- MESOPOTAMIA 243
led to join the midpoints of its sides in order, that is, to draw the diagonals of the
four copies of the original square. Since these diagonals cut the four squares in half,
they will enclose a square twice as big as the original one (Fig. 4). It is quite likely
that someone, either for practical purposes or just for fun, discovered this way of
doubling a square. If so, someone playing with the figure might have considered
the result of joining in order the points at a given distance from the corners of a
square instead of joining the midpoints of the sides. Doing so creates a square in
the center of the larger square surrounded by four copies of a right triangle whose
hypotenuse equals the side of the center square (Fig. 4); it also creates the two
squares on the legs of that right triangle and two rectangles that together are equal
in area to four copies of the triangle. (In Fig. 4 one of these rectangles is divided
into two equal parts by its diagonal, which is the hypotenuse of the right triangle.)
Hence the larger square consists of four copies of the right triangle plus the center
square. It also consists of four copies of the right triangle plus the squares on the
two legs of the right triangle. The inevitable conclusion is that the square on the
hypotenuse of any right triangle equals the sum of the squares on the legs. This is
the Pythagorean theorem, and it is used in many places in the cuneiform texts.
2.2. Plane figures. Some cuneiform tablets give the area of a circle of unit radius,
which we have called the two-dimensional ð, as 3. On the other hand (Neugebauer,
1952, p. 46), the one-dimensional ð was known to slightly more accuracy. On a
tablet excavated at Susa in 1936, it was stated that the perimeter of a regular
hexagon, which is three times its diameter, is 0;57,36 times the circumference of
the circumscribed circle. That makes the circumference of a circle of unit diameter
equal to
_1_ = M = 3 .125.
0; 57,36 8
That the Mesopotamian mathematicians recognized the relation between the
area and the circumference of a circle is shown by two tablets from the Yale Baby-
lonian Collection (YBC 7302 and YBC 11120, see Robson, 2001, p. 180). The first
contains a circle with the numbers 3 and 9 on the outside and 45 on the inside.
These numbers fit perfectly the formula A = <7^2 /(4ð), given that the scribe was
using ð = 3. Assuming that the 3 represents the circumference, 9 its square and
45 the quotient, we find 9/(4 · 3) = 3/4 = 0;45. Confirmation of this hypothesis
comes from the other tablet, which contains 1;30 outside and 11; 15 inside, since
(1; 302 )/(4 · 3) = (2; 15)/12 = 135/12 = 11.25 = 11; 15.