The History of Mathematics: A Brief Course

418 14. EQUATIONS AND ALGORITHMS

LM = rx

Ì Í — r 3 - Ãé

Ê Í = r 3 - r 2

KM = ri + r 2

LN = v^f - 2r 3 r,

FIGURE 2. Sawaguchi Kazuyuki's first problem.

seems to be inspired by the desire to do some complicated algebra rather than by
any pressing geometric need.
One impetus to the development of mathematics in Japan came with the arrival
of the Chinese "method of the celestial element" (tian yuan shu), used in China.
This name was given to the unknown in an equation by Li Ye in his 1248 treatise
Ceyuan Haijing (Sea Mirror of Circle Measurements, see Mikami, 1913, p. 81).^9
This term spread to Korea as ch 'onwonsul and thence to Japan as tengen jutsu.
This Chinese algebra became part of the standard Japanese curriculum before the
seventeenth century.
Fifteen problems were published by Sawaguchi Kazuyuki in his 1670 work
Kokon Sampd-ki (Ancient and Modern Mathematics). As an example of the great
difficulty of these problems, consider the first of them. In this problem there are
three circles each externally tangent to the other two and internally tangent to a
fourth circle, as in Fig. 2. The diameters of two of the enclosed circles are equal
and the third enclosed circle has a diameter five units larger. The area inside the
enclosing circle and outside the three smaller circles is 120 square units. The prob-
lem is to compute the diameters of all four circles. This problem, although it yields
to modern algebra, is complicated. In fact, Fig. 2 shows that the problem leads to
the simultaneous equations
5
Ð + 2 = ra,
27rr^2 + ?rr| + 120 = ÔÃÃ^2 , ,
4r^2 r 3 + 2rir 2 r 3 + + ÃÉÃ^2 = 4r 2 r^2 ,
where ô-÷, r 2 , and r 3 are the radii of the circles. The last of these relations results
from applying the Pythagorean theorem first to the triangle LMN to get LM, then
to KIM.

3.1. Seki Kowa. This problem was solved by Seki Kowa (Smith and Mikami,

1914, pp. 96-97). In case Seki Kowa's prowess in setting up and solving equations

was not clear from his solution of Sawaguchi Kazuyuki's first problem, remember

(^9) The same word was used in a rather different and obscure sense by Qin Jiushao a year earlier
in his Sushu Jiu Zhang (Libbrecht, 1973, pp. 345-346).