ChineseMathematics 83
which combines the theory of heavens and earth with a certain amount of trigonometry. Being
a manual for astronomers rather than a ‘textbook’, it did not have the same status as a founding
work for mathematicians.
The comparison ofThe Nine Chapterswith Euclid has been made so often that it is something of
a cliché, which is not to say that it is without importance. In a now largely outdated discourse,
exemplified by the quote from Boyer and Merzbach, a simple contrast was made between Euclid’s
use of proof and the axiomatic method as opposed to the supposedly basic practical orientation of
theNine Chapters. We shall see that the question is more complicated than that. Some initial points
which can be made are:
- Although much shorter than Euclid, theNine Chaptersis a substantial work, highly structured,
with each chapter organized around a particular type of problem, and with short but full
explanations for how the problems are to be solved. Like Euclid, the work appears to be the end
of a process of development of which we have no record; the various methods described must
have been worked out in the centuries which preceded the book’s final compiling. - Historically, theNine Chaptershas always been supplemented by commentaries, most partic-
ularly that of Liu Hui, which add a theoretical element which is missing from the bare text.
As the recent translation points out:
Liu was a unique mathematician, well-read in both science and literature, who wrote with great style, selecting
appropriate phrases from historical and literary classics in his descriptions of the relevant scientific subjects,
and showed succeeding generations how to solve problems and also how to justify and explain the rules used.
TheNine Chapterswould have remained a mere recipe book and not a complete classical mathematical textbook
without Liu’s work. (Shen et al. 1999, p. 5)
The work of commenting has continued through Chinese history since Liu, indeed in a modern
historicized style it is still ongoing. This (like the Euclid heritage) has had its positive and negative
aspects; it has provided a tradition, but has also allowed generations of mathematicians to
restrict their work within fairly narrow limits. - The detail given on the manipulation of counting-rods makes the book unique in its
arithmetical specificity. At key points (e.g. on extracting roots), the text goes into the pro-
cess of how to proceed with the rods with a precision which must have made clear to
readers (if not always to us) both how they should apply the procedures and why they
worked.
To illustrate some of these points, let us look again at the problem of the chickens cited in
Section 1. In the Islamic world and Europe, the method was to become known as the ‘method of
double false position’. The brief exposition and answer of the problem (with several similar ones) is
followed by the general rule.
The Excess and Deficit Rule.Display the contribution rates; lay down the [corresponding] excess and deficit below.
Cross-multiply by the contribution rates; combine them as dividend; combine the excess and deficit as divisor. Divide
the dividend by the divisor. [If] there are fractions, reduce them.
To relate the excess and the deficit for the articles jointly purchased: lay down the contribution rates. Subtract the
smaller from the greater, take the remainder to reduce the divisor and the dividend. The [reduced] dividend is the price
of an item. The [reduced] divisor is the number of people.
Liu’s commentary.Let the bottom terms cross-multiply the top, [combine and] then uniformize by the common
denominator...[Lay down] the contribution rates. Subtract the smaller from the greater, this is called the assumed