Despite the eye-catching title of this MAA publication, potential readers may be deterred by the fact it comes in the form of an e-book. Having read it, I’m greatly encouraged by the many advantages gained by preserving it in PDF form. In fact, the MAA has published several of its newest titles in this format, for which it claims the advantages of ‘ease of access, portability and flexibility of use’. Apart from the matter of ‘portability’, I can vouch for the other claimed qualities, which mean that one can indulge in the sacrilegious practice of annotating and highlighting relevant parts of any printed PDF pages — and yet still keep the original version in electronically pristine condition. Moreover, as with the other MAA e-books, there is the option of purchasing the printed version (which I haven’t seen).

The modest blurb on the MAA website says that *Lobachevski Illuminated* provides an historical introduction to non-Euclidean geometry, and that readers will be guided step-by-step through a new translation of Lobachevski’s groundbreaking book *The Theory of Parallels*. It also claims that Seth Braver’s extensive commentary locates Lobachevski’s work in its mathematical, historical, and philosophical context, thus granting readers a vision of the mysterious and beautiful world of non-Euclidean geometry as seen through the eyes of one of its discoverers. Although Lobachevski’s 170-year old text is challenging to read on its own, it is said that the author’s carefully arranged “illuminations” render this classic text accessible to any modern reader undaunted by high school mathematics. To my mind, there is little exaggeration in this description of the Braver’s e-book, but I could find no mention of who carried out the aforementioned ‘new translation’ of Lobachevsky’s tract — nor, indeed, a summary of how it differs from the previous one done by Bruce Halsted in 1891.

In total, the analysis of Lobachevski’s ‘*Theory of Parallels’* occupies about 200 pages, and the new translation of the original 22-page text is included as an appendix. But Lobachevski’s complete text appears twice within the pages of Seth Braver’s book because, in the main text, it is dissected into very many pieces that are interwoven with his detailed commentary. For example, while Lobachevski’s introduction to the concept of horocycle occupies only one third of a side of A4, Braver’s explanatory narrative on the horocycle extends over ten pages. Moreover, he has extended Lobachevski’s ideas and adhered to his way of working to provide the most comprehensive analysis of a synthetic derivation of horocycles that I have so far encountered.

Throughout *Lobachevski Illuminated*, the historical observations are interwoven with the mathematical explanation, and they range from the time of Euclid to Hilbert’s early 20^{th} century work on the foundations of Euclidean geometry. Naturally, the logical shortcomings of Euclid are examined in the light of Hilbert’s ideas, which are also invoked to reveal various deficiencies in Lobachevski’s thinking. However, the initial historical thrust focuses upon the period from John Wallis’s famous lecture of 1663 to the publication of Legendre’s *Éléments de Géométrie* in 1823. This is because the work of Legendre on the parallel postulate formed the starting point for Lobachevski’s development of non-Euclidean geometry. Seth Braver also discusses the closely guarded ideas of Gauss with respect to non-Euclidean geometry, and he makes due comparison of Bolyai’s ideas with those of Lobachevski (for example, their definitions of ‘parallel’ are almost the same and Bolyai’s L-curve is shown to be analogous to Lobachevski’s horocycle).

As suggested above, Seth Braver doesn’t just interpret the existing contents of Lobachevski’s *Theory of Parallels,* but he continually adds to it by way of making it more mathematically coherent. In this respect, his achievement is first rate and it is equalled by his eloquently inspiring literary style. By such means, he succeeds with his aim of taking the reader back 170 years into an approach to geometry that has long been buried under streamlined modern renditions of non-Euclidean geometry. As such, this is a masterly addition to the literature on the history of geometry, and I see it as an ideal companion to Jeremy Gray’s beautiful book on János Bolyai.

Peter Ruane’s career was centred upon primary and secondary mathematics education.