Extant Egyptian Mathematical Texts

 There are only a handful of Egyptian mathematical texts that are still in existence today. It is amazing how these ancient texts can survive for thousands of years so we can continue to study them and understand the minds of ancient Egyptians. Egyptians were more scientifically and mathematically advanced than one might think. The most famous mathematical papyri are the Recto of the Rhind Mathematical Papyrus, the Mathematical Leather Roll, and the Moscow Mathematical Papyrus. These extant papyri played an important role in applied mathematics during the time of ancient Egypt and the advancement of mathematics for humanity.

 Ancient Egypt was split into two parts: the Nile valley and the Northern Delta. The valley is hundreds of miles long but never more than about 10 miles wide. The southern border of Egypt was only a few miles wide because it consisted only of the width of the Nile and its bank (Reimer 13). In ancient Egyptian times, the North and South cardinal directions were mixed up because the Nile flowed towards the North and the ancient Egyptians considered that direction to be South. The Mesopotamians had a system remarkably similar to our decimal system except that they used base 60. When we want to express half, we write .5 because 5 is half of 10. When Mesopotamians wanted to write one half, they wrote something like .30 because 30 is half of 60 (Reimer 14). The Euphrates and Tigris Rivers ran through Mesopotamia and ancient Egypt had the Nile River running through it, so these two ancient civilizations traded goods and services with boats. Agriculture and trading food was also common because the soil near the rivers were really fertile.

 The Egyptian method of writing fractions sometimes involved using hieroglyphics. For example, 1/4 looked like an eye with 4 vertical lines under it. Egyptians also exclusively used unit fractions witch a bar written over the numbers (yellow highlighted numbers). The word unit refers to the 1 in the numerator of the fraction. There was no way to write 2/5: when Egyptians wanted to express this value, they wrote 3 15 because 1/3 + 1/15 = 2/5. The approximation of pi for Egyptians was expressed at 3 10 25 or 3 + 1/10 + 1/25 which is 3 7/50 = 157/50 or about 3.14 (Reimer 15). Ancient Egyptians used unit fractions for easier comparison which came in handy especially in trades and for division of goods to be fair. Pi has always just been out there waiting to be discovered. There is much debate on whether Egyptians discovered the true pi or just found an approximation. True pi is irrational and it is also likely that many ancient Egyptians used the number 3 in calculations for simplicity and that others used the approximation of 3.12 instead of 3.14 for mathematics involving circles.

 The hieroglyphic system of writing was a pictorial script where each character represented an object or an idea. Special symbols were used to represent each power of 10 from 1 to 10^7. Addition and subtraction posed a few problems for ancient Egyptians. When adding two numbers, one made a collection of each set of symbols that appeared in both numbers making it necessary to replace them with the next higher symbol. Subtraction was just the reversed process of addition. Egyptians represented fractions using unit fractions rather than our traditional form of fractions. There are fractions that have an implied “one” in the numerator. For example, 1/2 = 2. The line above the number represents a numerator of one. A reason that ancient Egyptians used unit fractions was to make it easier to compare different quantities. (Joseph 85). For example, if we needed to compare 1/7 and 13/89 to determine which quantity is larger, then this task can be quite difficult. After converting these into fractions with sums of powers of 60, we get 1/7 = 8/60 +34/60^2 + 17/60^3. And 13/89 = 8/60 + 45/60^2 + 50/60^3. Now we can tell that 13/89 is greater than 1/7 because the second number being added 45/60^2 is greater than 34/60^2. 1/7 = .1428… and 13/89 = .1460… As we can tell these two quantities are very close and 13/89 > 1/7.

 A scribe in ancient Egypt was the job of a math teacher or an accountant. A scribe would receive some amount of food and have to divide it out according to each worker’s share value. Assume that 7 workers get loaf of bread to share. The scribe realizes that one loaf divided between 7 men means that each man get 1/7, so the scribe would record it as 7. The next day, he needs to divide 4 loaves of bread for the 7 workers. Cutting each loaf into 7 pieces would make the slices too small and the workers would not appreciate that. The scribe gets an idea and decides to cut each loaf in half, so each worker gets one of the eight pieces and one is left over. Being an honest scribe, he does not keep the left over piece for himself but he cuts the remaining piece into 7 pieces. The size of these pieces are 1/7 of 1/2 which means that they are 1/14 of the loaf of bread. So, each worker gets 1/2 of a loaf and 1/14 of a loaf and the scribe records it as 2 14 (Reimer 16). Scribes played an important role in the ancient Egyptian society even though it was not common and most scribes had a higher socio-economic status. Scribes were responsible for dividing food and goods fairly to paid workers. However, not all workers were paid fairly and people had a certain value depending on what social class they were born into. There was also the complication of slavery where people were forced to work for no pay.

