Thoughts on Ancient Egyptian Mathematics
Kim R. W. Zahrt
Communicated by: Dr. Micheal Kinyon
Department of Mathematics
abstract
Do the experts, Egyptomologist, know the true history of Egypt, and furthermore, are they sure of the level of
Egyptian mathematical thought? This paper demonstrates two problems found on ancient papyri, and examines
the words of scholars to determine the answer to the above questions.
It has been said that Egyptian mathematics was only practical arithmetic and simple geometry. There are several
scholars who contend Egyptians lacked formal proofs and
stated rules, and therefore they lacked \the scientic attitude of mind" (Gillings 232). Thomas Eric Peet, a noted
professor of Egyptology, refers to the table of fractions in
the Rhind Mathematical Papyrus as being proof that the
Egyptians did not reach a scientic understanding of mathematics (ibid). Concerning geometry, Bartel L. Van der
Waerdan in Science Awakening II states \The Egyptians of
the Middle Kingdom had methods for calculating areas . .
. but there is no textual evidence of a geometry with constructions and proofs. Furthermore . . . the mathematical
texts were no longer copied after the Hyksos period" (41).
These and other scholars contend that the Egyptians did
not have the rigor of mathematics, and the value of studying Egyptian mathematics lies only in its place as history.
I disagree. The rst question that comes to mind is, how
did the Egyptians gure the labor, materials and plans for
building the Giza Pyramids? This occurred almost six hundred years before the earliest known mathematical artifact,
and shortly after the earliest known Egyptian writing. But
even more interesting is that the Zoser step pyramids of
Sakkara were built 200 hundred years before then (Gillings,
Appendix 13). It is my contention that due to the enormous time span of ancient Egyptian history (cira. 4241 1785 B.C.E.) and the scarcity of very early mathematical
artifacts, we can not be sure the Egyptians did not rise to a
theoretical level of ability in mathematics. Isn't it possible
they reached a high level of mathematical thought, and then
settled into this pragmatic style which the experts submit?
We must also keep in mind our own ethnocentric expectations concerning Greek style proofs and formulae. Perhaps
the Egyptians viewed formulae as infered by example. If
we continue to follow the suppositions of the experts, we
will surely miss the clue that leads to the bases of Egyptian
mathematical thought.
There are two prime sources which are considered the best
examples of Egyptian mathematics. These are the Rhind
Mathematical Papyrus and the Moscow Mathematical Papyrus. Several other sources exist such as the the Kahun
Mathematical Fragments and the Mathematical Leather
Roll at the British Museum, but these add nothing of substance to our knowledge of ancient Egyptian mathematics.
The author of the Rhind Mathematical Papyrus was an
Egyptian scribe named Ah'mose in circa. 1552 B.C.E.. In
the rst paragraph, he stated that his rendition was a copy
of a papyrus made (circa. 1844 - 1797 B.C.E.) (Clagett
113). Please note how late in Egypt's development this
occured. There is no indication of the extent of Egypt's
theoretical development in the previous 2000 years. The
Papyrus is a collection of practical mathematical problems
and solutions which a scribe of that time would be expected
to know. These math problems were originally numbered
by A.A. Eisenlohr, a German Egyptologist, and that basic
numbering scheme is well known today (Robbins 9). The
Papyrus also includes several tables of fractions.
The Egyptians primarily used unit fractions { one as the
numerator and any natural number as the denominator 12 ,
1
2
3
1
7 , 655 , etc., but included exceptions such as 3 and 4 . It
2
was as though the Egyptians had a preference for 3 since it
was used in many of the problems and had its own symbol
which was unrelated to 13 . Some argue that the Egyptians
derived 31 from 23 , unlike today where 13 is inferred and then
doubled to calculate 23 . The hieratic (the cursive form of
the hieroglyphic) of 43 is seen on the Palermo Stone, but 43
was not used in known computations (Gillings 183). The
Egyptians had no concept for zero and used no placeholder
or decimal point (Robbins 11). On the Papyrus, red auxiliaries (red ink)were used as reference numbers similar to the
modern use of natural number numerators (Gillings 81).
