The History of Mathematics: A Brief Course

4. JAPAN 253

U

FIGURE 9. Sawaguchi Kazuyuki's quadrilateral problem.

It is given that

z^3 -u^3 = 271

u^3 - v^3 = 217

v^3 -y^3 = 60.8

y^3 -w^3 = 326.2

w^3 -x^3 = 61.

Required, to find the values of u, v, w, x, y, z.
The fact that the six quantities are the sides and diagonals of a quadrilateral
provides one equation that they must satisfy, namely:

u^4 w^2 + x^2 (v^4 +w^2 y^2 - v^2 (w^2 ~x^2 + y^2 )) - (y^2 (w^2 + x^2 -y^2 ) + v^2 (-w^2 + x^2 + y^2 ))z^2

+ y^2 z^4 - u^2 (v^2 (w^2 + x^2 - y^2 ) + w^2 (~w^2 + x^2 + y^2 ) + (w^2 - x^2 + y^2 )z^2 ) = 0.

This equation, together with the five given conditions, provides a complete set
of equations for the six quantities. However, Seki K6wa's explanation, which is only
a sketch, does not mention this sixth equation, so it may be that what he solved
was the indeterminate problem given by the other five equations. That, however,
would be rather strange, since then the quadrilateral would play no role whatsoever
in the problem. His solution is discussed in Sect. 3 of Chapter 14. Whatever the
case, it is known that such equations were solved numerically by the Chinese using
a counting board. Here once again it is very clear that the motive for the problem is
algebraic, even though it does amount to a nontrivial investigation of the relations
among the parts of a quadrilateral.

4.2. Beginnings of the calculus in Japan. By the end of the seventeenth cen-
tury the wasanists were beginning to use techniques that resemble the infinitesimal
methods being used in Europe about this time. Of course, in one sense Zu Chongzhi
had used some principles of calculus 1000 years earlier in his application of Cava-
lieri's principle to find the volume of a sphere. The intuitive basis of the principle is
that equals added to equals yield equal sums, and a solid can be thought of as the
sum of its horizontal sections. It isn't really, of course. No finite sum of areas and
no limit of such a sum can ever have positive volume. Students in calculus courses
learn to compute volumes using approximating sums that are very thin prisms, but
not infinitely thin.