Left: Rhind Mathematical Papyrus (Source). Right: Otto E. Neugebauer (1899–1990) (Source: Mathematisches Institut, Georg-August-Universität, Goöttingen)

Neugebauer and Ancient Egyptian Mathematics

Published in

Cantor’s Paradise

12 min readMay 8, 2021

Universality is perhaps the most intrinsic characteristic of mathematics. The fact that a curious mind is all that is required to carry out mathematics has made the discipline approachable to people of different times, races, religions, etc. Since antiquity, mathematics, in one form or the other, has been practiced by various cultures globally.

However, the current Eurocentric view towards the history of mathematics belittles the valuable contributions of other civilizations in shaping the discipline into what it is today. The classical Eurocentric view towards the development of mathematics can be roughly described as follows -

600 BC-AD 400: Greek civilization acting as the cradle of mathematics.
AD 400 — AD 1500: The Dark Ages, a period devoid of any significant developments. In fact, many view this period as a regression.
AD 1500 — Present Day: Renaissance enlightenment, followed by rapid developments leading to the contemporary state of the subject.

As is obvious from the above, mathematics developed by cultures predating the Greek civilization is rarely given due acknowledgment. The same treatment is dealt with the mathematics belonging to non-European cultures, irrespective of the time. In fact, the following words of Morris Kline, a writer on the history of mathematics, best summarize this view -

[Mathematics] finally secured a new grip on life in the highly congenial soil of Greece and waxed strongly for a short period …. With the decline of the Greek civilization the plant remained dormant for a thousand years … when the plant was transported to Europe proper and once more embedded in fertile soil. — (Kline, 1953)

Thankfully, from the turn of the twentieth century onwards, historians of mathematics started giving some attention to mathematics stemming out of non-European roots. That has ensured the subject is now relatively well received at least among the scholarly circles pertaining to the history of mathematics, if not on a broader scale.

One such mathematics is that of the Egyptians belonging to the Middle and New Kingdom periods (2000 BC — 1000 BC approximately). One such historian who went to great lengths to bring the matter to general awareness was Otto E. Neugebauer. With the help of the two, this article aims at providing a taste of the discipline (history of ancient mathematics) to the reader. Here, we would be delving into —

Neugebauer’s life and works
How ancient Egyptians carried out basic arithmetical operations

Otto E. Neugebauer

Early years

Neugebauer was born on May 26, 1899, in Austria. Unfortunately, both of his parents died quite early, and he was brought up by his uncle. He was actively involved in the First World War, serving as an artillery lieutenant in the Austrian army on the Italian front. Later, when the war ended in 1918, he was detained as an Italian prisoner of war.

After being released from the prison camp in 1919, Neugebauer moved around quite a bit. From 1919–1921, he studied engineering and physics at the University of Graż. After a brief spell at Munich (where he studied mathematics and physics), he moved to Göttingen in 1922. There, he finally settled into mathematics, studying it with Richard Courant, Edmund Landau, and Emmy Noether.

Foray into Egyptian Mathematics

It was at Göttingen that Neugebauer developed his interest in the study of ancient Egyptian mathematics. Thanks to his interest in languages, Neugebauer was already familiar with Egyptian. Thus, it was only natural that his friend Harald Bohr (brother of the famous physicist Neils Bohr) once asked him to review a study on the Rhind Mathematical Papyrus (discussed in detail later). Upon studying the work, Neugebauer realized that this was where his true interest lay, in the history of ancient mathematics! … And he simply never looked back.

Neugebauer sought permission from Courant and Hilbert to work on the subject (ancient Egyptian mathematics) for his doctoral thesis, to which they consented. This ultimately led to his 1926 dissertation on Egyptian fractions, titled ‘Die Grundlagen der ägyptischen Bruchrechnung’ (The Fundamentals of Egyptian Calculation with Fractions).

Emigration from Germany

As was common among the German mathematicians and scientists of the time, Neugebauer’s career was impacted by the rise of the Nazi regime. His leftist political views obviously didn’t bode well for life under Nazi rule. In 1933, Courant was removed as the director of the mathematical institute in Göttingen, and Neugebauer was appointed as the acting director. But he soon resigned from his position upon being informed that he wasn’t being well received by the (Nazi influenced) student body.

Finally, a year later in 1934, Harald Bohr facilitated Neugebauer’s escape from Germany by extending an invitation for mathematics professorship at Copenhagen, Denmark. Neugebauer accepted the same and remained in Copenhagen till 1939.

