Due to the close geographical proximity and co-existing timeframe of ancient Egypt and ancient Greece, it is not surprising that their mathematics had several similarities. They each had two main number systems, same general uses of numbers, common materials, and similar schooling systems. The main difference that exists between the mathematics of each culture is the level of abstraction. Both civilizations used numbers for practical reasons, but there is evidence that the Greeks had a more abstract understanding of mathematics beyond the practical applications.
The number systems used in ancient Egypt were the hieroglyphic system and the hieratic system [5]. Almost parallel to these, the systems used in ancient Greece were the acrophonic system and the alphabetical system [7]. Both systems in each civilization used certain symbols for different amounts and were non-positional in nature, meaning the order of the symbols did not change the meaning of the number being represented. The Egyptian hieroglyphs and the Greek acrophonics both used a base of 10, but the acrophonic system also incorporated a sub-base of 5. The Egyptian hieratic system and the Greek alphabetical system were both more compact, requiring less symbols for a given number. Egyptians found a need for writing more quickly and efficiently when writing on papyrus became available [4]. Before that, the symbols had to be carved into stone, which was more time-consuming and presumably more difficult. The alphabetical system in Greece was based on their use of an alphabet for writing [7], which came from their cultural emphasis of education. Interestingly, before writing on papyrus, the Greeks relied on an oral tradition of passing along knowledge. This may be another reason why the city-states developed more independently of each other. One significant drawback to using papyrus was that it was not very durable, especially in dry heat. Thus, to preserve writings over a long period of time, they would have to be copied and recopied [6].
The practical uses of the number systems in both cultures were taxes, trading, keeping records, and taking measurements. Besides these, the Egyptians also used numbers in other areas specific to their civilization’s needs. They used numbers to create a calendar, which was used to predict the flooding of the Nile and therefore the different farming seasons [4]. Their calendar year was 365 days, later accounting for the extra 1⁄4 of a day, and was divided into twelve months [4], which is the basis of the calendar that we use to this day. Since religion was another important aspect to Egyptian culture, especially ideas about the after-life, there existed a great attention to detail and precise calculations for building their monuments and tombs [2]. Most likely by coincidence, the angle made by the base and one of the faces of the Great Pyramid has a secant very close to the golden ratio [4]. As evidenced by tablets and papyrus remnants, the Egyptians added and subtracted numbers by grouping and regrouping. They multiplied and divided using binary multiples.
Below is an example modeled after a problem given in [2] of what the process of division would have looked like in ancient Egypt, shown with modern numbers and symbols.
315 − 256 = 59 − 32 = 27 − 16 = 11 − 8 = 3
315 ÷ 8
8 1
16 2
32 4
64 8
128 16
256 32
512 64
315 = 32 × 8 + 4 × 8 + 2 × 8 + 1 × 8 + 3
= (32 + 4 + 2 + 1) × 8 + 3
= 39 × 8 + 3
Therefore 315 ÷ 8 = 39 3/8 = 39 + 1/4 + 1/8.
First, the divisor (8) is doubled with a record of its multiple in a separate column obtained by doubling the number 1 over and over. Then, the largest multiple of the divisor smaller than or equal to the dividend (315) is subtracted from the dividend. This process is then repeated with each difference until the number remaining is smaller than the divisor. Expanding these multiples and regrouping them allows the dividend to be written as the product of the divisor and some number plus the part that does not go in evenly. Last, the quotient is written as the total multiple of the divisor plus unit fractions of the remainder divided by the divisor.
