Reading proofs in Chinese commentaries 475
is, the rule of three, it guarantees precisely that the number by which one
multiplies and divides be an integer. 69 Th is leads to mixing together inte-
gers and non-integers in the computations in a dissymmetric way that is
quite specifi c to the procedure for the rule of three described in Th e Nine
Chapters. 70 Th e predominant role given to integers can be read in the way
in which algorithms are composed and in the specifi c concepts that are
introduced in correlation with this. To establish whether this feature actu-
ally plays a part in the proofs of correctness, as we suggested above, we
would have to observe how the concept of lü is actually put into play in
the commentaries, an issue that we leave for another publication. 71 Let us
turn instead to the relationship between transformation ii and multiplying
between quantities containing fractions.
Inverting the order of a division and a multiplication that
follow each other
We already hinted at the reasons for linking the ‘procedure for the fi eld with
the greatest generality’ and transformation ii. It is hence natural to seek,
in the commentary of the former, a proof of the validity of the latter. As in
the previous subsection, we shall fi rst examine how the correctness of the
algorithm for multiplying quantities of the type a + b/c is established. While
doing so, we shall naturally be led to connecting this proof to that of the
validity of transformation ii.
Let us recall the procedure given by Th e Nine Chapters , which was
already discussed in Part i of the chapter:
Procedure: The denominators of the parts respectively multiply the
integer corresponding to them; the numerators of the parts join these
(the results); multiplying makes the dividend. The denominators of the
parts being multiplied by each other make the divisor. One divides the
dividend by the divisor.
Liu Hui establishes the correctness of the procedure in two steps, each of
which relates to a step in the procedure. Th e commentary on the fi rst set of
operations reads as follows:
70 Such is not the case for the rule of three given by the Book of Mathematical Procedures.
Discussing this diff erence exceeds the scope of this chapter and I shall deal with it elsewhere.
71 As already indicated above, the nature of the data to which the operations of the various
algorithms are applied should also be systematically observed, if we were to be more precise
regarding the extension of the algorithms for which correctness is established.
69 Incidentally, it also allows that these numbers be prime with respect to each other.