New Books In Mathematics

From an undergraduate perspective, coming from the rigid proofs and concrete constructions of middle- or high-school courses, the broad discipline of geometry can be at once intimately familiar and menacingly exotic. For most of its history, and perhaps for many of the same reasons, geometers struggled to come to terms with the unsolved problems, unstated assumptions, and untapped generalizability contained in the "bible of mathematics", Euclid's Elements. In their recent text, Geometry: The Line and the Circle (MAA Press, 2018), Maureen T. Carroll and Elyn Rykken have produced a unified survey of Euclidean and many significant non-Euclidean geometries, one that draws from the patterns of historical development to immerse students into progressively new territory. Their book is organized around the Elements but soon (and often) detours into spherical, finite, and other geometries that bring the limitations of the classic text—and the contributions of subsequent geometers—to the fore. Throughout, they examine