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 C₀ with L and of radius 2R₀.

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 R₀, R₁, R₂ we have

By induction, then

Denoting 1/√R_n = u_n, we convert (2) to a simple linear relation

Since, for all n, u_n > 0, (3) shows that the process may continue as long as u₁ - (n-1)u₀ > 0, o, in other words, while

For R₀ = 10⁶mm and R₁ = 1 mm, u₁ / u₀ = 10³. Which means that it is impossible to construct circle C₁₀₀₁ and C₁₀₀₀ is the last one, thus solving the problem. (The solution from [Morsels] shows also that R₁₀₀₀ = R₀, giving another reason why the process can't continue beyond n = 1000.)

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

1. M. Hajja, On a Morsel of Ross Honsberger, The Math. Gazette, v. 93, n. 527, July 2009, pp. 309-312
2. R. Honsberger, Mathematical Gems, MAA, 1973, pp. 52-53, 153-155
3. R. Honsberger, Mathematical Morsels, MAA, 1978, pp. 218-219

Sangaku

1. Sangaku: Reflections on the Phenomenon
2. Critique of My View and a Response
3. 1 + 27 = 12 + 16 Sangaku
4. 3-4-5 Triangle by a Kid
5. 7 = 2 + 5 Sangaku
6. A 49^th Degree Challenge
7. A Geometric Mean Sangaku
8. A Hard but Important Sangaku
9. A Restored Sangaku Problem
   • A better solution to a difficult sangaku problem
   • A Trigonometric Solution to a Difficult Sangaku Problem
10. A Sangaku: Two Unrelated Circles
11. A Sangaku by a Teen
12. A Sangaku Follow-Up on an Archimedes' Lemma
13. A Sangaku with an Egyptian Attachment
14. A Sangaku with Many Circles and Some
15. A Sushi Morsel
16. An Old Japanese Theorem
17. Archimedes Twins in the Edo Period
18. Arithmetic Mean Sangaku
19. Bottema Shatters Japan's Seclusion
20. Chain of Circles on a Chord
21. Circles and Semicircles in Rectangle
22. Circles in a Circular Segment
23. Circles Lined on the Legs of a Right Triangle
24. Equal Incircles Theorem
    • Equal Incircle Theorem, Angela Drei's Proof
25. Equilateral Triangle, Straight Line and Tangent Circles
26. Equilateral Triangles and Incircles in a Square
27. Five Incircles in a Square
28. Four Hinged Squares
29. Four Incircles in Equilateral Triangle
30. Gion Shrine Problem
31. Harmonic Mean Sangaku
32. Heron's Problem
33. In the Wasan Spirit
34. Incenters in Cyclic Quadrilateral
35. Japanese Art and Mathematics
36. Malfatti's Problem
37. Maximal Properties of the Pythagorean Relation
38. Neuberg Sangaku
39. Out of Pentagon Sangaku
40. Peacock Tail Sangaku
41. Pentagon Proportions Sangaku
42. Proportions in Square
43. Pythagoras and Vecten Break Japan's Isolation
44. Radius of a Circle by Paper Folding
45. Review of Sacred Mathematics
46. Sangaku à la V. Thebault
47. Sangaku and The Egyptian Triangle
48. Sangaku in a Square
49. Sangaku Iterations, Is it Wasan?
50. Sangaku with 8 Circles
51. Sangaku with Angle between a Tangent and a Chord
52. Sangaku with Quadratic Optimization
53. Sangaku with Three Mixtilinear Circles
54. Sangaku with Versines
55. Sangakus with a Mixtilinear Circle
56. Sequences of Touching Circles
57. Square and Circle in a Gothic Cupola
58. Steiner's Sangaku
59. Tangent Circles and an Isosceles Triangle
60. The Squinting Eyes Theorem
61. Three Incircles In a Right Triangle
62. Three Squares and Two Ellipses
63. Three Tangent Circles Sangaku
64. Triangles, Squares and Areas from Temple Geometry
65. Two Arbelos, Two Chains
66. Two Circles in an Angle
67. Two Sangaku with Equal Incircles

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