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Geometry: the Line and the Circle VOL AMS / MAA TEXTBOOKS 44 AMS / MAA TEXTBOOKS VOL 44 Geometry: The Line and the Circle VOL AMS / MAA TEXTBOOKS 44 Maureen T. Carroll and Elyn Rykken Geometry: The Line and the Circle Geometry: Geometry: The Line and the Circle is an undergraduate text with a strong narrative that is written at the appropriate level of rigor for an upper-level survey or axiomatic course in geometry. Starting with Euclid’s Elements, the book connects topics in Euclidean and non-Euclidean geometry in an intentional and meaningful way, with historical context. The line and the circle are the principal characters driving the narrative. In every geometry considered—which include spherical, hyperbolic, and and Elyn Rykken Carroll T. Maureen taxicab, as well as finite affine and projective geometries—these two objects are analyzed and highlighted. Along the way, the reader contem- plates fundamental questions such as: What is a straight line? What does parallel mean? What is distance? What is area? There is a strong focus on axiomatic structures throughout the text. While Euclid is a constant inspiration and the Elements is repeatedly revisited with substantial coverage of Books I, II, III, IV, and VI, non-Euclidean geometries are introduced very early to give the reader perspective on questions of axiomatics. Rounding out the thorough coverage of axiomatics are concluding chapters on transformations and constructa- bility. The book is compulsively readable with great attention paid to the historical narrative and hundreds of attractive problems. For additional information and updates on this book, visit www.ams.org/bookpages/text-44 AMS/ MAA PRESS TEXT/44 4-Color Process 496 pages on 50lb stock • Backspace 1 11/16'' 10.1090/text/044 Geometry: The Line and the Circle AMS/MAA TEXTBOOKS VOL 44 Geometry: The Line and the Circle Maureen T. Carroll Elyn Rykken Committee on Books Jennifer J. Quinn, Chair MAA Textbooks Editorial Board Stanley E. Seltzer, Editor Bela Bajnok Suzanne Lynne Larson Jeffrey L. Stuart Matthias Beck John Lorch Ron D. Taylor, Jr. Heather Ann Dye Michael J. McAsey Elizabeth Thoren William Robert Green Virginia Noonburg Ruth Vanderpool Charles R. Hampton 2010 Mathematics Subject Classification. Primary 51-01. For additional information and updates on this book, visit www.ams.org/bookpages/text-44 Library of Congress Cataloging-in-Publication Data Names: Carroll, Maureen T., 1966– author. | Rykken, Elyn, 1967– author. Title: Geometry: The line and the circle / Maureen T. Carroll, Elyn Rykken. Description: Providence, Rhode Island: MAA Press, an imprint of the American Mathematical Society, [2018] | Series: AMS/MAA textbooks; volume 44 | Designed for an upper-level college geometry course. | Includes bibliographical references and indexes. Identifiers: LCCN 2018034790 | ISBN 9781470448431 (alk. paper) Subjects: LCSH: Geometry–Textbooks. | Geometry–Study and teaching (Higher) Classification: LCC QA445 .C2985 2018 | DDC 516–dc23 LC record available at https://lccn.loc.gov/2018034790 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary ac- knowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permit- ted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. © 2018 by the authors. All rights reserved. Printed in the United States of America. ⃝1 The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 23 22 21 20 19 18 To our parents for their love, support and encouragement Contents Note to the Instructor xi Outline of the book xii Designing a course using this text xiii Note to the Reader xvii Acknowledgments xix 1 The Line and the Circle 1 1.1 Introduction 1 1.2 Which came first? 2 1.3 What is a straight line, anyways? 