440 karine chemla
of the ‘procedure for the fi eld with the greatest generality’ shows that, before
dividing by the product of the denominators, the resulting ‘dividend’ cor-
responds to the ‘parts of the product’. 23 Note, however, that the order of the
division by the product of the denominators and the multiplication by the
height was implicitly inverted so that the meaning of the result could be
stated in this way. Th is transformation is valid. Its validity again rests on
the fact that the results of divisions are exact. Here too, this transforma-
tion is one that may be applied to the list of operations as such in order to
change it into another list. In other passages, Liu Hui brings to light and
comments on this inversion, which he calls by the name of ‘ fan ’ (inversion).
However, here the inversion is carried out tacitly. We shall come back to it
later. In conclusion, we see the operations involved here in determining the
‘meaning’ ( yi ) of the result of the fi rst part of the algorithm, the correctness
of which is to be established. Th ey depend in an essential way on relying on
the meaning of previously established algorithms.
Th e discussion above highlights an interesting fact. If we concentrate on
the fi rst section of the algorithm determining the volume of the truncated
pyramid with circular base, we can view it from two angles. When seeking
to uncover its ‘meaning’, it is necessary to restore the opposed operations
that cancel each other and consider algorithm 2. However, when using the
section as a list of operations for computing, it is more rational to delete the
unnecessary operations, as in algorithm 2′. Although both algorithms yield
the same result, the algorithm for computing diff ers from the algorithm
for shaping the meaning ( yi ) of the result. Th is is a crucial fact for proving
the correctness of procedures. Sometimes, the two algorithms coincide, in
which case the algorithm is transparent concerning the reasons for which it
is correct. Th e main reason for which it may not be transparent is due pre-
cisely to the very transformations that are applied to the list of operations as
such, and which interest us in relation to the second line of argumentation.
At this point of our argument, several remarks can be made on the way in
which Liu Hui deals with the algorithms found in Th e Nine Chapters. First,
23 Here, an element of argumentation can be retrospectively added to what was said earlier.
Th e ‘procedure for the fi eld with the greatest generality’ is not referred to by the name of the
operation in the commentary we are analysing. Th ree elements lead us, nevertheless, to the
conclusion that such is the procedure that is inserted. First, the situation described is exactly
the one for which the procedure was made: multiplying in general and multiplying integers
increased by fractions in particular. Th is is clearly the case envisaged by Liu Hui. In addition,
the list of operations to be followed corresponds exactly to that of the ‘procedure for the fi eld
with the greatest generality’. However, other procedures could be used, as is demonstrated
by the ‘procedure for more precise lü ’ (CG2004: 194–7). Lastly, the terms tong ‘make
communicate’ and jifen ‘parts of the product’ are specifi cally attached to the arithmetical
procedures given in Th e Nine Chapters to deal with integers increased by fractions.