442 karine chemla
about their motivations, when he accounts for why the order of a division
and a multiplication was inverted with regard to the order given by the
reasoning he off ered. Th e rewriting of lists of operations that the author(s)
of procedures undertook may hence be motivated, in his view, by the
actual handling of computations. Th is is how Liu Hui explains the form of
the beginning of the procedure. As we shall discuss below, several specifi c
features of the mathematics of ancient China can be correlated with this
concern. In our case, the fact which the commentator brings to light in this
respect is that the procedure off ered by Th e Nine Chapters has the property
of working uniformly for all the data. As mentioned above, this property
was stressed by Li Chunfeng as characterizing the ‘procedure for the fi eld
with the greatest generality’. It would then be transferred to the algorithm
for determining the volume of the examined truncated pyramid. Note that,
in contrast to the former, for which uniformity was obtained at the expense
of simplicity, in the latter case, no artifi cial step is necessary to guarantee
a uniform treatment of all the possible data. It is to be noted, however,
that uniformity is not a property shared by all the algorithms in Th e Nine
Chapters. Th e procedure given for dividing between quantities having frac-
tions, which will be discussed below, is a counterexample, in which the
latter cases are reduced to the former ones.
Th ese remarks lead to an observation that is essential for the argument
made in this chapter. If we observe the transformation between the fi rst
part of algorithm 1 and that of algorithm 2′, what was carried out was an
inversion in the order of divisions and multiplications. Th is transforma-
tion, accomplished in the algorithm as a list of operations, was actually
carried out and accounted for through a procedure dealing with quanti-
ties with fractions. A link is thereby established between a transformation
that operates on lists of operations as such and an algorithm for executing
arithmetical operations on quantities with fractions. Th is link will be more
generally the focus of Part ii of this chapter. Furthermore, as has already
been stressed, this decomposition of the transformation that leads from the
fi rst section of algorithm 1 to that of algorithm 2′ highlighted the necessity
of relying on the possibility of cancelling two opposed operations that were
placed one aft er the other. Th is is how the transformation appears to be
carried out in Liu Hui’s view. In Part ii , we shall also come back to this point.
Without entering into all the details, let us give a sense of what the fl ow
of computations on the surface for computing looks like for the algorithms
considered. We can represent the main structure of the initial section of
algorithm 1 – which amounts to that of algorithm 2 – as the following
sequence of states ( Figure 13.5 ).