Reading proofs in Chinese commentaries 453
Th e reason Liu Hui adduces for explaining why it was necessary to give
the result in the form of quadratic irrationals, when necessary, is fundamen-
tal for our purpose. Th e commentator fi rst considers a way of providing the
result as a quantity of the type integer increased by a fraction but discards it
as impossible to use. Th is leads him to make explicit the constraints that, in
his view, the result should satisfy. He writes (my emphasis):
Every time one extracts the root of a number-product 35 to make the side of a
square, the multiplication of this side by itself must in return (huan) restore (fu) (this
number-product).
Th is sentence is essential: the kind of result to be used is the one that
guarantees a property for a sequence of opposed operations. A link is
thereby established between the kinds of numbers to be used as results
and the possibility of transforming a sequence of two opposed operations.
More precisely, the result of the square root extraction must ensure that the
sequence of two opposed operations annihilates their eff ects and restores
the original data: their sequence can thereby be deleted.
Why is this important? To suggest answers to this question, one may
observe how the results of actual extractions are given in the commentaries.
It turns out that, when a commentator is seeking to establish a value, the
results of square root extraction are given as approximations. 36 However,
the fact that the operation inverse to a square root extraction restores ( fu )
the original number and the meaning of the magnitude to which the extrac-
tion was applied is used precisely in the context of an ‘algebraic proof in
an algorithmic context’. 37 Th is confi rms the link we suggested between the
35 Th e type of number for which one can extract the square root is a number that, from a
conceptual point of view, is a ‘product’. Th is corresponds to a specifi c concept in Chinese, ji ,
which can designate a number-product, an area, or a volume.
36 Th is is the case when the commentator discusses new values for expressing the relationship
between the circumference of the circle and its diameter. See the commentary aft er problem
1.33, CG2004: 178–85. However, this statement must be nuanced. Th ere is a context in which
Liu Hui uses quadratic irrationals as such in computations. Th is is in fact the passage that
allows interpretation of the obscure sentence by which Th e Nine Chapters introduces quadratic
irrationals. In it, the commentator seeks to assess with precision the ratio between the sphere
and the circumscribed cube that Zhang Heng (78–142) derived from his approximation for π ,
which states that the square on the circumference is to the square on the diameter as 10 is to 1.
As I suggested, the use of the irrationals here is driven by the aim of highlighting that Zhang
Heng’s algorithms were worse than that of Th e Nine Chapters. In the end, Liu Hui introduces
an approximation of a square root in the form of an integer to conclude the evaluation. See
Chemla and Keller 2002.
37 Th e text in question, that is, the commentary aft er problem 5.28, is discussed in Chemla
1997/8. An outline is provided below, in note 39.