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Publisher:

Dover Publications

Publication Date:

1982

Number of Pages:

288

Format:

Paperback

Price:

11.95

ISBN:

0-486-24315-X

Category:

Monograph

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

[Reviewed by , on ]

Richard J. Wilders

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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