Pappus' Sangaku

Few sangaku illustrate better the disparity of the independent development of mathematics during the Edo period (1603-1867) of self-imposed seclusion of Japan from the Western world than those related to arbelos and Pappus' chain of circles. The tool of inversion so useful in solving problems involving circles remained unkown to the Japanese until late 19 century.

Consider the sangaku 1.8.2 from the collection by Fukagawa and Pedoe:

Circles C₁(r) and C₂(r) touch at O in the circle O(2r) and also touch O(2r), C₁(r) touching at T. O₁(r₁) touches O(2r) internally and both C₁(r) and C₂(r) externally, O₂(r₂) touches O₁(r₁) and C₁(r) externally and O(2r) internally, and so on. Find r_n.

Solution

References

H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989

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Charles Babbage Research Center
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With the help of inversion the solution is immediate. In the generalization of Archimedes' formula

r_n = rk / (k²n² + k + 1).

where k is the ratio between the radii of the two inner circles, i.e. k = 1 in this case. Thus the answer to the problem is

r_n = r / (n² + 2),

exactly the expression that appears in the Gothic Arc diagram.

In the very first volume (1826) of Crelle's Journal, J. Steiner proved several theorem concerning arbelos, the above being one of them. The sangaku however is dated 1801 (and was written in the Mie prefecture.)

Sangaku

40617696