Detail
1Damaged and unreadable.
2Damaged and unreadable.
3A cedar mast.

\bar{3}\bar{5}

of

30=16

. Unclear.
4Area of a triangle.

\bar{2}

of

4 x 10=20

.
5Pesus of loaves and bread. Same as No. 8.
6Rectangle, area

=12,b=\bar{2}\bar{4}l

. Find

l

and

b

.
7Triangle, area

=20,h=2\bar{2}b

. Find

h

and

b

.
8Pesus of loaves and bread.
9Pesus of loaves and bread.
10Area of curved surface of a hemisphere (or cylinder).
11Loaves and basket. Unclear.
12Pesu of beer. Unclear.
13Pesus of loaves and beer. Same as No. 9.
14Volume of a truncated pyramid.

V=(h/3)(a2+ab+b2)

.
15Pesu of beer.
16Pesu of beer. Similar to No. 15.
17Triangle, area

=20,b=(\bar{3}\bar{15})h

. Find

h

and

b

.
18Measuring cloth in cubits and palms. Unclear.
19Solve the equation

1\bar{2}x+4=10

. Clear.
20Pesu of 1000 loaves. Horus-eye fractions.
21Mixing of sacrificial bread.
22Pesus of loaves and beer. Exchange.
23Computing the work of a cobbler. Unclear. Peet says very difficult.
24Exchange of loaves and beer.
25Solve the equation

2x+x=9

. Elementary and clear.

Other papyri

Other mathematical texts from Ancient Egypt include:

General papyri:

For the 2/n tables see:

See also

References

  1. 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.
  2. http://www.encspb.ru/en/article.php?kod=2804014273 Struve V.V., (1889–1965), orientalist :: ENCYCLOPAEDIA OF SAINT PETERSBURG
  3. Struve, Vasilij Vasil'evič, and Boris 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
  4. Williams, Scott W. Egyptian Mathematical Papyri
  5. .
  6. Book: Gillings, Richard J. . Mathematics in the Time of the Pharaohs . . 9780486243153 . 246–247.

Full text of the Moscow Mathematical Papyrus

Other references