Geometry: the Line and the Circle VOL AMS / MAA TEXTBOOKS 44

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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: 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 Maureen T. Carroll and Elyn Rykken 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.

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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

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© 2018 by the authors. All rights reserved. Printed in the United States of America. ⃝∞ 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 experience teaching junior/senior level advanced geometry and history of mathematics courses for over twenty years. We have learned that, independent of ability and mathematical maturity, most students have only faint memories of the synthetic Euclidean geometry that they studied in high school. This, coupled with the fact that mathematics 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 readers 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 emphasize 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. When we move beyond Euclidean geometry, we identify the behavior of our main characters in each of their new environments in order to keep track of any changes. Comparing and contrasting the nature of these fundamental figures in other geometries has the added benefit of challenging thereader’s preconceived notions of straightness and roundness, forcing a re-examination of basic geometry concepts. In particular, we encourage the reader to contemplate provocative questions such as: What is a straight line? What does parallel mean? What is distance? What is area? Euclid’s Elements is a mathematical achievement of historical significance. No other mathematics text has been published as many times or read by as many people as the Elements. Its longevity is due to its clarity, rigor and, most importantly, its supe- rior organization and development of geometry as an axiomatic system. It is the proper gateway to the study of axiomatic systems. Of course, the only way to fully appreciate and understand Euclidean geometry is to step outside of it in order to gain perspective. We do this by exploring Spherical, Taxicab, Hyperbolic, and finite affine and projective geometries. In fact, we take a detour halfway through Euclid’s first book in order to consider two of these geometries. This change of perspective at an early stage of our exploration of the Elements provides a natural way to expose hidden flaws in Eu- clid’s reasoning. It also sheds light on the importance of the axiomatic development

xi xii Note to the Instructor of mathematics, and creates an avenue to discuss the difference between axiomatic systems and their models, as well as the desirable properties of such systems. The history of geometry comprises the majority of the history of mathematics. Fit- tingly, this text includes discussions of many important historical figures and develop- ments in geometry, including the controversy surrounding Euclid’s fifth postulate, the impossible constructions of Greek antiquity, and the development of non-Euclidean, projective and finite geometries. It is important for students to understand that new mathematics does not arrive fully formed on the page, but rather, evolves as it is dis- covered and created by individuals, sometimes spanning centuries.

Outline of the book This book covers traditional Euclidean geometry with an axiomatic approach through the lens provided by the Elements. After introducing our two main characters in Chap- ter 1, we discuss Euclid’s definitions, Postulates, and Common Notions in Chapter 2. Chapter 3 presents Neutral geometry, covering the first 28 propositions of Book I along with I.31. We take a detour from the Elements to explore Spherical and Taxicab geometries in Chapters 4 and 5. These geometries force us to reconsider our preconceived ideas about straightness, distance and area. They also reveal the gaps in Euclid’s original set of axioms. Opening the door to other geometries early in the book allows us to consider axiomatic systems in general, and to introduce Hilbert’s axioms for plane geometry in Chapter 6 as a way to shore up the gaps in Euclid’s foundation. Here, we also detail the fundamental and desirable properties of axiomatic systems and the mathematicians who were the first to successfully navigate these metamathematical waters. After discussing Neutral, Spherical and Taxicab geometries, we head back to Eu- clid’s Book I in Chapter 7. By including Book I in its entirety, readers experience the beauty of Euclid’s reasoning and his modular approach to the systematic development of propositions. He meticulously avoids any use of the Parallel Postulate for as long as possible and builds new tools as allowed. It is only at the very end of Book I, after carefully working his way through triangle congruence, parallelism, area and quadrature, that we discover how these pieces fit together to achieve his ultimate goal, the proof of the Pythagorean Theorem and its converse. Geometric algebra is the topic of Book II in Chapter 8, where readers should experience a newfound appreciation for algebra as it greatly simplifies the propositions of this book. This chapter is particularly helpful for secondary education students since it includes the Law of Cosines. Euclid’s book ends with the quadrature of any polygon, but the chapter ends with a return to Spherical geometry as we consider which figures can be “squared” on a sphere. Chapter 9 briefly covers the topic of similarity (found in Book VI) and ends with a generalization of the Pythagorean Theorem. Chapter 10 explores Euclid’s third book where the focus is squarely on the circle, and our oft-neglected main character gets some long overdue “me time.” Chapter 11 covers Book IV which concerns concurrency points of a triangle, constructions of regular polygons, and results about inscribed and circumscribed circles. In Chapter 12 we shift to Hyperbolic geometry where we continue to re-examine the concepts of straightness, parallelism, distance and area. Understanding the useful- ness of models, we present three models for Hyperbolic geometry, ultimately focusing Designing a course using this text xiii on the Poincaré Half-Plane model. In Chapter 13, we give an axiomatic development of Hyperbolic geometry and prove the following surprising results: parallel lines are not everywhere equidistant, the angle sum of a triangle is less than two right angles, rect- angles and squares do not exist, AAA is a congruence scheme, and area is a function of angle sums rather than lengths. We follow this in Chapter 14 with an axiomatic development of finite affine and projective planes, including a lengthy discussion of the history of the development of projective geometry. Chapter 15 covers isometries of the Euclidean and hyperbolic planes. We finish the book with a return to Euclidean geometry to tell the storyof four classic construction problems of Greek antiquity and how the nineteenth-century solutions to these problems unambiguously marked the limitations of the Euclidean tools while simultaneously opening new paths ripe for mathematical exploration.

