470 karine chemla
in correlation with each other. In the case of a division, by a simultaneous
dissection of the units of the dividend and the divisor, one may get rid of the
fractions.
Just as in the context of problem 3.17, Liu Hui approaches the correlative
transformation of the values of lü with full generality, introducing the disag-
gregation of the basic units in parallel with the opposite operation, i.e. aggre-
gating units. Th e circumstances in which the latter operation can be used are
referred to by the technical expression of ‘reiterated superpositions’, which
had been introduced earlier, in the commentary on the simplifi cation of
fractions. Th ere, it designated the possibility that the numerator and denom-
inator could be represented as rectangular arrangements of units – ‘parts’ in
this case – having a side of the same length, equal to their greatest common
divisor, or their ‘equal number’ in the terminology of ancient China. 59 A s a
consequence, dividing both by the ‘equal number’ amounted to expressing
the fraction in terms of parts coarser than the original ones by a factor equal
to that number. In the context of the general discussion about lü s in the
commentary on ‘directly sharing’, disaggregation has been introduced. Th e
next sentence then refers to the units as ‘parts’, even though they may be of
a diff erent nature, and states: ‘If parts are reiterated superpositions, one sim-
plifi es.’ Th e concept of ‘reiterated parts’ and the operation of simplifi cation
that it helps justify are thus imported into a new and more general context.
Once the general considerations have been developed fully, the com-
mentary applies them to the case under discussion, namely, dividend and
divisor. In a fi rst step, following on the last statement, Liu Hui introduces
the new concept of ‘ lü s put in relation with each other’, precisely when he
identifi es the fi rst instance for it: ‘Divisor and dividend, divided by the
equal number (i.e. their greatest common divisor), are lü s put in relation
with each other.’
In a second step, Liu Hui translates the properties of lü s discussed above
for the specifi c case examined in this context. Dividend and divisor having
both parts, one disaggregates repeatedly their units in parallel, which
amounts to multiplying. Th e commentator writes with full generality:
‘Th erefore, if one disaggregates the parts, one necessarily makes the two
denominators of the parts both multiply divisor and dividend.’
Th e general prescription of disaggregating (formulated at the level
of reasons) leads, within our specifi c context, to specifi c operations (at
the level of computations), namely, two multiplications. Th inking of the
process in terms of disaggregating and joining, the procedure amounts to59 On these terms, see the glossary in CG2004.