Reading proofs in Chinese commentaries 455
types of proof. 39 Th e introduction of such quantities is hence related to a
specifi c perspective on lists of operations as such.
In conclusion, Liu Hui interprets the necessity of introducing frac-
tions and quadratic irrationals as deriving from the necessity of restoring
the original value when applying the inverse operation to the result of an
operation – this is the only motivation he brings forward. In other words,
for the results of divisions or square root extractions – which are conceived
39 Compare the discussion of the commentary placed aft er problem 5.28, mentioned above. In it,
the commentator successively applies the operation inverse to the last of the operations to the
results of a sequence of algorithms. Th is operation, he states, restores the meaning and value
of the last intermediary step. If we represent the sequence of operations as above, we have the
following pattern of reasoning. Th e algorithm known to be correct is the following one:
C > C^2 > C^2 h > V
multiplying multiplying dividing by 12
by itself by h
Th e question is to determine the meaning of the following sequence of operations applied to V :
V > > >?
multiplying dividing extracting the
by 12 by h square root
Th e meaning of the result of the fi rst two steps can be determined as follows:
C > C^2 > C^2 h > V > C^2 h
multiplying multiplying dividing multiplying
by itself by h by 12 by 12
then
C > C^2 > C^2 h > C^2
multiplying multiplying dividing
by itself by h by h
Th is is correct, because multiplying by 12 restores that to which the division by 12 had been
applied. Th ereaft er, dividing by h restores that to which multiplying by h had been applied.
Now, because of the property of square root discussed, we have
C > C^2 > C
multiplying extracting the
by itself square root
and the meaning of the result of the following algorithm is established
V > > C^2 >^12 V
h
= C
multiplying by 12 dividing by h extracting the square root
Th is is how the correctness of the inverse algorithm is established. In the case of problem 5.28,
the inverse operations successively applied are a multiplication, a division and a squaring.
At each step, the commentator stresses that ‘restoring’ was achieved. Note that the reasoning
implicitly put into play to express the meaning of the fi rst part of algorithm 2′ as ‘the parts of
the product of 3 truncated pyramids with square base’ in the passage discussed above can be
seen as similar to the one just described. Th ese examples show the relationship of the property
of numbers which permits restoration and the conduct of the second line of argumentation
with the operation of establishing the meaning ( yi ) of the result of a list of operations.