Reading proofs in Chinese commentaries 435
thereby appears as an operation of multiplication carried out on the three
lines that are the array of numbers yielded by the previous division. Th e
operation multiplies the content of the upper row by that of the lower row,
progressively adding the results to the middle row, where, in the end, the
fi nal result is to be read. Th is point is quite important. First, it reveals the
continuity between an array of positions read as a quantity ( a + b /3) and
the confi guration on which a computation is carried out on the surface.
In the same vein, an array of two lines will regularly be considered as a
quantity (a fraction) or as an operation (a division). We shall come back
to this feature on several occasions below. Second, this point shows the
material articulation between the operations of multiplication and division
on the surface for computing. Each of the operations can be applied to the
confi guration at which the other operation ends. Th e management of
positions on the surface hence appears to be highly sophisticated and
carefully planned to allow forms of articulation between the diff erent
computations.
It is from this point of view that we can best understand Li Chunfeng’s
interpretation of the name of the operation carried out by the procedure:
‘Field with the greatest generality’. 19 What explains such a name, in his
view, is that, in contrast to previous algorithms, this procedure unifi es
the three algorithms for multiplying either integers, or fractions, or even
quantities composed of integers and fractions. If we interpret integers as
being numbers of the type a + 0/ n (for any number n ), fractions as of the
type 0 + b/n , the ‘procedure for the fi eld with the greatest generality’ can be
uniformly applied to multiply any type of numbers. Furthermore, the ‘pro-
cedure for multiplying fractions’ is embedded in it. Note that the procedure
is quite complex in the case of multiplying integers. However, uniformity,
as stressed by Li Chunfeng, seems to be preferred over simplicity. 20 Th ese
remarks will prove useful below. In case the procedure Liu Hui devised for
19 In fact, Li Chunfeng explains the name ‘the greatest generality’, which is actually the name
given to the same operation in the Book of Mathematical Procedures. It may well be the case
that the original name of the procedure in Th e Nine Chapters was ‘the greatest generality’. We
shall see that the generality of the procedure is precisely the key point Li Chunfeng stresses
in his comment. Th e critical edition and the translation of this piece of commentary can be
found in CG2004: 172–3.
20 It is from this angle that one may understand why the description of an algorithm given in the
introduction of this chapter is oversimplifi ed. An algorithm may cover several types of cases
and include branchings to deal with them. In relation to this, practitioners of mathematics in
ancient China seem to have valued generality in algorithms, which led to writing algorithms
of which the text may be less straightforward than our fi rst description at the beginning of this
chapter. See Chemla 2003.