Moscow Mathematical Papyrus

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 12th Dynasty of Egypt, around 1850 BC. Around 5.5 m (18 ft) long and varying between 3.8 and 7.6 cm (1.5 and 3 in) wide, its format was divided by the Soviet Orientalist Vasily Struve in 1930 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 Moscow Papyrus contains many math problems without a specific order. It is famous for its geometry questions, like finding the surface area of a rounded shape and the space inside a cut-off pyramid. Other problems include figuring out parts of ships, solving puzzles with unknown numbers, and measuring how strong bread or beer is made from grain.

Some problems even ask how much work can be done, such as figuring out how many smaller logs equal a certain number of bigger logs, or how many sandals a shoe maker can finish. These exercises show the clever math skills of ancient people.

Main article: Rhind Mathematical Papyrus

Further information: Egyptian algebra

Two geometry problems

The Moscow Mathematical Papyrus includes two interesting geometry problems.

Problem 10 asks how to find the surface area of a special round shape, like half of a ball. The ancient Egyptians used a step-by-step method to calculate this, showing they had clever ways to work with shapes.

Problem 14 shows how to find the space inside a cut-off pyramid, where the top and bottom are flat squares of different sizes. The Egyptians used a clear process to work this out, and their answer matches what we know today.

Summary

The Moscow Mathematical Papyrus is an ancient Egyptian document filled with math problems. It includes questions about counting, shapes, and solving equations. Scholars study it to learn about how people in ancient times did mathematics.

Numbers written with special marks in the papyrus show fractions, like one-fourth, which were widely used in their calculations.

The contents of The Moscow Mathematical Papyrus

No.	Detail
1	Damaged and unreadable.
2	Damaged and unreadable.
3	A cedar mast. 3 ¯ 5 ¯ {\displaystyle {\bar {3}}\;{\bar {5}}} of 30 = 16 {\displaystyle 30=16} . Unclear.
4	Area of a triangle. 2 ¯ {\displaystyle {\bar {2}}} of 4 × 10 = 20 {\displaystyle 4\times 10=20} .
5	Pesus of loaves and bread. Same as No. 8.
6	Rectangle, area = 12 , b = 2 ¯ 4 ¯ l {\displaystyle =12,b={\bar {2}}\;{\bar {4}}l} . Find l {\displaystyle l} and b {\displaystyle b} .
7	Triangle, area = 20 , h = 2 2 ¯ b {\displaystyle =20,h=2\;{\bar {2}}b} . Find h {\displaystyle h} and b {\displaystyle b} .
8	Pesus of loaves and bread.
9	Pesus of loaves and bread.
10	Area of curved surface of a hemisphere (or cylinder).
11	Loaves and basket. Unclear.
12	Pesu of beer. Unclear.
13	Pesus of loaves and beer. Same as No. 9.
14	Volume of a truncated pyramid. V = ( h / 3 ) ( a 2 + a b + b 2 ) {\displaystyle V=(h/3)(a^{2}+ab+b^{2})} .
15	Pesu of beer.
16	Pesu of beer. Similar to No. 15.
17	Triangle, area = 20 , b = ( 3 ¯ 15 ¯ ) h {\displaystyle =20,b=({\bar {3}}\;{\bar {15}})h} . Find h {\displaystyle h} and b {\displaystyle b} .
18	Measuring cloth in cubits and palms. Unclear.
19	Solve the equation 1 2 ¯ x + 4 = 10 {\displaystyle 1\;{\bar {2}}x+4=10} . Clear.
20	Pesu of 1000 loaves. Horus-eye fractions.
21	Mixing of sacrificial bread.
22	Pesus of loaves and beer. Exchange.
23	Computing the work of a cobbler. Unclear. Peet says very difficult.
24	Exchange of loaves and beer.
25	Solve the equation 2 x + x = 9 {\displaystyle 2x+x=9} . Elementary and clear.

Other papyri

Other mathematical texts from Ancient Egypt include:

• Berlin Papyrus 6619
• Egyptian Mathematical Leather Roll
• Lahun Mathematical Papyri
• Rhind Mathematical Papyrus

General papyri:

• Papyrus Harris I
• Rollin Papyrus

For the 2/n tables see:

• RMP 2/n table