466 karine chemla
Whatever the literal interpretation of this expression may be, there is no
doubt that the result is understood as being of the kind of ‘parts’, that is, as
sharing the same identity as the numerator and the denominator – both of
which are a collection of ‘parts’. Th is identifi cation derives from interpreting
the ‘meaning’ of the multiplication, in terms of the situation in which it is
applied, as a disaggregation.
As we alluded to above, Liu Hui had already discussed the link between
multiplying and disaggregating parts in the context of the addition of frac-
tions. Th ere, aft er the numerator and denominator were both multiplied
by the same number n – an operation he called a ‘complexifi cation’ – the
fraction obtained was interpreted as composed of parts that were n times
fi ner. Moreover, in this other context, diff erent ‘sets of parts’ ( a/b , c/d ,.. .)
were ‘complexifi ed’ jointly, that is, in correlation with each other, in such a
way that their denominators became equal to ( bd.. .) and the parts compos-
ing them were identical. Liu Hui interpreted this joint transformation as
‘making the parts communicate’ and thereby allowing them to be added to
each other.
Th e same link between multiplication and disaggregation recurs here, but
in a slightly diff erent way. Th rough the multiplication, the units composing
the integers are interpreted to be dissociated into parts of the same size as
the fractional parts. Th is dissymmetric transformation of the integers alone
ensures that the parts forming the two elements of a quantity of the type
a + b/c are ‘made to communicate’ and can be added to each other. It will
prove interesting to distinguish here two dimensions in the interpretation
of the eff ect of the operation. On the one hand, with the disaggregation, Liu
Hui brings to light a ‘material meaning’ of the multiplication. On the other
hand, he recognizes in this transformation the operation of ‘making entities
communicate’. In diff erent contexts, the way in which this formal result is
achieved may diff er. However, from a formal point of view, the action is the
same. Th is is what accounts for the fact that the same name can be used to
refer to diff erent actual computations.
In fact, so far, the commentator has considered the operation of ‘making
entities communicate’, prescribed by the Classic for case 2 of the divi-
sion, only from the point of view of each quantity of the type a + b/c taken
separately. As above, each quantity is transformed by the operation into an
integral number of parts. However, in case 2 of the ‘procedure for directlycase under discussion, the numerator consists of an accumulation of layers of parts equal
in number to the denominator, in contrast to the state in which, aft er the division is carried
out, these layers are each transformed into a unit. Th e glossary in CG2004 discusses why the
technical term jifen can refer, in some circumstances, to ac and, in others, to ac + b.