456 karine chemla
as a kind of division – the fact that they are exact guarantees that inverse
operations which follow each other can be deleted from an algorithm. 40
Yielding exact results perhaps matters less to computations than to proofs:
it grounds the validity of one of our three fundamental transformations.
Such is the link that is established between the numbers with which one
works and the transformations that can be applied to sequences of opera-
tions. Because the evidence relating to quadratic irrationals is far less abun-
dant than the evidence involving fractions, for the remaining part of my
argumentation, I shall hence focus mainly on the latter.
So far, we can establish that the commentator Liu Hui ascribes the moti-
vation in question to Th e Nine Chapters , thereby demonstrating that he
himself makes the connection between the use of some quantities and the
validity of a transformation. Can we follow Liu Hui and attribute the same
idea to the author(s) of the Classic? Th e argumentation is delicate and dif-
fi cult to conclude with certainty. It is true that quantities such as fractions
and quadratic irrationals date to the time when Th e Nine Chapters was com-
piled. In fact, only fractions occur in the Book of Mathematical Procedures.
As for using such quantities in relation to proofs, so far, our terminus ante
quem is 263, when Liu Hui completed his commentary. Th e occurrences of
the term ‘restoring’ or ‘returning to’ ( fu ) the original value provide interest-
ing clues. Th e concept is not to be found in Th e Nine Chapters. However,
it is attested to in the Book of Mathematical Procedures , in contexts where
similar concerns can be perceived. Interestingly enough, there, fu occurs
only aft er the statement of an algorithm for carrying out division or root
extraction. Aft er these algorithms, a procedure is then prescribed that
aims at ‘returning to’ the original value. By contrast, fu never occurs in a
procedure solving a problem. It is always appended to another algorithm
and carries out the inverse operation. Th is is complementary to the idea
one may derive from the commentaries on Th e Nine Chapters that there is a
link between the way in which the results of division and root extraction are
given and an interest in the possibility of restoring the original value. 41 E v e n
40 Note that, so far, the link has been established only for multiplications and divisions by
integers. Th e more general case still awaits consideration.
41 S e e fu in my glossary (CG2004: 924–5). In the Book of Mathematical Procedures , one
occurrence of fu is to be found in the context of the operation of ‘detaching the length’, which
asks to determine the length of a rectangle when its area and its width are given (slips 160–3,
Peng Hao 2001 : 114). Th ere, the fi rst procedure deals with the case when both the area of
the rectangular fi eld and its width are integers. Th e inverse procedure distinguishes the case
when the result is an integer from the one in which it has a fraction. A second procedure
considers the case when both data are pure fractions. Th e algorithm that returns to the
original value is that of multiplying fractions. When the width consists of an integer increased
by a set of fractions, the operation called ‘small width’ is carried out by a general procedure,