xvi Preface I replace the SMSG postulate about the area of a triangle with a postulate about the area of a unit square, which is much more fundamental. After the treatment of Euclidean geometry comes a transitional chapter (Chapter 17), which summarizes the most important postulates that have been found to be equivalent to the Euclidean parallel postulate. In so doing, it sets the stage for the study of hyperbolic geometry by introducing such important concepts as the angle defect of a polygon. Then Chapters 18 and 19 treat hyperbolic geometry, culminating in the classification of parallel lines into asymptotically parallel and ultraparallel lines. Chapter 20 is a look forward at some of the directions in which the study of geometry can be continued. I hope it will whet the reader’s appetite for further advanced study in geometry. After Chapter 20 come a number of appendices meant to supplement the main text. The first four appendices (A through D) are reference lists of axioms for the axiomatic sys- tems of Hilbert, Birkhoff, SMSG, and this book. The next two appendices (E and F) give brief descriptions of the conventions of mathematical language and proofs they can serve either as introductions to these subjects for students who have not been introduced to rig- orous proofs before or as review for those who have. Appendices G and H give a very brief summary of background material on sets, functions, and the real number system, which is presupposed by the axiom system used in this book. Finally, Appendix I outlines an al- ternative approach to the axioms based on the ideas of transformations and rigid motions it ends with a collection of challenging exercises that might be used as starting points for independent projects. There is ample material here for a full-year course at a reasonable pace. For shorter courses, there are various things that can be omitted, depending on the tastes of the in- structor and the needs of the students. Of course, if your interest is solely in Euclidean geometry, you can always stop after Chapter 16 or 17. On the other hand, if you want to move more quickly through the Euclidean material and spend more time on non-Euclidean geometry, there are various Euclidean topics that are not used in an essential way later in the book and can safely be skipped, such as the material on nonconvex polygons at the end of Chapter 8 and some or all of Chapters 14, 15, or 16. Don’t worry if the course seems to move slowly at first—it has been my experience that it takes a rather long time to get through the first six chapters, but things tend to move more quickly after that. I adhere to some typographical conventions that I hope will make the book easier to use. Mathematical terms are typeset in bold italics when they are officially introduced, to reflect the fact that definitions are just as important as theorems and proofs but fit better into the flow of paragraphs rather than being called out with special headings. The symbol is used to mark the ends of proofs, and it is also used at the end of the statement of a corollary that follows so easily that it is not necessary to write down a proof. The symbol // marks the ends of numbered examples. The exercises at the ends of the chapters are essential for mastering the type of thinking that leads to a deep understanding of mathematical concepts. Most of them are proofs. Almost all of them can be done by using techniques very similar to ones used in the proofs in the book, although a few of the exercises in the later chapters might require a bit more ingenuity.
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