Reading proofs in Chinese commentaries 429
lines of argumentation. Th e fi rst line consists of establishing an algorithm,
for which Liu Hui proves that it yields the desired volume. Th e second line
amounts to transforming this algorithm as such into the algorithm the
correctness of which is to be proved. For this, Liu Hui applies valid trans-
formations to the algorithm taken as list of operations, thereby modifying
it progressively into other lists of operations, without aff ecting its result. In
the following, we shall make clear what such transformations can be. Th ird,
in doing so, the commentator simultaneously accounts for the form of the
algorithm as found in Th e Nine Chapters , by making explicit the motiva-
tions he lends to its author for not stating the algorithm as he or she most
probably fi rst obtained it, but instead changing it.
Th is whole process provides an analysis of the reasons underlying the
algorithm. Th e analysis is not developed merely for its own sake. It also
yields a basis on which the commentators devise new algorithms for
determining the volume of the truncated pyramid with circular base.
Accordingly, in a second shorter section of his exegesis, Liu Hui can make
use of the values he employs for the relationship between the circumfer-
ence and the diameter of the circle (314 and 100) to off er new algorithms.
Later on, Li Chunfeng will similarly rely on the values he selects for these
magnitudes to do the same. However, our analysis will concentrate on the
fi rst section of Liu Hui’s commentary.
Interestingly enough, a reasoning that has exactly the same structure and
the same wording is developed to account for the algorithm that Th e Nine
Chapters gives for the volume of the cone, aft er problem 5.25. On the one
hand, this similarity indicates that the text of the commentary analysed
here is reliable. On the other hand, such a fact shows that the proofs of
the correctness were established by the commentators in relation to other
proofs and not developed independently. Other phenomena lead to the
same conclusion. 12 Th is similarity relates to the fact that the proof had a
certain kind of generality – an issue to which we shall come back later. Let
us for now concentrate on how Liu Hui deals with the truncated pyramid
with circular base.
Th e fi rst step in Liu Hui’s reasoning is to make use of an algorithm for
which the correctness was established in the section placed immediately
before this one. Provided aft er problem 5.10, this algorithm allows the com-
putation of the volume of the truncated pyramid with square base when
one knows the sides of the upper square ( D s ) and lower square ( D i ) as well
as the height h (see Figure 13.2 ).^13
12 See Chemla 1991 and 1992 , for example.
13 Th e proof is analysed in Li Jimin 1990 : 304ff ., Chemla 1991 and Guo Shuchun 1992 : 132–5.