The History of Mathematics: A Brief Course

252 9. MEASUREMENT

4. Japan

The Wasanists mentioned in Chapter 3, whose work extended from 1600 to 1850,
inherited a foundation of mathematics established by the great Chinese mathemati-
cians, such as Liu Hui, Zu Chongzhi, and Yang Hui. They had no need to work
out procedures for computing the areas and volumes of simple figures. The only
problems in elementary measurement of figures that had not been solved were those
involving circles and spheres, connected, as we know, with the value of ð in various
dimensions. Nevertheless, during this time there was a strong tradition of geometric
challenge problems. It has already been mentioned that religious shrines in Japan
were frequently decorated with the solutions of such problems (see Plate 2). The
geometric problems that were solved usually involved combinations of simple figures
whose areas or volumes were known but which were arranged in such a way that
finding their parts became an intricate problem in algebra. The word algebra needs
to be emphasized here. The challenge in these problems was only incidentally geo-
metric; it was largely algebraic, as the book of Fukagawa and Pedoe (1989) shows
very convincingly. New geometry arose in Japan near the end of the seventeenth
century, with better approximations to ð and the solution of problems involving
the rectification of arcs and the computation of the volume and area of a sphere by
methods using infinite series and sums that approximate integrals.
We begin by mentioning a few of the challenge problems without giving their
solutions, since they are really problems in algebra. Afterward we shall briefly
discuss the infinitesimal methods used to solve the problems of measuring arcs,
areas, and volumes in spheres.

4.1. The challenge problems. In 1627 Yoshida Koyu wrote the Jinko-ki (Trea-

tise on Large and Small Numbers), concluding it with a list of challenge questions,

and thereby stimulated a great deal of further work. Here are some of the questions:

1. There is a log of precious wood 18 feet long whose bases are 5 feet and
   2^ feet in circumference. Into what lengths should it be cut to trisect the
   volume?


2. There have been excavated 560 measures of earth, which are to be used for
   the base of a building. The base is to be 3 measures square and 9 measures
   high. Required, the size of the upper base.


3. There is a mound of earth in the shape of a frustum of a circular cone.
   The circumferences of the bases are 40 measures and 120 measures and the
   mound is 6 measures high. If 1200 measures of earth are taken evenly off
   the top, what will be the height?


4. A circular piece of land 100 [linear] measures in diameter is to be divided
   among three persons so that they shall receive 2900, 2500, and 2500 [square]
   measures respectively. Required, the lengths of the chords and the altitudes
   of the segments.

These problems were solved in a later treatise, which in turn posed new math-
ematical problems to be solved; this was the beginning of a tradition of posing and
solving problems that lasted for 150 years. Seki Kowa solved a geometric problem
that would challenge even the best algebraist today. It was the fourteenth in a
list of challenge problems posed by Sawaguchi Kazuyuki: There is a quadrilateral
whose sides and diagonals are u, v, w, x, y, and æ [as shown in Fig. 9].