Squaring circles in the hyperbolic plane (Q1908682)

In the hyperbolic plane \(H^2\), a ``square'' will be a convex quadrilateral with four equal edges and four equal angles (which must be acute). Combining results known from J. Bolyai and N. M. Nestorovich (1948) as well as from number theory the author first establishes the Theorem (A): ``Suppose a square of corner angle \(\sigma\) and a circle of radius \(r\) in \(H^2\) have the same area \(\omega\), so that \(\omega \leq 2\pi\). Then both are constructible if and only if \(\sigma\) satisfies these conditions: \(0 \leq \sigma < \pi/2\), and \(\sigma\) is an integer multiple of \(2\pi/n\), \(n\) a positive integer such that the regular polygon of \(n\) sides can be constructed with compass and straightedge in the Euclidean plane \(E^2\)''. Besides two new results are proved: There can be no general construction in \(H^2\) that begins with the radius \(r\) of a circle and produces the corner angle \(\sigma\) of the square with matching area (Theorem B), or vice versa (Theorem C).