SQUARING CIRCLES IN THE HYPERBOLIC PLANE

The syndicated newspaper column of Marilyn vos Savant was particularly interesting one Sunday in November 1993 [Sa]. Ms. vos Savant announce d there that she had no faith whatsoever in the work of Andrew Wiles o n Fermat's Last Theorem. In stating her objections to the methodology of Wiles, she wrote that Janos Bolyai ''managed to 'square the circle' -but only by using his own hyperbolic geometry.'' The word ''using'' c reates the misleading impression that Bolyai used illicit methods to s quare the circle in the Euclidean plane. What Bolyai did, in fact, was to construct, using the correct intrinsic versions of the compass and straightedge, a square and a circle in the hyperbolic plane with the same area. In this article, I will exhibit all possible such examples (Theorem A). I will also show that the square and circle must be const ructed simultaneously: there cannot be a construction that begins with a circle of radius tau and produces the correct corner angle sigma fo r the square of equal area (Example B); neither can there be a constru ction beginning with a that produces the correct tau (Example C). Theo rem A, discovered independently by the present author, is contained in a 1948 article of Nestorovich [Ne1] that has received little attentio n in English-language publications. That article also has an example s imilar to those in Example B, but Example C is not considered there. I t may be, therefore, that Example C and the interpretation provided by Theorems B and C are new.