Not many books can be regarded as both a serious work of history and a mathematics textbook, but this is certainly one of them. As such, it provides a fascinating introduction to Euclidean and Non-Euclidean geometry — seamlessly interwoven with themes of an historical, philosophical, scientific and cultural nature. Also, given the clarity of the prose, the excellent standard of its organisation and the attractive presentation, it has to be said that this fourth edition of Marvin Greenberg’s book exemplifies expositional writing at its best.

In fact, since the first edition in 1972, the book has grown incrementally through three subsequent editions, so that this latest version (637 pp) is more than twice the length of the first (300 pp) and it is 120 pages longer than the third edition. Moreover, the 1972 version was one of the earliest publications to reflect the current belief that any aspect of mathematics can be effectively taught in a way that illustrates its historical development.

And yet, although its basic structure remains unaltered, the innovative aspects of this 4^{th} edition are so extensive as to require a six-page summary in the preface; but there is no change in the suggested readership, which is modestly described as consisting of the following groups:

- Prospective high school and college geometry teachers.
- General education and liberal arts students.
- Mathematics majors.

However, the book will also appeal to historians, and to the general reader seeking fresh insights into a range of geometrical ideas. Because of the diverse nature of the intended readership, and because of the richly varied subject matter, the author has described five abbreviated pathways through the book, each of which could form the basis of a course.

As for the mathematical content, the declared emphasis is on the elementary geometry of lines and circles, and the general approach in the first two-thirds of the book is synthetic. But this reference to ‘elementary’ geometry doesn’t mean that the treatment is rudimentary, because the mathematical content is always challenging and it is subjected to profound scrutiny in the light of the book’s general themes.

Take, for instance, the very first chapter (Euclid’s Geometry), which begins by describing the achievements of the Egyptians, the Babylonians and the Pythagoreans. It goes on to consider the work of the Platonists, the ideas of Eudoxus and the influence of Archimedes. The emergence of Euclid’s treatise is outlined, and readers are introduced to the role of axioms and undefined terms, as well as the concepts of incidence and betweenness. Inevitably, there is an introduction to the controversy surrounding the parallel postulate, and various attempts to prove its dependence upon earlier axioms are analysed. Following that, epistemic themes arise in the sections on the power and danger of diagrams. Constructions, and the notion of constructibility, are considered from an historical perspective that begins with Euclid’s elements and culminates with the findings of Descartes, Kepler and Gauss. At this juncture, the narrative extends to the topic of neusis constructions, which is new to the fourth edition. An example of this would be Archimedes’ use of a ruler (instead of a straightedge) for the trisection of any angle. Nine hundred years after that, Viète devised an axiom (mentioned on p. 33) in the vain hope that it would qualify such methods for inclusion within axiomatic Euclidean geometry. Anyway, I can’t make sense of this version of Viète’s axiom, given by Marvin Greenberg, and it doesn’t agree with that given in Witmer’s translation of Viète’s *Opera Mathematica* (from whence it came).

Since the emphasis in this book is on synthetic geometry, axiomatic methods are the main theme of the first seven chapters, and chapter 2 (Logic and Incidence Geometry) reviews elementary logic and proof. It then introduces the axioms of incidence and explains the concept of models for axiomatic systems, up to those for affine and projective planes. Of course, the powerful concept of isomorphism between models is defined, and it is used at various points in subsequent chapters. For example, in chapter 7 (Independence of the Parallel Postulate), it is shown that the Klein model for hyperbolic geometry is isomorphic to that of Poincaré. Amongst the new material in chapter 2 is a section on problems of consistency for axiomatic systems — an idea that is often revisited later in the book.

By this stage, the reader will fully understand why Hilbert formulated his axiomatic system, and this is carefully described in the partly re-written chapter 3. Herein, the discourse is enlivened by a most imaginative range of examples. For instance, the axiom of betweenness is illustrated in the context of affine planes over ordered fields, as well as ordered incidence planes with the hyperbolic property. There is also discussion of the axioms of Archimedes, Aristotle and Dedekind, which receive no mention in many books on the foundations of geometry.

