The History of Mathematics: A Brief Course

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138 6. CALCULATION

2.2. The Jiu Zhang Suanshu. This work is the most fundamental of the early
Chinese mathematical classics. For the most part, it assumes that the methods
of calculation explained in the Sun Zi Suan Jing are known and applies them to
problems very similar to those discussed in the Ahmose Papyrus. In fact, Problems
5, 7,10, and 15 from the first chapter reappear as the first four problems of Chapter
2 of the Sun Zi Suan Jing. As its title implies, the book is divided into nine
chapters. These nine chapters contain a total of 246 problems. The first eight of
these chapters discuss calculation and problems that we would now solve using linear
algebra. The last chapter is a study of right triangles. The first chapter, whose
title is "Rectangular Fields," discusses how to express the areas of fields given their
sides. Problem 1, for example, asks for the area of a rectangular field that is 15
bu by 16 bu.^3 The answer, we see immediately, is 240 "square bu." However, the
Chinese original does not distinguish between linear and square units. The answer
is given as "1 mu.'" The Sun Zi Suan Jing explains that as a unit of length, 1 mu
equals 240 bu. This ambiguity is puzzling, since a mu is both a length equal to 240
bu and the area of a rectangle whose dimensions are 1 bu by 240 bu. It would seem
more natural for us if 1 mu of area were represented by a square of side 1 mu. If
these units were described consistently, a square of side 1 linear mu would have an
area equal to 240 "areal". mu. That there really is such a consistency appears in
Problems 3 and 4, in which the sides are given in li. Since 1 li equals 300 bu (that
is, 1.25 mu), to convert the area into mu one must multiply the lengths of the sides
in li, then multiply by 1.25^2 · 240 = 375. In fact, the instructions say to multiply
by precisely that number.


Rule of Three problems. Chapter 2 ("Millet and Rice") of the Jiu Zhang Suanshu
contains problems very similar to the pesu problems from the Ahmose Papyrus.
The proportions of millet and various kinds of rice and other grains are given
as empirical data at the beginning of the chapter. If the Ahmose Papyrus were
similarly organized into chapters, the chapter in it corresponding to this chapter
would be called "Grain and Bread." Problems of the sort studied in this chapter
occur frequently in all commercial transactions in all times. In the United States,
for example, a concept analogous to pesu is the unit price (the number of dollars the
merchant will obtain by selling 1 unit of the commodity in question). This number
is frequently printed on the shelves of grocery stores to enable shoppers to compare
the relative cost of purchasing different brands. Thus, the practicality of this kind
of calculation is obvious. The 46 problems in Chapter 2, and also the 20 problems
in Chapter 3 ("Proportional Distribution") of the Jiu Zhang Suanshu are of this
type, including some extensions of the Rule of Three. For example, Problem 20
of Chapter 3 asks for the interest due on a loan of 750 qian repaid after 9 days if
a loan of 1000 qian earns 30 qian interest each month (a month being 30 days).
The result is obtained by forming the product 750 qian times 30 qian times 9 days,
then dividing by the product 1000 qian times 30 days, yielding 6| qian. Here the
product of the monthly interest on a loan of 1 qian and the number of days the loan
is outstanding, divided by 30, forms the analog of the pesu for the loan, that is, the
number of qian of interest produced by each qian loaned. Further illustrations are
given in the problems at the end of the chapter.


(^3) One bu is 600,000 hu, a hu being the diameter of a silk thread as it emerges from a silkworm.
Estimates are that 1 hu is a little over 2 meters.

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