For every field of study in mathematics, there is that one magisterial text – the source that all must partake of, the formal and forbidding tome that requires all of one’s attention. The original example of this is, of course, Euclid’s Elements. The original dense compilation of facts and procedures that all who would learn Geometry must study. The book under review is quite a different beast. It pays its dues to Euclid and his work remains at the center of things but the authors have also worked hard to create a rather different experience for its readers. Their approach is full of ideas, intuitions, pictures and stories. This is not a “text with seven seals” but rather a fun and masterful road to learning what geometry is actually about. This is likely an ideal text for use in training secondary level teachers who will teach this glorious subject.

The subtitle of this book indicates a framing device that the authors use to good effect. We are introduced to the two main players in the field – the Line and the Circle. They are cast as ways in which we understand the world – firstly the “stretched string” as representative of Line. Here lines (small pun intended) from Davis and Hersh’s The Mathematical Experience motivate us to consider that

…that human beings were forced to discover…in each new human life…the shortest path from here to there, the activity of going directly towards something.

According to this metaphor, the form of the line receives its strength from concerns of optimization – the demands of energy and muscular means produce this archetype of progress. The Circle, on the other hand, is introduced as representative of the Native American viewpoint. Black Elk tells us that

…the Power of the World always works in circles, and everything tries to be round. In the old days when were a strong and happy people, all our power came to us from the sacred hoop of the nation, and so long as the hoop was unbroken, the people flourished.

Next follows a fairly standard discussion of Euclid’s definitions and axioms with a quick history of Proclus, Euclid, Hippocrates and the whole idea of a book devoted to the “elements” of geometry. The treatment continues with Postulates and common notions and at each step, the prospective student is asked to think about the possible meanings of the former. Simple ideas of construction abound. Chapter 3 is a meaty blow-by-blow description of the major propositions of Euclid I all enhanced by colored text boxes (modeled after the work of Oliver Byrne) which help codify the proofs of the first 31st Propositions. Standards of proof are much emphasized and the power of the axiomatic method is laid out for the reader. In addition (and this is a strong feature of the book) the authors are careful not to let Euclid off the hook. The student is invited to learn to look deeply for unremarked principles or ideas that are being used without a clear definition. In fact, the careful return to first principle (with greater depth) continues to mark this presentation of geometry. As a case in point, the next chapter introduces Spherical geometry and Taxicab geometry to re-examine the nature of “straight” lines and circles. In a lovely turn of events, the “lines” of Spherical geometry become recognized as great circles! And lo and behold the metric neighborhoods of points in Taxicab geometry are linear (as the boundaries of square neighborhoods). The discussion goes much further than this, however. In order to consider “triangles” and convex “polygons” on the Sphere, we must introduce new restrictions on the angles. Given that every great circle through point A also passes through the antipodal point A’, we must take the shorter side of such a circle as a “side” of a triangular arrangement of points. The chapter ends with a quick overview of spherical trigonometry and a discussion and proof of Girard’s formula for the area of a spherical triangle in terms of angular excess. This is a nice touch and sets the stage for a comparison of area across different geometries which is continued in later chapters.

The discussion of axiom systems for differing geometries continues in the next few chapters featuring Taxicab geometry, a discrete geometry of 4 points, Hilbert’s axiom system and Godel’s Incompleteness theorem before turning back to Euclid’s non-Neutral geometry (geometry which uses the parallel postulate) and the first proof of the Pythagorean theorem (page 151 – phew!). The section on Taxicab geometry is fine but insists on using lattice points for all computations (which may cause confusion for the student – is the metric defined on RXR or on ZXZ?). Taxicab geometry supplies an example of a model where SAS congruence does not hold but the preceding Euclidean postulates do. Chords, tangents and secant lines are all examined and their inter-relations proven. The chapter on circles is a real gem and obviously close to the authors hearts. The exercises are rich and include some very nice challenges such as the Butterfly Theorem. The following chapter on Circles and Polygons is even nicer and features the Euler Line, Morley’s theorem on angle trisectors and Heron’s formula for the area of a triangle. Even the Nine Point Circle Theorem is given a full treatment. The construction of regular polygons leads up to Gauss and his seminal construction of the regular 17-gon. Perhaps the most enticing is the selection of trefoils, rounded quadrifoils and other compound drawings (to be constructed by the student) from Mabel Sykes’s 1912 book *A Source Book of Problems for Geometry, Based on Industrial Design and Architectural Ornament*.

The last stops on this journey involve hyperbolic geometry, finite and projective geometry, isometries and the transformational approach and a last look at the four famous problems of antiquity. The two chapters on hyperbolic geometry are a real treat and take special care with a development that is both intuitive and formal. The subject is approached historically and begun with a search for models that might make non-Euclidean geometry as obvious to the eye as plane Euclidean geometry is. This is accomplished by exploring the Beltrami-Klein model of the unit disc followed by the Poincare Disc and Half-Plane model (but not, curiously, Beltrami’s pseudosphere). Following the same plan as before the question to be answered is “What satisfies the roles of Line and Circles in these settings?” The integral formula for arc-length in the Half-plane is used to explore the nature of geodesics and culminates in an analytic expression for hyperbolic distance. The following chapter tackles the same issue from an axiomatic approach. Hyperbolic triangles and Saccheri quadrilaterals are studied in some depth and the discussion finishes with a very nice discussion of hyperbolic area and equi-decomposability. The chapter of Finite Geometries is also nice (I learned things about the Fano Plane and Fano himself that I never knew!) and the illustrations for the chapter on perspective and Projective geometry are quite beautiful. Just as before, history and intuition are present first followed by constructions followed by axiomatic development and important theorems. The chapter on transformations proves everything you could want to know – the three reflections theorem, an introduction to inversion and then hyperbolic translations and reflections in the Poincare Half-Plane. I continue to be impressed by the complete development of results and ideas in this text.

The flow of ideas can only help students as they think about the various levels of understanding geometry. From constructions to proofs of results to the larger meaning of these results within the overarching context of Geometry, this text is there to guide students and assist them in constructing their own mastery of the subject. It is a beautiful text with real depth and detail and will be of great value to anyone who wishes to know “Just what is geometry about, after all is said and done?” Highly recommended!

Jeff Ibbotson is the Smith Teaching Chair in the Mathematics Department at Phillips Exeter Academy