292
Chapter 8
Plate 8.3. In about the same year Jakob Steiner was inventing Steiner chains, Ikeda Sadasuke hung this problem at the Ushijima Cho¯meiji temple in Tokyo. The drawing is from Shiraishi Nagatada’s 1827 book Shamei Sanpu. (University of Aichi Education Library.)
One can well imagine that, if the radii of the circles in the loop are not just right, the circles will not fit properly and end up overlapping, or contain a gap at the end of the chain. Employing the method of inversion, which he largely invented, Steiner found the conditions on the radii of the smaller circles in the loop. However, in 1826, just about the time Steiner was inventing inversion, a sangaku was hung by Ikeda Sadakazu at the Ushijima Cho¯meiji temple in Tokyo. The tablet was lost, but recorded the following year in Shiraishi Nagatada’s book Shamei Sanpu, or Collection of Sangaku, from which plate 8.3 was taken. The problem asks, given two nonconcentric circles with a loop of fourteen smaller circles inscribed between them, show that 1 1 1 1 + = + . r1 r 8 r 4 r11