Reading proofs in Chinese commentaries 431
To determine the ‘meaning’ of the result, that is, that one obtains the
volume of the truncated pyramid with square base, Liu Hui has to rely on
both the algorithm established earlier and values corresponding to a value
of π. Such an operation of ‘interpretation’ corresponds to a key concept
used by the commentators in the course of proving the correctness of algo-
rithms: they refer to the ‘intention’ of an operation or a procedure, or its
‘meaning’, by the specifi c term of yi. In what follows, we shall pay particular
attention to the ways in which such a ‘meaning’ is determined.
Th e fi rst step in Liu Hui’s proof of the correctness of the investigated
algorithm belonged to what I have called above the ‘fi rst line of argumen-
tation’. Th e next step goes along both the second and the third lines. Th is
step makes us encounter the aspect of proof that is the main focus in this
chapter. I shall hence examine it in great detail.
Aft er having obtained the algorithm just examined, Liu Hui considers a
case:
Suppose that, when one simplifi es the circumferences of the upper and lower circles
by 3, none of the two is exhausted,...
Here, as is the rule elsewhere, the term ‘simplifying’ has to be interpreted
as meaning ‘dividing’. 15 In all extant mathematical documents from ancient
China, the result of a division is given in the form of an integer to which, if
the dividend is not ‘exhausted’ by the operation, a fraction is appended. Th e
numerator and denominator consist of the remainder of the dividend and
the divisor, respectively, both possibly simplifi ed when this was possible. As
a consequence, more generally, in these texts, fractions are always smaller
than 1.
With respect to the algorithm he has just established, Liu Hui then consid-
ers the case in which, aft er dividing the circumferences by 3, neither of them
yields an integer. In such cases, the next step of the algorithm would lead
to multiplying quantities composed of an integer and a fraction with each
15 To obtain evidence supporting this claim, the reader is referred to the glossary of Chinese
terms I composed (CG2004: 897–1035). Unless otherwise mentioned, all glosses of technical
terms rely on the evidence published in this glossary.
Divisions by 3 Multiplications, sum, Division by 3
Multiplication by hC (^) i > D (^) i = C (^) i /3 > CCisCi Cs h
33 3 3
⎡ ++ 22
⎣⎢
⎤
⎦⎥
() ( ). > CCisCi Cs
33 3 3
⎡ ++ 22
⎣⎢
⎤
⎦⎥
() ( ).h/3
C (^) s D (^) s = C (^) s /3