Designing a course using this text With a healthy amount of both Euclidean and non-Euclidean geometry, instructors are free to choose their focus based on the needs, abilities and mathematical maturity of their students. We outline an option for a course with a majority emphasis on Eu- clidean topics below, but this can easily be revised to form a minority Euclidean course. Regarding the Euclidean content of the book, rather than simply picking and choosing highlights of the first six books of Euclid’s Elements, we have included all of the propositions from the first two books, over half of the propositions from the third book,and nearly all of the fourth book. We do not, however, intend for the instructor to show all of these propositions in class. Our courses are designed to be more interactive than a standard lecture format course, as we find that it helps our students take ownership of the material. We recommend a plan where students rewrite Euclid’s propositions using modern notation and present their updated version to the class. Through this process, students gain skill in both writing and presenting proofs since they must carefully read the proofs in order to separate assumptions from conclusions, and to determine the key definitions, postulates and previous propositions needed in each argument. Allof our students, not simply those seeking a degree in secondary education, benefit from these oral presentations (as recommended by Mathematical Association of America [MAA] guidelines). For the construction propositions, we suggest that students re- produce these results using dynamic geometry software. This methodical approach to Euclidean geometry highlights the strength of an axiomatic development and has the added benefit of clearly distinguishing Neutral from non-Neutral propositions. Cer- tainly, some of these propositions can be skipped in the classroom altogether, and they are available in the book as a resource. While there is more content than can be covered in a semester, this book is, never- theless, designed to be used for a one-semester course in geometry. Its intended audi- ence is junior/senior undergraduate mathematics majors (be they seeking certification in secondary education or not). One could spend the entire semester on the first eleven chapters, but by carefully choosing topics, we expect that an instructor can cover these chapters in roughly eight or nine weeks. To get a sense of how to pace a course, we have included a breakdown of time that we typically spend on each of these chapters. We have taught the course three days a week with 50-minute periods, and two days a week with 75-minute periods. xiv Note to the Instructor

We fully expect students to read this book. In that spirit, we assign the reading of Chapter 1 before the first class, and we discuss Chapter 1, as well as Chapter 2,onthe first day. We then cover the first three propositions of Book I and introduce dynamic geometry software such as Geometer’s Sketchpad®or GeoGebra in class. We have found that a basic familiarity with geometric terms facilitates a smooth transition from understanding Heath’s translation of Euclid’s propositions and proofs to updating the proofs with revised notation. Consequently, for roughly a week we have the students present their updated versions of the propositions from Chapter 3, skipping the proofs of the construction propositions which are included in a lab assignment utilizing the software. For Chapter 4, we cover the basic ideas of Spherical geometry and then have the students explore which propositions of Neutral geometry still hold in Spherical geometry. Since this chapters lends itself to hands-on exploration, we have our students work in groups to explore the nature of lines, circles and triangles in this strange new uni- verse by utilizing strings, markers, tape and inexpensive plastic balls. The sections on Spherical trigonometry and uniquely Spherical constructions are optional. Likewise, for Chapter 5, we discuss only the basic ideas of Taxicab geometry before ceding the re- mainder of the class to student exploration to determine which propositions of Neutral geometry still hold in Taxicab geometry. We do not cover all of the proofs in Chap- ter 6 in the classroom, instead choosing to highlight desirable properties of axiomatic systems as well as Hilbert’s axioms. We spend over a week in Chapter 7, once again having students re-write and present select propositions up to I.46. We take the helm to present the Pythagorean Theorem and its converse. Chapters 8 and 9 take about a week, with “Quadrature on the Sphere” and “A Generalized Pythagorean Theorem” as optional sections. Students present select propositions from Chapter 10 for roughly a week, and we use dynamic software to discuss much of Chapter 11. • Chapters 1–3 (roughly two weeks) • Chapters 4–6 (roughly two weeks) • Chapters 7–9 (roughly two weeks) • Chapters 10–11 (roughly two weeks) After Chapter 11, there is considerable flexibility for an instructor to choose asub- set of the remaining five chapters according to his or her own interests. One ofthe authors typically spends the rest of the semester on Chapter 16 (one week), followed by Chapters 12 and 13 (roughly four weeks—skipping some of the proofs from Chap- ter 13), and finishing with Chapter 15 (roughly two weeks—omitting inversions and Hyperbolic isometries). The other author replaces Chapter 15 with Chapter 14. It is also possible to reserve more weeks for the last five chapters of the book by choosing only the highlights of Chapters 8, 9, 10 and 11.

Common Core State Standards for Mathematics [CCSS]. The following recommendations are taken from the Geometry Course report of the Geometry Study Group [GSG]. The group was charged by the MAA’s Committee on the Undergraduate Program in Mathematics [CUPM] with making recommendations about geometry in the undergraduate mathematics curriculum. Their report is part of the 2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences. Below, we briefly describe how this book addresses each recommended topic. Designing a course using this text xv

GSG writes: “To be prepared to teach a geometry course based on CCSS, future teachers should take a college geometry course in which definitions and proof are em- phasized. In addition, the course they take should include coverage of the following topics:” • Proof We emphasize reading, writing and presenting proofs throughout the book. • Transformations In Chapter 15, we prove that any isometry in Neutral geometry can be written as the composition of three or fewer reflections. We then study reflections, rotations, translations and glide reflections in the Euclidean plane. We also consider Eu- clidean inversions and their role as reflections in the Poincaré Half-plane model of the hyperbolic plane. • Parallel Postulate By separating Book I into Chapters 3 and 7, our book is clear on which results in Euclidean geometry depend on the Parallel Postulate. By presenting Spherical, Hyperbolic and projective geometries, we provide multiple two-dimensional geometries in which the Parallel Postulate does not hold. • Pythagorean Theorem In addition to Euclid’s proof, we include seven other well-known proofs of this famous theorem in our exercises, including proofs by Bhāskara, Leonardo da Vinci, U.S. President James A. Garfield and Thābit ibn Qurra. We also consider agener- alized version of this theorem. • Dynamic geometry software We encourage the use of dynamic geometric software throughout the textbook as it provides valuable insight into geometric relationships. We routinely use Geome- ter’s Sketchpad® or GeoGebra in our courses as well as Spherical Easel and Non- Euclid. • Historical perspectives We incorporate historical context throughout the book, particularly regarding the resolution of the controversy surrounding Euclid’s fifth postulate, the development of Hyperbolic, affine and projective geometries, the classic impossible constructions of Greek antiquity, and the development of the mathematics required to prove the impossibility of these constructions. • Real-life applications Chapters 11 and 14 include connections between art, architecture and geometry. Based on these recommendations, an appropriate course could include the following: • Chapters 1–3 • Chapter 4, sections 1–5 • Chapters 5–7 • Chapter 8, sections 1–2 (while algebraic in nature, helpful for future teachers) • Chapter 9, sections 1–2 • Chapter 10 • Chapter 11 (optional) • Chapters 12–13 xvi Note to the Instructor

• Chapter 14 (optional) • Chapter 15, sections 1–3 • Chapter 16 (optional)

