458 karine chemla
fact that a value was used earlier in the fl ow of computations, that it was
interpreted as having been expanded by an unnecessary factor, and that the
‘division in return’ compensates for this by cancelling the factor. In dealing
with the proof of the correctness, the commentary usually brings to light a
pattern in the way in which the algorithm is accounted for, thereby echoing
the formulation of the procedure in the Classic. Such divisions highlight an
interesting point, suggesting a hypothesis to account for why fu does not
occur in Th e Nine Chapters.
So far, we have shown that Liu Hui establishes a link between the intro-
duction of kinds of numbers expressing the results of divisions and root
extractions, on the one hand, and the fact that the sequence of a division
and the multiplication inverse to it restored the original value, on the other
hand. Th is link coordinated perfectly with situations we met in the example
analysed in Part i of this chapter, where this property was twice used to
explain why pairs of operations were deleted from the fi nal algorithm.
However, situations in which one ‘divides in return’ reveal other ways in
which the annihilation of the eff ects of a pair of two opposed operations
by each other can be put into play in an algorithm. In such cases, the two
operations do not both disappear from the algorithm. Th is is precisely why,
when prescribing one of them, Th e Nine Chapters can refer to the reason
for using it. By contrast, since the operation of ‘restoring’ is disclosed when
one accounts for an algorithm but not when one describes it, the fact may
explain why the term fu does not occur in the Classic.Establishing the validity of fundamental operations and the
arithmetical operations on parts
In fact, one of the divisions examined in Part i of the chapter is of the kind
of a ‘division in return’. When, in algorithm 3, a division by 9 is prescribed,
it echoes the fact that earlier in the computations, instead of multiplying
diameters, the algorithm multiplied their triple. 44 Liu Hui does not use spe-
cifi c terminology that would indicate its nature as a ‘division in return’. Like
Th e Nine Chapters , he more generally indicates the point only occasionally.
However, in this case, the division by 9 is part of the ‘procedure for the fi eld
44 Perhaps the distinction between the two types of situation is grasped by the distinction
which Liu Hui introduces between ‘backtracking’ ( huan ) and ‘compensating each other’
( xiang zhunzhe ). If this is the case, a relation would be introduced between various types of
cancellation of opposed multiplication and division. In any event, although the distinction
is important, the fundamental reason underlying the fact that the eff ects of the operations
eliminate each other is the same: it relies on the premiss that the exact results of division
are given.