Preface xvii Prerequisites Because this book is written as a textbook for an advanced undergraduate course, I expect readers to be conversant with most of the subjects that are typically treated at the beginning and intermediate undergraduate levels. In particular, readers should be comfort- able at least with one-variable calculus, vector algebra in the plane, mathematical induc- tion, and elementary set theory. It would also be good to have already seen the least upper bound principle and the notions of injective and surjective functions. Students who have had a course that required them to construct rigorous mathematical proofs will have a distinct advantage in reading the book and doing the exercises. But for students whose experience with proofs is limited, a careful study of Appendices E and F should provide sufficient opportunities to deepen their understanding of what proofs are and how to read and write them. Acknowledgments I am deeply indebted to my colleague John Palmieri, who has taught from two different draft versions of this book and contributed immeasurably to improving it. I owe just as much thanks to my students, many of whom—especially Natalie Hobson, Dan Riness, and Gina Bremer—have generously and patiently provided detailed suggestions for helping the book to serve students better. Of course, I owe enormous gratitude to a long list of authors who have written rigorous texts about Euclidean and non-Euclidean geometry that have educated and inspired me, especially Marvin Greenberg [Gre08], Robin Hartshorne [Har00], George Martin [Mar96], Richard Millman and George Parker [MP91], Edwin Moise [Moi90], and Gerard Venema [Ven05]. Additional Resources I plan to post some supplementary materials, as well as a list of corrections, on the website www.ams.org/bookpages/amstext-21. For the sake of future readers, I encourage all readers to make notes of any mistakes you find and any passages you think could use improvement, whether major or trivial, and send them to me at the email address posted on the website above. Happy reading! John M. Lee Seattle, WA November 27, 2012
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