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A Gateway to Modern Geometry: The Poincaré Half-Plane

Saul Stahl
Jones and Bartlett
Publication Date: 
Number of Pages: 
[Reviewed by
P. N. Ruane
, on

Saul Stahl’s book begins with a condensed overview of synthetic Euclidean geometry, which includes a good selection of theorems about plane shapes. As with most of the subsequent chapters, historical observations are interspersed with the expository narrative, and this first chapter concludes with a motivated introduction to Hilbert’s axioms. Therefore, to this extent, Euclidean geometry is viewed from a modern axiomatic perspective.

Next is a rather formal introduction to the topic of Euclidean Rigid Motions in which the notion of invariance is somewhat understated. By this I mean that, unlike the first chapter, none of the theorems pertaining to plane figures are derived by such methods. In other words, the treatment seems to be all process and no product.

But there is no such imbalance in the ensuing introduction to inversive geometry. And, although no historical or mathematical motivation is provided for the introduction of this topic, the treatment is concise, and notion of invariance comes across very clearly. Inversions are then used to derive many interesting properties of circles.

Omissions from the usual Kleinian hierarchy include affine and projective geometry. But the pay-off is a fast track into the heart of hyperbolic geometry, which is achieved by an elegant introduction to the Poincaré half-plane. From then on, the notions of hyperbolic length, angle and rigid motions are clearly spelled out.

In this book, you can always see the wood for the trees — both mathematically and historically. In this vein, chapter 5 compares the structure of Euclidean with that of hyperbolic geometry. The next three chapters explore further aspects of the hyperbolic universe and then, in chapter 9, heavier mathematical machinery appears in the form of Moebius transformations. The second half of the book reveals the charms of spherical and elliptic geometry, the differential geometry of surfaces, the unit disk and Beltrami-Klein models

Among the many attractive features of Saul Stahl’s book is that it doesn’t overwhelm the incipient geometer with a myriad of mathematical techniques. For instance, the chapter on differential geometry assumes no knowledge of the geometry of curves, and many of the ideas are conveyed by analogy. The various diagrams and illustrations also add to the book’s aesthetic appeal.

Obviously, this book is highly recommended as the basis of an introductory course on non-Euclidean geometry. It portrays geometry as a subject with an interesting history and exciting possibilities. Each section of the book concludes with a range of exercises, from the routine to the more challenging. Finally, the fact no solutions are provided is one thing, but the difficulty of accessing the online instructors manual is altogether more puzzling.

Peter Ruane finds that, as he approaches the boundary of this temporal universe, the days fly by more quickly, and his height is annually diminishing by a miniscule amount.


1.   Euclidean Geometry

2.   Euclidean Rigid Motions

3.   Inversions

4.   The Hyperbolic Plane

5.   Euclidean versus Hyperbolic Geometry

6.   The Angles of the Hyperbolic Triangle

7.   Hyperbolic Area

8.   The Trigonometry of the Hyperbolic Triangle

9.   Complex Numbers and Rigid Motions

10.  Absolute Geometry and the Angles of the Triangle

11.  Spherical Trigonometry and Elliptic Geometry

12.  Differential Geometry and Gaussian Curvature

13.  The Cross Ratio and the Unit Disk-Model

14.  The Beltrami-Klein Model

15.  A Brief History of Non-Euclidean Geometry

16.  Spheres and Horospheres