A History of Mathematics From Mesopotamia to Modernity

92 A History ofMathematics

208

B

C

15

E

135

South

D (centre)

A (the tree)

Fig. 5Following Chinese convention, south is at the top. B is the point where the walker catches sight of the tree. Other lettering

refers to the solution of the problem (later).

To give a particularly striking example: in hisCeyuan HaijingLi Zhi gives the problem:

135 puout of the south gate [of a circular city] is a tree. If one walks 15puout of the north gate and then turns east

for a distance of 208pu, the tree comes into sight. Find the diameter of the town. (Ceyuan Haijing, problem 11.18)

Li Zhi solves the problem by an equation of degree 4, with the result 240pu. The picture is shown

in Fig. 5; clearly, its analysis requires more than basic geometry—the ‘Pythagoras’ theorem, called

gouguby the Chinese, properties of tangents and similar triangles.

On the other hand, Qin Jiushao solves an extremely similar problem by an equation of degree 10

(see Libbrecht 1973, p. 134ff)—partly, it is true, by the artifice of using ‘x^2 ’ (as we would say) for the

diameter. There are very strong reasons to suppose that Li and Qin never met or communicated—

they lived as near contemporaries in mutually hostile parts of China. The two symbolize two kinds

of mathematician: for Qin, the world consists of watchtowers constructed on the walls of cities (see

Fig. 6), while for Li, people wander aimlessly round similar cities trying to catch sight of trees. Yet

the mathematics is much the same. What is the explanation for this sudden eruption of a ‘school’

of mathematicians who, working apparently independently, produced work which is both original

and in some ways related?

A part of the explanation is simple: it lies in our great ignorance. None of the writers, with

regard to the work which seems most striking, claims to be innovating, and some refer explicitly

to predecessors whose works are lost. The ‘Golden Age’ of the thirteenth century might therefore

appear less golden if we knew more of the ages which had preceded it. So, for example, Yang Hui

(like others) uses the ‘Pascal triangle’, but ascribes it to the eleventh-century writer Jia Xian whose

works have not survived. Even the striking notation for polynomials called the ‘tianyuan’ or celestial

element method was apparently copied from an earlier lost writer. In Libbrecht’s judgement:

[I]t is obvious that only a few names have been recorded, and that the greater part of Chinese mathematical works

have been lost. (Libbrecht 1973, p. 18)