460 karine chemla
dividend and divisor of a division to be carried out, or as the numerator
and denominator of a completed division. Both interpretations will be
used in the commentaries examined below. Th e fact that the operation of
division and the expression of a fraction are set up in the same way evokes
the identity of their representations in modern notations. However, two
diff erences should be stressed. First, in ancient China, the fundamental
concept of quantity was not that of a general fraction – a rational number,
if you will – but that of an integer increased by a fraction smaller than 1,
which is precisely the result of a division on the surface. Fractions were just
a component of them. Second, in our case, we do not have, on the surface,
notations for ‘objects’, but rather ‘operational notations’, i.e. notations on
which operations are carried out. Th e continuity just emphasized derives
from the fact that, following the fl ow of a division, we go from one to the
other and back again. Indeed, the application of the inverse multiplication
to the fi nal confi guration of a division restores the division one started with,
exactly as it was originally set up (see Figure 13.4 ). But, in the case of adding
up fractions, the corresponding numerator and denominator are placed on
the same line horizontally, in such a way that, in the end, the result of the
addition is yielded in three lines consisting of an integer, a numerator and
a denominator. 46
Seen from another angle, a numerator and a denominator compose a
quantity and are essentially dependent on each other. In ancient China,
they were both conceived of as constituted of the same ‘parts’ fen of a unit,
which could either be abstract or not. 47 Th e size of the part was determined
by the denominator, which amounted to the number of parts into which
the unit was cut. As for the numerator, it was understood as consisting of a
multiplicity of such parts. In contrast to this, a dividend and a divisor are, to
start with, separate entities, which happen to be brought into relation when
they become functions in the same operation of division. Th is operation of
bringing entities into relation with each other seems to have been deemed
essential in ancient China, as we shall see below. As regards the entities con-
sidered, at that point, they become linked in a way that makes them share
properties with the pair of a numerator and a denominator. Th is parallel is
regularly stressed by the commentaries.
Th e fi rst example of this kind is found in the commentary glossing
the name of the operation of ‘simplifying parts’ – the fi rst operation on46 Compare Li Jimin 1982b : 204–5, especially; Chemla 1996 , where I reconstruct operational
notations in a diff erent way.
47 When the fraction was appended to an integer, its numerator and denominator were made of
parts of the smallest unit used in the expression of the integer.