476 karine chemla
If ‘the denominators of the parts respectively multiply the integer corresponding to
them and the numerators of the parts join these (the results)’, one makes the bu^72
that are integral communicate and be incorporated in the numerator of the parts. In
this way, denominators and numerators all (contribute to) make the dividends.
Above, we already alluded to the main elements of this commentary. Let
us add only two remarks. First, we now see how the operation of ‘making
communicate’ that is used in this proof is precisely one that was analysed
in the commentary on ‘directly sharing’. Second, in the transformation of
in th
a
b
c⎡
⎣
⎢
⎡⎡
⎢
⎢⎢
⎢⎣⎣
⎢⎢
⎤
⎦
⎥
⎤⎤
⎥
⎥⎥
⎥⎦⎦
⎥⎥ into ac + b , the latter is designated as ‘dividend’. Th is is one of the severalsigns of the continuity, which we already stressed, between quantities of
the type a + b/c and division, from both a conceptual and a notational point
of view. Th is point will prove important below. As a commentary on the
remaining part of the procedure, Liu Hui states:
Th is is like ‘multiplying parts’.
In other words, he asserts that the algorithm is, from this point onwards,
analogous to the procedure for multiplying between ‘pure’ fractions, which,
in Th e Nine Chapters , is placed just before it. As was observed above, the
commentator refers the interpretation of some steps of the procedure to his
previous commentary. 73 Th ree points are worth noting.
First, in the same way as we showed previously how the procedure for
the truncated pyramid with circular base embedded, among other algo-
rithms, the ‘procedure for the fi eld with the greatest generality’, the latter is
now shown to embed another procedure. Th is embedding is, however, to
be distinguished from the one which accounts for the name of the opera-
tion, discussed in Part i of the chapter. Th e latter embedding related to the
fact that the ‘procedure for the fi eld with the greatest generality’ unifi ed
three procedures for multiplying diff erent types of numbers: it referred to
the algorithm as a list of operations. Th e new embedding manifests itself
in the proof: it brings to light that, among the three cases covered by the
algorithm, one of them is, in terms of reasons, more fundamental in that
the correctness of the general procedure relies on its correctness. Th ese two
cases show that algorithms may be built by making use of other algorithms(^73) Th is conclusion is reinforced by the commentary placed aft er the procedure, which repeats
one of the arguments given to account for the correctness of the algorithm for multiplying
fractions.
72 Th e commentary refers to the data of the problems aft er which the procedure is given. Th ey
are all lengths expressed with respect to the unit of measure bu.