Reading proofs in Chinese commentaries 465
on the other hand, share a common sub-procedure and, in the context
of division, which comes second in Th e Nine Chapters , the commenta-
tor states the conclusions of his previous analysis without developing the
reasoning again.
Th is stands in contrast to the luxury of details with which Liu Hui dis-
cusses the second operation to be considered within the context of division
in the following sentences. Th e Nine Chapters prescribes this operation
with the same term of ‘making them communicate’ as the one we discussed
above. Th e term is encountered here for the fi rst time in the Classic proper.
However, although the name is the same as the term already discussed, it
corresponds here to the prescription of diff erent computations. Following
Liu Hui in his analysis, we shall be able to make clear which prescription is
meant and why the same term can refer to diff erent operations according
to the context.
As above, Liu Hui translates what, in this context, ‘making them com-
municate’ amounts to in operational terms. He then brings to light the
‘meaning’ of the operation in terms of parts. He writes:
With the help of the denominator 53 ‘making them communicate’ is multiplying by
the denominator of the parts the integers (or: integral parts of the quantities) and
incorporating these (the results) into the numerators. By multiplying, one disag-
gregates the integers, thus making the parts of the product ( jifen ). Th e parts of the
product and the numerators hence communicate with each other, this is why one
can make them join each other.
Liu Hui hence makes explicit what the operation of ‘making them com-
municate’ means for the quantities at hand. In terms of computations,
( a + b / c ) and ( d + e / c ) are transformed into ( ac + b ) and ( dc + e ), respec-
tively. We recognize the result of the operation as indicated in Part i of this
chapter. However, in contrast to that previous occurrence, here the com-
mentator decomposes this transformation into elementary operations and
interprets their eff ects in such a way that he brings to light why Th e Nine
Chapters may refer to it as ‘make communicate’.
Th e fi rst operation consists of multiplying the integers a and d by c , thereby
transforming them into ac and dc. Th ese quantities are what is fi rst desig-
nated here as ‘parts of the product’, or parts yielded by a multiplication. 54
54 We have already discussed the expression jifen in Part i of this chapter. In this new context,
jifen could also be understood as ‘accumulated parts’, which would give ji an ordinary
meaning. As we suggested above, jifen may be interpreted as referring to what, for us, would
be a numerator, in a situation in which the numerator is larger than the denominator. In the
53 Note that, whether one is within the context of case 2 or aft er the equalization in case 3, only a
single value remains for all denominators.