438 karine chemla
Why backtrack, one may ask, when discussing these two operations,
if it leads us to start from where, in any case, our starting point already
was? Liu Hui’s next sentence makes clear where the relevance for this
‘detour’ lies. Indeed if the value obtained is the same, the sequence of twoopposed operations provides it with a new meaning ( yi ): C (^) i and C (^) s no longer
represent the circumferences, but as results of the operation of ‘making
communicate’, they are now interpreted as representing the diameters,
disregarding denominators, that is, with reference to other algorithms. Th is
passage reveals the importance the commentator grants to interpreting the
meaning of operations.
Cancelling opposed operations: another key operation for proof
Let us now consider the consequences of these remarks for algorithm
2 when considered as a list of operations. What was just analysed
implies that the fi rst section of the list of operations can be transformed
(transformation 3):
Division by 3 Make communicate Multiplications, sums, etc.
C (^) i > D (^) i =
a
b
i
i
3
3ai + bi = Ci –(...) >
C (^) s D (^) s =
a
b
s
s
3
3 as + bs= Cs
is transformed into:
Multiplications, sums, etc.
C (^) i (............) >
C (^) s
Th e fi rst two operations cancel each other, since their sequence amounts
to returning to the original values – and to the original set-up. Deleting both
operations from the list of operations does not change the value yielded by
algorithm 2, nor does this transformation change the meaning of the fi nal
result. Th is is the fi rst transformation of a list of operations qua list that we
encounter and it belongs to what I called the second line of argumenta-
tion. We shall meet with other transformations of this kind below. Th is
particular transformation is valid for the reasons stressed above. Taken as a
whole, algorithm 2, which computed the volume of the truncated pyramid