482 karine chemla
lianchu ‘dividing at a stroke’, bingchu ‘dividing together’. 81 Th ese terms refer
to the three fundamental transformations (those we designated by i , ii and
iii ) involved in the ‘algebraic proof in an algorithmic context’ as carried
out for establishing the correctness of the algorithms presented in Th e Nine
Chapters. In fact, the validity of these transformations rests on the fact that
the results of divisions and extractions of square root are given as exact. We
have seen that Liu Hui explicitly related the validity of the fi rst fundamental
transformation to this fact. Before we go further in concluding about the
two other transformations, let us introduce the general remark regarding
algebraic proof to which this fact leads us.
Such a type of proof can be characterized by the fact that it carries out
transformations on sequences of operations as such. What appears here is
that the validity of such transformations rests on the structural properties
of the set of quantities to which the variables and constants involved in the
formulas transformed may refer. As soon as it is stated, the remark sounds
obvious. My claim is that it can be documented that a fi rst version of this
fact came to be understood in ancient China, in relation with the conduct of
‘algebraic proof in an algorithmic context’. Th is claim, in turn, raises a his-
torical question regarding this range of issues on which I shall conclude the
chapter: how was the relationship between the validity of algebraic proof
and structural properties of the set of magnitudes on which it operated
historically discussed? It is clear that inquiring into this question should
elucidate a fundamental dimension of the history of algebraic proof.
Th e second level on which I would like to focus in concluding relates
to my argument regarding the validity of transformations ii and iii. In the
chapter, I argued that there was an interest, in ancient China, in illuminat-
ing the grounds on which this validity rested. Moreover, I suggested that the
question was dealt with in the commentaries on the algorithms for dividing
and multiplying quantities of the type a + b/c. It is to be noted that fractions
conceived as a pair of a numerator and a denominator, as well as quanti-
ties a + b/c , appeared in Asia, in the earliest known Chinese and Indian
books. In China, the fi rst extant document attesting to the arithmetic with
such numbers, that is, the Book of Mathematical Procedures , also exhibits
a concern for the problem of ‘restoring’ ( fu ) the original quantity that was
divided, when applying the inverse operation. Th e main point, however,
is that, to my knowledge, the pages that Chinese commentators devoted
to establishing the correctness of algorithms carrying out arithmetical
81 Note that, although multiplications also happen to be joined – for instance, in the
commentary following problem 6.10 – no specifi c term was coined for this transformation.
Th is dissymmetry between multiplication and division is remarkable.