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*Euclidean and Non-Euclidean Geometries: Development and History*, 4^{th} Edition, by Marvin Jay Greenberg, 2008, xxix+637 pp., ISBN-13 978-0-71679948-1, ISBM-10 0-7167-9948-0, Hardcover, $98.44, W.H. Freeman and Company, 41 Madison Ave., New York, NY 10010

For much of the last half of the twentieth century, college level mathematics textbooks, particularly calculus texts, have included short, marginal, historical blurbs; a short bio of Brook Taylor in the section on Taylor series, for example. Such inclusions can be interesting for the faculty member who has not had much exposure to the history of mathematics or the student with a pre-existing interest. As a student I found these excerpts tantalizing and they surely whetted my appetite for mathematics history. However, as a professor I have found them frustrating as they rarely say enough about the mathematics itself. Moreover it has been my experience that most students barely even notice them in their frenzy to find examples ‘just like the problems.’

In recent years there has been a movement to imbed history more directly into the text. Several textbooks integrate the history of the topic into their content in such a way that the history of the subject enlightens and enlivens the subject itself. Bressoud’s well known *A Radical Approach to Real Analysis* is one example. The volume under review is another.

A quick glance through the table of contents confirms this situation. The first chapter is titled “Euclid’s Geometry” and begins with sections on ‘The Pythagoreans’, ‘Plato’, and ‘Euclid of Alexandria’ before moving on to discussion of ‘The Axiomatic Method’, ‘The Parallel Postulate’, ‘The Danger in (and Power of) Diagrams’, and ‘Descartes’ Analytic Geometry and a Broader Idea of Constructions’. Clearly, the textual development follows the historical evolution. This pattern is continued in the remaining chapters.

Chapter 3 introduces the reader to ‘Hilbert’s Axioms’ beginning with a discussion of the flaws in Euclid. Chapter 4 develops neutral geometry – geometry without the parallel postulate. The fifth chapter, ‘The History of the Parallel Postulate’, includes sections on Proclus, Wallis, Saccheri, Legendre and Farkas Bolyai, and the sixth, ‘The Discovery of Non-Euclidean Geometry’, begins with sections on Janos Bolyai, Gauss and Lobachevsky. The seventh chapter, ‘Geometric Transformations’, begins with a section on ‘Klein’s Erlanger Program’. Every named individual actually appears in the text as an individual, not as a name attached to a theorem. The nature and relevance of his work in geometry is clearly delineated.

The biographical material is always kept short and relevant as befits a textbook. However there is a very smooth interweaving of the history and the mathematics. The history sets up the concepts to be defined and the problems to be addressed. Addressing the problems leads naturally back into the history of the topic. The text moves seamlessly from the history of geometry to geometry itself and back again, constantly weaving the two together in a style that I find not only enlightening but utterly absorbing.

There is a large collection of exercises at the end of each chapter. These are divided, in increasing level of difficulty, into ‘Review Exercises’, ‘Exercises’, and ‘Major Exercises’. There is also a list of ‘Projects’ that invites the student to further research on topics both historical and mathematical from each chapter.

If you prefer the standard Theorem-Proof-Examples-Short Discussion format that many mathematics texts adopt, you will probably find this book excessively wordy. I anticipate that this will be a common lament from students as well. The author discusses at length the mathematics community’s discomfort with the parallel postulate, our efforts to prove it from the other four axioms, why those efforts failed, and our difficulties in accepting the new geometries which were eventually built from the wreckage of those failures. In my view, however, this is one of the most exciting stories in mathematics. It is fascinating in itself and also highly relevant, reaching as it does from the ancient past to the modern world and into the future. It is a story that we should be teaching our students for, in the words of J. W. L. Glaisher, “I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history.”

Eugene Boman, Associate Professor of Mathematics, The Pennsylvania State University, Capital College, Middletown, PA.

Eugene Boman (Pennsylvania State University-Capital College), "Euclidean and Non-Euclidean Geometries: Development and History," *Convergence* (April 2010), DOI:10.4169/loci003354