ChineseMathematics 79
Clearly the first question is ‘easy’, the second slightly harder, although the original reader may not
have found it so. They belong in form to a very widespread tradition, which goes back to Egypt and
Babylon. At first sight, only the specificity of the questions (the three passes clearly relate to a quite
specific terrain and mode of tax gathering) reminds us to think about who was asking them, who
they were addressing, and what was the aim. But aside from the particular organization of society,
the well-known isolation of China meant that, while there were certainly cultural influences in
both directions, they seem to have been rare and poorly documented. Furthermore, the ideas and
aims of Chinese mathematics have elements which are hard to translate into our own terms. The
questions and answers may be similar to those in other societies, but is that simply coincidence?
How should the subject be studied?
The beginner may well be daunted. If we begin around 400bce(although there are no texts quite
that early in date), and make the conventional end with the arrival of ‘European’ mathematics
around 1600ce,^2 we have an unbroken history of nearly 2000 years. Over this period, there
are a succession of texts, similar in form—mainly lists of problems (like the ones above), with
solution and commentary. The texts will have their own technical terms, which are not only exotic
measures of length, but may refer to procedures within mathematics. Again, this helps historically
in clarifying that we are not dealing with ‘our’ culture, but it does not help our comprehension. The
non-Chinese reader is likely to know nothing of the Chinese language; it may be the second world
language, but its script and structure make it inaccessible to most. Even transliteration can be a
problem; while modern texts agree on using the now official ‘pinyin’ system, older ones will use
some other one, so that the student should be warned that the mathematician who was formerly
called Ch’in Chiu-shao is now Qin Jiushao (compare Peking and Beijing).^3 We shall try to provide
some orientation on history and background; fortunately, not more than a minimum is absolutely
necessary. For more, see the references in Section 2.
Added to these difficulties, classical Chinese mathematical texts can pose quite specific prob-
lems. Their language is compressed, so that the ‘translations’ which we have may be rather free
adaptations. Some translators, in fact, (particularly Jock Hoe 1977) have tried to circumvent this
by adopting a special telegraphic form of English which may help. Furthermore, they may be
dependent on the specific calculator’s skill of manipulating counting-rods, which for a long time
was central to all work. This will have to be considered in its place, particularly in relation to claims
argued forcefully by Lam and Ang (1992) that the use of the rods led to the Chinese invention of
the place-value system and of decimal fractions. The questions usually asked, which extend some
of these considerations, are:
- What is specifically ‘Chinese’ about Chinese mathematics?
- To what extent can similarities between Chinese mathematics and that of other cultures (Indian,
Islamic) be attributed to cultural diffusion and to what extent are they independent? Specifically,
one could consider the decimal system, Pascal’s triangle, and methods for root extraction.
Despite its frequently mentioned restrictions, there is great diversity in Chinese mathematics,
and this chapter can only discuss a part of it. Hopefully, you will be encouraged to read further.
Exercise 1. Explain the answers to the questions from theNine Chapters, given that 1dou= 10
sheng.
- The later period has been neglected, and is still less studied; we shall consider it very briefly at the end of the chapter.
- To help, the older form will sometimes be given in brackets, and quotes—for example Yijing (‘I Ching’).