Reading proofs in Chinese commentaries 461
fractions discussed in Th e Nine Chapters. Th ere, the commentary discusses
the reasons why, once fractions are introduced, it is a valid operation to
divide – or to multiply – both the numerator and the denominator by the
same number to transform the expression of the fraction. Th is property
is required in order for the ‘procedure for simplifying parts’ to be correct.
Th e validity of the operation is approached from the perspective that the
numerator and the denominator are constituted of parts of the same size.
Multiplying them by the same number n is interpreted as a dissection of
each part into n fi ner and equal parts – a process called ‘complexifi cation’,
and the operation opposite to the ‘simplifi cation’ that the commentator
introduces. Conversely, a simultaneous division of the numerator and the
denominator by n leads to uniting the parts composing them, n at a time,
and getting coarser parts. Th is does not change the quantity as such, but
just its inner structuring and its expression. Th us the commentary can con-
clude: ‘Although, hence, their expressions diff er, when it comes to making
a quantity, this amounts to the same.’ 48 Note that, from the point of view of
the operations involved, the reasoning establishes the validity of another
mode of inserting a multiplication and a division opposed to each other in
the course of an algorithm.
What is important is that, immediately aft erwards, this question of mul-
tiplying and dividing conjointly numerator and denominator is extended to
the case of dividend and divisor. Th e commentator writes: ‘Dividend and
divisor are deduced one from the other.’ Once the two entities are placed in
relation to each other, as functions of a division, the same reasoning then
applies. One can break up or assemble units in the same way. However,
the diff erence between the case of the fraction and the general case is that
dividend and divisor ‘oft en have (parts) that are of diff erent size’. Th e divi-
dend, for instance, may have an integer and a fraction. Its expression would
then include at least two types of units. Both terms of the division may also
have diff erent fractions. ‘Th is is why’, the commentator concludes, ‘those
who make a procedure (a procedure generalizing simplifi cation?) fi rst deal
with all the parts.’ Th is will require a technique, introduced immediately
aft erwards, related to the adding up of fractions. On this basis, the question
will be taken up again in the context of dividing between quantities having
fractions, for which all the necessary ingredients will be available. Th ereby,
the parallel between the pair of numerator and denominator and the pair of
dividend and divisor will be completed.
48 To be precise, part of the above discussion is held in the commentary on the algorithm
following the ‘procedure for simplifying parts’, i.e. the ‘procedure for gathering parts’, which
allows adding up fractions. Compare, respectively, CG2004: 156–7, 158–61.