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Two-Dimensional Geometries: A Problem-Solving Approach

C. Herbert Clemens
American Mathematical Society
Publication Date: 
Number of Pages: 
Pure and Applied Undergraduate Texts
[Reviewed by
Sr. Barbara Reynolds
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This delightful text tells the story of two-dimensional geometries in seven parts.  Parts I – IV are appropriate material for a pre-service course for high school mathematics teachers.  The material presented here would give future teachers a depth and breadth of geometric understanding that would allow them to teach for understanding.
Beginning with Euclid’s first four postulates and assuming only high school geometry, algebra, and trigonometry, Part I guides the reader to explore topics in neutral geometry (NG).  Euclid’s Fifth Postulate is introduced in Part II, and the student is led on a guided exploration of Euclidean geometry (EG) which includes trigonometry, dilations and similarity.  Clemens presents the bare bones – definitions, postulates, key theorems in a context of minimal yet engaging discussion. Then he invites the student to do geometry through a series of carefully structured exercises.  Altogether there are 171 exercises in this text (63 in the first two Parts), which are presented in the flow of the text – not as optional activities at the end of a chapter, but rather as integral to the flow of the text.  Part II ends with an outline of the proof of Ptolemy’s Theorem, and an exercise for the student to apply Ptolemy’s Theorem to prove the Addition Law of Sines.
Parts III – IV use linear algebra and multivariable calculus to explore topics in spherical geometry, Euclidean three-space, and transformations.  Spherical geometry is motivated via a study of volumes, magnification, and surface area.  
In Part V, Clemens guides the reader through changing coordinates, clearly demonstrating that one can change coordinates without changing the geometry.  He introduces K-geometry, where K is the curvature of the space.  Euclidean geometry is the K-geometry where K = 0, and spherical geometry is the space where K > 0.  This naturally lays the groundwork for the possibility that the curvature K might become negative.  In Part VI, Clemens returns to spherical geometry from an advanced standpoint.  The text concludes in Part VII with an overview of hyperbolic geometry (HG). 
For upper-division students who have a good background in linear algebra and multivariable calculus, this text will provide a delightful introduction to plane geometries – from plane Euclidean space, to the spherical plane, and finally to the hyperbolic plane.  As the course progresses, Clemens leads the student through carefully sequenced exercises to prove that there is a complete, simply connected, two-dimensional geometry of constant curvature, K, for each real number K.  
Clemens succeeds in telling a story of plane geometries so that various topics flow naturally, one topic building toward the next.  In reviewing this text, I imagined myself teaching a class based on this text and realized that I would greatly appreciate some kind of teaching notes as a guide.  Within the text, there is no indication of the relative difficulty of the exercises, that is, no starred (*) exercises marked as more challenging or as essential to the continued development of the story.  For the student, this invites attempting all the exercises and engaging in the process of “doing geometry.”  As an instructor, I would appreciate a roadmap that would help me to guide the students toward the highlights, so as to reach significant destinations within the journey.  Perhaps such resources for instructors could be developed and provided online.
Sr. Barbara Reynolds has been teaching mathematics at Cardinal Stritch University, Milwaukee, WI for 40 years.  Occasionally she gets the opportunity to teach geometry to preservice teachers.