Genres of Math: Arithmetic, Algebra, and Algorithms in Ancient Egyptian Mathematics

By contributing author E.L. Meszaros

As non-native readers of Egyptian hieratic and hieroglyphics, our understanding of the mathematics recorded in these languages must necessarily go through a process of translation. Such translation is both necessary to allow us to study these problems, but also precarious. If done improperly, it can prevent us from true understanding. One way that we approach translating Egyptian math problems is by grouping them into genres, using categorization to aid in our translation by thinking about problems as algebraic or geometric equations, crafting them into algorithms, or piecing together word problems from their prose. If the process of understanding Egyptian math problems relies so heavily on translation, and translation in turn is influenced by categorization, then we must consider how our processes of categorization impact our understanding of ancient Egyptian math. 

The necessity of translation for the modern study of ancient mathematics has been the source of a great schism within the community. In an infamous 1975 paper, Unguru argued that one of the unintentional consequences of translation was the attribution of algebraic thinking to these ancient cultures. Mathematicians and historians tend to translate the word problems of ancient Iraq or Egypt into the abstracted symbolic statements we are familiar with today. This has helped us to better understand ancient mathematical ideas, but has also done a disservice to the math itself. The process of abstraction manipulated the geometry or arithmetic of ancient math into algebra, a way of examining mathematical problems that Unguru argued these ancient cultures never used (78).

However, others have pushed back against Unguru. Van der Waerden suggests that Unguru has misunderstood “algebra” by attributing such importance to the symbolic representation of data. Rather, van der Waerden emphasizes the convenience of symbols as a way of interpreting, analyzing, and comparing data, rather than the structural language of understanding data (205). Freudenthal similarly takes umbrage with Unguru’s understanding of what algebra is. “Symbols,” he writes, “…are not the objects of mathematics…but rather they are part of the language by which mathematical objects are represented” (192).

We can compare the strict translation of an Egyptian word problem to its algebraic translation by looking at problem 14 of the Moscow Papyrus.

Prose English translation:
Method of calculating a /￣\.
If you are told a /￣\ of 6 as height, of 4 as lower side, and of 2 as upper side.
You shall square these 4. 16 shall result.
You shall double 4. 8 shall result.
You shall square these 2. 4 shall result.
You shall add the 16 and the 8 and the 4. 28 shall result. 
You shall calculate  ̅3 of 6. 2 shall result.
You shall calculate 28 times 2. 56 shall result.
Look, belonging to it is 56. What has been found by you is correct. (Translation by Imhausen 33)

Algebraic Translation:
V = 6 (2² + (2*4) + 4²)/3

Abstracted Algebraic Translation:
V = h (a² + ab + b²)/3
where
h (height) = 6
a (base a) = 2
b (base b) = 4
V = volume

The algebraic translations are at once easier to take in but also visibly shorter, clearly missing information that the prose translation contains.

As an alternative to these translation techniques, Imhausen proposes the use of algorithms. Imhausen suggests that we translate Egyptian mathematical problems into a “defined sequence of steps” that contain only one individual instruction (of the type “add,” “subtract,” etc.) (149). These algorithms can still represent math problems in multiple ways. A numerical algorithm preserves the individual values used within Egyptian problems, while a symbolic form abstracts the actual numbers into placeholders (152). 

Numeric Algorithmic Translation:
6
4
2

1. 4² = 16
2. 4 x 2 = 8
3. 2² = 4
4. 16 + 8 + 4 = 28
5.  ̅3 x 6 = 2
6. 2 x 28 = 56

Here the first three numerical values are the given bases and height from the problem. The unfamiliar ” ̅3″ is the standard way of writing a fraction of 3, namely 1/3, in ancient Egyptian math.

Symbolic Algorithmic Translation:
D₁
D₂
D₃

1. D₂²
2. D₂ x D₃
3. D₃²
4. (1) + (2) + (3)
5.  ̅3 x D1
6. (5) x (4)

Drawing out the scaffolding of the problem by defining such algorithms allows scholars to easily compare math problems. The abstraction into symbols, the removal of extraneous information, and the sequential rendering allow us to more easily notice variation or similarity between problems (“Algorithmic Structure” 153). Imhausen suggests that identifying the substructure encoded beneath the language of presentation allows us to compare individual math problems not only with each other, to generate groups of mechanisms for solving and systems of similar problems, but also to look cross-culturally. Breaking down problems from Mesopotamia, China, and India may reveal similarities in their underlying algorithmic structures (158). 

The generation of algorithmic sequences from Egyptian word-based math problems does not solve all of our translation problems, however. Any act of translation, no matter how close it remains to the original language, is a choice that necessitates forgoing certain options. It also allows for the insertion of biases on the part of the translator themselves—or rather, such insertion is unavoidable.

In the example from the Moscow Papyrus, for example, the initial given values of the frustum are not specifically identified. The images from the original problem are missing, as are the verbs for the mathematical operations. Imhausen herself points out that this algorithmic form reduces some interesting features. The verb “double” in the original problem, for example, is replaced with “x 2” in the algorithmic translation (75). Making these changes requires us to confront the choice between algorithmic structure and staying true to the source material. “Fixing” these differences allows us to more easily compare math problems, but also presumes that we know what was intended.

The translation of Egyptian math problems into schematic algorithmic sequences is, therefore, not without its own set of problems. While Imhausen claims that they avoid some of the pitfalls of translation into algebraic equations that have so divided the community (158), algorithm interpretations are still likely to present the material in a way that differs from how ancient mathematicians thought about their own material. However, when applied carefully, such mapping may provide valid interpretations of these texts and a focal point for comparison.

Thinking about the genre of translation, the use of algebraic or geometric or algorithmic tools to interpret ancient math, is important for a number of reasons. We have already seen that the choice of genre impacts ease of understanding. Modern scholars used to thinking about math problems in an algebraic format will, unsurprisingly, read algebraic translations more easily. But these choices also impact what aspects of the original we preserve — algebraic translations lose information about the order of operations and remove the language used to present the problem.

However, paying attention to generic classification can also prevent us from reading ancient math problems with the “Western” lens. While algebraic interpretations are an artifact of modern scholarship, they are also an artifact of European scholarship. Too often the idea of geometry is put forward as an entirely Greek invention, while algebra is thought of as belonging to Renaissance Europe. By privileging these ways of thinking about ancient math problems we may be inherently white-washing native Egyptian thinking. Prioritizing algebraic interpretations, even if they aid in understanding, work to translate Egyptian math into the more familiar “Western” vernacular. Instead, scholars should work with the unfamiliar and think about these math problems without filtering them through these modern concepts.

Regardless of who one sides with in the debate between algebra and arithmetic, prose and algorithm, we must be cognizant of the fact that categorizing ancient Egyptian math is a conscious choice that influences how these problems are understood. Much like the act of translation itself, categorization is a process that is inherently influenced by the biases—intentional or otherwise—of the scholar. There may be nothing wrong with thinking about Moscow 14 in terms of an algebraic equation as long as we understand that this is an act of translation from the original and, therefore, reflects a reduced understanding of the native problem itself and incorporates aspects of the translator’s biases.

Which is all to say: tread carefully, because even numbers are not immune to the bias of translation.

E.L. Meszaros is a PhD student in the History of the Exact Sciences in Antiquity at Brown University. Her research focuses on the language used to talk about science, particularly as this language is transmitted between cultures and across time.