- CHINA 251
r Ë
Ê Ë
2\/á^2 - h^22áFIGURE 8. Sections of the cube, double square umbrella, and
sphere at height h.Hence each doubling of the number of sides makes it necessary to compute a square
root, and the approximation of these square roots must be carried out to many
decimal places in order to get enough guard digits to keep the errors from accumu-
lating when you multiply this length by the number of sides. In principle, however,
given enough patience, one could compute any number of digits of ð this way.
One of Zu Chongzhi's outstanding achievements, in collaboration with his son,
was finding the volume enclosed by Liu Hui's double square umbrella. As Fu (1991)
points out, this volume was not approachable by the direct method of dissection and
recombination that Liu Hui had used so successfully.^10 An indirect approach was
needed. The trick turned out to be to enclose the double square umbrella in a cube
and look at the volume inside the cube and outside the double square umbrella.
Suppose that the sphere has radius a. The double square umbrella can then be
enclosed in a cube of side 2a. Consider a horizontal section of the enclosing cube at
height h above the middle plane of that cube. In the double umbrella this section
is a square of side 2\/a^2 — h^2 and area 4(á^2 - /i^2 ), as shown in Fig. 8. Therefore
the area outside the double umbrella and inside the cube is 4h^2.
It was no small achievement to look at the region in question. It was an
even keener insight on the part of the family Zu to realize that this cross-sectional
area is equal to the area of the cross section of an upside-down pyramid with a
square base of side 2a and height a. Hence the volume of the portion of the cube
outside the double umbrella in the upper half of the cube equals the volume of a
pyramid with square base 2a and height a. But thanks to earlier work contained
in Liu Hui's commentaries on the Jiu Zhang Suanshu, Zu Chongzhi knew that this
volume was (4a^3 )/3. It therefore follows, after doubling to include the portion
below the middle plane, that the region inside the cube but outside the double
umbrella has volume (8a^3 )/3, and hence that the double umbrella itself has volume
8a^3 - (8a^3 )/3 = (16a^3 )/3.
Since, as Liu Hui had shown, the volume of the sphere is ð/4 times the volume
of the double square umbrella, it follows that the sphere has volume (ð/4) • (16á^3 )/3,
or (4ðá^3 )/3.
(^10) Lam and Shen (1985, p. 223), however, say that Liu Hui did consider the idea of setting the
double umbrella inside the cube and trying to find the volume between the two. Of course, that
volume also is not accessible through direct, finite dissection.