A History of Mathematics- From Mesopotamia to Modernity

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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,
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