206
Chapter 6
r
rn
a r4
r1 r2
r3
b
Figure 6.17. Find ri in terms of a, b, r, and r 1.
As shown in figure 6.17, two spheres of radii a and b lie inside a large sphere of radius r, touching each other and sphere r internally. A loop of n spheres of radii r 1, . . . , rn circles the neck between a and b. Each of the spheres ri touches its nearest neighbors, as well as spheres a, b, and r. Find ri in terms of a, b, r, and r 1. We give a traditional solution on page 236 and discuss the problem further in chapter 8.
Problem 18 This problem was written on a tablet hung in 1804 at the Daikokutendo¯ temple, Koishikawa, Tokyo. It was proposed by a student of Ichino Shigetaka and recorded in the book Shamei Sanpu. ∧ ∧ As shown in figure 6.18, on a sphere of radius r, draw three circular arcs AB, BC, and ∧ CA, and let the straight-line segments AB = c, BC = a, and CA = b. Find the area S of the spherical triangle ABC in terms of a, b, c, and r.
A c
b r
B
Figure 6.18. Find the area of the spherical triangle ABC in terms of a, b, c, and r.
a
C