The above calculation clearly utilizes the distributive and commutative properties of current mathematics. We can also see that fractions were used. In fact, 81 of the 87 problems given in the Rhind papyrus, also known as the Ahmes papyrus after the scribe who wrote it, involve fractions [4]. However, these fractions were limited to unit fractions, ones that only had the number one in the numerator, except the fractions 2/3 and 3/4. The process for converting fractions into multiple unit fractions can be seen in the problem above where 3/8 is rewritten as the sum of 1/4 and 1/8. The geometric concepts of similarity, area, and volume were also present in Egyptian calculations [2]. The Egyptians were even able to work with simple algebraic systems up to two dimensions [4]. Unfortunately, several important Egyptian works were likely destroyed along with the Library of Alexandria that housed them [2]. It may be the case that some Egyptian achievements in the field of mathematics were lost or wrongfully attributed to other regions whom they came in contact with. There is no evidence of formulas in Egyptian mathematics. Instead, only solutions to specific problems were given and likely used with adjustments for other problems circumstantially [2].
The Greeks, on the other hand, are credited with an understanding of the abstractness of numbers. The Greek culture was centered around religion and philosophy. Unlike the Egyptians who did not seem to recognize the difference between approximations and exactness [2], the Greeks concentrated on accuracy and justified their findings with proof. One of the most important artifacts of ancient Greece is Euclid’s Elements. The oldest complete copy of the Elements is from 888 AD. It was copied and recopied several times from the original around 300 BC. The use of generalizations, theory, and proof are evident in this work. For example, the answer to the problem “Given two numbers, to find the least number which they measure.” [1] is quoted below.
“Given two numbers, to find the least number which they measure. Let A, B be the two given numbers; thus it is required to find the least number which they measure. Now A, B are either prime to one another or not. First, let A, B be prime to one another, and let A by multiplying B make C; therefore also B by multiplying A has made C. [VII. 16] Therefore A, B measure C I say next that it is also the least number they measure. For, if not, A, B will measure some number which is less than C. Let them measure D. Then, as many times as A measures D, so many units let there be in E, and, as many times as B measures D, so many units let there be in F; therefore A by multiplying E has made D, and B by multiplying F has made D; [VII. Def. 15] therefore the product of A, E is equal to the product of B, F. Therefore, as A is to B, so is F E. [VII. 19] But A, B are prime, primes are also least, [VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [VII. 20] therefore B measures E, as consequent consequent. And, since A by multiplying B, E has made C, D, therefore, as B is to E, so is C to D. [VII. 17] But B measures E; therefore C also measures D, the greater the less: which is impossible. Therefore A, B do not measure any number less than C; therefore C is the least that is measured by A, B. Next, let A, B not be prime to one another, and let F, E, the least numbers of those which have the same ratio with A, B, be taken; [VII. 33] therefore the product of A, E is equal to the product of B, F. [VII. 19] And let A by multiplying E make C; therefore also B by multiplying F has made C; therefore A, B measure C. I say next that it is also the least number that they measure. For, if not, A, B will measure some number which is less than C. Let them measure D. And, as many times as A measures D, so many units let there be in G, and, as many times as B measures D, so many units let there be in H. Therefore A by multiplying G has made D, and B by multiplying H has made D. Therefore the product of A, G is equal to the product of B, H; therefore, as A is to B, so is H to G. [VII. 19] But, as A is to B, so is F to E. Therefore also, as F is to E, so is H to G. But F, E are least, and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [VII. 20] therefore E measures G. And, since A by multiplying E, G has made C, D, therefore, as E is to G, so is C to D. [VII. 17] But E measures G; therefore C also measures D, the greater the less: which is impossible. Therefore A, B will not measure any number which is less than C.”
Two other famous Greek mathematicians are Archimedes and Apollonius. Archimedes lived in Syracuse and is the author of many mathematical works, many of which were written in palimpsest [6]. He axiomatically developed the law of the lever, some of the laws of hydrostatics, compared the areas of geometrical shapes, gave inequality parameters for the number pi, analyzed the quadrature of the parabola, proved how to calculate the sum of a geometric series, and much more [13]. Archimedes also designed a system of raising 10” to different powers. Apollonius was born in Perga but studied and worked mostly in Alexandria. He, similar to Archimedes, used a system of raising 10$, also known as the myriad, to different powers. Apollonius’ major contributions include the theory of deferent circles and epicycles and the discovery that conics could be represented using intersections with more arbitrary planes. He was the first to use the modern terms parabola, ellipse, and hyperbola [9].