Move to the USA

In 1939, Neugebauer accepted an offer of a professorship at the Brown University in the USA. He remained there until his retirement in 1969. During his tenure, he founded the History of Mathematics department at the University. One key thing to note here is that all the three universities mentioned above (Göttingen, Copenhagen, Brown) had strong Egyptology departments, which would have been of great utility to Neugebauer’s research work.

Though employed at Brown university, Neugebauer spent a considerable portion of his time at the Institute for Advanced Study at Princeton as well. In 1984, he shifted to the Institute permanently.

Pioneering works in the History of Mathematics / Science

Having begun his career as a mathematician, Neugebauer shifted to the history of ancient mathematics (a subject which was in its infancy at the time), particularly that of Egypt and Mesopotamia, and then went on to carry out comprehensive research in the history of ancient mathematical astronomy across various cultures. This should not come as a surprise, for mathematics was strongly coupled with astronomy in ancient times.

Neugebauer deployed a “technical cross-cultural” approach in his research, which emphasized the universal nature of mathematics. In other words, he held the belief that mathematics belonging to any culture and time can be unconditionally reinterpreted appropriately in modern symbolic notations.

Neugebauer concentrated on texts, with much of his work revolving around the compilation of source collections belonging to ancient mathematics and astronomy. Given his mathematical background, Neugebauer commanded respect in the mathematical circles of Europe and North America. And certainly, it would be no exaggeration to say that it was his presence and stature that led to the history of mathematics being taken seriously as a discipline.

Neugebauer received a large number of awards and honorary degrees. One of those was the 1986 Balzan Prize for History of Science, which perhaps best summarizes Neugebauer’s contributions. He has conferred the award

For his fundamental research into the exact sciences in the ancient world, in particular, on ancient Mesopotamian, Egyptian and Greek astronomy, which has put our understanding of ancient science on a new footing and illuminated its transmission to the classical and medieval worlds. For his outstanding success in promoting interest and further research in the history of science.

Ancient Egyptian Mathematics

In this section, we would be getting a glimpse of how ancient Egyptians used to deal with numbers. However, rather than taking a bird’s eye view of the landscape, we would be diving deep straight into something particular (which should help us form a nascent understanding of Egyptian mathematics in general).

Here, we would be studying a couple of examples from Neugebauer’s work on the Rhind Mathematical Papyrus (RMP), which formed a part of his doctoral thesis. Most of the information in this section is based on a recent research paper surveying Neugebauer’s work on the RMP.

Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus is one of the most well-known primary sources for understanding ancient Egyptian mathematics. It was written around 1300 BC in the Second Intermediate Period. The script employed is Hieratic, rather than Hieroglyph. The text is written in black and red ink. The document is 33 cm tall and over 5 m long.

The papyrus, which was probably placed in the burial tomb of its copyist, was purchased in Thebes (modern-day Luxor) in 1858 by Alexander Henry Rhind (a Scottish lawyer and excavator). After Rhind’s death in 1864, it was acquired by the British Museum. The papyrus is a collection of around 80 mathematical problems. The content spans various fields — arithmetic, geometry, mensuration, and some miscellaneous problems.

Neugebauer’s work on the RMP

As noted above, Neugebauer’s Ph.D. dissertation was centered around Egyptian calculations with fractions. The RMP was one of the key primary sources he perused towards the same. In this subsection, we would be touching upon a couple of arithmetical procedures which Neugebauer deep-dived into. But before that, let us first build a working understanding of ancient Egyptian mathematics.

The scribes of ancient Egypt used a decimal numerical notation without a position. It was complemented by a system of fractions, made up almost entirely of unit fractions (1/2, 1/3, 1/4, etc., this was because the Egyptians saw their fractions as n-th part of a whole). A two-column scheme (detailed below) was applied for carrying out the calculations. The elementary operations of the scheme were -

Addition
Doubling
Halving
Multiplying by 10
Dividing by 10
Inversion (if the numbers of a row read 3 and 5, then the next row could be written down as 1/5 and 1/3)
Taking 2/3rd of a number

General Multiplication Procedure

We now explore the two-column procedure employed by the Egyptians. Suppose we have to multiply two natural numbers, m, and n. For this, the Egyptians would make a table with two columns and multiple rows. To start with, one among m and n would be selected. Let us say m was selected. Then, the first row would comprise of 1 and m.

Now, all the subsequent rows’ entries would be double the previous row’s values. So, the second row would contain 2 and 2m. The third one would contain 4 and 4m, and so on.

Finally, from the first column, one would select the values which add up to n. Then, the sum of the corresponding values in the second column would give us our answer, m*n.