4 2 Euclid’s Elements: Definitions and Axioms 7 2.1 The Elements 7 2.2 Definitions 9 2.3 Postulates and common notions 12 3 Book I of Euclid’s Elements: Neutral Geometry 17 3.1 Propositions I.1 through I.8 18 3.2 Propositions I.9 through I.15 32 3.3 Propositions I.16 through I.28 and I.31 41 4 Spherical Geometry 59 4.1 What is a straight line, anyways? - Part 2 60 4.2 Triangles in Spherical geometry 65 4.3 Euclid’s axioms viewed in Spherical geometry 68 4.4 Neutral geometry on the sphere 71 4.5 Area in Spherical geometry 77 4.6 Trigonometry for spherical triangles 82 4.7 Uniquely spherical constructions 88 5 Taxicab Geometry 93 5.1 Points, lines, angles, distances and circles 94 5.2 Euclid’s postulates in Taxicab geometry 100 5.3 Congruence schemes in Taxicab geometry 101 5.4 The rest of Neutral geometry 102 6 Hilbert and Gödel 105 6.1 Axiomatic systems 106 6.2 A Four Point geometry 110 vii viii Contents 6.3 Hilbert’s axioms for Euclidean plane geometry 113 6.4 Spherical and Taxicab geometries 120 6.5 Gödel and consistency 123 7 Book I: Non-Neutral Geometry 127 7.1 Parallel lines 128 7.2 Propositions I.32 and I.33 132 7.3 Area 135 7.4 Propositions I.34 through I.41 138 7.5 Propositions I.42 through I.46 145 7.6 The Pythagorean Theorem 151 8 Book II: Geometric Algebra 159 8.1 Proposition II.1 through II.10 160 8.2 Propositions II.11 through II.14 165 8.3 Quadrature on the sphere 171 9 Book VI: Similarity 175 9.1 Book V: Ratio and proportion 176 9.2 Similarity 176 9.3 A generalized Pythagorean Theorem 184 10 Book III: Circles 189 10.1 Definitions 190 10.2 Tangency 193 10.3 Arcs, chords and angles 201 10.4 Area Propositions: III.35 through III.37 210 10.5 The circumference of a circle & 휋 216 11 Book IV: Circles & Polygons 221 11.1 Definitions 222 11.2 Circles & triangles 223 11.3 Circles & squares 236 11.4 Circles & pentagons 241 11.5 Constructing regular polygons 246 11.6 The area of a circle & 휋 251 12 Models for the Hyperbolic Plane 261 12.1 Historical overview 262 12.2 Models of the hyperbolic plane 266 12.3 Arc length & distance in the half-plane model 277 13 Axiomatic Hyperbolic Geometry 289 13.1 Parallel lines 291 13.2 Omega triangles 302 13.3 Saccheri quadrilaterals 307 13.4 Hyperbolic area 320 Contents ix 14 Finite Geometries 331 14.1 Four Point geometry - Part 2 332 14.2 Fano’s plane 335 14.3 Projective geometry 339 14.4 Affine planes 355 14.5 Transforming afine into projective 366 14.6 Open problem in finite geometry 370 15 Isometries 373 15.1 Rigid motions or isometries 374 15.2 Reflections 378 15.3 Isometries of the Euclidean plane 382 15.4 Inversions in the Euclidean plane 399 15.5 Isometries of the hyperbolic plane 406 16 Constructibility 415 16.1 Four famous problems of antiquity 416 16.2 Constructible numbers 419 16.3 Four counterexamples 429 16.4 The limits of geometry 435 Appendix A Euclid’s Definitions and Axioms 437 A.1 Definitions 437 A.2 Postulates 438 A.3 Common notions 438 Appendix B Euclid’s Propositions 439 B.1 Book I 439 B.2 Book II 442 B.3 Book III 443 B.4 Book IV 444 B.5 Book VI 445 Appendix C Visual Guide to Euclid’s Propositions 447 C.1 Book I 447 C.2 Book II 453 C.3 Book III 454 C.4 Book IV 456 Appendix D Euclid’s Proofs 457 D.1 Book I 457 Appendix E Hilbert’s Axioms for Plane Euclidean Geometry 461 Credits, Permissions and Acknowledgements 463 Bibliography 465 Notation Index 471 Index 473 Note to the Instructor This book is an introduction to Euclidean and non-Euclidean geometry designed for an upper-level college geometry course. Its content and narrative grew out of our ex- perience teaching junior/senior level advanced geometry and history of mathematics courses for over twenty years. We have learned that, independent of ability and math- ematical maturity, most students have only faint memories of the synthetic Euclidean geometry that they studied in high school. This, coupled with the fact that mathemat- ics students rarely have occasion to read primary sources in their major, led us to take full advantage of the unique opportunity available in geometry and introduce our read- ers to the bible of mathematics, Euclid’s Elements. With Euclid as a guide, the reader begins by travelling along the same path as millions of geometry students spanning multiple millennia, continents and languages. Before beginning our journey with Euclid, we introduce the two most important and familiar geometric objects as our main characters, the line and the circle. These characters coincide with the Euclidean tools provided by the axioms at the start of the Elements. For us, they serve as a narrative touchstone as we periodically check in on them as we make our way through the Elements, noting that, while some books em- phasize the line, others highlight the circle. In Book I, for example, the line has the starring role in the proposition statements, but the circle is doing a lot of the heavy lift- ing behind the scenes within the proofs.
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