By Wilhelm Magnus

ISBN-10: 0080873774

ISBN-13: 9780080873770

ISBN-10: 0124654509

ISBN-13: 9780124654501

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67a) reduces to the remark that in a right triangle, the product of the legs equals the product of the altitude and the base, both products being equal to the area of the rectangle whose sides are the legs. 67b), we use a transformation g of r*which maps A onto A’, the ray A B onto the ray A’B’, and which maps the side of the ray A B onto which (I lies onto that side of the ray A’B‘ on which C‘ lies. 17), and, since g preserves distances and angles, B is mapped onto B’ and C is mapped onto C’. Therefore, corresponding sides and angles in the triangles ABC and A‘B‘C‘ are equal in pairs.

Note that the axis is mapped onto itself by the substitution; in the case of a hyperbolic substitution, part of it represents a noneuclidean straight line, which shows that a hyperbolic substitution behaves along one straight line like a euclidean translation. 7 Appendix: Hilbert’s Axioms of Geometry 43 advantage of the model K z over the model Pz. In the publications of Felix Klein and his school, this language is used widely, although by no means exclusively. The advantages of the model Pz are twofold: Angles, which play a dominant role in noneuclidean geometry, are recognizable in the model, and, above all, the applications of noneuclidean geometry to the theory of R i e m a p surfaces make the presentation of noneuclidean motions as mappings of C onto itself the only feasible one.

22 Noneuclidean circles are the orthogonal trajectories of the noneuclidean rays through the center 2 0 . They are also euclidean circles. However, the euclidean and noneuclidean centers may be different points. Conversely, every euclidean circle within the upper halfplane (or, if we use the unit disk as a model, every euclidean circle within the unit disk) is also a noneuclidean circle. Proof: Map Zoonto the center of the unit disk and let Pobe a point on the positive real axis with the prescribed noneuclidean distance p from 2,.

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