Although earlier evidence of Egyptian mathematics exist,
the Rhind Mathematical Papyrus is the source most frequently quoted as an example of classical Egyptian mathematics (Gillings 260-261). It was discovered as fragments,
with some missing, in a building near the Ramesseum on
the banks of the Nile River near Thebes in 1858, and it was
then purchased by A. Henry Rhind. In 1865, after Rhind's
death, the papyrus was sold to the British Museum. The
Rhind Mathematical Papyrus is labeled BM 10057 and BM
10058 and is often referred to by these numbers. In 1922,
additional fragments of the Rhind Mathematical Papyrus
were found by P. E. Newberry (ibid). These are known
as the Smith Fragments after Edwin Smith, who purchased
them in 1862 - 1863 (Clagett 113).
In 1877 Eisenlohr released a version of the text from pirated
facsimiles originating at the British Museum. The British
Museum nally released for publication the Rhind Mathe-
Thoughts on Ancient Egyptian Mathematics
matical Papyrus facsimiles in 1898. Peet penned his translation and documentary in 1923. This was followed with
two volumes by Arnold Buum Chace, Chancellor of Brown
University, during the period of 1927 to 1929 (Robbins 9).
Other scholars have continued to analyze the Rhind Mathematical Papyrus. As recently as 1999, Marshall Clagett,
Professor of History, Emeritus, at the Institute for Advance
Study, Princeton, NJ, published a source book of ancient
Egyptian science in which the aptly titled third volume,
Ancient Egyptian Mathematics , apportioned a large part to
the Rhind Mathematical Papyrus.
Many math history classes study the documentation of this
Papyrus because it is thought to be a good representation
of the mathematical level of the ancient Egyptians and displays one of the earliest known forms of mathematics. The
following is Problem 28 on the Papyrus. I use this particular
problem because the steps used by the scribe produce the
correct answer no matter what beginning number is used.
Where as the think of a number problem was coined by Diophantus of Alexandria in the 3rd century C.E. (Gillings 181).
I have used Chace's translation of the problem and will use
Gillings' interpretation of Chace's translation as a guide in
explaining Problem 28.
Figure 1. Problem 28 as translated by A. B. Chace. Gillings,
Richard J. Mathematics in the Time of the Pharaohs. The MIT
Press. Cambridge (1975): 182.
Figure 2. Gillings interpretation of Problem 28. (ibid.)
91
First notice the two lines over the 3 and the single line over
the 10 in Gillings' modern interpretation (Fig. 2); these
stand for 32 and 101 . A single line over the 3 stands for 31 .
In modern mathematical notation this problem would look
like this:
Formula
Pick a number X
X + 23 X = Y
Y - 13 Y = Z
Z - 101 Z = X
Example
Let X = 27
(27) + 32 (27) = 45
(45) - 31 (45) = 30
(30) - 101 (30) = 27 = X
The rst part shows how the problem is set up, and the
second part is an example of how to solve the problem by
replacing X with any variable (27 was used here). By working though the problem, the answer (27) is found to be the
same as was started with. While it's not what we are use to
seeing, it is easy to infer that this was an example of a general pattern that can be represented by a modern formula.
The second most important document of Egyptian mathematics is the Moscow Mathematical Papyrus. W. S. Golenishchev, in a reply to L. S. Bull in 1927, tells of his purchase
1893
of the papyrus in 1892
3 or 4 . Clagett states that it is evident from this reply that the papyrus came from a tomb
near the place where the Rhind Mathematical Papyrus was
discovered. The rst time the Moscow Mathematical Papyrus was mentioned, it was merely noted by M. Cantor in
1894 in his Vorlesungen uber Geschichte der Mathematik ,
Vol. 1. B. A. Turae, the curator of the Moscow Museum
in 1917, was the rst to analyze some of the contents of the
papyrus (Clagett 205-206).
