Reading proofs in Chinese commentaries 459
with the greatest generality’. And the nature of the division as being ‘in
return’ is highlighted in the commentaries, precisely when they establish
the correctness of this other algorithm.
Th is brings us back to the thesis that we aim at establishing here: that is,
that the reasoning which accounts for the validity of the fundamental trans-
formations identifi ed in Part i may have to be read from the commentaries
on the procedures for carrying out arithmetical operations on numbers
with fractions. We saw that the simple fact of introducing fractions was
essential to accounting for the validity of the fi rst fundamental transforma-
tion. Computing with fractions proves essential for the validity of the other
two transformations.
When introducing transformation ii , I already stressed the link between
transforming sequences of operations (in that case, inverting the order of
division and multiplication) and describing algorithms for computing with
quantities having fractions (inserting the ‘procedure for the fi eld with full
generality’). In the remaining part of this chapter, I shall argue for my main
thesis by showing that the validity of transformations ii and iii can be inter-
preted as being treated in the commentaries dealing with the correctness of
algorithms given for multiplying and dividing between quantities having
fractions, respectively. To do so, we shall discuss them in the order in which
they are presented in Th e Nine Chapters , since, interestingly enough, it
appears to be also the relevant order of the underpinning reasons. We shall
hence deal fi rst with division in relation to transformation iii , and then turn
to multiplication in relation to transformation ii. Note that all the proce-
dures that allow the execution of arithmetical operations with fractions are
systematically provided in Chapter 1 of Th e Nine Chapters.
One point will appear to be central in this discussion: the relationship
between the pair numerator and denominator and the pair dividend and
divisor. 45 Let us then examine further this relationship as a preliminary
to the following subsections of this chapter. In Part i , we recalled that,
in ancient China, fractions, conceived of as a pair of a numerator and a
denominator, were introduced as the result of division. As we showed in
Figure 13.3 , dividend and divisor were arranged in an orderly fashion on the
surface for computing and, at the end of the division, what remained in the
position of the dividend and the divisor were read, respectively, as numera-
tor and denominator. Th e continuity between the two pairs of objects
is hence manifest from the point of view of the surface for computing.
One can choose to read the two lower lines on the surface either as the
45 In his discussion on fractions, Li Jimin 1990 : 62–91 stresses this relationship and discusses the
algorithms for dividing and multiplying that we analyse below.