in modern writing we showed the number 149 by this combi-
nation of symbols: 1 1 1 1 1 1 1 1 1 10 10 10 10 100. Moreover,
the individual components of the number could be shown in
any order that the scribe found pleasing or best suited to the
space available in the text. Th us, even reading a simple num-
ber would require performing a series of additions (though
no doubt this would have become more or less automatic for
experienced scribes).
Similarly, fractions could, in genera l, be represented only
by unit fractions; that is, 1 over the divisor. Th us, a more com-
plex fraction would have to be represented by a series of unit
fractions equal to it. For example,^2 ⁄ 15 would have been writ-
ten as^1 ⁄ 10 +^1 ⁄ 30. Fractions were denoted by drawing the sign
for mouth (an oval) over the numeral. Th ere were, however,
special signs for ⅔ and ¾.
Operations like multiplication also were carried out as a
series of additions. For instance, if a scribe wanted to square
13 (that is, 13 × 13), he might start by making a table like this,
doubling each row and also keeping track of how many 13s
were in each row:
13 1
26 2
52 4
104 8
Th en, knowing that 8 + 4 +1 = 13, he would add 104 +
52 + 13 to produce the answer of 169. With practice, much of
this work could be done automatically without writing out
the whole process.
Hieratic (or cursive) script was used for everyday writ-
ing and supplemented the formal hieroglyphic script found
in inscriptions and offi cial documents. In this style of writ-
ing, the numbers from 2 to 9 came to be represented by an
abbreviated sketch of what the group of upright strokes in hi-
eroglyphic form typically resembled. In this way something
like our individual numerals existed.
Egyptian mathematical notation and procedures were at
once highly practical and terribly cumbersome and inelegant.
Counting and mathematical problems recorded in Egyptian
sources were intended to provide solutions to the practical
problems of everyday life, including the distribution of provi-
sions, the valuations of commodities against precious metals
(because Egyptian civilization lacked coinage), and various
forms of farm accounting. It is easy to see that the reverent
awe in which the ancient Greeks and many later people held
Egyptian mathematical thought was quite undeserved. It was
based on the inherent diffi culty of reading the hieroglyphic
script and the glamour of the antiquity of Egyptian civiliza-
tion rather than any sophisticated or mysterious Egyptian
mathematical science.
In the Hellenistic and Roman periods (aft er 330 b.c.e.),
Greek scholars working in Egypt, especially at the Museum
in Alexandria, made tremendous advances in both theoreti-
cal and applied mathematics. For the most part, they drew on
Greek and Babylonian mathematical science, not that of na-
tive Egyptian civilization. Euclid (fl. 300 b.c.e) perfected the
concept of the formal mathematical proof. Claudius Ptolemy
(ca. 90–168 c.e.) created a system of mathematical astronomy
that is both ancestral to modern astronomy and remained
unchanged until the time of Copernicus in the 16th^ century.
However, Eratosthenes (ca. 276–ca. 194 b.c.e.) drew upon the
Egyptian tradition of geometry as land surveying in his cal-
culation of the circumference of the earth. He relied on the
skills of native surveyors to determine the distance between
the cities of Alexandria and Syene (modern-day Aswān);
since he also knew the number of degrees of arc that sepa-
rated them on the earth’s surface, he used that information to
calculate the total size of the whole earth.THE MIDDLE EAST
BY JUSTIN CORFIELD
Various numbering systems were used in Mesopotamia dur-
ing ancient times. Th e earliest was that of the Sumerians and
is k nown to have been used from at least 2800 to 2600 b.c.e. It
involved lines of signs listing numbers from left to right. Th is
system was developed by the Elamites, who occupied what
is now Iran, and at the same time by the Akkadians, whose
numeral notation system provided a basis for that developed
by the Babylonians. It is known that the Babylonian numer-
als had appeared between 1900 and 1800 b.c.e., though some
scholars think they may have been introduced in Akkad 500
years earlier. Th e Babylonian system was the fi rst that used a
place-value numeral, in which the value of a particular digit
depends not just on the character but also on where it ap-
pears in a number, as in the diff erence between 82 and 28.
Curiously, neither the Sumerians nor the Akkadians (from
whom the Babylonians drew their numerical system) used
place-value numerals.
Th e Babylonians had a sexagesimal system, that is, with
a base of 60. Th ey wrote numbers by making marks on soft
clay using a wedge-tipped stylus. A symbol similar to a letter
Y denoted the units. One of these symbols indicated 1, two for
2, and so on up to 9. To denote the “10s,” a symbol similar to
a left angle was used. Th us “<<<YY” meant 32. Th is system al-
lowed the Babylonians to compute complicated mathematical
sums, including linear equations, though they did not have a
digit for zero, which was denoted by its absence.
Th e reason for the algebraic achievements of the Baby-
lonians is unknown, because it is commonly believed that
all pre-Hellenic mathematics, including theirs, was largely
used for practical matters rather than for pure mathematics,
or mathematics engaged in for its own sake. One tablet in the
Plimpton Collection at Columbia University, New York, dating
to 1600 to 1900 b.c.e., was initially thought to have been a set of
business accounts, but some scholars have suggested that it was
part of a much larger tablet. Th e numbers on it are not listed in
a haphazard order but follow a pure mathematics inquiry.
Because of the Babylonian interest in mathematics and
astronomy, the base of 60 survives in the number of min-800 numbers and counting: The Middle East