472 karine chemla
each other. 61 In the context of dividing between quantities with fractions,
the last analysed sentence of Liu Hui’s commentary shows how the com-
mentator links the proposed transformation of units, the correctness of
which was established, to the application to both the dividend and the
divisor – both, in this case, themselves the results of a previous division – of
the same sequence of multiplications. In other cases, the concept of lü is
brought into play when accounting for an inversion in the order of a multi-
plication and a division is at stake. 62
Th is brings us back to the main question of this subsection: what is
the relationship between this development of Liu Hui’s and the validity
of our fundamental transformation iii? To bring the link to light, let us
consider one of the cases to which the ‘procedure for directly sharing’ can
be applied:(^) (a+b)/ =( + )/
c dacbdc
and let us look at this from the point of view of the surface for computing
( Figure 13.8 ). Th e set-up of the dividend (column 1) shows in which ways
it can be considered as the result of the division of ac + b by c (column 2).
Th e algorithm thus amounts to dividing by d the result of a division by c.
On the one hand, ac + b is that to which one returns when ‘making com-
municate’ the integer a and the numerator b – this property is guaranteed,
as Liu Hui stressed, by the fact that the results of division are given as
62 See, for instance, the second proof of correctness of the ‘procedure for multiplying parts’ or
the proof of the correctness of the ‘rule of three’ in CG2004: 170–1, 224–5.
61 Above, the introduction of specifi c quantities such as fractions or quadratic irrationals
was justifi ed by the necessity of having inverse operations cancel each other. Here, it is the
introduction of a concept, that of lü , that is to account for cancelling opposed multiplication
and division.
Figure 13.8 Th e division between quantities with fractions on the surface for
computing.
a
b
c
Two readings:
Dividend/Division
of ac + b by c
ac + b Dividend
d Divisor cd Dividing by the
product