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Foreword to the second edition
Chapter I Historical sketch of the foundations of geometry
Modern axiomatic construction of Euclidean
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absolute geometry affine geometry affine plane analogously anharmonic ratio assertion axiom III7 axioms of Euclidean axis of perspectivity belongs called cartesian realization center of perspectivity circle coincides complete quadrangle completes the proof congruent consequently consists construct continuity axiom contradiction corresponding curve defined denote Desargues disk distinct half-planes endpoints equal equation Euclidean geometry exists a motion fact fifth postulate fixed follows from axiom formulas function given harmonic conjugate Hilbert homogeneous coordinates horocycle horosphere incidence axioms infinitely distant point invariant isomorphism Klein interpretation Lemma lies line g linear Lobachevskian geometry Lobachevskian plane lying motion axioms motion H obtain Obviously order axioms pair of points parallel axiom pencil perpendicular points of intersection projective coordinates projective geometry projective plane projective transformation properties prove ray h right angles Saccheri satisfied second class segment AC sides similarly directed succession of triples Suppose system of axioms Theorem three points triangle ABC vectors zero