- MESOPOTAMIA 141
first approximation, turns up the number y| as the next approximation. Thus the
fraction ^| represents an approximation to = ^-. The method of approximating
square roots can be understood as an averaging procedure. In the present case, it
works as follows. Since 1 is smaller than \/2 and 2 is larger, let their average be
the first approximation, that is, |. This number happens to be larger than v^2,
but it is not necessary to know that fact to improve the approximation. Whether
it errs by being too large or two small, the result of dividing 2 by this number
will err in the other direction. Thus, since | is too large to be \/2, the quotient
3^2 = I is too small. The average of these two numbers will be closer to \/2 than
either number;^4 the second approximation to \/2 is then |(| + |) = Again,
whether this number is too large or two small, the number 17212 = ff w'l' err m
the opposite direction, so that we can average the two numbers again and continue
this process as long as we like. Of course, we cannot know that this procedure was
used to get the approximate square root unless we find a tablet that says so.
The writers of these tablets realized that when numbers are combined by arith-
metic operations, it may be of interest to know how to recover the original data
from the result. This realization is the first step toward attacking the problem of
inverting binary operations. Although we now solve such problems by solving qua-
dratic equations, the Mesopotamian approach was more like the Chinese approach
described above. That is, certain arithmetic processes that could be pictured were
carried out, but what we call an equation was not written explicitly. With every
pair of numbers, say 13 and 27, they associated two other numbers: their average
(13 + 27)/2 = 20 and their semidifference* (27 - 13)/2 = 7. The average and
semidiffcrence can be calculated from the two numbers, and the original data can
be calculated from the average and semidifference. The larger number (27) is the
sum of the average and semidifference: 20 + 7 = 27, and the smaller number (13)
is their difference: 20 - 7 = 13. The realization of this mutual connection makes it
possible essentially to "change coordinates" from the number pair (a, b) to the pair
((á + 6)/2,(á-6)/2).
At some point lost to history some Mesopotamian mathematician came to
realize that the product of two numbers is the difference of the squares of the average
and semidifference: 27 • 13 = (20)^2 - 7^2 = 351 (or 5, 51 in Mesopotamian notation).
This principle made it possible to recover two numbers knowing their sum and
product or knowing their difference and product. For example, given that the sum
is 10 and the product is 21, we know that the average is 5 (half of the sum), hence
that the square of the semidifference is 5^2 - 21 = 4. Therefore, the semidifference
is 2, and the two numbers are 5 + 2 = 7 and 5-2 = 3. Similarly, knowing that the
difference is 9 and the product is 52, we conclude that the semidifference is 4.5 and
the square of the average is 52 + (4.5)^2 = 72.25. Hence the average is v/72.25 = 8.5.(^4) The error made by the average is half of the difference of the two errors.
(^5) This word is coined because English contains no one-word description of this concept, which
must otherwise be described as half of the difference of the two numbers. It is clear from the
way in which the semidifference occurs constantly that the writers of these tablets automatically
looked at this number along with the average when given two numbers as data. However, there
seems to be no word in the Akkadian, Sumerian, and ideogram glossary given by Neugebauer to
indicate that the writers of the clay tablets had a special word for these concepts. But at the very
least, they were trained to calculate these numbers when dealing with this type of problem. In
the translations given by Neugebauer the average and semidifference are obtained one step at a
time, by first adding or subtracting the two numbers, then taking half of the result.