Reading proofs in Chinese commentaries 427
base (see Figure 13.1 below). 10 Th e problem in which Th e Nine Chapters
introduces this topic reads as follows: 11
(5.11) Suppose one has a truncated pyramid with circular base, the cir-
cumference of the lower circle of which is 3 zhang , the circumference
of the upper circle of which is 2 zhang , and the height of which is 1
zhang. One asks how much the volume is. Answer: 527 chi 7/9 chi.
Note the numerical values attached to the particular solid considered:
the circumference of the circle forming the base is 3 zhang. Th is detail will
prove important below. Let us stress the fact that Th e Nine Chapters uses
throughout the ratio of 3 to 1 for that of the circumference of a circle to its
diameter. Liu Hui opens his commentary by putting forward the hypothesis
that these were also the values used when the examined procedure was
shaped. He states: ‘Th is procedure presupposes that the circumference is 3
when the diameter is 1.’
Elsewhere, the commentator designates such values as lü s, thereby indi-
cating that they can be multiplied or divided by a same number without
their relative meaning, which is to represent a relationship between the
circumference and the diameter of the circle, being aff ected. We shall meet
this concept again below. To go back to problem 5.11 in Th e Nine Chapters ,
10 I translate the Chinese term yuanting as ‘truncated pyramid with a circular base’ on the basis
of an analysis of the structure of a system of terms designating solids in Th e Nine Chapters.
In the terminology of solids, three pairs of names work in a similar fashion: each of these
pairs contains two terms formed by prefi xing either fang (square, rectangle) or yuan (circle)
to the name of a given body. Th e designated solids correspond to each other, in that they
belong to the same genus. Th ey diff er only in that they have, respectively, either square or
circular sections. Th e relation between the terms in Chinese expressed a relation between
the designated solids. I hence translated these pairs as such, reproducing, in English, the
structure of the terminology of the Chinese. Th is leads to an interpretation of the second
term as designating a general kind of solid, two species of which are considered: the one with
square base and the one with circular base. Since fangting designates the ‘truncated pyramid
with square base’, yuanting was translated as ‘truncated pyramid with circular base’. For more
details, see Chapter D in CG2004: 103–4. On previous occasions (Chemla 1997/8; Chapter
A in CG2004: 36–8), I have already discussed this passage of Th e Nine Chapters and the
commentaries. Th e critical edition and the translation into French can be found in CG2004:
424–7. I come back to it again in this chapter to cast light on the proof from a new angle.
LD1987: 73, Li Jimin 1990 : 327–8 and Guo 1992 : 137–8 present an outline of Liu Hui’s proof.
11 A problem of Th e Nine Chapters is indexed by a pair of numbers: the fi rst number indicates
the chapter in the Classic in which the problem is placed. Th e second number indicates its
position in the sequence of problems of the chapter. We shall always translate the text of the
Classic in upper-case letters, in contrast to the commentaries, which are translated in lower-
case ones. In addition to indicating clearly to which part of the text a given passage belongs,
this convention imitates the way in which the diff erent types of text are presented in the
earliest extant documents.