250 9. MEASUREMENT
is rotated, the ratio of the volumes generated is 2:3, while that of the original
areas is ð : 4. Liu Hui noticed that the sections of the figure parallel to the base
of the cylinder do not all have the same ratios. The sections of the cylinder are
all disks of the same size, while the sections of the sphere shrink as the section
moves from the equator to the poles. He also formed a solid by intersecting two
cylinders circumscribed about the sphere whose axes are at right angles to each
other, thus producing a figure he called a double square umbrella, which is now
known as a bicylinder or Steinmetz solid (see Hogendijk, 2002). A representation
of the double square umbrella, generated using Mathematica graphics, is shown in
Fig. 7. Its volume does have the same ratio to the sphere that the square has to
its inscribed circle, that is, 4 : ð. This proportionality between the double square
umbrella and the sphere is easy to see intuitively, since every horizontal slice of this
figure by a plane parallel to the plane of the axes of the two circumscribed cylinders
intersects the double square umbrella in a square and intersects the sphere in the
circle inscribed in that square. Liu Hui inferred that the volume enclosed by the
double umbrella would have this ratio to the volume enclosed by the sphere. This
inference is correct and is an example of what is called Cavalieri's principle: Two
solids such that the section of one by each horizontal plane bears a fixed ratio to
the section of the other by the same plane have volumes in that same ratio. This
principle had been used by Archimedes five centuries earlier, and in the introduction
to his Method, Archimedes uses this very example, and asserts that the volume of
the intersection of the two cylinders is two-thirds of the volume of the cube in
which they are inscribed.^8 But Liu Hui's use of it (see Lam and Shen, 1985)
was obviously independent of Archimedes. It amounts to a limiting case of the
dissections he did so well. The solid is cut into infinitely thin slices, each of which
is then dissected and reassembled as the corresponding section of a different solid.
This realization was a major step toward an accurate measurement of the volume
of a sphere. Unfortunately, it was not granted to Liu Hui to complete the journey.
He maintained a consistent agnosticism on the problem of computing the volume
of a sphere, saying, "Not daring to guess, I wait for a capable man to solve it."
3.5. Zu Chongzhi. That "capable man" required a few centuries to appear, and
he turned out to be two men. "He" was Zu Chongzhi (429-500) and his son Zu Geng
(450 520). Zu Chongzhi was a very capable geometer and astronomer who said that
if the diameter of a circle is 1, then the circumference lies between 3.1415926 and
3.1415927. From these bounds, probably using the Chinese version of the Euclidean
algorithm, the method of mutual subtraction (see Problem 7.12), he stated that the
circumference of a circle of diameter 7 is (approximately) 22 and that of a circle
of diameter 113 is (approximately) 355.^9 These estimates are very good, far too
good to be the result of any inspired or hopeful guess. Of course, we don't have to
imagine that Zu Chongzhi actually drew the polygons. It suffices to know how to
compute the perimeter, and that is a simple recursive process: If sn is the length
of the side of a polygon of ç sides inscribed in a circle of unit radius, then
s^2 2n = 2 -y/4^n.(^8) Hogendijk (2002) argues that Archimedes also knew the surface area of the bicylinder.
(^9) The approximation ð ~ ø was given earlier by He Chengtian (370-447), and of course much
earlier by Archimedes. A more sophisticated approach by Zhao Youqin (b. 1271) that gives |||
was discussed by Volkov (1997).