Reading proofs in Chinese commentaries 471
(a+ )/( + )=( + )/( + )=( + )/( + )
b
c de
fca b cd
ec
ffca fb fcd ec(fi rst multiplication) (second multiplication)
which is equivalent to the algorithm as provided in Th e Nine Chapters :
(^) (a+b)/( + )=(+ )/( + )=( + )/(
c
d
e
f
a
bf
cf
d
ec
fc
fca fb fccd+ec)
(equalizing) (multiplying by the two
denominators)
Th ese are the operations applied to the divisor and the dividend, and this
is what is meant by the prescription of ‘making them communicate’, if we
follow Liu Hui’s interpretation. Th e values of the dividend and the divisor
are transformed correspondingly, and they both become integers, without
their relationship being altered. Here the analysis of the operation of
‘making communicate’ is completed, and the correctness of the procedure
for ‘directly sharing’ is established. 60
From the previous discussion, three points are worth stressing. Th e fi rst
two are important for a description of the practice of proof.
First, as we already emphasized, the proof is carried out in such a way
as to approach the phenomena with the greatest generality possible. In our
case, this leads to the introduction of some key abstract concepts such as lü.
Second, through the analysis that is conducted during the proof, a sim-
plifi cation of the algorithm is hinted at, since it is shown that dividend and
divisor can be simplifi ed before a division is to be carried out. Again this is
a recurrent feature in the commentators’ proofs: they off er a basis on which
to develop new algorithms.
Th ird, and more importantly for our purpose, the concept of lü that is
introduced is intimately related to the theme of this chapter. Th is is the
point where we go back to the main thesis for which we argue here.
Combining divisions that follow each other
In fact, identifying, in a given context, the property of entities to be lü s with
respect to each other is a way of establishing the validity of introducing into
the fl ow of computations multiplications and divisions that compensate
60 In a last paragraph, the commentator describes another procedure that articulates the diff erent
cases possible in a diff erent way; see Chemla 1992 for a discussion.