452 karine chemla
were conceived. Th ese suggestions seem to be natural on the basis of
the previous discussion. Indeed, on several occasions, we observed the
connection between transformations applied to a list of operations and
algorithms carrying out arithmetical operations on quantities with fractions.
We now need to focus on the latter procedures to analyse this connection
systematically.II Grounding the validity of the fundamental transformations
of lists of operations
Th e fi rst hint that the commentators link the validity of the fundamental
transformations to the kinds of numbers used in them is found when Liu
Hui accounts for why, in his view, Th e Nine Chapters introduces quadratic
irrationals. We shall hence follow him in his argumentation.Eliminating inverse operations that follow each other
Aft er problem 4.16, Th e Nine Chapters describes a general and abstract
‘procedure for extracting the square root’. 33 I n a fi rst part of the procedure,
an algorithm is provided for determining the root of an integer digit by
digit. It is followed, in a second part , by a procedure dealing with quantities
containing fractions, which reduces the problem to the case dealt with in
the fi rst part. Th e commentary in which we are interested discusses a state-
ment that concludes the fi rst part of the procedure and asserts:
If, by extraction, the (number) is not exhausted, that means that one
cannot extract the (its) root, hence, one must name it (i.e., the number)
with ‘side’.
Th ree historians, independent from each other, have established that,
here, Th e Nine Chapters was addressing the case when the number N , the
root of which is sought, was not exhausted when one had reached the digit
for the units in the square root. All concluded that Th e Nine Chapters was
prescribing, for such cases, that the result be given as ‘side of N ’, which is to
be interpreted as meaning ‘square root of N ’. 34
33 It relies on a numeration system that is place-valued and decimal. Th e introduction to
Chapter 4 in CG2004: 322–35 analyses its main features. Th e critical edition and the
translation of the piece of commentary discussed can be found in CG2004: 364–6.
34 Volkov 1985 ; Li Jimin 1990. As for me, references can be found in Chemla 1997/8 or CG2004.
Note that the Classic states, without providing any argument in favour of this assertion, that in
these cases the extraction cannot be carried out.