The History of Mathematical Proof in Ancient Traditions

Reading proofs in Chinese commentaries 477

in various ways, and the proof of the correctness of the former may as well
incorporate the proof for the latter according to diff erent modalities.
Second, interestingly enough, in their proofs, the commentaries regularly
refer to the proofs of algorithms placed just before in the Classic. 74
Th is seems to possibly provide an interpretation of the reasons why the
algorithms are presented in this order in Th e Nine Chapters.
Th ird, if we look at Figure 13.9 , we see that the part of the algorithm that
is applied to the elements placed on the surface for computations aft er the
fi rst step, that is, when divisions are restored, can be considered similar
to the algorithm applied to fractions: this is an essential prerequisite for
the proof of this section of the algorithm to be referred to that of the ‘pro-
cedure for multiplying parts’. Th is yields yet another hint of the fact that
practitioners of mathematics in ancient China saw continuity between the
notation of quantities and the set-up of operations. Th e commentary on
‘multiplying parts’, to which we shall now turn, starts by discussing pre-
cisely this point.
Th e algorithm referred to reads as follows:

Multiplying parts
Procedure: The denominators being multiplied by one another make the
divisor; the numerators being multiplied by one another make the divi-
dend. One divides the dividend by the divisor.

Th e opening sentence of the commentary relates the pair of a numerator
and a denominator to that of a dividend and a divisor. Liu Hui writes:

74 Th e second proof of the correctness of the ‘procedure for multiplying parts’ refers explicitly to
‘directly sharing’. See CG2004: 170–1. We shall show below that the fi rst proof also needs to
rely on ‘directly sharing’.

Figure 13.9 Th e multiplication between quantities with fractions on the surface for
computing.

a integer

b numerator

c denominator

ca + b

cc

(ca + b).(c′a′ + b′)(ca + b). (c′a′ + b′)

cc′

Dividend

Divisor:

a′ integer

b′ numerator

c′ denominator

c′a′ + b′

c′

c′a′ + b′

c′

c′a′ + b′

c′

Th e order of the

operations

was inverted