Reading proofs in Chinese commentaries 469
Th e consequences of such a state are made explicit in the commentary
following problem 3.17, in which Liu Hui asserts:
Every time one obtains lü s, that is that, since when one refi nes (the units in which
they are expressed), one refi nes them all and, when one makes them coarser, one
makes them all coarser, the two quantities are transformed in relation to each other
(literally, interact with each other) and that is all. 57
Once the relationship is set, for instance, in our case, by the fact that ‘divi-
dend’ and ‘divisor’ are ‘put in relation to each other’ as quantities of given
units, any modifi cation of the value of one that comes from a systematic
dissection of its units – or a reunion of them – must be refl ected in a dis-
section – or reunion – for the units of the other for the relationship to be
maintained. 58 Th is is where the property of numerator and denominator is
seen in a more general perspective. Th is is also the point where a parallel
is established between the commentary on the ‘procedure for simplifying
parts’ and our context. Th e next sentence of Liu Hui’s commentary on the
‘procedure for directly sharing’ states the same property with respect to lü s:
‘If there are parts, one can disaggregate; if parts are reiterated superposi-
tions, one simplifi es.’
However, in contrast to the former statement, this quote makes precise
in which circumstances one may fi nd it useful to ‘disaggregate’ the units of
both terms, or ‘simplify’ them – that is, carry out a systematic aggregation
of their units. Th e disaggregation is to be used when the values put in
communication have ‘parts’, that is, contain fractions. Previously, being
in communication allowed the integer and the fractions to enter together
into the same operation of addition. Here, being in communication further
implies that, when modifi ed, the values are transformed simultaneously.
Th is latter property is used to transform the values of the lü s into integers
57 See CG2004: 306–7, 797, n. 73. In that case, the commentary brings to light that, in order to
account for the procedure, one must understand that the lü s chosen to express the relationship
between two diff erent kinds of silk are given in diff erent units of weight. By virtue of their
quality of being lü s, they nevertheless change in relation to each other. Note that there can
be more than two quantities, the set of which constitutes lü s. In Chemla 2006 , I discuss
source material from the Book of Mathematical Procedures which documents the process
of introduction of the concept of lü , as encapsulating parallel sequences of computations
carried out on quantities that occur within a dividend and a divisor. Th e way in which the
transformations encapsulated are described echoes in many ways Liu Hui’s commentary here.
58 Since the lü s express this relationship, the nature of the units of the quantities involved can
be forgotten, even though this is by no means mandatory. Th is corresponds to what is found
in the text, where in most cases, the values of lü s are expressed by abstract numbers. In some
sense, introducing the concept of lü is a way of addressing the possibility of carrying out an
abstraction with respect to units.