The Moscow Mathematical Papyrus was broken down into
25 problems by W. W. Struve in 1930, although other formats are in existence (Clagett 207). Several of the problems
are unreadable or unclear, but they seem to be of the same
type as found in the Rhind Mathematical Papyrus. The
14th problem, the truncated pyramid, is probably studied
more than any other problem on this papyrus. A truncated
pyramid has a square base, and it tapers to a square top. In
other words, a normal pyramid with the top cut o. Turae
believed the Egyptians used a formula to nd the volume. In
modern terms, it would be V = ( h3 )(a2 + ab + b2 ), where h
is the height, and a and b are the sides of the corresponding
squares. Clagett went on to say:
The author [Turae] believed that, if his explanation of the problem was correct, it presented
a new and interesting fact, that Egyptian mathematics yielded a problem not yet found in Euclid (207).
Below I have used Clagett's translation to demonstrate
Problem 14.
K. Zahrt
92
multiply, corresponding to the Lines above from Clagett's
translations. Using Line 4 of Col XXVII as an example:
First they used two or more columns. In the right column
they placed the multiplicand (4), then in the left column
they placed the rst multiplier (1). They then doubled
each multiplier going down the column, along with its corresponding multiplicand (they also were able to use 10, 12 ,
2
1
3 , 10 as multipliers). When they found a combination of
multipliers that added up to the original multiplier (4), they
marked them with a \n" (in this case, it's just 4), then they
added the right side numbers that corresponded (just 16
here, so Total = 16). This is similar to the binary systems
used in computers today.
Figure 3. The Truncated Pyramid Problem (Clagett 395-396).
Note: the Egyptians read from right to left, so the rst part of the
problem is on the right. The upper images which are facsimiles
of the original are written in the hieratic, and the lower images
are Clagett's translation into the hieroglyphic.
Column XXVII:
Line 4
1
4
2
8
n4
16
Total 16
Column XXVIII:
Line 4
1
6
23
4
n 13
2
Total 2
Line 5
Line 4
1
n2
Total
1
2
n4
n8
n16
Total
4
8
8
2
4
8
16
32
56
Line 6
1
n2
Total
2
4
4
I have included these two example problems to show that
the methods the Egyptians used to solve the problems can
be universalized or used with any numbers. These point to
possible theorems or functions.
Figure 4. Clagett's Translation of Problem 14 (ibid. 221).
Clagett's Translations simplied:
[Col. XXVII]
Line 1) States what is going to take place.
Line 2) The height (h ) is established.
Line 3) The lower (a ) and upper (b ) squares' sides are dened.
Line 4) First square the lower side (a 2 ) which is 16 in the
example.
Line 5) It asks us to double the side a , but it is thought that
this meant multiplying a by b .
Line 6) Then square the upper side (b 2 ) which is 4.
[Col. XXVIII]
Line 1) Then, we are asked to add 16
Line 2) and 8 and 4 (a 2 + ab + b 2 )
Line 3) 28 is the result of this polynomial.
Line 4) Next we take 13 of 6h which is 2, and multiply it by
28. Notice: h3 (a 2 + ab + b 2 )
Line 5) Lo and behold the answer.
Next I have included the method that the Egyptians used to
Many experts have examined Problem 14 on the Moscow
Mathematical Papyrus, and several have tried using it to
justify opinions that the Egyptians did not have an understanding of theoretical geometry. Clagett presents the theories of Gunn and Peet, Gillings, Van der Waerden, and
R. Engelbach, to show how the Egyptians might have gured the volume of the truncated pyramids without the use
of formulae (Clagett 83-90). The fact that many experts
nd it necessary to prove the Egyptians used various practical methods (Peet thinks they made models in the mud
fibid 88g) causes skepticism of their interpretations. Why
else would they dwell on this simple problem? Clearly it
is shown in Problem 14, Egyptians knew how to solve the
volume given any parameters.
A millennium had past from the time of these early papyri
and the advent of the Greek culture. Notice that Diophantus is credited with naming the think of a number problems
almost two thousand years after the Rhind Mathematical
Papyrus presented one. It is interesting to note that in De
vita Pythagorica Chapter 4, Iamblichos states that Pythagoras spent 22 years studying astronomy and geometry with
the Egyptians. Furthermore, Van der Waerden states that
we must choose not to believe Herodotus, Aristotle, Eudemos and Proklos in order to believe that Thales did not bring
true geometry from the Egyptians to the Greeks (Van der
Waerden 40-41). Does this sound like a civilization bounded
by simplicity?
