448 karine chemla
expectation is all the more natural when we know that in a second part of
his commentary, Liu Hui relies on his proof to yield a new algorithm that
makes use of his own values for π. However, such is not the case. Th e com-
mentator cancels operations that follow each other. Th is seems to indicate
that he takes care not to modify arbitrarily the order in which the reasoning
led to establishing the operations constituting an algorithm. Such a detail
reinforces the hypothesis that he is working on lists of operations as such,
being careful to make explicit the transformations applied to them and the
motivations for using them. 30 Th ere is, however, another way of accounting
for this detail, i.e. that Liu Hui thinks that he recovers the reasoning fol-
lowed by the author(s) of the Classic.
By transformation 6, a list of operations was remodelled into another list,
equivalent in that it yielded the same result. Transformation 7 continues
along the second line of argumentation, even though it consists of applying
a diff erent operation to algorithm 5. Liu Hui goes on as follows:
We thus only multiply the lü of the square, 4, by the denominator 9, hence we obtain
36, and we divide at a stroke.
Liu Hui designates the two factors by which one should still divide to end
algorithm 5, i.e. 4 and 9, by the part they were shown to play in the reason-
ing ( lü of the square, denominator). Instead of carrying out the divisions
successively, transformation 7 suggests ‘dividing in combination’ ( lianchu ),
which I translated as ‘dividing at a stroke’. Th is implies transforming the end
of algorithm 5 into the multiplication of the two divisors by each other and
dividing by the product.
With the expression of ‘dividing at a stroke’, we meet with a technical
term that recurs regularly in the commentaries but is not to be found in Th e
Nine Chapters. We may account for this by noticing that it is a designation
of the division typical of the mode of proving the correctness of algorithms
on which the chapter concentrates.
Two successive divisions were accounted for, each being shown to be
necessary for its own reasons. As above, Liu Hui had to dissociate them
to bring to light the meaning of the result of the algorithm he shaped.
However, viewing the list of operations as a means for computing leads
to modifying the way of carrying them out, namely, by transforming the
end of algorithm 5. Liu Hui thereby accounts for the form of the algorithm
given by Th e Nine Chapters , by highlighting that the two operations were30 Th is conclusion should be nuanced by the remark made above concerning the change in the
order of the multiplication by h and the division by 9.