xiv Preface subject that engendered the breakthrough in the nineteenth century that led to our modern conception of the axiomatic method (and thus of the very nature of mathematics). Un- til very recently, virtually every high-school student in the United States took a geometry course based on some version of the axiomatic method, and that was the only time in the lives of most people when they learned about rigorous logical deduction. The primary purpose of this book is to tell this story: not exactly the historical story, because it is not a book about the history of mathematics, but rather the intellectual story of how one might begin with Euclid’s understanding of the axiomatic method in geometry and progress to our modern understanding of it. Since the axiomatic method underlies the way modern mathematics is universally done, understanding it is crucial to understanding what mathematics is, how we read and evaluate mathematical arguments, and why mathematics has achieved the level of certainty it has. It should be emphasized, though, that while the axiomatic method is our only tool for ensuring that our mathematical knowledge is sound and for communicating that soundness to others, it is not the only tool, and probably not even the most important one, for dis- covering or understanding mathematics. For those purposes we rely at least as much on examples, diagrams, scientific applications, intuition, and experience as we do on proofs. But in the end, especially as our mathematics becomes increasingly abstract, we cannot justifiably claim to be sure of the truth of any mathematical statement unless we have found a proof of it. This book has been developed as a textbook for a course called Geometry for Teachers, and it is aimed primarily at undergraduate students who plan to teach geometry in a North American high-school setting. However, it is emphatically not a book about how to teach geometry that is something that aspiring teachers will have to learn from education courses and from hands-on practice. What it offers, instead, is an opportunity to understand on a deep level how mathematics works and how it came to be the way it is, while at the same time developing most of the concrete geometric relationships that secondary teachers will need to know in the classroom. It is not only for future teachers, though: it should also provide something of interest to anyone who wishes to understand Euclidean and non-Euclidean geometry better and to develop skills for doing proofs within an axiomatic system. In recent years, sadly, the traditional proof-oriented geometry course has sometimes been replaced by different kinds of courses that minimize the importance of the axiomatic method in geometry. But regardless of what kinds of curriculum and pedagogical methods teachers plan to use, it is of central importance that they attain a deep understanding of the underlying mathematical ideas of the subject and a mathematical way of thinking about them—a version of what the mathematics education specialist Liping Ma [Ma99] calls “profound understanding of fundamental mathematics.” It is my hope that this book will be a vehicle that can help to take them there. Many books that treat axiomatic geometry rigorously (such as [Gre08], [Moi90], [Ven05]) pass rather quickly through Euclidean geometry, with a major goal of devel- oping non-Euclidean geometry and its relationship to Euclidean geometry. In this book, by contrast, the primary focus is on Euclidean geometry, because that is the subject that future secondary geometry teachers will need to understand most deeply. I do make a serious excursion into hyperbolic geometry at the end of the book, but it is meant to round out the

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