436 karine chemla
the circumscribed truncated pyramid dealt only with integers or fractions,
other procedures could be used to multiply. However, given the fact that
there are cases in which ‘none of the circumferences is exhausted’ by the
division by 3, the most general procedure must be used.
Th e second element important for us in Liu Hui’s commentary on the
fi rst step of the ‘procedure for the fi eld with the greatest generality’ is the
intention he reads in the fact that the operation be used. Multiplying an
integer by the corresponding denominator, as he interprets, intends to
‘make’ the integers ‘communicate’ ( tong ) with the numerators. In other
words, the units of the integer a and those of the numerator (expressed by
the denominator) are made equal, which allows adding up the transformed
integer and the numerator. As is oft en the case, the reason brought to light
for employing an operation is expressed in the form of an operation (‘make
communicate’). Th e former operation can be prescribed by directly making
use of the latter name, which thus refers to both the operation to be carried
out and the intention motivating its use. Th e result, in our case 3 a + b , is
designated as the ‘parts of the product’ ( jifen ). It is ‘parts’, here a number
of ‘thirds’, in that it is composed of units, the size of which is defi ned by a
denominator. In what follows, we shall meet with these terms again. 21
We are now in a position to go back to the list of operations established
by Liu Hui for computing the volume of the truncated pyramid circum-
scribed to the one considered in problem 5.11.Inserting an algorithm: a key operation for proof
As Liu Hui envisaged, it is possible that none of the upper and lower
circumferences is ‘exhausted’ by the division by 3. Th us, in order to carry
out the various multiplications required by algorithm 1, one needs to
make use of the ‘procedure for the fi eld with the greatest generality’. Th e
insertion of this procedure in algorithm 1 (transformation 2) yields algo-
rithm 2, which, qua list of operations, can be represented by the following
list of operations:21 For the interpretation of the terms, see my glossary (CG2004). In fact, jifen ‘parts of the
product’ refers to the numerator in our sense, when its value is greater than that of the
denominator. One may view the numerator as a dividend, when looking at it from an
operational point of view, and as ‘parts of the product’, when considering it as constituting
a quantity. To be more precise, the commentator introduces the expression of ‘parts of the
product’ ( jifen ) in relation to the operation of ‘making communicate’, when the latter is fi rst
used in Th e Nine Chapters , that is, when commenting on the procedure for dividing between
quantities with fractions. We shall analyse this operation and the commentary on it below.