Chapter 4 (Neutral Geometry) has undergone major revision and contains the surprising observation that Euclid’s first 29 propositions made no use of his controversial parallel postulate. Therefore, up to that point, Euclid was effectively operating within neutral geometry, and the author gives several examples of such results. But the full range of Hilbert’s thirteen axioms are employed to establish a variety of theorems on angles, triangles etc, and his parallel postulate is shown to be equivalent to that of Euclid. The chapter concludes with recently added material on the significance of Saccheri and Lambert quadrilaterals with respect to neutral geometry, and they are discussed again in the last chapter on hyperbolic geometry.

Historical commentary pervades all of the preceding mathematical narrative, but chapter 5 (History of the Parallel Postulate) and chapter 6 (The Discovery of Non-Euclidean Geometry) are specifically devoted to historical matters. In particular, the work of Proclus, Wallis, Legendre, Clairaut and Taurinus is central to the discourse of the fifth chapter, which has a new section on Clavius’ axiom. Moreover, while the pioneering contributions of Bolyai, Gauss and Lobachevsky still constitute main thrust of chapter 6, it has been thoroughly revised and now includes, for example, discussion of Hilbert’s hyperbolic axiom of parallelism.

The book’s second major mathematical theme, geometric transformations, is the purpose of chapter 9, which begins with historical notes on Klein’s Erlanger Programme, followed by a compact introduction to Euclidean motion geometry and similarities. What is unusual about the treatment here is that the author immediately shows the power of such methods by applying them to the solution of geometric problems and the derivation of Euclidean theorems. Subsequent to this, there is the general classification of motions and a delineation of Klein’s sub-geometries of real plane projective geometry (affine, hyperbolic, elliptic, parabolic and Euclidean).

Chapter 10 explores further results in hyperbolic geometry, including area and the pseudosphere. Amongst other things, it considers Riemann’s view of the hyperbolic plane as being *a complete, simply connected, two-dimensional Riemannian manifold of constant negative curvature.* As for the discussion on hyperbolic trigonometry, the definitions of the hyperbolic functions are geometrically unmotivated, compared to the way in which Klein introduced them in the section on goniometric functions in [1]; but this is a rare example of instrumental pedagogy to be found in this book.

In a simplistic attempt to summarise the book as a whole, I would say that the main arena for mathematical discussion is spanned by axiomatic and hyperbolic geometry, with the theories of inversion and projective geometry being used to define models for the hyperbolic plane. Elliptic, and other Riemannian geometries do receive some mention in the main text, but they are otherwise treated more fully in Appedix A, whilst Appendix B examines Hilbert’s geometry without real numbers.

As for the general themes that permeate the book, the ‘philosophical’ aspects are more or less epistemic or metamathematical, and chapter 8 (Philosophical Implications and Fruitful Applications) encapsulates the range of such ideas. It contains sections on the geometry of physical space, the nature of mathematics and foundational matters that include discussion of Gödel’s incompleteness theorems. The ‘fruitfulness’ of hyperbolic geometry is vindicated largely with respect to other areas of mathematics, such as the uniformization of compact orientable surfaces, Thuston’s geometrization conjecture for 3d manifolds and tessellations of the hyperbolic plane.

Finally, each of the ten chapters concludes with a rich collection of exercises, ranging from true/false questions for purpose of review to problems that are mathematically demanding. Other tasks require the reader to undertake further reading and/or research on historical issues For example, here’s one mathematical exercise from the very first chapter:

Write a paper explaining in detail why it is impossible to trisect an arbitrary angle or square a circle using compass and straightedge (see Jones, Morris and Pearson 1991; Eves 1972; or Moise 1990). Explain how arbitrary angles can be trisected if in addition we are allowed to draw a parabola or a hyperbola or a conchoid or a limaçon (see Peressini and Sherbert, 1971).

In total, the exercises occupy more than one quarter of the whole book, and many of them serve as prerequisite reading for of subsequent chapters.

However, the expansive range of mathematical ideas contained in Marvin Greenberg’s superb exposition, and the historical and philosophical perspectives from which they emerge, can only be appreciated by ownership of this latest edition. Go forth and purchase!

**Reference**

**[1]** Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra and Analysis , by Felix Klein (Dover, 2004).

Peter Ruane has led an altogether different life since retiring (ten years ago) from a career in primary and secondary mathematics teacher education; but Marvin Greenberg’s book will greatly enrich his remaining years.