The Recto of the Rhind Mathematical Papyrus (RMP) is one of the most important sources in mathematics of Egyptian history and it originates from the Middle Kingdom (2040BC – 1640 BC). Only a few mathematical texts from Ancient Egypt are extant or still in existence. The mathematical texts were recorded on papyrus could only survive in absolutely dry conditions. It is also called the Ahmes papyrus named after Ahmose who is the scribe that copied it in 1650 BC. The RMP is 6 feet wide and 8 feet long and it is one of the largest extant mathematical texts (Gillings 442). Ahmes tells us his material is derived from an earlier document belonging in the Middle Kingdom. The knowledge may ultimately be from Imhotep (2560 BC) who was an architect and physician to the Pharoah Zoser of the Third Dynasty. The opening statement in the Ahmes Papyrus claims that it has “rules for enquiring into nature and knowing all that exists.” (Joseph 83). Mathematics has been philosophical to humanity for thousands of years and it describes the very laws and principles of our universe. I view mathematics as order, symmetry, and beauty in the universe that is observable and also abstract in nature.

Ahmose used the RMP to make a table called the 2/n table of odd numbers ranging from 3 to 101. Each division is expressed as the sum of 2, 3, or 4 unit fractions with no denominator as large as 1000 and even numbers were preferred over odd numbers. Ahmose’s answer to 2 ÷ 95 was written as 1/60 + 1/380 + 1/570 which equals 2/95. Ahmose did not show how he got his 50 answers but he took up a lot of space to prove he was correct. However, he left one clue to how he got 2 ÷3 5 by writing the number 6 in red and the numbers 7 and 5 in black. I think the read 6 means that 6 ÷ 6 is the multiplier for the fraction 2/35 (Gillings 444). For example, 2/35 = 2/35 x 6/6 = 12/(7 x 5 x 6) = 7 + 5/(7 x 5 x 6) = 1/(5 x 6) = 1/(7 x 6) = 1/30 + 1/42. Which was expressed as unit fractions: 30 42 because 1/30 + 1/42 = 2/35. Let’s simplify Ahmose’s method with smaller unit fractions. For example, 2/3 = 1/3 + 1/3 or 3 3. 2 /5 = 1/3 + 1/15 or 3 15. Let 2/n be expressed as the sum of 2, 3, or 4 unit fractions where n is an odd integer so that 3 < n < 101. Egyptians would refer to the RMP table to add two 2/n values together. Let n = a + b where b is a number in the standard table. So, they would add the values 2/a + 2/b = 2/(a + b) and convert their answer into the sum of unit fractions. If a = 3 and b = 5, then we are trying to find 2/3 + 2/5. The RMP table says 2/3 = 3 3 and 2/5 = 3 15, so we add those values together. For example, 1/3 + 1/3 + 1/3 + 1/15 = 3/3 + 1/15 = 1 + 15 which would be expressed as 1 15. If a = 15 and b = 25, then we are trying to find 2/15 + 2/25. The RMP table says that 2/15 = 10 30 and 2/25 = 15 75, so we add those values together. For example, 1/10 + 1/30 + 1/15 + 1/75 = 1/10 + 1/15 + 1/30 + 1/75 which would be expressed as 10 15 30 75 because Egyptians preferred the denominators to be in order from least to greatest if the unit fractions cannot be added together or reduced.

Today the RMP is kept in the British Museum in London. The papyrus was bought in Egypt by the British lawyer, Alexander Henry Rhind in 1858. The first edition of the Rhind Papyrus was published in 1877 by the German Egyptologist, August Eisenlohr who gave a hieroglyphics translation. Due to interest of mathematicians and historians, an abbreviated version without photos was published in 1979. Finally, after the RMP was cleaned and restored, a little booklet with color photos was published in 1979. The RMP contains 64 problems and several tables; the numbering of problems today includes up to 87 problems introduced by Eisenlohr. Numbers 7 to 20 are just calculations that are more associated with the 2/n table (Imhausen 57). There are many different interpretations of the Ahmes Papyrus but one thing that most historians can agree on is that it is about the sum of unit fractions and that ancient Egyptians used the 2/n table as a reference for fractions with odd denominators. Not every division is going to end up nice and even and the ancient Egyptians realized that, so that is why they made a practical table for odd fractions.

