Geometry: The Line and the Circle

Maureen T. Carroll, Elyn Rykken

• Publisher: American Mathematical Society
• Year: 2018
• Print ISBN:
  9781470448431
• eBook ISBN:
  9781470450632
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

Geometry: The Line and the Circle is an undergraduate text with a strong narrative that is written at the appropriate level of rigor for an upper-level survey or axiomatic course in geometry. Starting with Euclid's Elements, the book connects topics in Euclidean and non-Euclidean geometry in an intentional and meaningful way, with historical context.The line and the circle are the principal characters driving the narrative. In every geometry considered—which include spherical, hyperbolic, and taxicab, as well as finite affine and projective geometries—these two objects are analyzed and highlighted. Along the way, the reader contemplates fundamental questions such as: What is a straight line? What does parallel mean? What is distance? What is area?There is a strong focus on axiomatic structures throughout the text. While Euclid is a constant inspiration and the Elements is repeatedly revisited with substantial coverage of Books I, II, III, IV, and VI, non-Euclidean geometries are introduced very early to give the reader perspective on questions of axiomatics. Rounding out the thorough coverage of axiomatics are concluding chapters on transformations and constructibility. The book is compulsively readable with great attention paid to the historical narrative and hundreds of attractive problems.The authors have provided a supplemental book of laboratory projects offering guided explorations to accompany topics found in the book. These projects use GeoGebra, a free interactive application. Download the GeoGebra Labs and a zip file of GeoGebra Lab starter files.

Table of contents

• Cover
• Title page
• Copyright
• Contents
• Note to the Instructor
• Note to the Reader
• Acknowledgments
• Chapter 1. The Line and the Circle
• Chapter 2. Euclid’s Elements: Definitions and Axioms
• Chapter 3. Book I of Euclid’s Elements: Neutral Geometry
• Chapter 4. Spherical Geometry
• Chapter 5. Taxicab Geometry
• Chapter 6. Hilbert and Gödel
• Chapter 7. Book I: Non-Neutral Geometry
• Chapter 8. Book II: Geometric Algebra
• Chapter 9. Book VI: Similarity
• Chapter 10. Book III: Circles
• Chapter 11. Book IV: Circles & Polygons
• Chapter 12. Models for the Hyperbolic Plane
• Chapter 13. Axiomatic Hyperbolic Geometry
• Chapter 14. Finite Geometries
• Chapter 15. Isometries
• Chapter 16. Constructibility
• Appendix A. Euclid’s Definitions and Axioms
• Appendix B. Euclid’s Propositions
• Appendix C. Visual Guide to Euclid’s Propositions
• Appendix D. Euclid’s Proofs
• Appendix E. Hilbert’s Axioms for Plane Euclidean Geometry
• Credits, Permissions and Acknowledgements
• Bibliography
• Notation Index
• Index
• Back Cover