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Mathematics in the Time of the Pharaohs

Richard J. Gillings
Publisher: 
Dover Publications
Publication Date: 
1982
Number of Pages: 
288
Format: 
Paperback
Price: 
11.95
ISBN: 
0-486-24315-X
Category: 
Monograph
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Richard J. Wilders
, on
01/27/2015
]

This is a Dover reprint (1982) of a book originally published in 1972. It is a detailed, well-written account of the idiosyncratic and yet very sophisticated mathematics of ancient Egypt. As the author points out, this text would not have been possible, or even conceivable, without the wonderful work of those who deciphered the two writing systems (hieroglyphic and hieratic) with which the ancient scribes recorded their work.

The Hindu-Arabic numeration system (which is how virtually universal) serves not only to name numbers but to facilitate computation. The invention (discovery?) of zero was a key to this system. Civilizations such as ancient Egypt — lacking such a system — created elaborate mechanisms for performing calculations we now perform with ease. The mathematicians of ancient Egypt relied on twice-times tables (e.g., 3, 6, 12, 24…) as well as the ability to compute 2/3 of many numbers as the core of their computational algorithms. Beginning with a description of the Egyptian numeration system Gillings’s book leads the reader through these ideas with care and precision. Here are a few highlights.

Multiplication by Duplation

The Egyptian hieroglyphic numeration system is based on a different symbol for each power of ten. These were then used multiple times (up to 9) as needed. This system is demonstrably not suited for computation. We suspect addition was performed using tables, though as of the date of the publication of Gillings’s book none had been found. Multiplication was performed by creating a table of duplicates for one of the numbers to be multiplied. The scribe then wrote the second number as a sum of powers of 2 and added the respective duplicates to obtain the required product. For example to multiply 23 by 11 we create the table of duplicates of 23

/    1    23
/    2    46
     4    92
/    8    184

We then note that \(11 = 8 + 2 + 1\) and so \(23\times 11 = 23\times(8 + 2 + 1)=23 + 46 + 92\).

This procedure implicitly assumes that any positive integer can be expressed uniquely as a sum of powers of two — a remarkable insight indeed.

Unit Fractions

With the exception of 2/3 (discussed above) all fractions were written as the sum of unit fractions (1/n) with repetitions not allowed. Thus, they would write \(\frac34 = \frac12+\frac14\): three fourth parts is the half and the quarter. Gillings does a nice job of describing how the scribes might have proceeded to generate the tables of unit fraction decompositions. A unit fraction was distinguished from its corresponding whole number by putting an over line above it: the third was denoted \(\overline{3}\).

Geometric and Arithmetic Sequences

Sequences and series also show up in ancient Egyptian writings — often in the form of clever puzzles. Here is an example (from page 173):

Divide \(10\) hekats of barley among \(10\) men so that the difference of each man and his neighbor is \(\overline{8}\) hekats. There are two solutions provided. In the first, the smallest share is computed, while in the second the largest share is computed. In each case we begin with the average share per man which is \(1\) hekat. There are \(9\) increments from smallest to largest or \(9\) decrements from largest to smallest. From the average to the largest the scribe uses \(9\) increments of one-half of the given increment. Thus, the largest share is \( 1+ 9\times\overline{16} = 1 \overline{2}\overline{16}\). We then subtract \(\overline{8}\) repeatedly from this number to get the terms in our series of payments. As Gillings points out (page 175) this technique produces our standard form for the sum of an arithmetic sequence as the number of terms times the average of the first and last terms.

Pyramids and Truncated Pyramids

The scribes could also compute volumes of a large class of solid objects. Because they did not have any algebraic notation to use, the solutions border on the poetic. Here is a complete solution as translated by Gillings based on an earlier German translation (page 188)

Method of calculating a truncated pyramid

It is said of thee, a truncated pyramid of \(6\) ellen in height
Of \(4\) ellen of the base, \(2\) of the top.

Reckon thou with this \(4\), squaring. Result \(16\).
Double thou this \(4\). Result \(8\).
Reckon thou with this \(2\), squaring. Result \(4\).
Add together this \(16\), with this \(8\), and with this \(4\). Result \(28\).
Calculate thou \(\overline{3}\) of \(6\). Result \(2\)
Calculate thou with \(28\) twice. Result \(56\)
Lo! It is \(56\)! Thou has found rightly.

I’m not sure how we know that this translation captures the spirit of the original, but it is fun to read. What’s more, if we replace the three given values with parameters, the instructions result in the formula \(V=\frac{h}{3}\left(a^2+ab+b^2\right)\) for the volume of a truncated pyramid with base \(a\), top \(b\) and height \(h\).

I think this book is worthy of inclusion in the library of anyone with an interest in the history of mathematics. In particular, if you teach history of mathematics this is a fine reference. Of course, the original came out in 1972, so there has been a lot of work done since then. To catch up on that I would recommend The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, edited by Victor J. Katz.


Richard J. Wilders is Marie and Bernice Gantzert Professor in the Liberal Arts and Sciences at North Central College where he teaches courses in the history of science and of mathematics in addition to the standard undergraduate mathematics courses.

