- CHINA 413
numbers in the abstract and illustrating them geometrically with lines. It does
seem strange that Euclid clings to what appears to be a completely unnecessary
geometric representation of a number. According to Klein, the mystery is solved if
we recognize that specific numbers were always intended, even though an abstract
symbol (a letter or two letters) was used for them. Klein (p. 124) cites Tannery in
arguing that these letters did not represent general, unspecified numbers, because
they were not amenable to being operated on.
2. China
The development of algebra in China began early and continued for many centuries.
The aim was to find numerical approximate solutions to equations, and the Chinese
mathematicians were not intimidated by equations of high degree.2.1. Linear equations. We have already mentioned the Chinese technique of
solving simultaneous linear equations and pointed out its similarity to modern ma-
trix techniques. Examples of this method are found in the Jiu Zhang Suanshu
(Mikami, 1913, pp. 18-22; Li and Du, 1987, pp. 46-49). Here is one example of the
technique.There are three kinds of [wheat]. The grains contained in two, three
and four bundles, respectively, of these three classes of [wheat], are
not sufficient to make a whole measure. If however we add to them
one bundle of the second, third, and first classes, respectively, then
the grains would become one full measure in each case. How many
measures of grain does then each one bundle of the different classes
contain?The following counting-board arrangement is given for this problem.
1 2 1st class
3 1 2nd class
4 1 3rd class
1 1 1 measuresHere the columns from right to left represent the three samples of wheat. Thus
the right-hand column represents 2 bundles of the first class of wheat, to which one
bundle of the second class has been added. The bottom row gives the result in each
case: 1 measure of wheat. The word problem might be clearer if the final result
is thought of as the result of threshing the raw wheat to produce pure grain. We
can easily, and without much distortion in the procedure followed by the author,
write down this counting board as a matrix and solve the resulting system of three
equations in three unknowns. The author gives the solution: A bundle of the first
type of wheat contains ^ measure, a bundle of the second ^ measure, and a bundle
of the third ^ measure.2.2. Quadratic equations. The last chapter of the Jiu Zhang Suanshu, which
involves right triangles, contains problems that lead to linear and quadratic equa-
tions. For example (Mikami, 1913, p. 24), there are several problems involving a
town enclosed by a square wall with a gate in the center of each side. In some cases
the problem asks at what distance from the south gate a tree a given distance east
of the east gate will first be visible. The data are the side s of the square and the