A Sushi Morsel
Twice in his books Ross Honsberger discussed the following problem:
The first solution in [Gems] is rather a brute force use of coordinate geometry. The second solution [Morsels], which Honsberger attributes to C. Stanley Ogilvy, is based on the observation that, if L is seen as the circle of infinite radius, the problem is reduced to the study of the chain of circles in arbelos or as a particular case of Steiner's chains. The problem is then elegantly reduced to the sequence of circles in a strip of paper by inversion in the circle centered at the point of tangency of C_{0} with L and of radius 2R_{0}.
The earlier discussion comes with no reference; in the later one, Honsberger points to the book Mathematical Games and Pastimes by A. P. Domoryad (Pergamon Press, 1964, problem 19, p. 242).
In a recent article published in The Mathematical Gazette, M. Hajja gives an elementary solution to the problem that makes use of a known result concerning three mutually tangent circles standing on a straight line:
Assuming the radii of the three circles in the diagram are successively R_{0}, R_{1}, R_{2} we have
(1)  1/√R_{2} = 1/√R_{1}  1/√R_{0}. 
By induction, then
(2)  1/√R_{n} = 1/√R_{1}  (n  1)/√R_{0}. 
Denoting 1/√R_{n} = u_{n}, we convert (2) to a simple linear relation
(3)  u_{n} = u_{1}  (n1)u_{0}, n > 1. 
Since, for all n, u_{n} > 0, (3) shows that the process may continue as long as
(4)  n < 1 + u_{1} / u_{0}, i.e, n ≤ u_{1} / u_{0} 
For R_{0} = 10^{6}mm and R_{1} = 1 mm, u_{1} / u_{0} = 10^{3}. Which means that it is impossible to construct circle C_{1001} and C_{1000} is the last one, thus solving the problem. (The solution from [Morsels] shows also that R_{1000} = R_{0}, giving another reason why the process can't continue beyond
The point of this note is to observe that the problem and the simple solution above both have a long history. Relation (1) is the subject of an 18th century sangaku. Relation (2) appeared in its generality in a 1789 sangaku and in several modifications during the 19th century. This may be one of those cases where Wasan  the traditional Japanese geometry  beat the West to a punch.
References
 M. Hajja, On a Morsel of Ross Honsberger, The Math. Gazette, v. 93, n. 527, July 2009, pp. 309312
 R. Honsberger, Mathematical Gems, MAA, 1973, pp. 5253, 153155
 R. Honsberger, Mathematical Morsels, MAA, 1978, pp. 218219
Sangaku

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