Figure 0.1. Section dependency chart Note to the Reader

Good stories have conflict and resolution. The story of geometry is no exception. The characters in this story are geometric objects known since childhood: lines, circles, triangles and squares, to name a few. What conflict could these characters possibly gener- ate? Are we not confident in our deeply ingrained understanding of these fundamental figures? Perhaps you recall a few core facts about triangles—say, the Pythagorean The- orem for right triangles or the 180-degree angle sum of any triangle. While we will revisit these and other well-known results, we will also visit geometric lands where these bedrocks no longer hold, where in some worlds lines are circles, and in others, circles are squares. How is this possible? To paraphrase Walt Whitman, geometry is large; it contains multitudes. Mathematics, by its very nature, is logical and systematic and, yet, it can still produce results that astonish. One famous case involves Georg Cantor (1845–1918) who, while exploring the nature of infinite sets, documented his incredulity upon discov- ering that intuition had led him astray, writing to a friend, “Je le vois, mais je ne le crois pas!” (“I see it, but I don’t believe it!”) Cantor was surprised by the conflict that arose when his findings contradicted his expectation that the infinite would play bythe same rules as the finite, and yet, he was delighted by the resolution his mathematical reasoning provided. In the same spirit, we aim to present you with a few surprises in geometry that run counter to your intuition. As Cantor’s story illustrates, surprise requires expectation, and expectation comes from experience. Thus, we need to build our experience and examine our pre-existing assumptions. We start with this fundamental question: How many geometries are there? If you think there is only one then you are in fine company. For 2000 years, there was only one geometry to study, and an ancient Greek mathematician named Euclid was its primary expositor. His book, the Elements, is the most famous and most published mathematics book of all time. Translated into many languages, it was standard reading for students through the centuries and, fittingly, we have chosen it as the starting point for our explorations. It was only relatively recently in the history of mathematics that Euclid’s geometry was found to be just one of many interesting and equally valid geometric worlds. What triggered this revolution? While our two- dimensional figures appear unambiguous, their properties and very nature aremore elusive than suggested by first glance. As we will see, a quest to resolve basic questions about the nature of parallel lines was responsible for this seismic paradigm shift. As we embark on this trip, we first take a closer look at Euclid’s geometry, itsori- gins and its axioms. The trip is all-inclusive; though we have prompted you here for your geometric recollections, we will provide all of the definitions, theorems and proofs necessary for the journey (even the Pythagorean Theorem). At the start, the words of

xvii xviii Note to the Reader

Euclid’s propositions and proofs will be familiar, but the style will be unlike others you have encountered. They are verbose, lacking most of the symbolic language and notation in use today. To better understand a Euclidean proposition and its proof, we suggest that you rewrite it in your own words using standard symbols and notation. Mathematics is a language, and the act of translating this language is a good way to learn how to read it and write it. Austrian Stefan Zweig (1881–1942), the author upon whom The Grand Budapest Hotel is based, shared this view. As a young writer he spent several years translating the works of French masters as an improvised apprenticeship in the literary arts. In doing so, he learned the structure of a good book without the pressure of creating the characters, plot or narrative. We echo Zweig’s advice to young writers to translate a seasoned author’s work into your language, as this is a reliable method of learning Euclid’s geometry and the art of writing proofs. As a translator, you do not have to create the mathematics, but you will come to understand the logical structure necessary to write clear, correct proofs. Finally, to state the obvious, we wrote this book to be read by you. To that end, we have included a considerable amount of commentary, history and explanation to help guide you through the story of geometry. Even with the additional narrative, reading a mathematics book is neither an easy nor a passive endeavor. To understand the mathematics you will need to read and then reread the axioms, definitions, theorems and proofs. We find it best to be an active reader with pencil and paper at the ready.Most importantly, as we journey to other strange worlds, keep an open and agile mind and be prepared to abandon preconceived notions as we reconsider and revise our assumptions about geometry and its most familiar objects. Acknowledgments

Before we extend our thanks to the organizations and individuals whose efforts were directly related to the preparation of this book, we would like to acknowledge the ripple effect created by two professional programs sponsored by the Mathematical Associa- tion of America over two decades ago. We thank Christine Stevens and Jim Leitzel for starting Project NExT, and Victor Katz and V. Frederick Rickey for creating the In- stitute in the History of Mathematics and Its Use in Teaching. Both programs helped shape our early careers, and most importantly, we owe our meeting to the former. As mathematicians the world is our office. With that in mind, we thank thecon- structors and supporters of the 165-mile long Pennsylvania Delaware & Lehigh Trail for providing the beautiful path where we logged thousands of miles bicycling in quiet contemplation or lively discussion about geometry and all book-related matters. To Muhlenberg College and the University of Scranton we extend our sincere thanks for sabbaticals, awards, grants and funding in support of our scholarship. We are grateful to our students who have carefully read various drafts of the manuscript over the years: Elyn’s Advanced Geometry students from 2013, 2015 and 2017, and Maureen’s Geometry students from 2014, 2016 and 2018. Special thanks go to Myles Dworkin and Eric Jovinelly for their insights and suggestions, and to Danny Clark, Jes- sica Hollister and Rob McCloskey for combing the manuscript for typographical errors. Finally, thanks to the reviewers and editors at the MAA, in particular Steve Kennedy and Stan Seltzer, and to the American Mathematical Society production staff, in particular Christine Thivierge, Peter Sykes and Becky Rivard for helping us to make our vision of this book a reality.

xix

Credits, Permissions and Acknowledgements

Figure 1.1: Circles in a Circle by Vasily Kandinsky (1923) courtesy of the Philadelphia Museum of Art, Object Number 1950-134-104, The Louise and Walter Arensberg Col- lection, 1950. Excerpt from “The Stretched String” in Section 1.2: Excerpt of the essay from In- ner Issues of The Mathematical Experience by Philip J. Davis, Reuben Hersh and Elena Anne Marchisotto, is reprinted by permission from Springer Nature ©2012 [30]. Excerpt from Black Elk Speaks: The Complete Edition in Section 1.2: Excerpt from the book [90] by John G. Neihardt is reprinted by permission of the University of Nebraska Press ©2014 by the Board of Regents of the University of Nebraska. Figure 1.2: National Museum of the American Indian, Washington, D.C., photograph in the Carol M. Highsmith Archive, Library of Congress, Prints and Photographs Divi- sion. Figure 2.1: CALVIN AND HOBBES ©1991 Watterson. Reprinted with permission of UNIVERSAL Uclick. All rights reserved. Euclid’s Elements: Common notions, postulates, selected definitions, propositions and proofs from Books I-VI of Sir Thomas Heath’s translation of the Elements [40] ap- pearing in Chapters 2, 3, 7-11, and Appendices A, B and D, are reprinted with the permission of Dover Publications. Figure 3.1: Proposition I.1 of Oliver Byrne’s The Elements of Euclid [19] courtesy of the University of Toronto Libraries. Figure 4.1: Photograph courtesy of the NASA Earth Observatory, earthobservatory.nasa .gov Figure 6.1: Hilbert photograph from the Archives of the Mathematisches Forschungsin- stitut Oberwolfach Archives. Gödel photograph from the Kurt Gödel Papers, the Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ, on deposit at Princeton University. Photographer unknown. Hilbert’s axioms: As found in Section 6.3 and Appendix E, Hilbert’s axioms are reprinted by permission of Open Court Publishing Company, a division of Carus Publishing Company, Chicago, IL, from Foundations of Geometry [74] by D. Hilbert (trans. L. Unger), ©1971 by Open Court Publishing Company.

