If a book of this quality can remain out of print for over twenty years, then nothing is safe in the world of mathematical literature. Originally published in the USA by John Day Co in 1966, it was the first book of its kind to provide an accessibly broad description of geometries ancient and modern. Subsequent publications with similar coverage (such as Greenberg) have evolved over several editions, but they are suited to a more mathematically sophisticated readership.

Appealing to readers of high school standard and above, *A New Look at Geometry* portrays the evolution of geometrical ideas from the Babylonians to the work of Bolyai and Lobachevski and beyond. Its basic themes include the relationship between physical space and the geometries that represent it. In particular, it discusses Euclidean geometry in relation to Newtonian physics and non-Euclidean geometry with respect to relativity.

The ideas of Descartes and Fermat form the starting point for the development of another of the book’s major themes — the relationship between algebra and geometry. Coordinates, vector algebra, isometries and symmetry groups are shown to be effective geometric tools and, following later introductions to non-Euclidean and projective geometry, the importance of Klein’s Erlanger Programme is considered. But since this is an introductory survey of geometries that is intended for non-specialists, Adler has kept the mathematical technicalities to a minimum

Perhaps the most dominant story in this book is how the theory of parallels, the theory of curved surfaces and the ‘geometry of position’ converged into an integrated whole. Saccheri, Lambert and Legendre are the main characters in the drama of Euclid’s parallel postulate, while the work of Gauss, Bolyai and Lobachevsky characterise its denouement. The relevance of Hilbert’s axioms to Euclidean and non-Euclidean geometry is also clarified. Projective geometry is approached algebraically and synthetically, and the chapter ‘Calculus and Geometry’ offers a highly intuitive survey of elementary differential geometry that leads to the notion of Gaussian curvature and the concept of a manifold.

But why review an out-of-print book such as this? The main reason for doing so is the hope that readers of *MAA** Reviews* will request that a publisher (such as Dover) will re-establish it as an equal to any of the introductory works on geometry that are currently in print.

I think Irving Adler is now 98 years of age. He is a literary polymath who has written 56 books (some under the pen name Robert Irving). These are about mathematics, science, and education, and he was the co-author of 30 more, for both children and adults. His books have been published in 31 countries in 19 different languages. There is a fascinating biography of him available on Wikipedia.

**Editor's Note:** Ruane's review above was based on the original edition and posted here in 09/03/2011. It served as motivation for Dover to produce this new printing, which comes with an introduction by none other than Peter Ruane.

Irving Adler died on October 5, 2012. See the obituary from *The Washington Post*.

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