Reading proofs in Chinese commentaries 441
the commentator aims at accounting for the algorithm as described in the
Classic – this is part of what we called the third line of argumentation and
is interwoven with the fi rst two lines. For instance, in this case, he seems to
be attempting to account for the reason why the algorithm does not begin
with a division by 3, or, more directly, for why the algorithm is not transpar-
ent, in the sense just introduced. 24 Th is question will lead him to formulate
motivations which explain the transformation of the algorithm he obtained
into the algorithm actually provided by Th e Nine Chapters , which yields the
same result.
Second, the reason Liu Hui adduces for that is the possibility that the
division by 3 may introduce results with fractions. Here this detail reveals
a key dimension in his expectations towards Th e Nine Chapters. If we recall
the data of problem 5.11, the circumference of the lower circle is 3 zhang.
However, the case Liu Hui considers, to reconstitute the motivations of the
author(s) of the procedure, is one in which ‘none’ of the two circumferences
is ‘exhausted’ by the division by 3. Th is indicates that he believes the authors
considered other cases than that of the problem in Th e Nine Chapters in
order to shape the procedure. Hence the commentator does not imagine
that the Classic provides algorithms for solving only the particular problem
aft er which they are given. He expects the algorithm to have been generally
established and consequently he accounts for the correctness of the general
algorithm as well as its form. 25 To be more precise, Liu Hui seems to be
considering that, in their shaping of the procedure, the author(s) of the pro-
cedure took into account all cases in which the data for the circumferences
would be integers. His reasoning would otherwise have been formulated in
a diff erent way. Such hints regarding the types of numbers that may consti-
tute data for a given algorithm would be extremely important to gather if
we want to understand better what generality meant in ancient China and
how the possibility of covering cases with diff erent types of numbers was
handled. It would be all the more important in the context of the argument
I want to make in this chapter, for establishing a link between the ‘algebraic
proof in an algorithmic context’ and the refl ection about numbers.
Th ird, it appears that the commentator believes that, when possible, the
author(s) of procedures avoided unnecessarily complex computations, in
particular computations with fractions. He regularly repeats this hypothesis
25 Th is is also what is shown by other passages of his commentary; see Chemla 2003.
24 On the basis of additional evidence, Chemla 1991 argues in favour of the hypothesis that Liu
Hui seeks to read reasons accounting for its correctness in the statement of an algorithm. He
succeeds in doing so for the algorithm which computes the volume of the truncated pyramid
with square base.