468 karine chemla
Th e key point that Liu Hui stresses is that by the very fact that these
quantities are taken as dividend and divisor, they are ‘put in relation’. By
this act, a relationship is established between them, which has operational
consequences. Here, the commentator fi rst introduces the concept of lü
which precisely characterizes the situation created: ‘Whenever quantities
are given/put in relation with each other, one calls them lü .’
In the case we examine, dividend and divisor are ‘put’ in relation, as
quantities of given, but distinct, units. It is the context of an operation that
shapes this relationship. Th e values expressing the relation between the cir-
cumference and the diameter of a circle are also lü s. However, by contrast
to the former quantities, they are rather ‘given’ in relation with each other.
In this case, it is a situation that brings them into relation. Liu Hui, meeting
here with a phenomenon that, from a formal point of view, will turn out to
be quite widespread, discusses it from a much more general angle, which
will thus prove useful and relevant in several other sections of his com-
mentary. Th is is a recurring and important feature of the commentator’s
proofs and one that makes them diffi cult to interpret: he systematically
brings to a given context a more general outlook from which to address
the correctness of a given operation, and thereby introduces a concept and
an argument that will be shown to recur in diff erent contexts. 56 In fact, the
concept of lü had already been introduced by Th e Nine Chapters in relation
to the prescription of the rule of three, at the beginning of Chapter 2. Th e
commentary will regularly, and more generally, bring to light in all kinds
of mathematical situations that quantities are lü s and use this property for
establishing the correctness of a procedure.
Once the concept is introduced, Liu Hui states the consequence for the
entities that it qualifi es: ‘ Lü s, being by nature in relation to each other,
communicate.’ We hence meet with a second occurrence of the term ‘com-
municate’ in the context of the commentary on the ‘procedure for directly
sharing’, an occurrence which echoes the wording of the procedure itself.
Th is time, it refers to the fact that the dividend and divisor are brought into
communication, even though this operation is grasped from a more general
point of view.56 On this feature of proofs, see Chemla 1991. Th e same phenomenon is shown to happen for
the operations of ‘homogenizing’ and ‘equalizing’, which are introduced in the commentary on
adding up fractions. We saw above another dimension of the relationship between the conduct
of a proof and the search for generality when we stressed the parallel between the proofs of
the correctness of the algorithms for the truncated pyramid with circular base and the cone,
respectively. On the concept of lü , see Li Jimin 1982a , Guo Shuchun 1984 , Li Jimin 1990 :
136–61, Guo Shuchun 1992 : 142–99, and the entry in the glossary in CG2004.