- CHINA^415
0 á a á
Í Í - a^3 Í-a^3 Í - a^3
0 -- á^2 —> 3á^2 -- 3á^2
0 a 2á 3á
1 1 1 1
Next, given any approximation á, the approximation is improved by adding
an adjustment 6, and the rows are then recomputed, again, first working from the
bottom to the second row, then from the bottom to the third row, and finally, from
the bottom to the fourth row:
a a + b a + b a + b
Í-a^3 Í-{a + b)^3 Í-(a + b)^3 Í-(a + b)^33a^2 —-» 3a^2 + 3ab + b^2 —-> 3(o + b)^2 —» 3(a -I- 6)^2
3a 3a + b 3a+ 26 3(a + 6)
11 11
By introduction of a counting board ruled into squares analogous to the reg-
isters in a calculator, the procedure could be made completely mechanical. Using
an analogous procedure, one can take fifth roots, seventh roots, and so on, with
increasingly messy computations, of course. Composite roots can be reduced to
prime roots, but since the generalization of this method works so well, there is
really no need to do so. The sixth root, for example, can be taken by extracting
the square root of the cube root, or it could be extracted directly following this
method.
2.4. The numerical solution of equations. The Chinese mathematicians of
800 years ago invented a method of finding numerical approximations of a root
of an equation, similar to a method that was rediscovered independently in the
nineteenth century in Europe and is commonly called Horner's method, in honor
of the British school teacher William Horner (1786-1837).^6 The first appearance
of the method is in the work of the thirteenth-century mathematician Qin Jiushao,
who applied it in his 1247 treatise Sushu Jiu Zhang (Arithmetic in Nine Chapters,
not to be confused with the Jiu Zhang Suanshu).
The connection of this method with the cube root algorithm will be obvious.
We illustrate with the case of the cubic equation. Suppose in attempting to solve
the cubic equation px^3 + qx^2 + rx + s = 0 we have found the first digit (or any
approximation) á of the root. We then "reduce" the equation by setting ÷ — y + a
and rewriting it. What will the coefficients be when the equation is written in terms
of y? The answer is immediate; the new equation is
py^3 + Spay^2 + 3po?y + pa^3
+ qy^2 + 2qay + qa^2
+ ry + ra
+ s = 0.(^6) Besides being known to the Chinese mathematicians 600 years before Horner, this procedure
was used by Sharaf al-Tusi (ca. 1135-1213), as discussed in Section 5 below, and was discovered
by the Italian mathematician Paolo Ruffini (1765-1822) a few years before Horner published it.
In fairness to Horner, it must be said that he applied the method not only to polynomials, but to
infinite series representations. To him it was a theorem in calculus, not algebra.