463 464 Credits, Permissions and Acknowledgements

Figure 7.35: Photograph of Babylonian tablet Plimpton 322 by C. Proust courtesy of the Rare Book & Manuscript Library at Columbia University. Used with permission. isaw.nyu.edu/exhibitions/before-pythagoras Figure 10.1: Olympic rings used with permission from the United States Olympic Committee. Figure 11.35: Close-up of 17-point star on statue of Gauss from Benutzer:Brunswyk via Wikimedia Commons, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license Figure 11.36: Homage to Carl Friedrich Gauss by Eugen Jost. Plate 23, p. 74, from BEAUTIFUL GEOMETRY by Eli Maor and Eugen Jost [86]. Copyright ©2014 by Prince- ton University Press. Reprinted by permission. Artist contact: [email protected] Figure 12.1: M.C. Escher’s Circle Limit III, ©2013 The M.C. Escher Company-The Netherlands. All rights reserved. www.mcescher.com Figure 12.3: Mercator projection, derived from NASA Earth Observatory Blue Marble series composite photograph, by Daniel R. Strebe, December 16, 2011, via Wikimedia Commons, lines enhanced by Elyn Rykken, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. Figure 12.4: Mollweide projection, derived from NASA Earth Observatory Blue Mar- ble series composite photograph, by Daniel R. Strebe, August 15, 2011, via Wikime- dia Commons, licensed under the Creative Commons Attribution-Share Alike 3.0 Un- ported license. Figure 12.8: M.C. Escher’s Circle Limit IV, ©2013 The M.C. Escher Company-The Netherlands. All rights reserved. www.mcescher.com Figure 14.6: Albrecht Dürer’s illustration, Wikimedia Commons. Figure 14.7: Detail of Hans Holbein the Younger’s The Ambassadors (1533), Wikime- dia Commons. Figure 14.8: Leonardo da Vinci’s The Last Supper (1498), Wikimedia Commons, lines added by Maureen T. Carroll. Section 14.5: Photograph of Waterloo tube station, London, courtesy of Daniel Carroll. All rights reserved. Figure 15.1: Photograph of Thomas Jefferson memorial, Washington, DC, courtesy of Kathy Wahl. All rights reserved. Figure 16.1: Detail of Raphael’s The School of Athens (1510-11), Wikimedia Commons. Figure 16.8: Crockett Johnson’s Heptagon from Its Seven Sides (1973), negative number 2008-2545, appears courtesy of the Division of Medicine & Science, National Museum of American History, the Smithsonian Institution. Every effort has been made to contact copyright holders and obtain permissions for the use of copyright material. Notification of any corrections for future reprints would be greatly appreciated. Bibliography

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Notation Index

∠퐵 refers to the angle at 퐵, page 24 ∠퐴퐵퐶 the angle with vertex 퐵, or the measure of the angle depend- ing on the context, page 11 훿(△퐴퐵퐶) defect of △퐴퐵퐶 in Hyperbolic geometry, page 322 훿(푃) defect of polygon 푃 in Hyperbolic geometry, page 323 ⟷ 퐴퐵 line between 퐴 and 퐵, page 11 퐴Ω⃗ ray with endpoint 퐴 sensed parallel to line with ideal point Ω, page 299 퐴퐵⃗ ray starting at 퐴, page 11 퐴퐵 finite line segment from points 퐴 to 퐵, page 11 퐴퐴퐴˜ Proposition VI.4: angle-angle-angle triangle similarity, page 179 퐴퐴˜ Corollary 9.1: angle-angle triangle similarity, page 180 푆퐴푆˜ Proposition VI.6: side-angle-side triangle similarity, page 179 푆푆푆˜ Proposition VI.5: side-side-side triangle similarity, page 179 휙 golden mean, page 165 Π푛 projective plane of order 푛, page 352 휋푛 affine plane of order 푛, page 360 right angle, page 38 △퐴퐵퐶 ≅ △퐷퐸퐹 △퐴퐵퐶 is congruent to △퐷퐸퐹, page 24 △퐴퐵퐶 ∼ △퐷퐸퐹 △퐴퐵퐶 is similar to △퐷퐸퐹, page 176 △퐴퐵퐶 a triangle with vertices at 퐴, 퐵 and 퐶, page 21 △퐴퐵Ω omega triangle in Hyperbolic geometry, page 302 퐴퐵ˆ minor arc of a great circle between 퐴 and 퐵, page 191 퐴퐶퐵ˆ arc of great circle between 퐴 and 퐵 passing through 퐶, page 63 퐴 point, page 11 퐴퐵 ∥ 퐶퐷 퐴퐵 is parallel to 퐶퐷, page 53 퐴퐵 finite line segment from points 퐴 to 퐵, page 11 퐴푟푒푎(△퐴퐵퐶) Area of △퐴퐵퐶, page 78 퐴푟푒푎(푃) area of polygon 푃, page 323 퐴푟푒푎(푅) Area of region 푅, page 136 퐶퐷 ⟂ 퐴퐵 퐶퐷 is perpendicular to 퐴퐵, page 34 푑(퐴, 퐵) distance between points 퐴 and 퐵, page 94

