464 karine chemla
One should not forget, however, that the modern symbolism erases the
fact that the two fractions are fractions of units that are of a diff erent nature.
Th e fi nal case (case 3) is, in turn, reduced to the previous one by the
operation of ‘equalizing’, tong’.^52 Th is operation relates to the fractional
parts of the quantities, making them share the same denominator (‘equal-
izing them’ in terms of parts). Th e resulting transformation for cases such
as that of problem 1.18 can be represented as follows:(a+ )/( + )(+ )/( +b
cde
fg
habfh
cfhdech+gfc
cfh+ =
Once all fractions share the same denominator, we are brought back
to case 2, and the problem is solved as above, by ‘making’ integers and
fractions ‘communicate’. Such is the complete procedure, the correctness
of which Liu Hui sets out to establish in his commentary. Note that the
procedure for solving case 3 contains that for solving case 2 which, in turn,
embeds that for solving the fundamental case. Liu Hui develops the proof
with respect to the whole procedure, that is, the one solving case 3, address-
ing the operations in the order in which they are carried out in this case.
In the fi rst section , Liu Hui thus addresses the operation that occurs last
in the text, i.e. that of ‘equalizing’.
He does so by reference to the algorithm for adding up fractions, which
he has discussed previously (aft er problem 1.9). Th e commentator quotes
the fi rst steps of this other procedure for computing bfh , ech, gfc , on the
one hand, and cfh on the other hand, thereby providing a translation of
‘equalizing’ into operational terms. It thus appears that, to divide in case 3,
the operations to be applied fi rst are the same as those by which one starts
adding up fractions. In parallel, Liu Hui recalls his interpretation of the
‘meaning’ of these steps: he had shown that the latter computed a denomi-
nator equal for all fractions whereas the former homogenized the numera-
tors so that the value of the original fractions might be preserved. Liu Hui
thereby refers the discussion for establishing the ‘meaning’ of the operation
that Th e Nine Chapters calls here ‘equalizing’ to this other commentary
of his, where he showed how the corresponding steps ensured that one
‘makes’ parts corresponding to diff erent denominators ‘communicate’. Th e
algorithms for adding up fractions, on the one hand, and dividing in case 3,52 To make things simpler, I mark the transcription of the term in pinyin with an apostrophe, to
distinguish it from the term that has the same pronunciation tong ‘make communicate’. For all
these terms, I refer the reader to my glossary in CG2004. I argue there that the operation to
which ‘equalizing’ corresponds diff ers slightly, whether one considers Th e Nine Chapters or its
commentaries.