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What is a good reference for the following result which I believe is proved by Tchebotarev.

There are exactly 5 types of Lunes that are squarable. (Hippocrates produced three and then two more were given by Euler).

edited Mar 27 2012 at 12:25

Joseph O'Rourke
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asked Mar 26 2012 at 23:22

Chebolu
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@Chebolu: I added what I think are more appropriate tags (but left your original ag.algebraic-geometry tag). – Joseph O'Rourke Mar 27 2012 at 12:26

Joseph O'Rourke · Answer 1 · 2012-03-27 00:04:13Z UTC

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"The Problem of Squarable Lunes." M. M. Postnikov, translated from the Russian by Abe Shenitzer. The American Mathematical Monthly. Vol. 107, No. 7 (Aug.-Sep., 2000), pp. 645-651. JSTOR link.

See also the MathPages web page entitled "The Five Squarable Lunes," and the Wikipedia page "Lune of Hippocrates," from which this image of the "lunes of Alhazen" is taken:
Lunes
"The two blue lunes together have the same area as the green right triangle."

edited Mar 27 2012 at 0:04

answered Mar 26 2012 at 23:57

Joseph O'Rourke
43.1k●1●49●214

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Note, however, that the Postnikov proof involves (p. 647) a statement whose proof "is beyond the level of this book." – John Stillwell Mar 27 2012 at 0:15

@John: Interesting; I didn't notice that. The statement is that a certain single-variable polynomial of potentially high degree cannot, except in one special case, be reduced to the solution of quadratic equations. – Joseph O'Rourke Mar 27 2012 at 13:48

Quadrature of the Lune

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