푑퐸(푃, 푄) distance between 푃 and 푄 in Euclidean geometry, page 95

푑퐻(푃, 푄) distance between 푃 and 푄 in Hyperbolic geometry, page 269

471 472 Notation Index

푑푆(푃, 푄) distance between 푃 and 푄 in Spherical geometry, page 121

푑푇(푃, 푄) distance between 푃 and 푄 in Taxicab geometry, page 95 퐹ℓ reflection across line ℓ in the Euclidean plane, page 378 퐺퐴퐵 glide reflection along 퐴퐵 in the Euclidean plane, page 393 퐻퐹ℓ hyperbolic reflection through line ℓ in the hyperbolic plane, page 406

퐻퐺퐴퐵 hyperbolic glide reflection by 퐴퐵⃗ in the hyperbolic plane, page 408 퐻푅퐴,훼 hyperbolic rotation about point 퐴 by angle 훼 in the hyperbolic plane, page 406 퐻푇퐴퐵 hyperbolic translation mapping 퐴 to 퐵 in the hyperbolic plane, page 407 퐼퐴,푟 inversion in the circle with center 퐴 and radius 푟 in the Eu- clidean plane, page 399 푙푒푛푔푡ℎ( 퐴퐵ˆ ) length of arc of circle 퐴퐵ˆ , page 218 푃 ≡ 푃′ polygon 푃 is equivalent to polygon 푃′ by finite decomposition, page 323 푅퐴,훼 rotation about point 퐴 by angle 훼 in the Euclidean plane, page 386 푇 or 푇 translation in the Euclidean plane by vector 퐴퐵⃗, page 383 퐴퐵 퐴퐵⃗ AAΩ angle-angle congruence for omega triangles, page 306 AAAH angle-angle-angle congruence in Hyperbolic geometry, page 313 AAAS angle-angle-angle congruence in Spherical geometry, page 91 AAS Proposition I.26: angle-angle-side triangle congruence scheme, page 24 ASΩ angle-side congruence for omega triangles, page 305 ASA Proposition I.26: angle-side-angle triangle congruence scheme, page 24 HL hypotenuse-leg congruence scheme for right triangles, page 52 I.16Ω Exterior Angle Theorem for Omega Triangles, page 304 SAS Proposition I.4: side-angle-side triangle congruence scheme, page 24 SASAS side-angle-side-angle-side congruence scheme for convex quadrilaterals, page 32 SSS Proposition I.8: side-side-side triangle congruence scheme, page 24 WLOG Without loss of generality, page 38 Index

36∘−72∘−72∘ triangle, 241 angle at the center, 192 17-gon, 248, 251, 431 angle at the circumference, 192 36 Officer Problem, 370 angle in a segment, 192 angle of parallelism, 264, 294 퐴퐴˜ triangle similarity, 180 on the left, 293 AAΩ, 306 on the right, 293 퐴퐴퐴˜ triangle similarity, 179 antipodal points, 61 AAAH, 313 apothem, 253 AAAS, 91, 182 arc length, 277 AAS, 24, 50 Archimedean point, 228 Abbott, Edwin, 2 Archimedes, 228, 249, 252, 432, 433 Abu Nasr Mansur, 88 Archimedes’ Axiom, 115 Abū al-Wafū’al Būzjānı̄, 221, 240 area, 136 adjacent angles, 53 axioms, 136 affine planes, 355–363 Heron’s formula, 225, 235 order, 360 Hyperbolic geometry parallel class, 357 polygon, 323 Alberti, Leon Battista, 339, 340 triangle, 322 algebraic number, 426 parallelogram, 141 Almagest, 203 rectangle, 137 alternate exterior angles, 53 regular polygon, 253 alternate interior angles, 53 spherical polygon, 81 altitude, 40, 176, 227 spherical triangle, 78 analytic geometry, 94, 354 trapezoid, 141 angle, 9, 115, 437 triangle, 141 acute, 9, 437 Aristotle, 21 adjacent, 53 ASΩ, 305 alternate exterior, 53 ASA, 24, 50 alternate interior, 53 Ascher, Marcia, 2 bisector, 33 asymptotically parallel, 268 exterior, 41, 53 axiomatic method, 8 exterior region, 290 axiomatic system interior, 53 categorical, 111 interior point, 117 complete, 108 interior region, 290 consistent, 107 obtuse, 9, 437 independent, 108, 111 opposite, 53 model, 106 plane, 9, 437 axioms, 8 rectilineal, 9, 437 affine planes, 356 reflex, 192 Archimedes’, 115 right, 9, 38, 437 betweenness, 114 supplementary, 36 Characteristic, 262, 289 vertex, 11 congruence, 115 vertical, 37 continuity, 115

473 474 Index

Fano, 336 angle in a segment, 192 Four Line, 335 center, 10, 438 Four Point, 110, 332 central angle, 191 Hilbert’s, 113 centre, see also center incidence, 114 chord, 191 line completeness, 115 circumference, 216 order, 114 circumscribed about polygon, 222, 223 parallels, 115 congruence, 191 Pasch’s, 114 diameter, 10, 191, 438 projective planes, 347 great, 61 Three Point, 113, 335 greater circumference, 191 greater segment, 191 base, 25 Hilbert’s definition, 116 Beltrami, Eugenio, 262, 265–268 inscribed angle, 192 between, 114 inscribed inside polygon, 222 bicentric quadrilateral, 236 less circumference, 191 biconditional statement, 34 less segment, 191 bijective, 376 major arc, 191 Bolyai, Farkas, 264 minor arc, 191 Bolyai, János, 109, 264 Neutral geometry, 214 Bose, Raj C., 371, 372 non-Neutral geometry, 214 boundary, 9, 437 radius, 191 Brianchon, Charles-Julien, 231, 340, 344 secant, 191 Bruck, Richard, 371 sector, 192 Bruck-Ryser-Chowla Theorem, 371 segment, 191 Brunelleshi, Fillipo, 339, 340 subtend, 191 Bussey, W. H., 336, 370 tangent, 192 Butterfly Theorem, 216 Circle Equivalence Theorem, 204 Byrne, Oliver, 17, 19 circle of inversion, 399 circles Calvin and Hobbes, 7 angle of intersection, 403 Cardinal, Douglas Joseph, 4 arc congruence, 205 Carnot, Lazare, 340, 344, 355 congruent, 191 categorical, 111, 333, 363 equal, 191 center externally tangent, 192 circle, 10, 438 internally tangent, 192 of gravity, 228 orthogonal, 403 of mass, 228 tangent, 192 pentagon, 245 Circular Continuity Principle, 118 regular 푛-gon, 245 circumcenter, 226, 227 rotation, 386 circumcircle, 223 triangle circumference, 216 centroid, 229 earth, 219 circumcenter, 226, 227 circumscribed incenter, 224, 225 circle, 223 orthocenter, 227 polygon, 222 center of inversion, 399 clockwise orientation, 379 central angle, 191 collapsbile compass, 14 regular 푛-gon, 245 collinear, 121, 230 central projection, 344 common notions, 12 centroid, 228, 229 compass, 14 Characteristic Axiom, 262, 289 complete, 108 Chasles, Michel, 340, 344 completing the square, 162 chord, 191 concurrent lines, 223 Chowla, Sarvadaman, 371 concyclic, 231 Cicero, 249 conditional statement, 47 circle, 10, 437 conformal, 271 휋, 216 congruent, 114 angle at the circumference, 192 arcs, 205 Index 475

