Preface for the Instructor and Reader I never intended to write a textbook and certainly not one in geometry. It was not until I taught a course to future high school teachers that I discovered that I have a view of the subject which is not very well represented by the current textbooks. The dominant trend in American college geometry courses is to use geometry as a medium to teach the logic of axiomatic systems. Though geometry lends itself very well to such an endeavor, I feel that treating it that way takes a lot of excitement out of the subject. In this text, I try to capture the joy that I have for the topic. Geometry is a fun and exciting subject that should be studied for its own sake. Though the primary target audience for this text is the future high school teacher, this text is also suitable for math majors, both because of the challenging problems throughout the text, and because of the quantity of material. In particular, I think this would make an excellent text for an undergraduate course in hyperbolic geometry. To the Student In theRepublic,Plato (ca. 427 - 347 B.C.) wrote that his ideal State should be ruled by philosophers educated first in mathematics. He believed that the value of mathematics is how it trains the mind, and that its practical utility is of minor importance. This philosophy is as valid now as it was then. A modern education might include vocational or technical training (such as engineering, medicine, or law), but at its core, there are the English and mathematics courses which make up a liberal education. Though mathematics has rather surprising utility, for many students, the most important lesson to be learned in their math classes is how to think analytically, creatively, and rigorously. Keep this in mind as you read this book. Recognize that the exercises are a fundamental and integral part of the text. This is where the most important lessons are learned. You will not solve them all, perhaps not even most, but I hope that the exercises you do solve will leave you with a feeling of satisfaction. Recommended Courses For a college geometry course for future high school teachers, the basic course outline that I recommend and usually teach is: (Section 1.1 - 1.12: Light on Sections 1.3 and 1.4); (Section 1.13 - 1.15: Optional); (Section 3.1 - 3.7: Section 3.7 is optional); (Section 4.1 - 4.4: Integrate with Chapter 3); (Section 5.1 - 5.5: Section 5.3 is optional); (Section 6.1 - 6.2, 6.4 - 6.6: Cover quickly and sparingly); (Section 7.1-7.4, 7.6 - 7.13, 8.1 - 8.2, 8.4 - 8.5: Use an overhead). Chapter 2 on Greek astronomy provides some interesting material which can be mixed in with Chapter 1, or used on 'optional' days, such as the Wednesday before Thanksgiving. I usually begin integrating Sketchpad (Chapter 4) after I have completed the first few sections on constructions (Chapter 3). A laptop and computer projector come in handy. Polyhedra (Chapter 5) might be considered optional, but I think it can be very valuable for a future high school teacher. In particular, Exercise 5.14 should not be missed, both as a class project and again as an exercise. These are lessons which can be easily brought into the high school classroom and have the potential to be memorable. I usually skip most of Chapter 6, and only introduce the 'crutch,' the concepts of parallel and ultraparallel lines, and the concept of asymptotic triangles. The beginning of Chapter 7 poses a bit of a dilemma. Most of my students are not familiar enough with path integrals and differentials to understand the arguments of Sections 7.2 and 7.3. I could not see a way of introducing the Poincare upper half plane model that avoids these arguments or something as difficult. I usually ask those students to accept these results and not worry too much if they do not understand the proofs. If I reach Chapter 8, it is usually covered during the last week of classeBaragar, Arthur is the author of 'Survey of Classical and Modern Geometry With Computer Activities', published 2000 under ISBN 9780130143181 and ISBN 0130143189.