Title: Geometry: The Line and the Circle (AMS/MAA Textbooks)

Author: Maureen T. Carroll

Publisher: American Mathematical Society

Language: English

Publication Date: December 20, 2018

ISBN-10: 1470448432

ISBN-13: 978-1470448431

Geometry: The Line and the Circle is a comprehensive undergraduate textbook designed for upper-level survey or axiomatic geometry courses. Balancing rigorous mathematical theory with a compelling historical narrative, the text explores the fundamental roles of the line and the circle across various geometric landscapes. Beginning with the foundational works of Euclid’s Elements, the book traces the evolution of geometric thought while bridging the gap between classical Euclidean foundations and modern non-Euclidean systems.

The curriculum introduces diverse geometric models early on—including spherical, hyperbolic, and taxicab geometries, as well as finite affine and projective geometries—to provide readers with a broader perspective on axiomatic structures. Throughout the journey, the author encourages deep reflection on essential questions: What defines a straight line? What is the nature of parallelism? How are distance and area measured? By highlighting how the line and the circle behave within different frameworks, the text provides a unified approach to understanding the core of geometric study.

While Euclid remains a constant inspiration—with substantial coverage of Books I, II, III, IV, and VI—the book rounds out its thorough exploration with concluding chapters on transformations and constructibility. Highly readable and meticulously researched, the text includes hundreds of attractive problems and illustrations that bring the historical and mathematical narrative to life, making it an essential resource for students and educators alike.