polygons, 24 Escher, M.C., 261, 270, 272 quadrilaterals, 32 Euclid, 6, 7 triangles, 24 Euclidean construction, 14 AAS, 50 Euclidean distance

ASA, 50 푑퐸(푃, 푄), 95 SAS, 24, 119 Eudoxus, 8, 252 SSS, 29, 119 Euler line, 229, 230 consistent, 107 Euler, Leonhard, 155, 229, 370, 417 constructible number, 419 exterior angle, 41, 53 constructions Exterior Angle Theorem, 41 impossible, 418 omega triangles, 304 neusis, 432–434 extreme and mean ratio, 176 Continuum Hypothesis, 106 converse, 27 Fano plane, 335–338 convex, 12, 79, 150 Fano, Gino, 335, 336, 345 convex polygon, 323 Fermat primes, 248 counterclockwise orientation, 379 Fermat, Pierre de, 248, 342, 354 Crossbar Theorem, 117, 291 Feuerbach point, 231 cyclic polygon, 192 Feuerbach, Karl Wilhelm, 231 figure, 9, 437 Dürer, Albrecht, 221, 243, 339, 340 height, 176 Da Vinci, Leonardo, 156, 339, 341 rectilineal, 10, 438

푑퐸(푃, 푄), 95 finite affine planes

푑퐻(푃, 푄), 269 order, 360

푑푇(푃, 푄), 95 finite geometry dart, 41 affine, 355–363 Dedekind’s Axiom, 289 Fano plane, 335–338 defect Four Line, 335 polygon, 323 Four Point, 110, 332 triangle, 322 intersecting lines, 332 triangulation, 322 parallel lines, 333 Delian problem, see also doubling the cube projective, 347–353 Desargues, Girard, 339, 342, 350, 354, 368, 379 Three Point, 113, 335 Descartes, René, 93, 159, 342, 354 Young’s, 339 descriptive geometry, 343 finite projective planes diameter, 191 order, 352 direct proof, 27 Finlay, Archibald, 135 dissection, 164 fixed point, 375 distance, 64 Flatland, 2 point to line, 44 Four Line geometry, 335 distance function, 94 Four Point geometry, 110, 112, 332 distributive property, 161 Fuss, Nicolaus, 236 divergently parallel, 268 divinely proportioned, 165 Gödel’s Incompleteness Theorem, 125 Dorodnov, A.V., 417 Gödel, Kurt, 105, 106, 124 Double Elliptic geometry, 76 Garfield, James, 157 doubling the cube, 416, 417, 429 Gauss’s Theorem, 248 dual statements, 348 Gauss, Carl Friedrich, 14, 109, 248, 264, 431 duality, 347–349 geodesic, 279 Dupin, Charles, 344 geometric algebra, 159 geometry, 135 earth circumference, 219 descriptive, 343 Elements, 7 affine, 355–363 ellipse, 100 affine planes, 356 Elliptic geometry, 76, 263, 347 analytic, 94, 354 endpoints, 114 Double Elliptic, 76 equidistant, 34, 314 Elliptic, 76, 263, 347 equivalent by finite decomposition, 323 Fano, 335–338 Eratosthenes, 219, 418 Fano plane, 336 476 Index

finite, see also finite geometry area, 320 Four Line, 335 polygon, 323 Four Point, 110, 112, 332 triangle, 322 Hyperbolic, 289 asymptotically parallel, 268 Neutral, 13 Beltrami-Klein model, 268 non-Euclidean, 109 Characteristic Axiom, 289 projective, 339–353 circle, 285 projective planes, 347 defect Spherical, 59 polygon, 323 synthetic, 94, 354 triangle, 322 Taxicab, 93 triangulation, 322 Three Point, 113, 335 distance, 282 Young’s, 339 divergently parallel, 268 Gergonne, Joseph, 340, 344, 355 glide reflection, 408 glide reflection, 393, 408 HAA, 263, 309 gnomon, 160 horolation, 407 Goldbach’s Conjecture, 106 ideal point, 268 golden mean (휙), 165, 177, 215 omega triangle, 302 golden rectangle, 177 orthogonal circles, 270 golden triangle, 223, 241 parallel displacement, 407 Goodwin, Edwin, 98 Poincaré Disc model, 270 Graeco-Latin square, 371 Poincaré Half-plane model, 272 great circle, 61 polygon group, 374 area, 323 defect, 323 HAA (hypothesis of the acute angle), 309 triangulation, 321 Halsted, George Bruce, 262 reflections, 406 Hardy, G. H. , 190 rotation, 406 Heath, Sir Thomas, 9, 262 Saccheri quadrilateral, 308 height, 176 sensed parallel, 268 heptadecagon(17-gon), 248, 251, 431 summit angles, 263, 308 heptagon, 431 translation, 407 Hermes, Johann Gustav, 249 triangle Hermite, Charles, 426 AAAH, 313 Herodotus, 135 angle sum, 311 Heron of Alexandria, 225, 228 area, 322 Heron’s formula, 225, 235 associated Saccheri quadrilateral, 311, 313 hexagon, 247 defect, 322 Hilbert’s axioms, 113 ultraparallel, 268 congruence, 115 hyperbolic reflections, 406 continuity, 115 hyperbolic rotation, 406 incidence, 114 hyperbolic translation, 407 order, 114 hypotenuse, 52 parallels, 115 hypothesis of the acute angle (HAA), 263, 309 Hilbert, David, 105, 107, 110, 267, 350 hypothesis of the obtuse angle (HOA), 263, 309 Hippocrates of Chios, 8, 146, 416 hypothesis of the right angle (HRA), 309 HL, 52 HOA (hypothesis of the obtuse angle), 309 ideal point, 268, 298, 366 Holbein, Hans the Younger, 341 infinity, 299, 317 horizon line, 340, 367 ideal triangle, 298 horn angle, 196 identity, 374, 375 horolation, 407 iff, 34 HRA (hypothesis of the right angle), 309 implication, 27, 47 hyperbola, 100 impossible constructions, 418 Hyperbolic distance doubling the cube, 429

