Preface xv study of Euclidean geometry and place it in its proper perspective, not to be the main focus of the book. Organization The book is organized into twenty chapters. The first chapter introduces Euclid. It seeks to familiarize the reader with what Euclid did and did not accomplish and to explain why the axiomatic method underwent such a radical revision in the nineteenth century. The first book of Euclid’s Elements should be read in parallel with this chapter. It is a good idea for an aspiring geometry teacher to have a complete copy of the Elements in his or her library, and I strongly recommend the edition [Euc02], which is a carefully edited edition containing all thirteen volumes of the Elements in a single book. But for those who are not ready to pay for a hard copy of the book, it is easy to find translations of the Elements on the Internet. An excellent source is Dominic Joyce’s online edition [Euc98], which includes interactive diagrams. The second chapter of this book constitutes a general introduction to the modern ax- iomatic method, using a “toy” axiomatic system as a laboratory for experimenting with the concepts and methods of proof. This system, called incidence geometry, contains only a few axioms that describe the intersections of points and lines. It is a complete axiomatic system, but it describes only a very small part of geometry so as to make it easier to con- centrate on the logical features of the system. It is not my invention it is adapted from the axiomatic system for Euclidean geometry introduced by David Hilbert at the turn of the last century (see Appendix A), and many other authors (e.g., [Gre08], [Moi90], [Ven05]) have also used a similar device to introduce the principles of the axiomatic method. The axioms I use are close to those of Hilbert but are modified slightly so that each axiom focuses on only one simple fact, so as to make it easier to analyze how different interpretations do or do not satisfy all of the axioms. At the end of the chapter, I walk students through the process of constructing proofs in the context of incidence geometry. The heart of the book is a modern axiomatic treatment of Euclidean geometry, which occupies Chapters 3 through 16. I have chosen a set of axioms for Euclidean geometry that is based roughly on the SMSG axiom system developed for high-school courses in the 1960s (see Appendix C), which in turn was based on the system proposed by George Birkhoff [Bir32] in 1932 (see Appendix B). In choosing the axioms for this book, I’ve endeavored to keep the postulates closely parallel to those that are typically used in high- school courses, while maintaining a much higher standard of rigor than is typical in such courses. In particular, my postulates differ from the SMSG ones in several important ways: I restrict my postulates to plane geometry only. I omit the redundant ruler placement and supplement postulates. (The first is a theo- rem in Chapter 3, and the second is a theorem in Chapter 4.) I rephrase the plane separation postulate to refer only to the elementary concept of intersections between line segments and lines, instead of building convexity into the postulate. (Convexity of half-planes is proved in Chapter 3.) I replace the three SMSG postulates about angle measures with a single “protrac- tor postulate,” much closer in spirit to Birkhoff’s original angle measure postulate, more closely parallel to the ruler postulate, and, I think, more intuitive. (Theorems equivalent to the three SMSG angle measure postulates are proved in Chapter 4.)
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