30 2. MATHEMATICAL CULTURES Ialso of importance both for the individual and for the state. Surveying is of use in
any society where it is necessary to erect large structures such as dams and bridges,
and where land is often flooded, requiring people to abandon their land holdings
and reclaim them later. These considerations at least provide a reason for people
to regard mathematics as useful in practice. However, the preface, written by the
commentator Zhao Shuang, gives a different version of the motive for compiling
this knowledge. Apparently a student of traditional Chinese philosophy, he had
realized that it was impossible to understand fully all the mysteries of the changing
universe. He reports that he had looked into this work while convalescing from an
illness and had been so impressed by the acuity of the knowledge it contained that
he decided to popularize it by writing commentaries to help the reader over the
hard parts, saying, "Perhaps in time gentlemen with a taste for wide learning may
turn their attention to this work" (Cullen, 1996, p. 171).
Here we see mathematics being praised simply because it confers understanding
where ignorance would otherwise be; it is regarded as a liberal art, to be studied
by a leisured class of gentlemen scholars, people fortunate enough to be free of
the daily grind of physical labor that was the lot of the majority of people in all
countries until very recent times.
The Jiu Zhang Suanshu. Another ancient Chinese treatise, the Jiu Zhang Suanshu,
meaning Nine Chapters on the Mathematical Art,^6 has been partly translated into
English, with commentary, by Lam (1994 )· A corrected and commented edition was
published in Chinese in 1992, assembled by Guo (1992). This work has claim to be
called the classic Chinese mathematical treatise. It reflects the level of mathematics
in China in the later Han dynasty, around the year 100 CE. The nine chapters
that give this monograph its name contain 246 applied problems of a sort useful
in teaching how to handle arithmetic and elementary algebra and how to apply
them in commercial and administrative work. Unfortunately, these chapters have
no prefaces in which the author explains their purpose, and so we must assume
that the purpose was the obvious one of training people engaged in surveying,
administration, and trade. Some of the problems have an immediately practical
nature, explaining how to find areas, convert units of length and area, and deal
with fractions and proportions. Yet when we analyze the algebraic parts of this
work, we shall see that it contains impractical puzzle-type problems leading to
systems of linear equations and resembling problems that have filled up algebra
books for centuries. Such problems are apparently intended to train the mind in
algebraic thinking.
The Sun Zi Suan Jing. The most elementary of the early treatises is the Sun Zi Suan
Jing, or Mathematical Classic of Sun Zi, even though its date is several centuries
later than the Jiu Zhang Suanshu. This work begins with a preface praising the
universality of mathematics for its role in governing the lives of all creatures, and
placing it in the context of Chinese philosophy and among the six fundamental arts
(propriety, music, archery, charioteership, calligraphy, and mathematics).
The preface makes it clear that mathematics is appreciated as both a practical
skill in life and as an intellectual endeavor. The practicality comes in the use of
compasses and gnomons for surveying and in the use of arithmetic for computing
weights and measures. The intellectual skill, however, is emphasized. Mathematics
(^8) Chinese titles are apparently very difficult to render in English. Martzloff (1994) translates this
title as Computational Prescriptions in Nine Chapters.