426 karine chemla
which the refl ection about proof developed in ancient China still awaits
further study. In this chapter, I shall focus on further highlighting and ana-
lysing two key operations that are fundamental constituents of the practice
of proof documented by our commentators. Th e fi rst part presents in some
detail an example illustrating the two features on which we shall concen-
trate: on the one hand, determining the ‘meaning’ of a computation or of a
sub-procedure; on the other hand, carrying out what I called an ‘algebraic
proof within an algorithmic context’ – what I mean by this expression
will become clear with the example. In the case of the former feature, our
analysis will provide an opportunity to examine the modalities according
to which the ‘meaning’ of a sequence of computations can be determined.
As for the latter feature, aft er having brought to light fundamental trans-
formations characteristic of this part of the proof, I shall present evidence
in favour of the hypothesis that there existed an interest in ancient China
regarding what could guarantee the validity of these transformations. In
particular, in Part ii of this chapter, I shall explain why the commentaries
on the algorithms carrying out the arithmetical operations on fractions can
be read as related to this concern. Th is explanation will lead us to examine
the algorithms that Th e Nine Chapters contains for multiplying and divid-
ing fractions. Beyond the fact that the proof of their correctness further
illustrates how the commentators proceeded in their proofs, we shall show
why they can be considered as belonging to the set of fundamental ingredi-
ents grounding the ‘algebraic proof in an algorithmic context’. Bringing this
point to light will require that we view algorithms from the two distinct per-
spectives by which they were worked out in ancient China. Not only should
we read algorithms, as the commentators did, as pure sequences of opera-
tions yielding a magnitude, but we should also consider them as prescrip-
tions of computations, carried out on the surface, on which the calculations
were executed, and yielding a value. 9 In conclusion, we shall be in a position
to raise some questions on the nature and history of algebraic proof.I Two key operations for proving the correctness of algorithms
Th e serng and the fi rst key components of the proofs
Th e main example in the framework of which we shall follow the third-
century commentator Liu Hui in his proof of the correctness of an
algorithm deals with the volume of the truncated pyramid with circular
9 On this opposition, see Chemla 2005.