A History of Mathematics From Mesopotamia to Modernity

96 A History ofMathematics

a=39 a=39

b=25

c=30

h 2

h 1

b=25

Fig. 8The ‘pointed field’, from Qin Jiushao’s problem.

takes the numbers

A=[b^2 −(c/ 2 )^2 ]×(c/ 2 )^2 ; B=[a^2 −(c/ 2 )^2 ]×(c/ 2 )^2

(in this case,A=90,000,B=29,160,000). He then shows that the area satisfies the equation

−x^4 + 2 (A+B)x^2 −(B−A)^2 = 0

In fact,x=

√

A+

√

B, which is a root of the equation.

Feeding the numbers in, this equation becomes:

−x^4 +763,000x^2 −40,642,560,000= 0

In the ‘Western’ world, which for our purposes at the time means Islam, the idea of applying such

exotic methods to a simple problem would have been rejected out of hand. Equations involving

only powers ofx^2 wereknown, and had been dealt with by the simple method of treating them as

quadratics in the variable ‘square of thing’ orx^2 , but the Chinese notation with its negative signs

would have posed difficulties.

What Qin does, and this is again odd if we suppose that he knew the answer, is to embark on a

sophisticated approximation procedure. This makes sense if you are trying to find (e.g.)

√

2 to three

or more decimal places, but forx=840 it looks like overkill. The idea is to find the figures of the

answer one at a time. If we know (say that the answer has three figures and the first is eight, then

the equationf(x)=0 becomesf( 800 +y)=0. Qin’s method is to use a simple way of working out

the coefficients ofyin the new polynomialg(y)=f( 800 +y), which he demonstrates by a sequence

of rod-number diagrams. The method which he used has been known since its ‘rediscovery’ in the

nineteenth century as the Ruffini–Horner procedure.

It is now certain that some Chinese mathematicians (e.g. Qin) and some Islamic ones

(e.g. al-Samaw’al, rather earlier) knew and practised this procedure, and the question of who

might have borrowed from the other has become something of a crux in the question of what could

have been transferred. It is not our intention to go further into the details of the procedure (see

Martzloff 1995, p. 232ff and Libbrecht 1973, p. 180ff for the Chinese version and Rashed 1994,