Reading proofs in Chinese commentaries 483
operations with quantities containing fractions are unique to China, by
contrast to other ancient traditions. If this were confi rmed, there would
appear to be a correlation between the latter proofs, on the one hand, and
the use of ‘algebraic proofs in an algorithmic context’, on the other hand.
In Part ii of this chapter, however, my argument was based only on
internal considerations. One of the most important facts that grounded the
argument was the continuity of concepts and notations on the surface for
computing, such as operations like division or multiplication on the one
hand and quantities such as fractions or numbers of the type a + b/c , on the
other. Th e same confi guration of numbers on the surface for computing
could be read as the set-up of a division, or the result of a division, that is,
a fraction. Moreover, applying a multiplication by c to the confi guration in
three lines representing a + b/c – hence read as the set-up of a multiplica-
tion – could restore the division that had yielded it. Th e key element for this
continuity is that of a position on the surface in which one could place and
operate on a component of a quantity or a function of an operation. Th e
surface served as a medium articulating these mathematical objects. In this
way, arithmetical operations on fractions were transformed into sequences
of operations, and the algorithms carrying them out were established on
the basis of interpretations and transformations of these sequences of
operations.^82 A link was thereby established between transforming lists of
operations and operating on fractions.
I suggested reasons for considering that this was the way in which
the commentators understood it. On the one hand, in the commentary
following problem 5.11, we saw how the inversion of the order of a division
and a multiplication was carried out by making use of the ‘procedure of the
fi eld with the greatest generality’. On the other hand, when Li Chunfeng
interprets the name of the operation for dividing between quantities of
the type a + b/c , he refers to the division of a quantity itself yielded by a
division.
Th ere is, however, another angle from which to consider the relationship
between the fundamental transformations i , ii , iii and the proofs of the cor-
rectness of algorithms for arithmetical operations on quantities of the type
a + b/c. Most of the technical terms listed above, by which the commenta-
tors refer to these transformations, are introduced precisely in relation to
commentaries discussing the necessity of using quantities like fractions or
quadratic irrationals ( fu ), on the one hand, and establishing the algorithms
82 One example for this is how, if there are parts in the dividend and the divisor, ‘directly sharing’
is explained to be equivalent to ‘multiplying’ both quantities by the two denominators.