Math in Ancient Egypt
Part of the pyramid’s enchanting effect on the imagination is how the ancient Egyptians managed to build it. Completed in 2560 BCE, after just twenty years from the start of construction, and long before any type of machinery to aid. The Egyptians used their extensive knowledge of mathematics to achieve a number of astonishing feats for daily life and as aspects that showed the greatness of their civilization. Despite the Great Pyramid being the most visible of the ancient Egyptian achievements, it was not the only to arise out of the Egyptians understanding of mathematics. In fact, it was partly due to their understanding of practical applications of mathematics that allowed them to rise to a level capable of undertaking such a monumental feat in the first place.
Fortunately, several artefacts have been found throughout history that give insight into how the Egyptians went about studying mathematics, the importance it played in their lives, and the sort of problems they were looking to solve. Among these artefacts are three papyri, another important Egyptian discovery that aided in their ability to record information. The papyri of interest are today known as the Rhind Papyrus, the Berlin Papyrus, and the Moscow Papyrus. Although there are certainly more papyri in existence today containing work from the ancient Egyptians, these three in particular paint an exquisite insight into the minds of ancient Egyptian mathematicians. An interesting aside is that the Egyptians did not truly have what would be known as a mathematician today, though it is convenient to refer to them as such rather than various profession names they would have worked under.
All three of the papyri feature mathematical equations with solutions, giving an idea of what the Egyptians were looking to solve and how they did so. It also showcases that the Egyptians were more concerned about the practicality of mathematics, rather than exploring theory. Even the Egyptian numbering system was built upon very practical measures. Being a system of base ten, which was derived from human’s ten fingers. This followed similar measurements from body parts employed by the Egyptian workers, such as a palm being the width of one hand. Similarly to their writing system, the Egyptians also employed Hieroglyphs for their numerals. Although they had clearly defined symbols for each representation of values base ten, from one to a million, the Egyptians did not develop a concept of place value. Of course this did not inhibit the Egyptians from performing operations with their numbers, having developed a unique style of multiplication suited for such, as one example.
Written Evidence about Math in Ancient Egypt
Named after Alexander Henry Rhind, a Scottish collector who purchased the papyrus in 1858 AD, approximately 3500 years after it was initially written, the Rhind Papyrus remains one of the most important artefacts to understanding ancient Egyptian mathematics today. As such, it is often described as
“a kind of instruction manual in arithmetic and geometry, and it gives us explicit demonstrations of how multiplication and division was carried out at that time. It also contains evidence of other mathematical knowledge, including unit fractions, composite and prime numbers, arithmetic, geometric and harmonic means, and how to solve first order linear equations as well as arithmetic and geometric series. “ (Mastin, 2010)
Contained within the papyrus are numerous problems involving algebra and geometry that directly related to aiding with the daily life of ancient Egyptians. In addition to this, what the papyrus is most famously known for is the table of numbers of the type 2n where n = 3,5,7,…,101. This functioned as a reference for the decomposition of these fractions. The way that Egyptian multiplication worked made this endeavor much easier than it may seem. The way multiplication worked under the Egyptian system was to continually multiply one number by two, while composing a binary, or power of two, column alongside. The numbers which added to the second number were then gathered, and the corresponding rows from the doubled column added for the result. As this can be difficult to visualize for those not familiar with the system, below is an example of a basic multiplication: Mastin. (2010)
A similar system can be seen in the way computers today perform operations. Despite the fact that modern computers use a similar system to the ancient Egyptians, the format of the table also suggests that the author had brute forced the calculations, reinforcing the idea that the Egyptians did not develop any theoretical mathematics and instead focused on the applied.
The Berlin Papyrus was published in two seperate pieces, both by Hans Schack-Schackenburg, in 1900 and 1902. The papyrus contains two problems, and suggests that the Egyptians had an understanding of the Pythagorean theorem, centuries before it was discovered by Pythagoras and given its namesake. Again, the Egyptians had less of a focus on theory and more on practical solutions. Thus, the problems on the Berlin Papyrus were solved for an explicit number. The first problem from the Berlin Papyrus states “You are told the area of a square of 100 square cubits is equal to that of two smaller squares, the side of one square is 12+ 14of the other. What are the sides of the two unknown squares?” (math tourist) In the form of a system of equations, it would look like x2 + y2 = 100, 4x -3y =0. There are a few interesting pieces happening here. The first is that it contains a second-order equation, which provides direct proof that the Egyptians were able to calculate non-linear quantities, such as those needed for the construction of the pyramids. Another is that the solution is x = 6, y = 8, for a triple of (6,8,10) or 2(3,4,5), with (3,4,5) being the most basic Pythagorean triple, a set of numbers to devise a right triangle.
