Maureen T. Carroll, Elyn Rykken
American Mathematical Society
Geometry: The Line and the Circle is a textbook for undergraduate students that has a compelling narrative and is written at the right level of rigor for an advanced-level survey or axiomatic course in geometry. The PDF version of this book can be found here. Beginning with Euclid's Elements, this ebook integrates various concepts in non-Euclidean geometry and Euclidean geometry in a way that is purposeful and meaningful, while also providing historical background. The line and the circle are the main characters in this story, and they are responsible for its progression. In every geometry that is taken into consideration, which includes hyperbolic, spherical, and taxicab geometries, as well as finite affine and projective geometries, these two items are brought to the forefront and examined. Along the process, the reader mulls over fundamental questions such as, “What exactly does it imply when something is parallel?” What is meant by the term “straight line”? What does the term “area” mean? What exactly is the meaning of distance? Axiomatic structures receive a great deal of attention over the entirety of the textbook. Euclid is a consistent source of motivation, and the Elements are referred to numerous times, with substantial coverage of Books I, II, III, IV, and VI. However, non-Euclidean geometries are presented at an early stage in order to provide the reader with a different point of view on issues pertaining to axiomatics. Axiomatics is covered in great detail throughout the final chapters, which focus on constructibility and transformations respectively. The book is written in a way that makes it impossible to put down, with significant focus placed on the historical story and a large number of intriguing puzzles.