PREFACE
Introduction
Hieroglyphic and Hieratic Writing and Numbers
The Four Arithmetic Operations
ADDITION AND SUBTRACTION
MULTIPLICATION
DIVISION
FRACTIONS
The Two-Thirds Table for Fractions
PROBLEMS 61 AND 61B OF THE RHIND MATHEMATICAL PAPYRUS
TWO-THIRDS OF AN EVEN FRACTION
AN EXTENSION OF RMP 61B AS THE SCRIBE MAY HAVE DONE IT
EXAMPLES FROM THE RHIND MATHEMATICAL PAPYRUS OF THE TWO-THIRDS TABLE
The G Rule in Egyptian Arithmetic
FURTHER EXTENSIONS OF THE G RULE
The Recto of the Rhind Mathematical Papyrus
THE DIVISION OF 2 BY THE ODD NUMBERS 3 TO 101
CONCERNING PRIMES
FURTHER COMPARISONS OF THE SCRIBE'S AND THE COMPUTERS DECOMPOSITIONS
The Recto Continued
EVEN NUMBERS IN THE RECTO: 2 – 13
MULTIPLES OF DIVISORS IN THE RECTO
TWO DIVIDED BY THIRTY-FIVE: THE SCRIBE DISCLOSES HIS METHOD
Problems in Completion and the Red Auxiliaries
USE OF THE RED AUXILIARIES OR REFERENCE NUMBERS
AN INTERESTING OSTRACON
The Egyptian Mathematical Leather Roll
THE FIRST GROUP
THE SECOND GROUP
THE THIRD GROUP
THE FOURTH GROUP
THE NUMBER SEVEN
LINE 10 OF THE FOURTH GROUP
THE FIFTH GROUP
Unit-Fraction Tables
UNIT-FRACTION TABLES OF THE RHIND MATHEMATICAL PAPYRUS
PROBLEMS 7 TO 20 OF THE RHIND MATHEMATICAL PAPYRUS
Problems of Equitable Distribution and Accurate Measurement
DIVISION OF THE NUMBERS 1 TO 9 BY 10
CUTTING UP OF LOAVES
SALARY DISTRIBUTION FOR THE PERSONNEL OF THE TEMPLE OF ILLAHUN
Pesu Problems
EXCHANGE OF LOAVES OF DIFFERENT PESUS
Area and Volumes
THE AREA OF A RECTANGLE
THE AREA OF A TRIANGLE
THE AREA OF A CIRCLE
THE VOLUME OF A CYLINDRICAL GRANARY
THE DETAILS OF KAHUN IV
Equations of the First and Second Degree
THE FIRST GROUP
SIMILAR PROBLEMS FROM OTHER PAPYRI
THE SECOND AND THIRD GROUPS
EQUATIONS OF THE SECOND DEGREE
KAHUN LV
"SUGGESTED RESTORATION OF MISSING LINES OF KAHUN LV 4, AND MODERNIZATION OF OTHERS"
Geometric and Arithmetic Progressions
GEOMETRIC PROGRESSIONS: PROBLEM 79 OF THE RHIND MATHEMATICAL PAPYRUS
ARITHMETIC PROGRESSIONS: PROBLEM 40 OF THE RHIND MATHEMATICAL PAPYRUS
KAHUN IV
"Think of a Number" Problems"
PROBLEM 28 OF THE RHIND MATHEMATICAL PAPYRUS
PROBLEM 29 OF THE RHIND MATHEMATICAL PAPYRUS
Pyramids and Truncated Pyramids
THE SEKED OF A PYRAMID
THE VOLUME OF A TRUNCATED PYRAMID
The Area of a Semicylinder and the Area of a Hemisphere
Fractions of a Hekat
Egyptian Weights and Measures
Squares and Square Roots
The Reisner Papyri: The Superficial Cubit and Scales of Notation
APPENDIX 1
The Nature of Proof
APPENDIX 2
The Egyptian Calendar
APPENDIX 3
Great Pyramid Mysticism
APPENDIX 4
"Regarding Morris Kline's Views in Mathematics, A Cultural Approach"
APPENDIX 5
The Pythagorean Theorem in Ancient Egypt
APPENDIX 6
The Contents of the Rhind Mathematical Papyrus
APPENDIX 7
The Contents of the Moscow Mathematical Papyrus
APPENDIX 8
A Papyritic Memo Pad
APPENDIX 9
"Horus-Eye Fractions in Terms of Hinu: Problems 80, 81 of the Rhind Mathematical Papyrus"
APPENDIX 10
The Egyptian Equivalent of the Least Common Denominator
APPENDIX 11
A Table of Two-Term Equalities for Egyptian Unit Fractions
APPENDIX 12
"Tables of Hieratic Integers and Fractions, Showing Variations"
APPENDIX 13
Chronology
APPENDIX 14
A Map of Egypt
APPENDIX 15
The Egyptian Mathematical Leather Roll
BIBLIOGRAPHY
INDEX

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