Axiomatic Geometry (Pure and Applied Undergraduate Texts) (Sally: Pure and Applied Undergraduate Texts, 21)

The story of geometry is the story of mathematics itself: Euclidean geometry was the first branch of mathematics to be systematically studied and placed on a firm logical foundation, and it is the prototype for the axiomatic method that lies at the foundation of modern mathematics. It has been taught to students for more than two millennia as a model of logical thought. This book tells the story of how the axiomatic method has progressed from Euclid's time to ours, as a way of understanding what mathematics is, how we read and evaluate mathematical arguments, and why mathematics has achieved the level of certainty it has.It is designed primarily for advanced undergraduates who plan to teach secondary school geometry, but it should also provide something of interest to anyone who wishes to understand geometry and the axiomatic method better. It introduces a modern, rigorous, axiomatic treatment of Euclidean and (to a lesser extent) non-Euclidean geometries, offering students ample opportunities to practice reading and writing proofs while at the same time developing most of the concrete geometric relationships that secondary teachers will need to know in the classroom. The Story Of Geometry Is The Story Of Mathematics Itself: Euclidean Geometry Was The First Branch Of Mathematics To Be Systematically Studied And Placed On A Firm Logical Foundation, And It Is The Prototype For The Axiomatic Method That Lies At The Foundation Of Modern Mathematics. It Has Been Taught To Students For More Than Two Millennia As A Mode Of Logical Thought. This Book Tells The Story Of How The Axiomatic Method Has Progressed From Euclid's Time To Ours, As A Way Of Understanding What Mathematics Is, How We Read And Evaluate Mathematical Arguments, And Why Mathematics Has Achieved The Level Of Certainty It Has. It Is Designed Primarily For Advanced Undergraduates Who Plan To Teach Secondary School Geometry, But It Should Also Provide Something Of Interest To Anyone Who Wishes To Understand Geometry And The Axiomatic Method Better. It Introduces A Modern, Rigorous, Axiomatic Treatment Of Euclidean And (to A Lesser Extent) Non-euclidean Geometries, Offering Students Ample Opportunities To Practice Reading And Writing Proofs While At The Same Time Developing Most Of The Concrete Geometric Relationships That Secondary Teachers Will Need To Know In The Classroom. -- P. [4] Of Cover. 1. Euclid -- 2. Incidence Geometry -- 3. Axioms For Plane Geometry -- 4. Angles -- 5. Triangles -- 6. Models Of Neutral Geometry -- 7. Perpendicular And Parallel Lines -- 8. Polygons -- 9. Quadrilaterals -- 10. The Euclidean Parallel Postulate -- 11. Area -- 12. Similarity -- 13. Right Triangles -- 14. Circles -- 15. Circumference And Circular Area -- 16. Compass And Straightedge Constructions -- 17. The Parallel Postulate Revisited -- 18. Introduction To Hyperbolic Geometry -- 19. Parallel Lines In Hyperbolic Geometry -- 20. Epilogue: Where Do We Go From Here? -- Appendices. John M. Lee. Includes Bibliographical References (pages 451-453) And Index. The story of geometry is the story of mathematics Euclidean geometry was the first branch of mathematics to be systematically studied and placed on a firm logical foundation, and it is the prototype for the axiomatic method that lies at the foundation of modern mathematics. It has been taught to students for more than two millennia as a model of logical thought. This book tells the story of how the axiomatic method has progressed from Euclid's time to ours, as a way of understanding what mathematics is, how we read and evaluate mathematical arguments, and why mathematics has achieved the level of certainty it has. It is designed primarily for advanced undergraduates who plan to teach secondary school geometry, but it should also provide something of interest to anyone who wishes to understand geometry and the axiomatic method better. It introduces a modern, rigorous, axiomatic treatment of Euclidean and (to a lesser extent) non-Euclidean geometries, offering students ample opportunities to practice reading and writing proofs while at the same time developing most of the concrete geometric relationships that secondary teachers will need to know in the classroom. Euclid......Page 20 Incidence Geometry......Page 42 Axioms for Plane Geometry......Page 72 Angles......Page 102 Triangles......Page 122 Models of Neutral Geometry......Page 142 Perpendicular and Parallel Lines......Page 160 Chapter 8+......Page 174