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Hello, I am looking for a source that discusses and teaches hyperbolic geometry from a synthetic approach (As opposed to the common analytinc approach in the poincare disk). I am looking for something more in spirit with eucld's elements or hilbert's geometry book.

Thank you

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This question should probably be community wiki. – Dan Petersen Oct 8 '12 at 18:37
You are probably right, Fixed. – Blade Oct 9 '12 at 11:25

I'd say your best bet is with works from the early 20th century, when this sort of thing was in fashion:

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I'll look into them all, but of those three, which would you recommend as the 'best'? – Blade Oct 10 '12 at 0:42
Depends for what. Carlslaw is the most illustrated and elementary, then Sommerville (which has exercises, and on p. 27 refers "the reader who wishes to study the development of non-euclidean geometry from a set of axioms" to Coolidge). Coolidge is more dry and more complete, with e.g. several chapters (IX, X, XVI) on line geometry in hyperbolic space: complexes, congruences, Malus-Dupin theorem, etc. – Francois Ziegler Oct 10 '12 at 2:32

Hartshorne's "Geometry: Euclid and Beyond" is very nice.

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Download the article at MARVIN and look at the references.

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Cederberg's A Course in Modern Geometries does some of this.

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