Reading proofs in Chinese commentaries 447
Although the transformation seems comparable to transformation 3
discussed earlier, it is worth noticing that Liu Hui refers to the two in dif-
ferent terms. Earlier, the commentator spoke of ‘backtracking’ and in cor-
relation with this he stressed the fact that the values of the circumferences
had been restored while their meaning had changed. In contrast to this, Liu
Hui stresses here the fact that the two operations ‘compensate each other’
( xiang zhunzhe ). Th e emphasis is placed on the cancellation of their eff ects
as operations. Th is gives a hint of the subtlety of the formulation of the
reasoning.
Th e validity of this transformation is not to be taken for granted. It is
again guaranteed by the fact that, in ancient China, the result of a division
was given exactly, that is, as an integer increased by a fraction. We shall
show below that the commentator links these two facts.
Th e quoted sentence makes use of another expression, which requires
further analysis: the argument given for establishing the conclusion that
the two operations ‘compensate each other’ is formulated in the form that
‘the two denominators are equal’. Why is the word ‘denominator’ ( mu ) used
here? Th ere appears no reason explaining in which sense the ‘3’ with which
one multiplies can be considered as a ‘denominator’. Let us stress that, in the
other passage in which the same reasoning is developed, aft er problem 5.25,
the same term recurs, which indicates that this is not due to an error in the
transmission of the text. Th ese occurrences seem to imply that this term mu
has another technical meaning that I was unable to elucidate. Th is is why,
before it is found out, I translate the term in the usual way. However, con-
sequently, a very striking fact must be noted: in the commentaries, there is
only one other occurrence of this term with exactly this same use, and this
usage is found in the commentary establishing the correctness of the algo-
rithm for multiplying fractions. 28 Th is hint again links the line of argumen-
tation we are examining with the algorithms for carrying out arithmetical
computations with fractions. Th e point is worth noting, in relation to the
argument to be developed in Part ii of this chapter.
Another detail casts some light on the way in which Liu Hui operates. If
we observe the list of operations that Liu Hui is transforming, we can see
that it fi rst enumerates a division by 9, where the ‘3’s’ involved stand for π ;
second, a division by 3 corresponding to the computation of the volume of
the circumscribed pyramid; and, thirdly, a multiplication by 3, where the ‘3’
again stands for π. One might have expected that the proof would cancel
a multiplication and a division by 3 that would both be linked to π.^29 Th e
28 S e e mu ‘denominator’ in my glossary, CG2004.
29 I am indebted to Anne Michel-Pajus for this remark.