480 karine chemla
Th e problem in Th e Nine Chapters asks to compute the product of a / b
by c / d. On the basis of the previous observation, this operation is shown to
amount to a · c / d : b , which, in its turn and for the same reasons, amounts to
ac : d : b. Liu Hui can hence interpret the ‘meaning’ of the fi rst prescribed
operation (computing ac ) and can establish that it must be followed by two
divisions for the desired result to be obtained. Th e commentator has thus
produced an algorithm yielding the result required by the Classic. Th e last
step needed to prove the correctness of the procedure given by Th e Nine
Chapters is to transform the algorithm obtained ( a·c : d : b ) into the one for
which the correctness is to be proved. Such a transformation comes under
the rubric of the second line of argumentation in an ‘algebraic proof in an
algorithmic context’, which we introduced in Part i of the chapter. Liu Hui
concludes his proof by transforming the former algorithm into the latter, as
follows: ‘Consequently, one makes “the denominators multiply each other”
and one divides at a stroke ( by their product) ( lianchu ).’
In other words, the commentator here applies transformation iii , the
validity of which was, as I argued above, dealt with in the commentary on
‘directly sharing’.Conclusions
Th e analysis developed in this chapter invites drawing conclusions on
several levels.
First, the passages examined illustrate how the earliest known com-
mentators on Th e Nine Chapters fulfi lled the task of establishing the cor-
rectness of algorithms. As we suggested in the introduction, this branch
of the history of mathematical proof has not yet been deeply explored. We
see how the Chinese source material calls for its development. Two issues
are at stake here. We need to understand the part played by proving the
correctness of algorithms in the overall history of mathematical proof, and
in particular in the history of algebraic proof. Moreover, on this basis, we
must determine how we should locate Chinese sources in a world history
of mathematical proof.
Whatever conclusion we may reach in this latter respect, it remains true
that Liu Hui’s and Li Chunfeng’s commentaries provide source material for
the analysis of the fundamental operations involved in proving the correct-
ness of an algorithm not only in ancient China but also in general. In our
limited survey of proofs from the Chinese source material, several funda-
mental operations appeared.