The Foundations of Geometry, by David Hilbert, is part of the HackerNoon Books Series. Read this book online for free on HackerNoon! - by hackernoon nonfiction mathematicalbooks
Group I: Axioms of connectionGroup III: Axiom of Parallels Group V: Axiom of Continuity Independence of the axioms of parallels. Non-euclidean geometry Independence of the axiom of continuity. Non-archimedean geometryAn algebra of segments, based upon Pascal’s theoremCHAPTER IV - THE THEORY OF PLANE AREAS.Parallelograms and triangles having equal bases and equal altitudesCHAPTER V - DESARGUES’S THEOREM.
Introduction of an algebra of segments based upon Desargues’s theorem and independent of the axioms of congruenceThe associative law of multiplication and the two distributive for the new algebra of segmentsThe totality of segments, regarded as a complex number systemSignificance of Desargues’s theoremThe commutative law of multiplication for an archimedean number systemProof of the two propositions concerning Pascal’s theorem. Non-pascalian geometry.
The demonstration, by means of the theorems of Pascal and Desargues, of any theorem relating to points of intersectionGeometrical constructions by means of a straight-edge and aAnalytical representation of the co-ordinates of points which can be so constructedCriterion for the possibility of a geometrical construction by means of a straight-edge and a transferer of segments
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