Reading proofs in Chinese commentaries 479
cates the reason why it must be used: earlier in the fl ow of computations,
one multiplied by a magnitude which was n times larger than it ought to
be – in most cases, by a numerator instead of the corresponding fraction –
therefore a division by n is needed to cancel this unwarranted dilation. 76 I n
our case, Liu Hui’s statement is an answer to the question of determining
the product of a / b by ‘something’ (let us call this ‘something’ X ) – one may
note the generality of the question considered. Th e reasoning appears to be
that, since a·X is equivalent to [( b · a ) : b ]· X or b · a / b · X , then a / b. X is hence
equivalent to a·X : b. If we pause a moment here, we can observe that what is
dealt with is precisely our transformation ii. A division followed by a multi-
plication, that is, a / b · X , which Liu Hui emphasized as equivalent to ( a : b )·X,
has been replaced by a multiplication followed by a division, a·X : b. Th e
way in which the commentator discusses the issue highlights the link he
reads between multiplying fractions (multiplying the result n / c by X ) and
what we called transformation ii – transforming the sequence ( n : c )· X into
nX : c. 77 In addition, the discussion has not yet specifi ed the quantity X. Its
result holds for any such quantity. Th is is yet another case where the proof
does not limit itself to the context in which it is developed, but highlights
the most general phenomenon possible.
In relation to the context in which Liu Hui develops this discussion, the
next step turns to the consideration of a specifi c value for X , that is, the
numerator c of the fraction c / d to be multiplied by a / b. He writes: ‘Now, “the
numerators are multiplied by one another”, hence the denominators must
each divide in return.’
76 In all observed cases, the ‘division in return’ eliminates a factor that is an integer. Note that
the beginning of Liu Hui’s commentary can be read as addressing the validity of such a
division: dividing, by a factor, a quantity that resulted from a multiplication by this very factor
eliminates from it this factor.
77 Th e commentary on the procedure solving problem 6.3 also stresses that the sequence
of multiplying by a and dividing by b can be carried out as multiplying by a/b , that is,
multiplying by a and dividing in return by b. Th e commentary on the procedure of ‘suppose’,
at the beginning of Chapter 2 , establishes the correctness of the algorithm carrying out the
rule of three in two ways. On the one hand, aft er having shown that a sequence of a division
and a multiplication yields the correct result, the commentator ‘inverts their order’ ( fan ) to
obtain the algorithm as described in Th e Nine Chapters. On the other hand, he transforms the
lü s expressing the relationship between the things to be changed one into the other, the former
into 1 and the latter into a fraction, by which the reasoning shows one must multiply to carry
out the task required. Th is, says Liu Hui, corresponds to ‘with the numerator, multiplying
and with the denominator, dividing in return’. A link is thereby established between the
operation of ‘inverting the order’ fan of a division and a multiplication and that of multiplying
by a fraction. Note how using the concept of lü and its operational properties is essential for
bringing this link to light. Th e commentary on the procedure solving problem 6.10 puts into
play all the elements examined so far.