Outside of the mathematical realm, there are other noted
scholars of Egyptian history who have made comments that
tend to lead one to doubt the opinions of the mathematical and science experts. Cyril Aldred in The Egyptians
states, \The bureaucrat would also require [the scribe] to
solve problems connected with such matters as the distribution of ration or seed-corn, the number of bricks required to
Thoughts on Ancient Egyptian Mathematics
build a given structure and the number of men . . ." (Aldred 90). Furthering this thought, James Henry Breasted
in A History of Egypt stated, \Education thus consisted
solely of the practically useful equipment for an ocial career" (Breasted 100). Is it not possible that by the time of
the known mathematical artifacts, theoretical study was no
longer valued? After all, almost 2000 years had elapsed since
the beginning of Egyptian civilization. What would be the
value of studying theory when the Egyptians already had
methods that worked perfectly well? Fred Gladstone Bratton in A History of Egyptian Archeology comments, \In no
other place can one see such evidence of high culture, such
splendid examples of technical skill and artistry. Greece,
the only rival, reckons her cultural history in centuries, but
Egypt reckons hers in millenniums" (Bratton 25). Finally,
John A. Wilson in The Culture of Ancient Egypt remarks:
The idea of a rule-of-thumb pragmatism, with
no trace of a reasoned and reasoning experimentalism, and the idea that a single word might
mean both 'celestial glory' and mundane 'utility' are so foreign to modern thought that they
should be argued more extensively.(Wilson 69)
Much of the debate on the Egyptian's theoretical mathematic ability is based on conjecture. Van der Waerden, in
the statement above, alludes to scarcity of ancient mathematical text with his words \no textual evidence." Struik
suggests the same by using the phrase \available texts."
Can our meager archeological ndings be the sum total of
the Egyptian's knowledge of mathematics? I would rather
doubt it. Egypt has survived conquerors, cultures, kings,
pharaohs, and tomb robbers for over 3000 years and still
ended up with the greatest library in the world at Alexandria. This refutes that they were interested only in the
simple and practical.
93
It is important to note that the works we do have were found
near the ancient government seats.
Is it not possible that the ancient authors were employed by
the government, and perhaps this practical style of mathematics was all that was needed by the bureaucracy? Recall
that Turae believed the Egyptians solved the truncated
pyramid problem which the Greeks did not. This problem
could be a clue that the Egyptians had the ability to assess
universal formulae. The sparse archeological evidence and
the great span of Egyptian civilization fuel my continued
doubts about the critics' views on the simplistic nature of
ancient Egyptian mathematics.
references
Aldred, Cyril. The Egyptians. Revised and Enlarged Edition.
New York: Thames and Hudson, Inc., 1984.
Bratton, Fred Gladstone. A History of Egyptian Archeology. New
York: Thomas Y. Crowell Company, 1969.
Breasted, James Henry. A History of Egypt: From the Earliest
times to the Persian Conquest. New York: Charles Scribner's
Sons, 1937.
Clagett, Marshall. Ancient Egyptian Science: A Source Book. Vol
1, Tome 1 & 2, Vol. 2 - 4. Copied for its Memiors Series by
the American Philosophical Society. Philadelphia: 1999.
Gillings, Richard J. Mathematics in the Time of the Pharaohs.
The MIT Press. Cambridge (1975): 232.
Robbins, Gay and Charles Shue. The Rhind Mathematical Papyrus. New York: Dover, 1987.
Van der Waerden, Bartel L. Science Awakening II. Noordho International Publishing. Leyton, The Netherlands: Noordho
International Publishing, 1974.
Wilson, John A. The Culture of Ancient Egypt. Chicago: The
University of Chicago Press, 1965.
Kim is a junior majoring in Secondary Education with primary in Math and supporting in Physics with a Computer En-
dorsement. He is scheduled to graduate in 2001. A version of this paper was written for M380, History of Mathematics.
\I became interested in this topic after reading Neugebauer's opening sentence in A history of Ancient Mathematical Astronomy, Part Two, Book III, when he states, 'Egypt has no place in a work of the history of mathematical
astronomy.' I thought, \Wow, this man is biased!"