432 karine chemla
other. Th is operation implies inserting at this point the algorithm that Th e
Nine Chapters gave for multiplying not only such quantities, but also any two
quantities – integers, fractions, integers with fractions – the correctness of
which has been established in the fi rst of Th e Nine Chapters. Let us examine
this algorithm in detail before considering the modalities of its insertion.Th e general procedure for multiplying
Th is algorithm, like the others, has two faces. On the one hand, it is a list of
operations, the text of which is recorded in Th e Nine Chapters. On the other
hand, the operations it prescribes were carried out on a surface on which
quantities were represented with counting rods in ancient China. 16 For the
sake of my argument, it will prove useful to have some knowledge about the
way in which computations were physically handled on this surface. At fi rst
sight, it may seem strange that such details are necessary, since we deal with
proofs and not with actual computations. However, the relation between
the two will become clearer below.
On the surface, the execution of division and multiplication started from
the basis of a fi xed layout of their operands, which evolved throughout the
fl ow of computations. At the beginning of a multiplication, the multiplicand
was set in the lower row of the space in which the operation was executed,
while the multiplier was placed in its upper row. At the end of the computa-
tion, the multiplier had disappeared, leaving the result in the middle row
of the surface and the multiplicand in the lower row. In contrast, division
started with the dividend placed in the middle row, in opposition to the
divisor, put in the lower row. At the end of the computation, the quotient
had been obtained in the upper row. Under the quotient, either the place
of the dividend had been left empty, which indicated that the result was an
integer, or there was its remainder, in which case the result had to be read
as integer (upper row) plus numerator (middle row) over denominator
(lower row). Let us illustrate this description by what the computations for
the algorithm yielding the volume of the circumscribed truncated pyramid
must have looked like. Figure 13.3 shows a sequence of three successive
states of the surface for computing. We indicate a separation between the
rows for the sake of clarity. In fact, we have no idea whether or not there were
marks on this surface. In the fi rst state, on the left -hand side, the circumfer-16 Although they do refer to the fact that computations were carried out on such a surface, the
earliest extant texts discussed in this chapter contain very little information regarding how
these computations were handled. Th e argumentation supporting the way in which I suggest
recovering them is provided in Chemla 1996.