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The celebrated Bolyai-Gerwin theorem states that

"In the Euclidean plane, two simple polygons are scissors-equivalent if and only if they have the same area."

By scissor-equivalent, we meant that one polygon can be dissected by straight lines and the remaining pieces can be placed in a way (no overlapping allowed) that they form the another one.

I am aware that the same holds for the spherical and hyperbolic 2-dimensional geometries. I would like to ask the community for a reference on the proof of Bolyai-Gerwin theorem for the spherical and for the hyperbolic cases. Any reference will be really appreciated.

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In Hartshorne's book, apparently Gerwien did the spherical case himself. I should point out that Hartshorne is doing the axiomatic approach as much as possible. Earlier contributors, certainly before Hilbert got involved, were thinking in terms compatible with the real numbers.

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    Do you know, by chance, if is Gerwien's paper in english? Or if someone exposed Gerwien's proof in a recent work, seminar etc? Thank you very much. – 2017-02-11
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    Maybe I should say that I don't have access to the papers from were I am. Thanks again. – 2017-02-11
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    @matgaio Gerwien (1833) is in German, only about 7 pages. So is Finzel (1912), about 23 pages. I don't have either article, or know of relevant seminars. – 2017-02-11