Reading proofs in Chinese commentaries 451
- ii. Inverting the order of divisions and multiplications
We saw that this very inversion had been carried out tacitly in the com-
mentary we examined but it is made explicit in other commentaries and
referred to by the technical term fan.^32 Moreover, I underlined the fact that
the transformation between algorithm 1 and algorithm 2′ could be con-
ceived of as belonging to this type.
- iii. Combining divisions
Now, several questions present themselves with respect to these trans-
formations, which appear to be the fundamental transformations needed
to argue along the line of argumentation examined. First, how were they
conceived of? Moreover, what guaranteed their validity? Furthermore, did
the commentators consider this question and in which ways? Addressing
these issues is essential to determine in which sense, in these commentaries,
we may have an ‘algebraic proof in an algorithmic context’. As announced in
the introduction, I shall argue that a link was established in ancient China
between the validity of these fundamental transformations and the kind
of numbers with which one operated. Moreover, in what follows, I intend
to show that the commentaries on the algorithms provided by Th e Nine
Chapters for carrying out arithmetical operations on numbers containing
fractions can be interpreted as addressing the question of the validity of the
fundamental transformations, in the ways in which these transformations
32 See the commentaries on the ‘procedure of suppose’ (rule of three), at the beginning of
Chapter 2 ; the procedure for unequal sharing, at the beginning of Chapter 3 ; the procedures
following problems 5.21 and 5.22.
Dividing by 9 Multiplying by hC (^) i , C (^) s (...) > ( C (^) i C (^) s + C (^) i 2 + C (^) s 2 )/9 > [( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) /9]· h
has been transformed into
Multiplying by h Dividing by 9
C (^) i , C (^) s (...) > ( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h > ( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h /9
Dividing by 9 Dividing by 4
( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h > ( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h /9 > [( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h /9]/4
has been transformed into
Dividing by 36
( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h > ( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h /36