The subject known as non-Euclidean geometry is studied in some undergraduate geometry courses, and often in a course on differential geometry. From these exposures, one might get the impression that the subject ends with the models of Poincaré that seal the achievements of Gauss, Bolyai, and Lobachevsky that show the independence of Euclid’s Parallel Postulate. As this book demonstrates, this impression is far from the truth—the geometries of the sphere and the hyperbolic plane offer a rich, and currently active area of research. The editors have collected a satisfying collection of surveys that make the point well. In order to capture all the variety packed into the collection, I will present a snapshot of each chapter.

The book is in three parts: the first focuses on area in spherical and hyperbolic geometries. The source of inspiration for these chapters is the work of Leonhard Euler (1707–1783) on the geometry of a sphere. There are some other actors, however, and we meet Anders Lexell (1740–1784), a friend and brief successor to Euler in St. Petersburg. Lexell posed the following problem for the sphere: given two distinct points A and B, determine the locus of points P for which the area of the triangle ABP is constant. Of course, the analogous problem may be posed for the hyperbolic plane, and Euler’s solution to Lexell’s problem leads to an area relation that can be proved in hyperbolic geometry: see Chapter 2. Another successor of Euler was F.T. von Schubert (1758–1825) who proved that a spherical triangle on a given base and fixed altitude has minimal area when its altitude meets the base at the midpoint, and maximal area is given at the point that gives the minimal area for the antipodal base. The analogous problem can be posed in hyperbolic geometry using hypercycles (equidistant curves from a line) where it can be shown that the isosceles triangle achieves the unique maximum: see Chapter 3.

Reaching further into the past for inspiration, the next few essays develop an analogue of Ceva’s theorem for spherical and hyperbolic geometries, and the notion of constructibility on the sphere: see Chapters 4, 5, 6. Post-Euler, we meet J.-M. de Tilly (1837–1906) whose Géométrie abstraite is based on kinematic arguments—points moving along a geodesic. Chapter 7 develops these methods for the non-Euclidean cases. Kinematics and statics return in Chapter 12 in which the geometry of bar-and-joint frameworks is developed. Chapters 8–10 review the Gauss-Bonnet theorem and the geometry of the sphere as cartography in which Euler’s theorem about perfect maps is a highlight. Chapter 11 bridges many centuries to present the non-Euclidean analogues of Heron’s formula, Ptolemy’s theorem, Brahmagupta’s formula, and more, before turning to three dimensions to obtain volume formulas for orthoschemes, polyhedra with adjacent edges orthogonal. Some of the results in this chapter are very recent.

Part 2 concerns Projective Geometries and opens with a translation of the American Journal of Mathematics paper *Beitraäge zur Nicht-Euklidschen Geometrie I* by Eduard Study (1862–1930). Study considered the geometry of lines outside the model of the hyperbolic plane in the projective plane. The projective plane with metrics determined by a choice of absolute gives an approach to unifying geometries that was championed by F. Klein (1849–1925) in his famous Erlanger Programm. Study’s essay presents a version of the reverse triangle inequality that is well known in relativity theory. Chapter 15 on conics develops these ideas much further, organized by a choice of two quadratic forms on a real vector space of dimension three. Spherical and hyperbolic conics are then the intersection of a quadratic cone in the projective plane with a unit sphere or a one-sheeted hyperboloid. Their expected properties of dualities, focal lines, and polarity can be derived from the projective setting, but their particularities are novel and attractive. Chapter 16 takes the reader into higher dimensions where there are spherical, Euclidean, and hyperbolic spaces, but also Minkowski, de Sitter and anti-de Sitter spaces. The organizing notion is a non-degnerate bilinear form together with the associated isometries. In this context, it is possible to talk about transitions between geometries where two geometries might share a third as a limit.

Part 3 explores more exotic places: in Chapter 17 we learn about Hermitian geometry on complex projective spaces where the metric is based on the usual Hermitian inner product, made projective in the manner of Study. The final chapter returns to the Lexell problem in the context of the axiomatic geometry of metric planes. The essays are preceded by an excerpt from Poincaré’s paper *Sur les hpothèses **fondamentales de la géométrie* of 1887. Poincaré organizes the classical two-dimensional geometries according to the behavior of the angle sum of a triangle. In the excerpt, he changes direction and asks the reader to consider order-two surfaces, that is, degree two algebraic surfaces, which he dubs quadratic geometries. He observes that the ellipsoid gives us spherical geometry, the two-sheeted hyperboloid the geometry of Lobachevsky, and the elliptic paraboloid gives Euclidean geometry. But what of a one-sheeted hyperboloid? Poincaré observes that some distinct pairs of points are distance zero apart, that there are two kinds of lines on the surface for which there are no congruences between the species of line, and that a real rotation through 180 ◦ to make a line congruent to itself does not exist. The spirit of Poincaré to face up to ideas “contrary to the habits of our spirit” makes these essays a wonderful addition to the literature on non-Euclidean geometry. Though the level of technical demand among the essays varies widely, students of geometry will be introduced to a rich landscape of ideas and methods worthy of further development.

John McCleary is Professor of Mathematics at Vassar College.