The History of Mathematics: A Brief Course

3. JAPAN 417

9

FIGURE 1. A quartic equation problem.

This problem is obviously concocted so as to lead to an equation of higher
degree. (The diameter of the town could surely be measured directly from inside,
so that it is highly unlikely that anyone would ever need to solve such a problem
for a practical purpose.) Representing the diameter of the town as x^2 , Qin Jiushao
obtained the equation^8

x^10 + 15x^8 + 72x^6 - 864x^4 - 11664x^2 - 34992 = 0.

The reasoning behind such a complicated equation is difficult to understand.

Perhaps the approach to the problem was to equate two expressions for the area of

the triangle formed by the center of the town, the tree, and the traveler. In that

case, if the line from the traveler to the tree is represented as b + y/a(x^2 + a), the

formula for the area of a triangle in terms of its sides is used, and the resulting area

is equated to 5(0+ (x^2 )/2)6, the result, after all radicals are cleared, will be an

equation of degree 10 in x, but not the one mentioned by Qin Jiushao. It will be

axw + (a^2 - ib^2 )x^8 - 8a6x^6 - 8a^262 x^4 + 16a6^4 x^2 + 16a^264 = 0.

One has to be very unlucky to get such a high-degree equation. Even a very

simplistic approach leads only to a quartic equation. It is easy to see (Fig. 1) that

if the diameter of the town rather than its square root is taken as the unknown,

and the radius is drawn to the point of tangency, trigonometry will yield a quartic

equation. If the radius is taken as the unknown, the similar right triangles in Fig. 1

lead to the cubic equation 2r^3 + 3r^2 — 243. But, of course, the object of this

game was probably to practice the art of algebra, not to get the simplest possible

equation, no matter how virtuous it may seem to do so in other contexts. In any

case, the historian's job is not that of a commentator trying to improve a text. It

is to try to understand what the original author was thinking.

3. Japan

The Japanese mathematicians showed themselves to be superb algebraists from the
beginning. We have already mentioned (Section 4 of Chapter 9) the quadrilateral
problem of Sawaguchi Kazuyuki, which led to an equation of degree 1458, solved
by Seki Kowa. This problem, like many of the problems in the sangaku plaques,

(^8) Even mathematicians working within the Chinese tradition seem to have been puzzled by the
needless elevation of the degree of the equation (Libbrecht, 1973, p. 136).