454 karine chemla
introduction of certain kinds of numbers and the line of proof that made
use of transformations carried out on lists of operations.
In fact, the commentary further bears witness to the fact that the link is
not merely established for such quantities. Once Liu Hui has introduced
the constraint that the result of a square root extraction must satisfy for the
cases in which the number N is not exhausted, he examines more closely
two results for root extraction in the form of an integer increased by a frac-
tion – one by defect and one by excess. It is revealing that his analysis of the
values concerns how they behave when one applies the inverse operation
to them but this is not what is most important for us here. Th e statement
by which he concludes his investigation is essential for the comparison it
establishes. Liu Hui writes:
One cannot determine its value (i.e. the value of the root). Th erefore, it is only when
‘one names it (i.e. the number N ) with “side” ’ that one does not make any mistake
(or, that there is no error). Th is is analogous to the fact, when one divides 10 by
3, to take its rest as being 1/3, one is hence again able to restore (fu) its value. (My
emphasis)
Th e mention of this other ‘restoring’ in the context of the commentary
on square root extraction reveals that for quantities of the type of an integer
increased by a fraction, it was a property that was also deemed essential.
Indeed, the comparison made here between square root extraction and
division further confi rms the link I seek to document. In his commentary,
Liu Hui manifests his understanding that, as kinds of numbers, quadratic
irrationals and integers with fractions diff er. 38 However, he stresses here
the analogy between them precisely from the point of view that introduc-
ing them as results in both cases allows two opposed operations applied in
succession to cancel their eff ects. In Part i of this chapter, we saw how this
cancelling led to deleting such a sequence of operations from the algorithm
that was being shaped. It is hence tempting to conclude that, as with quad-
ratic irrationals, Liu Hui linked the introduction of fractions to possibilities
of transforming lists of operations as such.
Th is hypothesis is supported by the fact that the ‘restoring’ made possible
by the introduction of fractions is also evoked and used within the context
of ‘algebraic proofs’ of the type we study. Th is is easily established by notic-
ing that the concept of fu ‘restoring’ introduced here occurs only in such
contexts. Th is fact confi rms, if it were necessary, the correlation between
this property shared by various kinds of numbers and the conduct of such38 See Chemla and Keller 2002.