This is the definitive presentation of the history, development and philosophical significance of non-Euclidean geometry as well as of the rigorous foundations for it and for elementary Euclidean geometry, essentially according to Hilbert. Appropriate for liberal arts students, prospective high school teachers, math. majors, and even bright high school students. The first eight chapters are mostly accessible to any educated reader; the last two chapters and the two appendices contain more advanced material, such as the classification of motions, hyperbolic trigonometry, hyperbolic constructions, classification of Hilbert planes and an introduction to Riemannian geometry.
Chapter 1 Euclid’s Geometry
Very Brief Survey of the Beginnings of Geometry
The Pythagoreans
Plato
Euclid of Alexandria
The Axiomatic Method
Undefined Terms
Euclid’s First Four Postulates
The Parallel Postulate
Attempts to Prove the Parallel Postulate
The Danger in Diagrams
The Power of Diagrams
Straightedge-and-Compass Constructions, Briefly
Descartes’ Analytic Geometry and Broader Idea of Constructions
Briefly on the Number ð
Conclusion
Chapter 2 Logic and Incidence Geometry
Elementary Logic
Theorems and Proofs
RAA Proofs
Negation
Quantifiers
Implication
Law of Excluded Middle and Proof by Cases
Brief Historical Remarks
Incidence Geometry
Models
Consistency
Isomorphism of Models
Projective and Affine Planes
Brief History of Real Projective Geometry
Conclusion
Chapter 3 Hilbert’s Axioms
Flaws in Euclid
Axioms of Betweenness
Axioms of Congruence
Axioms of Continuity
Hilbert’s Euclidean Axiom of Parallelism
Conclusion
Chapter 4 Neutral Geometry
Geometry without a Parallel Axiom
Alternate Interior Angle Theorem
Exterior Angle TheorelĂS
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