 The Egyptian Mathematical Leather Roll (EMLR) remained unrolled until 1927 because of its brittle condition. It contains two identical tables with 26 additions of unit fractions similar to those in the RMP. For example, there are 7 equalities: 1) 9 18 = 6, 2) 21 42 = 14, 3) 4 12 = 3, 4) 10 40 = 8, 5) 7 14 28 = 4, 6) 18 27 54 = 9, 7) 25 15 75 200 = 8. (Gillings 446). Notice these equalities are written in unit fractions where the bars above the numbers represent one in the numerator. For example 9 18 = 6 in modern notation is 1/9 + 1/18 = 1/6. Many extant papyri from ancient Egypt are in fragile conditions because they remained in dry locations for thousands of years before they were discovered. The mathematical papyri were not discovered anywhere near the Nile River or any river because the dryness of the desert away from these rivers is the reason why the EMLR and other texts were preserved for so long.

 The EMLR is from the same period as the Ahmes Papyrus which is a table that has 26 decompositions into unit fractions. “The teacher scribes were simply showing their student scribes how to apply certain procedures correctly.” (Joseph 84). A scribe in ancient Egypt was the job of a teacher or an accountant. The instructors were expected to teach advanced calculations to their students and the accountants were expected to work out labor requirements, food rations, land allocation, and grain distributions (Gillings 445). The Egyptian method of duplation is a way of multiplication that requires previous knowledge of addition and knowing how to “double” numbers or multiply them by two. For example, we would write 28 x 46 and create two columns where we would double the numbers in each column. There are marks next to certain numbers that represent which numbers we add together. We would choose 4 + 8 + 16 because it equals 28. Therefore, we add the number in the corresponding column, 184 + 368 + 736 = 1, 288.

 In a society that did not use money, there was a need for more accurate calculations with fractions to figure out transactions, division of food, dividing land, and mixing ingredients for bread or beer. Two important features of Egyptian calculations with fractions include that to calculate 1/3, a scribe would first find 2/3 of that number then take half of that result and Egyptian mathematics had no compound fractions; all fractions were decomposed into a sum of unit fractions (Joseph 82). The Egyptians also had common procedures for solving mathematics problems just like we do but there way was different than ours. Ancient Egyptians did not think about fractions the way we do; they actually thought of fractions like 2/3 as the sum of 1/3 plus 1/3 because 2/3 simply did not exist in the form we think of today. In their minds it was simpler to compare unit fractions this way and I can see that it would be easier to tell which fraction is greater simply by looking at the denominator. We often have to take the extra step to convert fractions or find a benchmark fractions for tricky comparisons.

 In the construction of the 2/n table, every multiplication involving unit fractions usually led to the problem of how to double unit fractions. Doubling a fractions with an even denominator was an easy task and all they had to do was take half of the denominator. However, double fractions with an odd denominator was a more difficult arithmetical task. Thus the 2/n table was born where n was an odd integer from 3 to 101. This table in the RMP was extremely useful and does not contain a single arithmetical error, which is impressive for the time period. There are about 28,000 different combinations of unit fractions that can be generated for 2/n, n = 3, 4, 5, … , 101. It is possible to discover the following guidelines for the sets of values that were chosen: 1) a preference for small denominators where none are greater than 900, 2) a preference for combinations with fewer unit fractions where none contain more than 4, 3) a preference for even numbers in the denominator, especially for the first unit fraction (Joseph 83). There is controversy on whether this table actually contains an error or not according to some Egyptian historians. According to the guidelines and patterns that they have observed and assumed, there are about 5 sums of unit fractions that would not fits the “rules.” However, the mathematics is correct and when you reduced the unit fractions, we will come to the conclusion that they are the same number and that the calculations are in fact accurate. There is many cultural and contextual reasons that ancient Egyptians could have changed these sums or chose different numbers that we simply do not know.