The Moscow Papyrus was recovered in 1892 by Vladimir Golenishchev, who purchased the papyrus in Egypt and later brought it to Russia where it still remains in Moscow to this day. Studies estimate that the material discussed on the papyrus dates to around 1850 BC, making it the oldest of the three papyrus presented here, and the oldest mathematical papyrus uncovered thus far. The papyrus contains several interesting problems, again, relating highly to that of daily life of ancient Egyptians. Examples of these problems are calculating the size of a ship’s parts, computing the surface area and volume of a frustum, measuring the strength of beer, calculating worker productivity, and computing the areas of triangles. The problems relating to solving the volume of a frustum are of particular interest as they provide evidence that the ancient Egyptians actually knew how to calculate the volume of the pyramids before they were built, meaning they could plan ahead and know the amount of required materials. Shown is the original picture from the Moscow Papyrus with a modern interpretation: Allen. (2001)
As shown through evidence from the Rhind, Berlin, and Moscow papyri, the Egyptian’s mathematics were almost always concerned with solving a practical problem. This is not to suggest that their work was not advanced for their time, but rather that many of the people working on these problems did not have the luxury to explore topics which would not have immediate benefit. Despite the fact that many modern technologies have rendered methods the Egyptians used obsolete, the foundational knowledge laid by the ancient Egyptians can be seen.
Nile and Land Measurement
If someone did not immediately think of the Great Pyramid of Giza upon first hearing the word “Egypt”, they may instead think of the Nile River. Although the Nile runs through several countries, its role in developing ancient Egypt to an unrivaled civilization has entwined the two in history. The Nile provided the ancient Egyptians with a resource scarce for the desert climate, fresh water. This promoted a rich agricultural region along the Nile which needed to be divided for work. This was a common occurrence, needing to be done every time the Nile flooded, which erased the previous boundaries but provided the soil with a fresh supply of nutrients. Thus, the Egyptians had a great need for a reliable and accurate way to survey the land. By utilizing ropes of a predetermined length, with the aid of surveying tools to ensure a straight line, the Egyptians could accurately and quickly assess these new boundaries so work necessary for the survival of the people could be underway as soon as possible.
Despite the heavy focus of mathematics in Egyptian society on practical applications, the ancient Egyptians were a deeply religious society, with nearly every aspect of their culture being influenced by their religion. A large focus of this was on death, thus presenting an interesting intersection between the Egyptian’s understanding of mathematics with their religion. For example, the pyramids were actually built to be tombs; the pyramid shape was actually evolved from an earlier iteration known as the mastaba, which closer resembled a frustum. From the Moscow Papyrus, the Egyptians had an understanding for calculating the volume of such shape. Perhaps then, these calculations had a practical use of contributing to something that was integral to Egyptain society. Either way, such large and distinct structures were sure to grab the attention of any foreign people who happened to explore nearby.
Influence on Greek Culture
The Egyptians were not an isolated culture, and thus shared their goods and knowledge through trade, especially with neighboring societies such as Greece. While the Egyptians may not have developed elegant theories of their own, their influence on Greek mathematics is prominent. In fact, as explained by the Greek geographer Strabo
“The flooding of the Nile repeatedly takes away and adds soil, altering the configuration of the landscape and hiding the markers that separate one person’s land from that of someone else. Measurements have to be made over and over again, and they say that this is the origin of geometry” (Violatti, 2013)
In another example, the renowned Greek mathematician Pythagoras, who was by many accounts obsessed with triangles, himself spent time in Egypt. Pythagoras later developed his theorem based on the 3:4:5 triangle of which the Egyptians were very familiar with. Not only this, but the Greeks first viewed the Pythagorean theorem through shapes, not a relationship of numbers as the Egyptians had a rudimentary understanding of. There are numerous examples of famous Greek scholars proclaiming such admiration for the Egyptians.
Egyptian mathematician names may not be as familiar as those of the Greeks that came after them, but without the Egyptians, perhaps the Greeks would not be either. Though they did not develop any intricate theorems for understanding some universal truth about mathematics, the Egyptians did discover many applied elements of these properties. Many Egyptian methods seem simple with today’s understanding of mathematics, this combined with their often rudimentary tools would perhaps paint the Egyptian’s work as being simple. Though, due to the lack of developed theoretical mathematics, the work developed in ancient Egypt goes largely unnoticed to those learning about history and the development of many ideas used to this day. For this same reason, it has also caused many Egyptian methods to become obsolete, as today it is far easier to understand a general theorem or equation and apply it as needed than to brute force one for a specific task. The Egyptian mathematical experience can be represented as “The ancient Egyptians would not have had a proof of this, nor would have appreciated that a proof was necessary. They would just know from practical experience that it could always be done” (O’Connor, 2000). The Egyptians may not have explored mathematics for the beauty of mathematics, but they were able to invoke their own beauty through mathematics nonetheless.