84 A History ofMathematics
difference, which is taken to be the lesser assumption. Then combine the excess and deficit to be the determined
dividend. Therefore dividing the determined dividend by the lesser assumption then gives the divisor to be the number
of people, and reducing the dividend gives the item price. (Shen et al. 1999, pp. 359–60)The first point to note is that the general rule is already, one might say, enough; a far broader
account of what needs to be done than the pre-Greek texts which proceeded only by example. The
explicit rules of procedure seem to lead to something like a matrix:
96
11 16
with contribution rates at the top, and excess and deficit below. Cross-multiplying and adding
gives the ‘dividend’( 9 × 16 + 6 × 11 = 210 ), while simply adding excess and deficit gives
the ‘divisor’ (15). We are not told where to place these, unfortunately. These are not our
answers, but they become what we want after dividing by a third number, the ‘difference of
the contribution rates’, in our case 9−6 or 3. Dividend then goes to price, divisor to number
of people.
What does Liu’s commentary add? In this particular case, not very much (there are better
examples). The original has already laid down the basis of a technical language (‘divisor’, ‘dividend’,
and so on); Liu’s concern is to refine this, by introducing extra explanatory terms such as ‘the lesser
assumption’. He clearly feels, if the word ‘therefore’ means anything, that his scheme makes clear
why the solution works. Yet it is not, in any sense, a proof.
In a carefully argued essay, Karine Chemla (1997) analyses one of Liu’s more substantial com-
mentaries—the one on addition of fractions, after problem 9 in chapter 1. Her argument is that in
explaining that the commentary contains a ‘proof ’ we risk simply finding what we are looking for.
The sentences in the commentary which count as proof are only a part of the story; they break off
at a point where, typically, Liu quotes a much more general idea, from theYijing: ‘Things of one kind
come together’. What follows is a discourse on whyqi(‘homogenizing’) andtong(‘uniformizing’)—
the basic procedures in adding fractions—work in the way that they do. What kind of mathematics
is it?
In other words, fractions with a common denominator can be added even if the numerators are quite different, while
fractions with different denominators cannot be added even if the numerators are close to each other...Multiplying
[the denominators] means fine division and reducing means rough division; the rules of homogenizing and uni-
formizing are used to get a common denominator. Are they not the key rules of arithmetic? (Shen et al. 1999,
p. 72)
What Chemla is suggesting is that in attempting to correct an unhistorical judgement—‘Chinese
mathematics had no proofs’—one may fall into an equally unhistorical claim: that the closely
argued commentaries of Liu are equivalent to ‘proofs’ in the Greek tradition. Arguably, they were
not, and Liu’s aim was a very different one: to explain for his readers how the parts of theNine
Chaptersworked and came together as a coherent whole.Exercise 2.(a) In the general case of ‘Excess and Deficit’, suppose the price is x. If y people pay a each
the excess is b; while if they pay a 1 each the deficit is b 1. What are the formulae for x and y, and what
role do the ‘dividend’, the ‘divisor’, and the ‘lesser assumption’ play in finding them? (b) Use either the
previous exercise or the method from theNine Chaptersto solve problem 7.3: Now jade is purchased
jointly; everyone contributes^12 , the excess is 4; everyone contributes^13 , the deficit is 3. Tell: the number of
people, the jade price, what is each?