 The Moscow Mathematical Papyrus (MMP) is the second largest extant mathematical source text. The length is 5.44 meters and the height is 8 centimeters. It is one big piece of papyrus with 38 columns of text and 9 little fragments (Joseph 84). The MMP was written in about 1850BC and it contains 112 mathematics problems with solutions. It was highly brittle with sections missing. The MMP was copied by a scribe who remains unknown; it shows little order of topics covered and it is not very different from the Ahmes Papyrus. It has 25 problems with 2 notable accomplishment of Egyptian Mathematics: 1) the formula for the volume of a truncated square pyramid and 2) the solution to the problem of finding the curved surface area of a hemisphere (Joseph 85). The scribe who coped the MMP was most likely not a mathematics teacher but perhaps he was a student or just really interested in mathematics. Either way I would not label this person as “incompetent” as Joseph does because we have no idea who this person is and they will remain anonymous.

 Number 14 in the MMP shows how to find the volume of a truncated square pyramid. For example, a truncated pyramid that has 6 cubits for vertical height by 4 cubits on the base by 2 cubits on the top and we need to find how many cubits fit inside the shape. The following are steps of the Egyptian method: First, square 4 to get 16 (4^2 = 16). Second, square 2 to get 4 (2^2 = 4). Third, Take 4 twice to get 8 (4 x 2 = 8). Fourth, add 16, 8, and 4 to get 28 (16 + 8 + 4 = 28). Fifth, take 1/3 of 6 to get 2 (6 ÷ 3 = 2). Sixth, take 28 twice to get 56 (28 x 2 = 56). Behold, the answer is 56 cubits! At the end of the MMP is a drawing of a trapezoid to illustrate their method. Each side of the trapezoid is labelled to correlate with the example above (Joseph 94). The formula for the volume of a truncated square pyramid is V = a^2h/3. Egyptologists and others who study the mathematical history of ancient Egypt have theorized 3 different ways how Egyptians arrived at the correct formula. The first theory is that the truncated pyramid was cut up into smaller and simpler solids whose volumes were estimated and then added together. The second theory is that they discovered that the volume of a truncated pyramid can be obtained as the product of the height and the Heronian mean of the areas of bases. Heron was an Alexandrian mathematician whose work contained detailed treatment of the volume of prisms, pyramids, cones, and other solids. The third theory was that it was calculated as the difference between the original complete pyramid and a smaller one removed from its top (Imhausen 59). In my opinion the third theory seems the most plausible because the calculations were too accurate to just have been approximations of smaller solids added together. I do not personally think they would have thought of the product of the height and the areas of the bases because that is a more modern way to think of it. Therefore, the volume being calculated as the difference between the complete pyramid and the smaller one removed from the top seems the most likely and it makes a lot of sense to solve this in a simpler manner.

 Number 10 in the MMP shows how to find the surface area of a curved surface on a semi-cylinder or hemisphere. For example, a semi-cylinder that has a diameter of 4 + 1/2 and dimension and we need to find how many cubits can fit on top of the surface of the shape. The following is the Egyptian solution: First, take 1/9 of 9 since the object is half to get 1 (9 ÷ 9 = 1). Second, take 1/9 of 8 (which is the remainder) to get 2/3 + 1/6 + 1/18 (8 ÷ 9). Third, find the remainder of 8 after taking 2/3 + 1/6 + 1/8 to get 7 + 1/9. Fourth, multiply 7 + 1/9 by 4 + 1/2 to get 32. Behold the answer is 32 cubits! The formula for the surface area of a hemisphere is A = 2(pi)rh where pi = 256/81, h = 1/2d which is the height, r is the radius, d is the diameter (Joseph 114). “The rule could have arisen from the empirical observation that in weaving a hemispherical basket whose radius is approximately equal to its height, the quantity of material required to make a circular lid is approximately half that required for the basket itself.” (Joseph 118). So, basically to find the surface area of the lid of a basket that is shaped like a half-sphere is about half of the area of the basket itself. This seems like a pretty logical conclusion for ancient Egyptians to make and it is not the most impressive mathematical discovery they have made. However, it was a necessary observation that had a practical application.

