János Bolyai’s treatment of non-Euclidean geometry burst upon the mathematical scene in 1832 as an appendix (in Latin), entitled *The Science Absolute of Space*, to an elementary mathematical work of his father Farkas. Its impact, like that of the contemporaneous treatment of the subject by Nikolai Ivanovich Lobachevsky, was essentially nil. Yet today, we recognize Bolyai as being the co-creator of the subject of non-Euclidean geometry, a subject whose philosophical impact in the last decades of the nineteenth century was enormous and whose mathematical impact since the time of Einstein’s discovery of general relativity has also been significant.

George B. Halsted’s English translation of Bolyai’s work has been available since 1895 and is included, along with his English translation of Lobachevsky’s *The Theory of Parallels*, in the English translation of Roberto Bonola’s classic work, *Non-Euclidean Geometry*, which first appeared in 1912. Yet Jeremy Gray’s fresh look at Bolyai’s work, which includes not only Halsted’s translation but also a facsimile copy of the original Latin, will be welcomed by today’s readers for its succinct summary of the history of the parallel postulate up to the time of Bolyai as well as thereafter, its insights into Bolyai’s thought processes, and its detailed explanation of Bolyai’s mathematical ideas.

The book is illustrated profusely with diagrams that make Bolyai’s arguments clear, as well as brief essays on some subtle points. For example, Gray explains how Bolyai constructed a surface in a non-Euclidean 3-space on which the parallel postulate is true, thus giving him a method of relating problems in non-Euclidean geometry to problems in Euclidean geometry. He discusses Bolyai’s trigonometrical formulas which enabled him to derive a number of basic results dealing with the angle of parallelism. He presents Bolyai’s construction which squares the circle in non-Euclidean geometry. And he discusses the remarkably similar presentation of non-Euclidean geometry in the work of Lobachevsky.

As Gray explains, it was only after the death of Gauss, and the realization that he too had thought that a new geometry was possible, that mathematicians began seriously to consider the ideas of Bolyai and Lobachevsky. Riemann took up Gauss’s idea of curvature of a surface and extended it to *n*-dimensional space in his *Habilitationsschrift* of 1854. He then noted that there were three possibilities for a two-dimensional surface of constant curvature, ones where the curvature was zero (a plane or a cylinder), positive (a sphere), and negative (although here he could give no example). But it was only when Riemann’s collected papers were published after his untimely death that Eugenio Beltrami, and later Felix Klein and Henri Poincaré, found methods of exhibiting a model of a (non-Euclidean) plane of constant negative curvature in a Euclidean disk. These models did much to convince mathematicians that non-Euclidean geometry existed and was equally as consistent as Euclidean geometry.

With the basic story concluded, Gray continues by discussing the effects of the discovery of non-Euclidean geometry on mathematical education in various European countries and the strong opposition to its validity by several prominent mathematicians. Yet even as mathematicians showed that any contradiction in non-Euclidean geometry would be accompanied by a contradiction in Euclidean geometry, logical problems with the latter began to surface. Although these were generally resolved through new sets of axioms, such as Hilbert’s, Gray presents one of the infamous proofs that all triangles are isosceles and challenges us to find the error in it. (This is a great problem to give your geometry students.)

As a fascinating conclusion to his work, Gray shows us that Bolyai came tantalizingly close to resolving the "existence" question of non-Euclidean geometry himself by stating, if not proving, an infinitesimal distance formula for it, a formula which, with only a bit more work, could have shown skeptical readers how to represent non-Euclidean geometry in a Euclidean half-plane. And finally, Gray shows us what is now thought to be the only reliable portrait of János Bolyai.

For those looking for a modern treatment of non-Euclidean geometry either for their own enjoyment or to share with students in a college geometry course, I highly recommend Gray’s book. The price is right; the history is accurate, if brief; the mathematical discussions, even of highly technical questions, are clear; the illustrations are extremely helpful; and the Latin original as well as the English translation of Bolyai’s appendix will enable the reader to compare Gray’s interpretations to Bolyai’s original.

Victor J. Katz is well-known as the author of *The History of Mathematics: An Introduction*. He teaches at the University of the District of Columbia.