푑퐻(푃, 푄), 269 squaring the circle, 429 Hyperbolic geometry, 109, 262, 289 trisecting the angle, 429 AAAH, 313 incenter, 224, 225 angle of parallelism, 264 incidence geometry, 332 Index 477 incident, 106, 332 Lobachevsky, Nikolai, 109, 264 incircle, 223 lune, 77, 416 independent, 108 injective, 375 Möbius, August, 340, 345 inscribed major arc, 63, 191 circle, 222 congruent, 205 polygon, 222 mapping, 375 inscribed angle, 192 marked straightedge, 432 interior angle, 53 Masaccio, 339, 340 Intersecting Chords Theorem, 210 median, 30, 227 intersecting lines, 332 Menger, Karl, 93 Intersecting Secants Theorem, 210, 215, 231 Mercator, Gerardus, 267 invariants, 346 Mersenne, Marin, 342 inverse, 374, 376 metamathematics, 106 inversion, 399 method of exhaustion, 252 center, 399 metric, 94 circle, 399 midpoint, 30 radius, 399 Minkowski, Hermann, 93 involution, 89, 379, 399 minor arc, 63, 191 isometry, 375 congruent, 205 orientation preserving, 380 mirror symmetry, 381 orientation reversing, 380 model, 106 isosceles trapezoid, 239 abstract, 123 concrete, 123 Johnson, Crockett, 135, 432, 433 conformal, 271 Mollweide, Karl, 267 Kepler, Johannes, 339, 342 Monge, Gaspard, 339, 342 Khayyam, Omar, 20 Morley’s Theorem, 190 kite, 31 Morley, Christopher, 199, 227 Klein, Felix, 76, 266, 268, 335, 340, 345, 374 Morley, Frank, 190 multilateral, 10, 438 Lam, Clement, 372 multiplication table, 392 Lambert, Johann, 310 Latin squares, 370 푛-gon orthogonal, 371 spherical, 79 Law of Cosines, 160, 167 necessary, 47 law of trichotomy, 30 neusis construction, 432–434 left-sensed parallel, 293 Neutral geometry, 13, 109 leg, 52 Newton, Isaac, 253, 355 Leibniz, Gottfried, 253, 355 Nicomedes, 432, 434 lemma, 28 Nine-point circle, 231, 232 length, 64 Nine-point Circle Theorem, 232 lie on, 114 non-Euclidean geometry, 109, 262 Lindemann, Ferdinand von, 105, 146, 426 nonagon, 431 line, 9, 114, 437 noncollinear, 116, 377, 378 at infinity, 367 straight, 4, 9, 437 omega triangles, 302 Line Completeness Axiom, 115 AAΩ: angle-angle congruence, 306 line of reflection, 378 ASΩ: angle-side congruence, 305 line of symmetry, 381 congruent, 305 line segment, 11, 114 Crossbar Theorem, 303 spherical, 68 I.16Ω: Exterior Angle Theorem, 304 lines interior, 302 concurrent, 223 isosceles, 307 intersecting, 332 one-point perspective, 341 parallel, 10, 333, 438 one-to-one, 375 perpendicular, 9, 437 onto, 375 reflection, 378 opposite angles, 53 Liouville, Joseph, 426 order, 352, 360 478 Index

affine planes, 360 polar, 72 projective planes, 352 polar orientation points, 72 preserving, 379 triangle, 88 reversing, 379 poles, 72 orthocenter, 227 polygon, 11 orthocentric set, 228 circumscribed about circle, 222 orthogonal circles, 270 convex, 12, 150, 323 orthogonal Latin squares, 371 cyclic, 192 orthogonal projection, 343 defect, 323 exterior, 117 Pacioli, Luca, 165 height, 176 Pappus of Alexandria, 26, 342 Hyperbolic geometry parallel, 128, 333, 355 area, 323 parallel class, 357 inscribed inside circle, 222 parallel displacement, 407 interior, 117 Parallel Postulate, 14, 128, 262 regular, 14, 150, 184, 185, 246 independence, 109 15-gon, 247 parallel projection, 343 17-gon, 248, 251, 431 parallelogram, 12, 133, 138 pentadecagon, 247 about the diameter, 147 heptagon, 431 contained by, 160 hexagon, 247 gnomon, 160 nonagon, 431 rectangular, 160 pentagon, 243–245 Pascal, Blaise, 339, 342 spherical, 82 Pasch’s Axiom, 114, 291 similar, 176 Pasch, Moritz, 110, 116 spherical, 79 Peano, Giuseppe, 110, 335 regular, 82 pentadecagon or 15-gon, 247 triangulation, 321 pentagon, 243, 245 Poncelet, Jean-Victor, 231, 340, 344, 355, 368 center, 245 pons asinorum, 25 Perelman, Grigori, 106 postulates, 12 perpendicular primitive terms, 106 uniqueness, 42 Proclus, 8, 26 perpendicular bisector, 34 Proclus’ Axiom, 129 perspective transformation, 345 projection, 266 perspectivity, 345 central, 344 휋, 216, 251 orthogonal, 343 Pieri, Mario, 110, 335 parallel, 343 Plücker, Julius, 340, 345, 355 projective geometry, 339–353 plane, 9, 114, 437 projective planes, 347 punctured, 399 finite, 347 plane separation theorems order, 352 line, 116, 290 real, 368 polygon, 117, 290 projective properties, 346 Plato, 5, 8 projective transformation, 346 Playfair’s Axiom, 116, 129, 262 proof Playfair, John, 116 by contradiction, 27 Plimpton 322, 155 by dissection, 164 Poincaré Conjecture, 106 by superposition, 25 Poincaré, Henri, 106, 266, 270 direct, 27 point, 9, 114, 437 reductio ad absurdum, 27 at infinity, 366 Ptolemy’s Theorem, 203, 210, 406 of concurrency, 223 Ptolemy, Claudius, 202, 203, 243 of intersection, 332 punctured plane, 399 points Pythagoras, 8 antipodal, 61 Pythagorean spiral, 419 collinear, 121, 230 Pythagorean Theorem, 151 noncollinear, 116, 377, 378 da Vinci’s proof, 156 Index 479

