Reading proofs in Chinese commentaries 473
exact. On the other hand, as Liu Hui shows, the ‘procedure for directly
sharing’ amounts precisely to multiplying c by d to divide ac + b by both of
them at a stroke (column 3). 63
Such a reasoning would be only the observation of an equivalence, were
it not indicated by precisely the name given to the operation of division
between any two quantities in Th e Nine Chapters , i.e. jing fen. I suggest
understanding that the original meaning of this name was ‘directly sharing’.
Th ere are two pieces of evidence to support this interpretation. First, the
procedure for carrying out the same division in the Book of Mathematical
Procedures has the same name, except for the fact that the character jing
is written with a homophone that means ‘directly’. 64 Secondly, when the
seventh-century commentator Li Chunfeng comments on the name of the
procedure in Th e Nine Chapters , his interpretation is in conformity with
how the name is written in the Book of Mathematical Procedures. Since this
interpretation is quite important for our purpose, let us read it:
Directly sharing. Your servant, Chunfeng, and the others comment respectfully:
As for ‘Directly sharing’, from ‘Gathering parts’ onwards, 65 (the procedures) all
made the (quantity of ) parts homogeneous with each other, but this one directly
seeks the part of one person. 66 One shares that which is shared by the number of
persons , this is why one says ‘Directly sharing’.
Th e most important statement for us here is the one I italicized: the
operation is interpreted as dividing a quantity that is understood as itself
being yielded by a ‘sharing’ or, in other terms, a division. Li Chunfeng thus
also reads the operation as we suggest doing, that is, as dealing with the
succession of two divisions. He thereby links, on the one hand, dealing with
operations that follow each other, and, on the other hand, how arithmetical
operations are carried out on quantities having fractions. In doing so, Li
Chunfeng probably seeks to account not only for the name of the operation,
but also for why the style of the algorithm breaks with the description of
all the others before it. However, this interpretation fi ts with what the Book
65 Th at is, all the procedures for adding up fractions, subtracting them, comparing them and
determining their average. Th ese procedures are all interpreted by the commentators as
making the number of parts, that is, the numerators, homogeneous to each other, before
applying the operation in question. Compare CG2004: 166–7.
66 One may understand that the division is prescribed directly, without having made the
fractions fi rst homogeneous in any respect.
63 It is interesting that the operation of ‘making communicate’ that Th e Nine Chapters prescribes
is, for one part, the very operation that restores what can be interpreted as the original
dividend. For another part, this reading provides an interpretation of ‘dividing at a stroke’ in
terms of ‘making communicate’, which can be shown to be meaningful.
64 Compare slip 26, Peng Hao 2001 : 48.