This expansive, copiously illustrated textbook provides rigorous and detailed treatments of Euclidean and hyperbolic plane geometries. Harvey (Univ of Virginia's College at Wise) begins with a thorough development of neutral geometry (i.e., geometry without the assumption of the parallel postulate or its negation) and then discusses Euclidean and hyperbolic geometry. Fifteen of the book's 47 chapters are devoted to Euclidean ... Read moretransformations and applications and 12 to hyperbolic topics, including the pseudosphere, upper-half plane, Poincare disk models, hyperbolic trigonometry, hyperbolic area, and tilings. Throughout, Harvey maintains a careful yet friendly tone and assumes mostly minimal prerequisites: facility with mathematical proof, some multivariable calculus, linear algebra, differential equations (needed only for the calculation of geodesics), and familiarity with complex numbers. The author succeeds in elevating Euclidean geometry in particular to the level of advanced undergraduate study and in so doing presents it as a counterpart to a first analysis course. At the same time, some readers may prefer to move on to new ideas and results more quickly; to that end, Harvey gives an indication of several paths through the book in his preface. - Choice To my mind, this book stands out from the crowd partly because of the way in which its imaginatively devised illustrations are used to stage the introduction of basic concepts and to guide the reader through various steps in a proof. It's not just a matter of pretty pictures, however, because Matthew Harvey's written commentary is easy going and yet mathematically precise. He has written a truly lovely book, which is now top of my reading list as an introduction to geometry at this level. - Peter Ruane Show Less