The History of Mathematics: A Brief Course

2. MESOPOTAMIA 403

t

y

t

xy = A

1

1

1 A

1

1

I

FIGURE 2. A scenario that may "fit" a text from cuneiform tablet AO 6670.

reciprocal of the length and multiply it by the product of length and width, which
must be given in the problem as the area. The mystery is then pushed into the first
two instructions. What product is being "taken twice"? Does taking a product
twice mean multiplying by 2, or does it mean cubing? Why is the number 1 being
subtracted? Perhaps we should go back to the original statement and ask whether
"as much as area" implies an equation, or whether it simply means that length
and width form an area. What does the word them refer to in the statement,
"Let them be equal"? Is it the length and the width, or some combination of
them and the area? Without knowing the original language and seeing the original
text, we cannot do anything except suggest possible meanings, based on what is
mathematically correct, to those who do know the language.
We can get a geometric problem that fits this description by considering Fig. 2,
where two equal squares have been placed side by side and a rectangle of unit
length, shown by the dashed line, has been removed from the end. If the problem
is to construct a rectangle on the remaining base equal to the part that was cut
off, we have conditions that satisfy the instructions in the problem. That is, the
length ÷ of the base of the new rectangle is obtained numerically by subtracting 1
from twice the given area. This scenario is fanciful, however, and is not seriously
proposed as an explanation of the text. Another scenario that "explains" the text
can be found in Problem 13.4.

2.2. Higher-degree problems. Cuneiform tablets have been found that give the

sum of the square and the cube of an integer for many values of the integer. These

tablets may have been used for finding the numbers to which this operation was

applied in order to obtain a given number. In our terms these tablets make it

possible to solve the equation x^3 + x^2 = a, a very difficult problem indeed. In fact,

given a complete table of x^3 + x^2 , one can solve every cubic equation ay^3 + by^2 +

cy = d, where b and c are nonnegative numbers and a and d are positive. (See

Problem 13.5.)

Neugebauer (1935, p. 99; 1952, p. 43) reports that the Mesopotamian math-

ematicians moved beyond algebra proper and investigated the laws of exponents,

compiling tables of successive powers of numbers and determining the power to

which one number must be raised in order to yield another. Such problems occur

in a commercial context, involving compound interest. For example, the tablet

AO 6484 gives the sum of the powers of 2 from 0 to 9 as the last term plus one less

than the last term, and the sum of the squares of the first segment of integers as

the sum of the same integers multiplied by the sum of | and | of the last term.