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is valued because it trains the mind. "If one neglects its study, one will not be able
to achieve excellence and thoroughness" (Lam and Ang, 1992, p. 151).
As in the quotation from the commentary on the Zhou Bi Suan Jing, we find
that an aura of mystery and "elitism" surrounds mathematics. It is to be pursued by
a dedicated group of initiates, who expect to be respected for learning its mysteries.
At the same time, it has a practical value that is also respected.Liu Hui. The Hai Dao Suan Jing. The fall of the Han Dynasty in the early third
century gave rise to three separate kingdoms in the area now known as China. The
north-central kingdom is known as the Kingdom of Wei. There, in the late third
century CE, a mathematician named Liu Hui (ca. 220-280) wrote a commentary
on the final chapter of the Jiu Zhang Suanshu. This chapter is devoted to the
theorem we know as the Pythagorean theorem, and Liu Hui's book, the Hai Dao
Suan Jing (Sea Island Mathematical Classic), shows how to use pairs of similar
right triangles to measure inacessible distances. The name of the work comes from
the first problem in it, which is to find the height of a mountain on an offshore
island and the distance to the base of the mountain. The work consists of nine
problems in surveying that can be solved by the algebraic techniques practiced in
China at the time. A translation of these problems, a history of the text itself, and
commentary on the mathematical techniques can be found in the paper by Ang
and Swetz (1986).
Liu Hui wrote a preface explaining that because of the burning of the books
400 years earlier, the few ancient texts still around had deteriorated, but that a
minister of agriculture named Zhang Cang had produced a revised and corrected
edition. However, most historians think that the Jiu Zhang Suanshu was written
around 200 BC, after Shih Huang Ti ordered the burning of the books.
Zu Chongzhi and Zu Geng. According to Li and Du (1987, pp. 80-82), fifth-century
China produced two outstanding mathematicians, father and son. Zu Chongzhi
(429-501) and his son Zu Geng (ca. 450-520) were geometers who devised a method
resembling what is now called Cavalieri's principle for calculating volumes bounded
by curved surfaces. The elder Zu was also a numerical analyst, who wrote a book
on approximation entitled Zhui Shu (Method of Interpolation), which became for a
while part of the classical curriculum. However, this book was apparently regarded
as too difficult for nonspecialists, and it was dropped from the curriculum and lost.
Zu Geng continued working in the same area as his father and had a son who also
became a mathematician.Yang Hui. We now leave a considerable (700-year) gap in the story of Chinese
mathematics. The next mathematician we wish to mention is Yang Hui (ca. 1238-
1298), the author of a number of mathematical texts. According to Li and Du (1987,
pp. 110,115), one of these was Xiangjie Jiuzhang Suan Fa (Detailed Analysis of the
Mathematical Rules in the Jiu Zhang Suanshu), a work of 12 chapters, one on each
of the nine chapters of the Jiu Zhang Suanshu, plus three more containing other
methods and more advanced analysis. In 1274 and 1275 he wrote two other works,
which were later collected in a single work called the Yang Hui Suan Fa (Yang
Hui's Computational Methods). In these works he discussed not only mathematics,
but also its pedagogy, advocating real understanding over rote learning.
Zhu Shijie. Slightly later than Yang Hui, but still contemporary with him, was Zhu
Shijie (ca. 1260-1320). He was still a young man in 1279, when China was united