90 A History ofMathematics
These are details. The method, it could be said, has not changed even if questions about different
grades of paddy are less frequent. There is, though, a more subtle difference. Confronted with such
a question, a modern textbook would call the paddy yieldsx,y, andz, form three equations and
thenwrite a matrix to solve them; in the history of European mathematics,xs precede matrices
by about 200 years. In theNine Chapters, there is no intermediary between the paddy and the
‘matrix’. There is indeed what one could call abstraction; but instead of our kind, which consists of
replacing unknowns by symbols, it inputs data directly into a solution diagram. This particular kind
of abstraction seems to have been peculiarly Chinese. It was clearly tied to the use of counting rods;
and so, we could guess, it travelled less well than some other Chinese mathematical inventions.Exercise 4.Follow through the calculations, and check that they give the right answer; either by using
matrices (if you know them), or using counting rods (if you can find or make them), or any other way.7. The Song dynasty and Qin Jiushao
There is no reason to doubt that the last half of the thirteenth century was the culminating point of Chinese
mathematics. (Libbrecht 1973, p. 13)
In later generations scholars were very proud of themselves and, considering [mathematics] inferior, did not teach
[or discuss] them. Only calculators and mathematicians were able to manage multiplication and division, but they
could not comprehend square-root extraction or indeterminate analysis. In case there were calculations to be per-
formed in the government offices, one or two of the clerks might participate but the position of the mathematicians
was never held in esteem; their superiors left things to them and let them do as they pleased; [but] if those who did
computations were only that sort of man, it was merely right that they should be disdained. (Qin Jiushao,Shushu jiu
zhang, preface, in Libbrecht 1973, p. 60–1)Subsequent historians have referred to theSong dynasty (960–1279)as a ‘golden age’ for culture
in many respects, and for mathematics in particular. To mathematicians such as Qin Jiushao,
who complained of their treatment as the nerds of the Chinese hierarchy, it did not appear so.
The dynasty lost dynamism over a short period, and its territory shrank to the southern half of
China, the north being controlled by a rival ‘barbarian’ dynasty, the Jin. In the thirteenth century
which Libbrecht describes as the ‘culminating point’ the Mongols under Chinggis Khan fought a
50-year war and finally overthrew the Song rulers. They ruled under the name of theYuan dynasty
from 1260 to 1368. The outstanding mathematics for which the period is known is distinctly
enigmatic. We have works from four apparently unrelated writers from the thirteenth/fourteenth
centuries: the prolific but fairly elementary Yang Hui, and three more surprising mathematicians
often sharing common concerns, but working in isolation, often with no official position to provide
them with problems or support. Their work is bothinnovative, in that it is clearly different in kind
from what has appeared before, and at the same timetraditional, in that the models which it draws
on are supplied by the classics.
- Li Zhi(1192–1279), who lived in the north, worked under the Jin rulers and later the Mon-
gols, and wrote the eccentric text calledCeyuan Haijing(‘Mirror comparable with the ocean’),
apparently dated 1248. This is entirely devoted to obtaining equations from problems of a geo-
metrical type about a town whose plan is drawn at the outset (Fig. 4). The problem is always to
find the town’s radius; the answer is always 120.