Let us work out an example, 17 * 13. Starting with 17, we would get the following table -

Now, as you can see, the cells marked with orange slashes add up to 13 (1+4+8). So, the Egyptians would calculate the final answer as the sum of the corresponding values in the second column, i.e., 17+68+136 = 221.

In fact, this procedure is not restricted to just multiplication. It can be (and indeed was) easily adjusted for division as well. For carrying out division, we start with 1 and the divisor in the first row. Then, as before, we go on doubling the values. However, this time around, we stop when the sum of certain values in the second column becomes equal to the dividend. The quotient is then obtained as the sum of the corresponding values in the first column!

Again, to learn with the help of an example, let’s solve for 150/15.

As before, the relevant rows have been marked with orange slashes. The values in the second column for these two rows add up to 150 (30+120), which is our dividend. Therefore, the quotient would be equal to the sum of the corresponding entries in the first column, i.e., 2+8=10.

Calculations With Fractions

As already noted above, the Egyptians’ arithmetic wasn’t confined to natural numbers. They were well familiar with the concept of fractions. However, they primarily operated with unit fractions only, i.e., 1/2, 1/3, etc.

So, what happened when they encountered quantities which were not unit fractions? — They tried to express such quantities as the sum of unit fractions. The most basic among these types of numbers are fractions of form 2/(2n+1), i.e., a fraction with 2 as a numerator and an odd number as denominator.

From the RMP, Neugebauer studied the decomposition of such fractions into unit fractions. But before we deal with that, a word on the notation employed for fractions. The fact that ancient Egyptians restricted their scope to unit fractions allowed them to denote the fractions as the corresponding integer accompanied with a dot.

However, here we shall be employing Neugebauer’s notation, wherein—

A unit fraction is denoted by a bar at the top of the corresponding integer.
Juxtaposition is used to denote the addition of fractions. That is,

Now, let’s come back to the question of decomposing fractions of form 2/(2n+1) into unit fractions. Again, here we wouldn’t deal with the problem in general. Rather, we would study an example from the RMP which Neugebauer worked on as part of his thesis.

The example is concerned with expressing 2/7 as a sum of unit fractions. Below, we lay down the relevant section of the RMP, staying as true as possible to the primary material’s content and positioning.

Calculation for 2/7 in the RMP. As mentioned before, both black and red inks were used in the RMP.

Now, let’s break down the above. As is clearly evident, there are three distinct portions here —

The text on the top right: Here, we see that the scribe has written down 1/4 and 1/28 with red ink. This is indicative of the fact that the scribe already knew the answer, i.e., 2/7 = 1/4 + 1/28. Thus, the whole exercise is about verification, not derivation. Naturally, one is bound to wonder how did the scribe have this knowledge beforehand. Unfortunately, there are no hints anywhere in the RMP to come up with a plausible supporting hypothesis. So, leaving that aside, we next notice the other numbers mentioned on the top right. These are 1, 1/2, 1/4, and 1/4. Observe that these all sum up to give 2.
Now, let us examine the left section. Here, the scribe has carried out the division of 2 by 7 using the standard two-column method. The first row starts with 1 and the divisor (7). The next row is half of the above one — 1 becomes 1/2 and 7 becomes (3+1/2). Again halving the values, we get 1/4 and (1+1/2+1/4). Now let’s come to the last line, which requires some explanation. Here, the values are 1/28 and 1/4 — how did the scribe know that 7/28=1/4? A plausible explanation is provided by the presence of two factors — the standalone 4 in the bottom left corner, and the calculation occupying the middle portion of the image.
These two factors taken together suggest that the scribe suspected (probably based on prior experience) that 4 multiplied by 7 gives 28. The scribe then verified this by multiplication (4*7) using the standard two-column procedure, which can be seen in the middle of the image. Having carried out the verification, the scribe then wrote down the final row of the division table, which consists of 1/28 and 1/4. Here, one may reason that the scribe could have used inversion on the last row of the multiplication table to arrive at the last row of the division table (which is legitimate given that both the multiplication and division tables start with the same set of values, i.e., 1 and 7).
Finally, we see that the scribe marked off the last two rows with a backward slash (\). The entries in the second column for these two rows add up to 2 (the dividend). Hence, the quotient would be obtained by summing up the corresponding values in the first column, which are 1/4 and 1/28. Therefore, 2/7 = 1/4 + 1/28. Hence proved!

That would be it for now. This was just a glimpse into the plethora of ancient Egyptian mathematics. Hope you enjoyed it! Below is a highly recommended talk on the same by Dr. Christopher Hollings from the Mathematical Institute of the University of Oxford.