144 6. CALCULATIONFor computations these cumbersome numerals were supplanted centuries ago by
the Hindu-Arabic place-value decimal system. Before that time, computations had
been carried out using common fractions, although for geometric and astronomical
computations, the sexagesimal system inherited from the Middle East was also
used. It was through contacts with the Muslim culture that Europeans became
familiar with the decimal place-value system, and such mathematicians as Gerbert
of Aurillac encouraged the use of the new numbers in connection with the abacus.
In the thirteenth century Leonardo of Pisa also helped to introduce this system
of calculation into Europe, and in 1478 an arithmetic was published in Treviso,
Italy, explaining the use of Hindu-Arabic numerals and containing computations
in the form shown in Fig. 3. In the sixteenth century many scholars, including
Robert Recorde (1510-1558) in Britain and Adam Ries (1492-1559) in Germany,
advocated the use of the Hindu-Arabic system and established it as a universal
standard.
The system was elegantly explained by the Flemish mathematician and engineer
Simon Stevin (1548-1620) in his 1585 book De Thiende (Decimals). Stevin took
only a few pages to explain, in essentially modern terms, how to add, subtract,
multiply, and divide decimal numbers. He then showed the application of this
method of computing in finding land areas and the volumes of wine vats. He wrote
concisely, as he said, "because here we are writing for teachers, not students." His
notation appears slightly odd, however, since he put a circled 0 where we now have
the decimal point, and thereafter indicated the rank of each digit by a similarly
encircled number. For example, he would write 13.4832 asl3©4®8©3©2@.
Here is his explanation of the problem of expressing 0.07 -ô- 0.00004:When the divisor is larger [has more digits] than the dividend, we
adjoin to the dividend as many zeros as desired or necessary. For
example, if 7 © is to be divided by 4 (5), I place some 0s next
to the 7, namely 7000. This number is then divided as above, as
follows:? 1
7 0 0 0 (1 750®
ß i i t
Hence the quotient is 1750©. [Gericke and Vogel, 1965, p. 19]
Except for the location of the digits and the cross-out marks, this notation
is essentially what is now done by school children in the United States. In other
countries—Russia, for example—the divisor would be written just to the right of
the dividend and the quotient just below the divisor.
Stevin also knew what to do if the division does not come out even. He pointed
out that when 4 © is divided by 3 © , the result is an infinite succession of 3s
and that the exact answer will never be reached. He commented, "In such a case,
one may go as far as the particular case requires and neglect the excess. It is
certainly true that 13 © 3 © 3± ©, or 13 © 3 © 3 © 3^ © , and so on are exactly
equal to the required result, but our goal is to work only with whole numbers in
this decimal computation, since we have in mind what occurs in human business,
where [small parts of small measures] are ignored." Here we have a clear case in
which the existence of infinite decimal expansions is admitted, without any hint
of the possibility of irrational numbers. Stevin was an engineer, not a theoretical
mathematician. His examples were confined to what is of practical value in business