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.