140 6. CALCULATIONexample, to do a long division with remainder, say, ^p, he would look for the next
number after 22 that divides 750 evenly (25) and write
750 _ 750 , (750\ 3
22 ~ 25 ' V 25 I 22'
that is,1 =30(1+ A) =30 + i=34i
Beyond these simple operations, he also codifies the methods of taking square and
cube roots, and he states clearly the Rule of Three (Colebrooke, 1817, p. 283).
Brahmagupta names the three terms the "argument," the "fruit," and the "requisi-
tion," and points out that the argument and the requisition must be the same kind
of thing. The unknown number he calls the "produce," and he gives the rule that
the produce is the requisition multiplied by the fruit and divided by the argument.4. Mesopotamia
Cuneiform tablets from the site of Senkerch (also known as Larsa), kept in the
British Museum, contain tables of products, reciprocals, squares, cubes, square
roots, and cube roots of integers. It appears that the people who worked with
mathematics in Mesopotamia learned by heart, just as we do, the products of all the
small integers. Of course, for them a theoretical multiplication table would have to
go as far as 59 ÷ 59, and the consequent strain on memory would be large. That fact
may account for the existence of so many written tables. Just as most of us learn,
without being required to do so, that ^ = 0.3333..., the Mesopotamians wrote
their fractions as sexagesimal fractions and came to recognize certain reciprocals,
for example ^ = 0; 6,40. With a system based on 30 or 60, all numbers less than 10
except 7 have terminating reciprocals. In order to get a terminating reciprocal for 7
one would have to go to a system based on 210, which would be far too complicated.
Even with base 60, multiplication can be quite cumbersome, and historians
have conjectured that calculating devices such as an abacus might have been used,
although none have been found. H0yrup (2002) has analyzed the situation by
considering the errors in two problems on Old Babylonian cuneiform tablets and
deduced that any such device would have had to be some kind of counting board,
in which terms that were added could not be identified and subtracted again (like
pebbles added to a pile).
Not only are sexagesimal fractions handled easily in all the tablets, the concept
of a square root occurs explicitly, and actual square roots are approximated by
sexagesimal fractions, showing that the mathematicians of the time realized that
they hadn't been able to make these square roots come out even. Whether they
realized that the square root would never come out even is not clear. For example,
text AO 6484 (the AO stands for Antiquites Orientales) from the Louvre in Paris
contains the following problem on lines 19 and 20:
The diagonal of a square is 10 Ells. How long is the side? [To find
the answer] multiply 10 by 0;42,30. [The result is] 7;5.Now 0; 42, 30 is |§ + = ^ = 0.7083, approximately. This is a very good
approximation to l/\/2 « 0.7071, and the answer 7; 5 is, of course, 7^ = 7.083 =
10 · 0.7083. It seems that the writer of this tablet knew that the ratio of the
side of a square to its diagonal is approximately The approximation to \/2 that
arises from what is now called the Newton Raphson method, starting from § as the