- THE VALUE OF CALCULATION 145
and engineering, and he made no attempt to show how to calculate with an actually
infinite decimal expansion.
Stevin did, however, suggest a reform in trigonometry that was ignored until
the advent of hand-held calculators, remarking that, "if we can trust our experience
(with all due respect to Antiquity and thinking in terms of general usefulness), it
is clear that the series of divisions by 10, not by 60, is the most efficient, at least
among those that are by nature possible." On those grounds, Stevin suggested that
degrees be divided into decimal fractions rather than minutes and seconds. Modern
hand-held calculators now display angles in exactly this way, despite the scornful
remark of a twentieth-century mathematician that "it required four millennia to
produce a system of angle measurement that is completely absurd."
8. The value of calculation
One cannot help noticing, alongside a few characteristics that are unique to a given
culture, a large core of commonality in all this elementary mathematics. All of
the treatises we have looked at pose problems of closely similar structure. This
commonality is so great that any textbook of arithmetic published in the modern
world up to very recent times is almost certain to repeat problems from the Jiu
Zhang Suanshu or the Ahmose Papyrus or the Brahmasphutasiddhanta almost word
for word. Thus, where Brahmagupta instructs the reader to "multiply the fruit
and the requisition and divide by the argument in order to obtain the produce,"
Greenleaf (1876, p. 233) tells the reader to "find the required term by dividing the
product of the second and third terms by the first." As far as clarity of exposition is
concerned, one would have to give the edge to Brahmagupta. The number of ways
of solving a mathematical problem is, after all, quite small; it is not surprising if
two people in widely different circumstances come to the same conclusion.
Of course, when looking at the history of mathematics in the late nineteenth
century, we tend to focus on the new research occurring at that time and overlook
mere expositions of long-known mathematics such as one finds in the book of Green-
leaf. But what Greenleaf was expounding was a set of mathematical skills that had
been useful for many centuries. Like the Jiu Zhang Suanshu and the Sun Zi Suan
Jing, his book contains discussions of the relations of various units of measure to
one another and a large number of examples, both realistic and fanciful, showing
how to carry out all the elementary operations. His job, in fact, was harder than
that of the earlier authors, since he had to explain the relation between common
fractions and decimal fractions, exchange rates for different kinds of currency, and
many principles of commercial and inheritance law. If we now tend to regard such
books as being of secondary importance in the history of mathematics, that is only
because such a high superstructure has been built on that foundation. When we
read the classics from Egypt, India, China, and Mesopotamia, on the other hand,
we are looking at the frontier of knowledge in their time. It is a tribute to the au-
thors of the treatises discussed in this chapter that they worked out and explained
in clear terms a set of useful mathematical skills and bequeathed it to the world.
For many centuries it could be said that the standard mathematical curriculum had
a permanent value. Only in very recent years have the computational skills needed
in commerce and law been superseded by the higher-level skills needed for deciding
when and what to compute and how to interpret the results.