Reading proofs in Chinese commentaries 457
more interesting is that, although in the Book of Mathematical Procedures
the aim of restoring is achieved for division, the results of which are always
exact, this requirement is not fulfi lled for root extraction. Th e procedure
provided for the latter operation gives only approximate results. In other
words, we reach an interesting conclusion: the concern for ‘restoring’, which
is explicit for both division and root extraction in the Book of Mathematical
Procedures , that is, already as early as the second century bce , apparently
existed before the solution satisfying it did for root extraction. Th is seems
to indicate that the need for ‘restoring’ motivated the introduction of a new
algorithm for root extraction and the introduction of quantities that would
ensure that the result be always exact, as we fi nd them in Th e Nine Chapters ,
and not the converse. Th ese remarks thus lend support to Liu Hui’s thesis
that, in Th e Nine Chapters , the introduction of quadratic irrationals and
fractions aimed at ensuring that opposed operations cancel each other.
We see how the evidence from the Book of Mathematical Procedures helps
to avoid misinterpreting the fact that neither the concept of ‘returning to’
( fu ) the original value nor the related one of ‘backtracking’ ( huan ) occur
in Th e Nine Chapters. Th is absence cannot be explained by the fact that
these concerns appear only at a later date. Nor, in fact, should the absence
be explained by the hypothesis that Th e Nine Chapters was merely a set of
recipes without any interest in accounting for the correctness of the algo-
rithms. I have already alluded to the fact that the commentator regularly
manifests his expectation that the procedures given by Th e Nine Chapters
be transparent on the reasons underlying them. 42 In addition to this, with
respect to the point under discussion, if the term fu ‘restoring’ does not
occur in Th e Nine Chapters , the Classic makes use of a technical expres-
sion that clearly belongs to a set of cognate terms and betrays the same
concern: baochu ‘dividing in return’. 43 For a division to be prescribed in this
way indicates the reason why it is carried out: the expression points out the
again followed by an algorithm explicitly aiming at ‘returning to’ the original value. In this
context, there are several occurrences of fu (slips 165–6, Peng Hao 2001 : 116). However,
the text of the procedure for doing so appears to be corrupted. Th e last occurrence of fu is
the most interesting for us. It is to be found in the Book of Mathematical Procedures , aft er a
procedure giving approximations for extracting square roots (slips 185–6, Peng Hao 2001 :
124–5). Th e case considered in the paradigm to which the procedure is attached is that of
an integer that is not a perfect square. Th e result is given as an approximation by an integer
increased by a fraction. However, it is asked to return to the original value. Th e end of the slip
reads: ‘one restores it like in the procedure for detaching the width’. In other words, not only
is the concern of fu common to the two contexts of division and root extraction, but also the
procedures for carrying it out.
42 See notes 3 and 24.
43 See, for instance, the second part of the algorithm for square root extraction.