Axiomatic Geometry (Pure and Applied Undergraduate Texts)

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Axiomatic Geometry (Pure and Applied Undergraduate Texts)

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  • 製本 Hardcover:ハードカバー版／ページ数 469 p.
  • 言語 ENG
  • 商品コード 9780821884782
  • DDC分類 516

Full Description

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.

Contents

Preface
Euclid
Incidence geometry
Axioms for plane geometry
Angles
Triangles
Models of neutral geometry
Perpendicular and parallel lines
Polygons
Quadrilaterals
The Euclidean parallel postulate
Area
Similarity
Right triangles
Circles
Circumference and circular area
Compass and straightedge constructions
The parallel postulate revisited
Introduction to hyperbolic geometry
Parallel lines in hyperbolic geometry
Epilogue: Where do we go from here?
Hilbert's axioms
Birkhoff's postulates
The SMSG postulates
The postulates used in this book
The language of mathematics
Proofs
Sets and functions
Properties of the real numbers
Rigid motions: Another approach
References
Index