Lobachevski Illuminated provides an historical introduction to non-Euclidean geometry. Within its pages, readers will be guided step-by-step through a new translation of Lobachevski's groundbreaking book, The Theory of Parallels. Extensive commentary situates 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, Seth Braver’s carefully arranged 'illuminations' render this classic accessible to any modern reader (student, professional, or layman) undaunted by high school mathematics.

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 Source: MAA Reviews Online