Methods for Euclidean Geometry is a college geometry textbook with a unique mission. Instead of treating the subject as a distinct unit in the math curriculum, the authors integrate a variety of mathematical disciplines to engage and enlighten the reader. The text begins with an early history of geometry and then introduces and explains the basics of an axiomatic system. After this fairly extensive introduction, there are eleven chapters of content, ranging from lines and polygons to conics and inversions. The content is accompanied by numerous practice and supplemental exercises; solutions are included.
The authors’ unique approach is first found in the chapter on the coordinate system. In high school, coordinates are not usually introduced until after geometry, so this approach may be new to many students. I found the section especially interesting, as the algebraic solutions often brought a new perspective to old problems.
Unlike many textbooks, this one hunts down and proves propositions that are often taken for granted. Take for instance solving systems of equations using the substitution and elimination methods. Rarely do texts help students understand why these methods give the same unique solution. In situations like this the authors prompt students to ask the questions “why” and “how” and then help answer them.
The content in the thorough explanations should excite and enthuse students. Questions leading to deeper understanding are sprinkled throughout the work. The chapter on inversion feels like reading about a magic trick. Curves are transformed into straight lines and distinct circles become concurrent; all the while certain distances never change. This text asks and then demands that students engage with the material.
The text would be most useful in a college geometry course. As in many college texts, the material is presented well, but briefly with no examples. Although bright students could work with the exercises to increase their understanding, the combination of class and text would be best. Despite being intended for an undergraduate audience, this book would also be a useful resource for high school teachers with college math experience. The infrequently found proofs shine light on many questions left unanswered in high school mathematics.
With its treatment of history, unusual proofs and various methods of finding solutions, this text strives to teach the whole picture. Methods for Euclidean Geometry does a wonderful job exploring geometry through fresh new eyes.
Ruth Doherty is a Teaching Fellow in Mathematics at Phillips Academy in Andover, Massachusetts. She teaches geometry and algebra to high school students and also tries to give them a taste for higher mathematics (currently using tic-tac-toe on a torus and the fourth dimension).
1. Early History
2. Axioms: from Euclid to Today
3. Lines and Polygons
5. Length and Area
10. Complex numbers
12. Affine Transformations
14. Coordinate Method with Software
Hints to Chapter Problems
Solutions to Chapter Problems