Reading proofs in Chinese commentaries 467
sharing’, this transformation is carried out on the dividend and the divisor
jointly and the denominators c will both be forgotten. Th e correctness of the
procedure can only be established aft er this other aspect has been accounted
for. In the next section of the commentary, Liu Hui turns to address the
transformation. Again, it will be dealt with in terms of ‘making communi-
cate’ and this expression will take on new concrete meanings. Indeed, the
argument will show why communication is established not only between
components of the same quantity ( a and b/c ; d and e/c ) but also between
the dividend and the divisor, contained in the middle and lower parts of
the surface for computing. Th is is how the procedure can be concluded by
a division between integers. Th is remark suggests that, in so doing, Liu Hui
is still deploying his interpretation of the meanings he reads in the term
‘making communicate’, a phrase used here by Th e Nine Chapters.
Th e correlative transformation of the dividend and the divisor recalls the
commentary on the ‘procedure for simplifying parts’. We noted above that,
in this commentary, Liu Hui had compared the two situations from the
point of view that fi rst numerator and denominator, and then dividend and
divisor could be transformed in relation to each other. Pointing out a con-
trast between the two pairs, the commentator had stressed that dividend
and divisor could involve ‘parts’ of diff erent size. We highlighted the fact
that, in the context of the addition of fractions, he showed how to transform
distinct fractional parts into parts of the same size. However, one aspect of
the diff erence between the two situations has not yet been discussed. We
meet it here for the fi rst time, and it appears that it is precisely this diff er-
ence that Liu Hui addresses now. Following him, we can explain the diff er-
ence as follows. Th e simplifi cation or complexifi cation of a fraction implies
considering the quantities expressed with respect to the same part jointly.
However, the terms of a division can have parts that diff er not only in size,
but also in nature. In our case, the dividend contains parts of cash while
the divisor has parts of persons. It is interesting that, for the operations
discussed previously (adding up, subtracting, comparing, computing the
average), the data of the problems were all abstract and, in correlation with
this, these operations can be applied only to terms that are homogeneous
with each other in this respect – they only need to be homogenized with
regard to their size. For division, in contrast, the terms can furthermore
be of a diff erent nature. 55 Th is is what the problem shows and what is dealt
with from a theoretical and, most importantly, general point of view now.
55 Li Chunfeng’s commentary on the name of the operation, ‘directly sharing’, may address
this diff erence. We shall come back to it below. Note that the same remark holds true for
‘multiplying parts’.