- CHINA^405
performed. Srinivasiengar (1967, p. 39) gives the solution using symbols for the
unknown values of the animals, but does not assert that the solution is given this
way in the manuscript itself.
4. China
With the exception of the the Zhou Bi Suan Jing, which is mostly about geometry
and astronomy, algebra forms a major part of early Chinese mathematical works.
The difficulty with finding early examples of problems leading to algebra is that the
earliest document after the Zhou Bi Suan Jing, the Jiu Zhang Suanshu, contains not
only many problems leading to systems of linear equations but also a sophisticated
method of solving these equations, fully equivalent to what we now call Gaussian
elimination (row reduction) of matrices and known as fang cheng or the rectangular
algorithm. Li and Du (1987, pp. 46-47) discuss one example involving the yield
of three different kinds of grain, in which a matrix is triangularized so that the
solution can be obtained by working from bottom to top. Our discussion of this
technique, like the discussion of quadratic equations, is reserved for Chapter 14.
4.1. The Jiu Zhang Suanshu. In Chapter 6 of the Jiu Zhang Suanshu we find
some typical problems leading to one linear equation in one unknown. This type
of problem can be solved using algebra, but does not necessarily require algebraic
reasoning to solve, since the answer lies very close to the surface. For example
(Mikami, 1913, p. 16), if a fast walker goes 100 paces in the time required for a
slow walker to go 60 paces, and the slower walker has a head start of 100 paces,
how many paces will be required for the fast walker to overtake the slow one? The
instruction given is to multiply the head start by the faster speed and divide by
the difference in speeds. That will obviously give the number of paces taken by
the faster runner. The author says nothing about the number of paces that will
be taken by the slower runner, but he probably noticed that that number could be
obtained in two ways: by subtracting 100 (the head start) or by multiplying by the
slower speed instead of the faster. This equivalence, if noticed, would give some
insight into manipulating expressions for numbers.
Chapter 7 contains the kind of excess-deficiency problems discussed in Section 2
of Chapter 6. The solutions are described in some detail, so that we can judge
the extent to which they are to be considered algebra. For example, an unknown
number of people are buying hens. If each gives nine (units of money), there will be
a surplus of 11 units. If each gives six, there will be a deficit of 16. The instructions
for solution are to arrange the data in a rectangle, cross-multiply, and add the
products. In other words, form the number 9 · 16 + 6 · 11 = 210. If this number is
divided by the difference 9 — 6, the result, 70, represents the total price to be paid.
Adding the surplus and deficit gives 27, and when this is divided by 9 - 6, we get
9, the number of people buying. This solution is far too sophisticated and general
to be an early method aimed at one specific problem. It is algebra proper.4.2. The Suanshu Shu. Li and Du (1987, pp. 56-57) describe a set of bamboo
strips discovered in 1983-1984 in three tombs from the western Han Dynasty con-
taining a Suanshu Shu (Arithmetical Book) and dated no later than the first half
of the second century BCE. This work contains instructions on the performance of
arithmetical operations and some applications that border on algebra. For exam-
ple, one problem is to find the width of a field whose area is 1 mu (240 square hu)