Maureen T. Carroll, Elyn Rykken
American Mathematical Society
Geometry: The Line and the Circle (PDF) is an undergraduate textbook 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 ebook connects topics in non-Euclidean and 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 hyperbolic; spherical; and taxicab; as well as finite affine and projective geometries these 2 objects are highlighted and analyzed. Along the way; the reader contemplates fundamental questions such as: What does parallel mean? What is a straight line? What is area? What is distance? There is a strong focus on axiomatic structures throughout the textbook. 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 ebook is compulsively readable with great attention paid to the historical narrative and hundreds of attractive problems.