Education in ancient Egypt and ancient Greece were fairly similar, though Greece seemed to place a bit more emphasis on knowledge as a discipline in and of itself. In both cultures, it was custom for upper class males to go to school in their early years to learn the basics of reading writing, mathematics, and morals [12]. After that, in Egypt, a young man would typically follow the trade of his father. In Greece, however, the young man would either learn under a hired sophist or attend another school such as Pythagoras’s school, Plato’s Academy, or Aristotle’s Lyceum [8]. There was more variety available for higher learning in Greece due to its independent city-states, each with their own set of emphasis. For example, under Pythagoras, one would study the science of numbers from a religious viewpoint that the human soul can grow closer to the divine through philosophical thought. A student of Plato would study mathematics as an entry into the philosophical reasoning required for work as a politician or statesman. At Aristotle’s Lyceum, a student would study a broader curriculum more similar to what would be taught at universities in later centuries. In all three schools, the teaching style was casual and conversational, allowing for inquiry-based learning among groups of students [8].
The social structures of the two civilizations may be their largest overall difference. The Egyptians had a strong centralized government with a very clear class system consisting, from top to bottom, of a Pharaoh, government officials, soldiers, scribes, merchants, artisans, farmers, and servants. Their religious beliefs fueled the massive building projects, which required many laborers [3]. The governing structure in Greece was much less centralized, with some of the first ideas of democracy surfacing in Athens [10]. Each city-state, or polis, had its own unique structure and interests. For example, Sparta focused on having a strong military, while Athens focused more on education and the arts [11]. It is most likely from these different structures that we find more of an emphasis on learning and thinking logically in Greece than in Egypt, leading to more abstract versus practical mathematical thinking.
References
[1] Euclid’s Elements: All Thirteen Books Complete in One Volume: the Thomas L. Heath Translation, Green Lion Press, Santa Fe, N.M., 2002.
[2] D. Allen, “Egyptian Mathematics,” 2003, https://www.math.tamu.edu/~dallen/masters/egypt_babylon/egypt.pdf.
[3] Independence Hall Association, “Egyptian Social Structure,” https://www.ushistory.org/civ/3b.asp.
[4] J. O’Connor & E. Robertson, “An overview of Egyptian mathematics,” December 2000, https://mathshistory.st-andrews.ac.uk/HistTopics/Egyptian_mathematics/.
[5] J. O’Connor & E. Robertson, “Egyptian numerals,” December 2000, https://mathshistory.st-andrews.ac.uk/HistTopics/Egyptian_numerals/.
[6] J. O’Connor & E. Robertson, “How do we know about Greek mathematics?,” October 1999, https://mathshistory.st-andrews.ac.uk/HistTopics/Greek_sources_1/.
[7] J. O’Connor & E. Robertson, “Greek number systems,” January 2001, https://mathshistory.st-andrews.ac.uk/HistTopics/Greek_numbers/.
[8] J. O’Connor & E. Robertson, “The teaching of mathematics in Ancient Greece,” May 2000, https://mathshistory.st-andrews.ac.uk/Education/greece/.
[9] M. Greenberg, Euclidean and non-euclidean geometries: Development and history, W.H. Freeman and Company, New York, 1993.
[10] Maryville University, “The Lives and Social Culture of Ancient Greece,” https://online.maryville.edu/social-science-degrees/social-culture-ancient-greece/.
[11] National Geographic Society, “Greek City-States,” March 15, 2019, https://www.nationalgeographic.org/encyclopedia/greek-city-states/.
[12] R. Anthes, Affinity and difference between Egyptian and Greek sculpture and thought in the seventh and sixth centuries B. C., Pro. of the Amer. Phil. Soc. 107 (1963) 60–81.
[13] S. Williams, “Greece: Archimedes and Apollonius,” April 1993, https://mathed.byu.edu/~williams/Classes/300F2011/PDFs/PPTs/Greece%202.pdf.
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