Reading proofs in Chinese commentaries 463
kinds of units – cash and persons. In correlation with these changes, the
algorithm described is of a type that breaks with previous procedures.^51 L e t
us translate how it reads before providing an interpretation:
Directly sharing.
Procedure: One takes the quantity of persons as divisor, the quantity
of units of cash as dividend and one divides the dividend by the divisor.
If there is one type of part, one makes them communicate. (here comes a
commentary by Liu Hui that we shall analyse below) If there are several types
of parts, one equalizes them and hence makes them communicate. (second
part of Liu Hui’s commentary)
Th e procedure hence presents itself as one that covers all possible
(rational) cases for the data. Th e organization of the set of problems distin-
guishes between cases when the data are both integers (case 1), cases when
they both contain only one type of denominator (case 2), and cases where
there appear several distinct denominators (case 3; problem 1.18 illustrates
which situations may occur in this case).
Th e fundamental case is case 1. It is solved by the fi rst operation pre-
scribed by the procedure: a simple division.
For problems falling in the category of case 2 (that of problem 1.17), the
data can be of the type either ( a + b / c ) and d , or ( a + b / c ) and ( d + e / c ). In
the second case, the procedure suggests applying the operation of ‘making
communicate’. Let us stress that the operation of ‘making communicate’ is
used here for the fi rst time by Th e Nine Chapters. In Part i of this chapter,
we encountered the operation in the context of Liu Hui’s commentary.
Th ere, we saw that this operation was applied to quantities such as ( a + b / c )
and ensured that a and b shared the same units, thus transforming ( a + b / c )
into ac + b. For the case considered here, it transforms the units of the two
integers a and d accordingly, so that the number of units obtained ( ac and
dc , respectively) share the same size as the corresponding numerators. Th e
quantities ( a + b / c ) and either d or ( d + e / c ) are thereby transformed into
ac + b and cd (or cd + e ). Th e problem is thus reduced to the fi rst case, and
the procedure is concluded by a division. In modern symbolism, the proce-
dure can be represented as follows:
(^) ()/( )/
()( ( )/( )
a
b
c
dacbdcab
c
de
c
ac b dc e/)
51 Th e previous procedures all prescribed operations involving numerators and denominators to
yield the result. Clearly the description of the procedure to come is of a diff erent style.