Spherical, 83 Richelot, Friedrich Julius, 249 Bhāskara’s proof, 164 Richmond, Herbert W., 251 dissection proof, 164 Riemann Hypothesis, 106 Garfield’s proof, 157 Riemann, Bernhard, 69, 268 proof using Ptolemy’s Theorem, 210 right triangle similarity proof, 187 hypotenuse, 52 Thābit ibn Qurra’s proof, 156 hypotenuse-leg (HL) congruence, 52 Pythagorean triples, 155 leg, 52 right-sensed parallel, 293 Q.E.D., 18 rigid motion, 375 Q.E.F., 18 Roberval, Gilles, 342 quadrable, 146 rotation, 386 quadrature, 145, 146, 169 rotational symmetry, 387 quadrilateral, 10, 438 Ryser, Herbert J., 371 bicentric, 236 congruent, 32 Saccheri quadrilateral, 307 convex base, 308 SASAS congruence, 32 summit, 308 dart, 41 summit angles, 308 kite, 31 Saccheri, Girolamo, 262, 307 oblong, 10, 438 SAS, 24, 119 rhomboid, 10, 438 푆퐴푆˜ triangle similarity, 181 rhombus, 10, 438 SASAS, 32 semiperimeter, 239 secant, 191 spherical, 79 sector, 192 square, 10, 438 Segre, Corrado, 335 tangential, 236 semicircle, 10, 191, 438 trapezia, 10, 438 semiperimeter, 225, 239 quadrisect, 251 sensed parallel, 268, 296 rays, 299 radius, 14 sensed parallel to ℓ, 295 radius of inversion, 399 side, 25 ray, 11, 115 similar, 176 real projective plane, 368 polygons, 176 rectangle, 12, 57, 132 triangles, 176 area, 137 퐴퐴퐴˜, 176 spherical, 171 푆퐴푆˜, 176 rectangular parallelogram, 160 푆푆푆˜, 176 reductio ad absurdum, 27 similarity, 175 reflection, 378 spherical reflectional symmetry, 381 excess, 79 reflex angles, 192 line segment, 68 regular 푛-gon lune, 77 center, 245 푛-gon, 79 central angle, 245 polar triangle, 88 regular polygon, 150, 184, 185, 246 polygon, 79 15-gon, 247 quadrilateral, 79 17-gon, 248, 251, 431 triangle, 67 apothem, 253 Spherical geometry, 59 area, 253 AAAS, 91, 182 heptagon, 431 line segment, 68 hexagon, 247 major arc, 63 nonagon, 431 minor arc, 63 pentadecagon, 247 polar points, 72 pentagon, 243–245 poles, 72 spherical, 82 Pythagorean Theorem, 83 relatively consistent, 123 triangle rhomboid, 138 AAAS, 91, 182 rhombus, 57 spherical rectangle, 171 480 Index spherical square, 171 base, 25 squarable, 146 centroid, 228, 229 square, 10, 150 circumcenter, 226, 227 spherical, 171 congruence, 24 squaring the circle, 146, 416, 429 equilateral, 10, 438 SSA, 56 golden, 241 SSS, 24, 29, 119 Hyperbolic geometry 푆푆푆˜ triangle similarity, 180 area, 322 Star Trek Theorem [III.20], 201 incenter, 225 Steiner, Jakob, 340, 345 isosceles, 10, 438 straightedge, 14 median, 227 subgroup, 392 obtuse, 10, 438 subtend, 191 orthocenter, 227 sufficient, 47 polar, 88 summit angles, 263 right, 10, 52, 438 superposition, 25, 101, 373 scalene, 10, 438 supplementary angles, 36 semiperimeter, 225 surd, 421 side, 25 surface, 9, 437 side-angle-side (SAS) congruence, 24 plane, 9, 437 side-angle-side (푆퐴푆˜) similarity, 176, 181 surjective, 375 side-angle-side (SAS) congruence, 119 Sykes, Mabel, 221, 222, 240 side-side-side (푆푆푆˜) similarity, 176, 180 symmetry side-side-side (SSS) congruence, 24, 29, 119 line, 381 similar, 176 mirror, 381 spherical, 67 reflectional, 381 Triangle Inequality, 45, 74, 94 rotational, 387 triangulation, 321 synthetic geometry, 94, 354 border triangulation, 321 defect, 322 tangent star triangulation, 321 circles, 192 subdivision, 325 line to circle, 192 trilateral, 10, 438 tangential quadrilateral, 236 trisecting the angle, 33, 416, 418, 429 Tarry, Gaston, 370 Tschebotaröw, N.G., 417 Taxicab distance

푑푇(푃, 푄), 95 ultraparallel, 268, 293, 296 Taxicab geometry, 93 ultraparallel to ℓ, 295 Thābit ibn Qurra, 156 unmarked straightedge, 14 Thales of Miletus, 8, 205 Thales’ Theorem, 206 vanishing point, 340, 366 converse, 234 Veblen, Oswald, 336, 370 Theon of Smyrna, 418 vector, 383 Three Point geometry, 113, 335 Vertical Angle Theorem, 37, 39 transcendental number, 426 vertical angles, 37 transformation, 375 von Staudt, Karl, 340, 345, 368 translation, 382 Waldo, Clarence, 98 transversal, 52 Wallenius, Martin Johan, 417 trapezoid, 11 Wantzel, Pierre, 33, 430 area, 141 Wilansky, Albert, 347 isosceles, 239 WLOG, 38 triangle, 10, 438 36∘−72∘−72∘, 241 Young’s geometry, 339 acute, 10, 438 Young, John Wesley, 339 altitude, 40, 227 angle-angle (퐴퐴˜) similarity, 180 angle-angle-angle (퐴퐴퐴˜) similarity, 176, 179 angle-angle-side (AAS) congruence, 24, 50 angle-side-angle (ASA) congruence, 24, 50 area (Heron’s formula), 225, 235 AMS / MAA TEXTBOOKS VOL 44

Geometry: The Line and the Circle VOL AMS / MAA TEXTBOOKS 44

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