SafekipediaMoscow Mathematical Papyrus
Adapted from Wikipedia · Discoverer experience
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 interesting math problems without a specific order. It is especially famous for its geometry questions. For example, it shows how to find the surface area of a rounded dome and the space inside a cut-off pyramid.
Other problems include figuring out the length of parts of a ship, solving puzzles where you need to find an unknown number, and calculating how strong bread or beer is based on the grain used. There are also questions about how much work different jobs might produce, like turning logs into smaller sizes or making shoes. Overall, the papyrus gives us a glimpse into the math skills of ancient Egypt.
Main article: Rhind Mathematical Papyrus
Two geometry problems
The Moscow Mathematical Papyrus includes two interesting geometry problems.
Problem 10 asks for the surface area of a hemisphere. The ancient Egyptians gave a step-by-step way to find this area, using simple arithmetic and a special number to stand in for π.
Problem 14 asks for the volume of a frustum, which is what you call a pyramid when the top part has been cut off. The papyrus shows how to calculate this volume using the lengths of the top and bottom squares and the height of the frustum.
Summary
Richard J. Gillings described the main ideas found in the Moscow Mathematical Papyrus. The papyrus uses special symbols to show fractions, like putting a line over the number 4 to mean one-fourth. These kinds of fractions were frequently studied in ancient Egyptian math.
| 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 the Berlin Papyrus 6619, Egyptian Mathematical Leather Roll, Lahun Mathematical Papyri, and the Rhind Mathematical Papyrus. There are also general papyri such as the Papyrus Harris I and the Rollin Papyrus. For tables showing fractions, you can look at the RMP 2/n table.
This article is a child-friendly adaptation of the Wikipedia article on Moscow Mathematical Papyrus, available under CC BY-SA 4.0.
Images from Wikimedia Commons. Tap any image to view credits and license.