 Number 18 in the MMP concerns that calculation of the area of a piece of linen garment measuring 5 cubits and 5 palms by 2 palms. A cubit approximately measures the length of the forearm and is about 1 foot and six inches in modern measurements. A palm is simply the measure of the width of the human palm. For example, a strip of 5 1/5 cubits in height has been cut from a garment or rectangular cloth by division. We want to calculate the total height and width of the garment. We transform this 5 to 2 into a part and a plus. A garment is ancient Egypt was a rectangular fringed cloth to be wrapped around the body. 3 numbers were given as data: 5, 1/5, and 2 where 2 is the total height. Conditions of the problem are repeated by using the formula, mj dd n = k which means “if one says to you.” Provided with its total height in reference to the garment, two numbers are involved in an operation of calculation: the first is 5 and the second is 2 palms but the horizontal line in the text is probably the letter n meaning number. The number is presumably 53 because 51 + 2 = 53. The divisor of 2 that gives 1/5 is 10, thus 10 x 5 = 50, which is then added to the end of line 5. The number 10 results from 50, which then the multiplier allows the 5 to transform into the “part” of cloth to be added to 2 (Miatello 82). The garment measure 53 cubits in length: the width of the cloth without considering the fringe is 97.5 centimeters which corresponds to 52 fingers or 13 palms. The data in the text is formatted like the following: D1) 5 quantity making part of the strip, D2) 1/5 quantity making part of the strip, D3) 2 quantity making the total length, and D4) 1 is the length of the fringe. Therefore, D3: D2 = 10, (1) x D1 = 50, (2) + D4 = 51, and (3) + D3 = 53 (Miatello 84). So, the ratio of the total length to the 1/5 part of the strip equals 10 cubits. The 5 part of the strip is equal to 50 cubits. The length of the fringe plus 2 equals 51 cubits. And the total length plus 3 equals 53 cubits.

 The MMP was bought in Russia in the middle of the last century by W. Golenischeff in 1892 and 1894. It was brought to the Museum of Fine Arts in Moscow, which is where it is now. The discovery of its missing fragments by the New York Historical Association in 1922 helped to restore to its original form although two parts remain with their separate owners (Imhausen 60). Moritz Cantor mentioned the MMP in his book, Vorlesungen Uber Geschicte der Mathematik. The edition edited by Wasili Struve included black and white photos of the extant text, hieroglyphic translation, and commentary. The papyrus contains 25 problems; the first 3 are very badly damaged. It include fascinating problems of Egyptian mathematics: the volume of a truncated pyramid and the calculation of the surface areas of a half sphere or the surface of a semi-cylinder (Joseph 86). Finding the volume of a truncated pyramid was one of the more impressive mathematical achievements of the ancient Egyptians because it helped their architecture become so geometrically symmetrical and the structure so sound that it lasted thousands of years! We can still enjoy the beauty of the Great Pyramids today even though people slaved to build them for Pharoahs.

 The Mathematical Leather Roll is a lot like the Rhind Mathematical Papyrus in that they both contained problems with unit fractions. In fact, it is said that the EMLR is derived from the RMP and copied from another scribe. The Moscow Mathematical Papyrus is also very similar to the RMP or the Ahmes Papyrus because it has problems dealing with the addition and multiplication of fractions. The MMP also shows that ancient Egyptians figured out how to calculate the volume of a truncated square pyramid and the surface area of a hemisphere. The RMP was bought in Egypt by the British lawyer, Alexander Henry Rhind in 1858 and he also acquired the Mathematical Leather Roll.With observational evidence, we can come to the conclusion that the ultimate source of extant Egyptian mathematics comes from the Ahmes Papyrus. Fractions were a huge part of ancient Egyptian society because there was a barter economy that included a lot of trade and division of goods.

References

• Gillings, R. (1980). The Recto of the Rhind Mathematical Papyrus and the Egyptian Mathematical Leather Roll. Academic Press, pp.442-447. Available at: https://ac.els-cdn.com/0315086079900314/1-s2.0-0315086079900314-main.pdf?_tid=8474bcd9-576c-450b-85e6-f3ca0f5d01e0&acdnat=1540154400_dc66be000093bdc56eb721238b41819e [Accessed 21 Oct. 2018].PDF.
• Imhausen, A. (2016). Mathematics in Ancient Egypt: A Contextual History. Princeton University Press, pp.57-81. Print.
• Joseph, G. (2011). The Crest of the Peacock: Non-European Roots of Mathematics. 3rd ed. Princeton: Princeton University Press, pp.81-109. Print.
• Miatello, L. (2012). A Longstanding Enigma: Problem 18 of the Moscow Mathematical Papyrus. Vol. 48. Journal of the American Research Center in Egypt, pp.81-89. Available at: https://www.jstor.org.ezproxy.indstate.edu/stable/pdf/24555440.pdf?refreqid=excelsior%3A4915300ffab44f17943797118c45e8c9 [Accessed 21 Oct. 2018]. PDF.
• Reimer, D. (2014). Count like an Egyptian: A Hands-On Introduction to Ancient Mathematics. Princeton: Princeton University Press, pp.1-80.