Geometry: The Line and the Circle

502 Pages·2018·6.62 MB·English

by  Maureen T. Carroll

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 is an undergraduate text with a strong G
narrative that is written at the appropriate level of rigor for an upper-level
e
survey or axiomatic course in geometry. Starting with Euclid’s Elements,
o
the book connects topics in Euclidean and non-Euclidean geometry in an
m
intentional and meaningful way, with historical context.
e
The line and the circle are the principal characters driving the narrative.
t
In every geometry considered—which include spherical, hyperbolic, and
M r
taxicab, as well as finite affine and projective geometries—these two a y
u
objects are analyzed and highlighted. Along the way, the reader contem- re : T
plates fundamental questions such as: What is a straight line? What does e
n h
parallel mean? What is distance? What is area? T
. e
C
There is a strong focus on axiomatic structures throughout the text. While
a
Euclid is a constant inspiration and the Elements is repeatedly revisited rr L
o
with substantial coverage of Books I, II, III, IV, and VI, non-Euclidean ll in
a
geometries are introduced very early to give the reader perspective n
e
d
on questions of axiomatics. Rounding out the thorough coverage of E
axiomatics are concluding chapters on transformations and constructa- ly a
n n
bility. The book is compulsively readable with great attention paid to the
R
historical narrative and hundreds of attractive problems. y d
k
k
e t
n h
e

C
i
r
c
l
e
For additional information
and updates on this book, visit
www.ams.org/bookpages/text-44 A
M
S
/
M
A
A

P
TEXT/44 R
E
S
S
4-Color Process 496 pages on 50lb stock • Backspace 1 11/16''
Geometry: The Line
and the Circle

AMS/MAA TEXTBOOKS
VOL 44
Geometry: The Line
and the Circle
Maureen T. Carroll
Elyn Rykken
CommitteeonBooks
JenniferJ.Quinn,Chair
MAATextbooksEditorialBoard
StanleyE.Seltzer,Editor
BelaBajnok SuzanneLynneLarson JeffreyL.Stuart
MatthiasBeck JohnLorch RonD.Taylor,Jr.
HeatherAnnDye MichaelJ.McAsey ElizabethThoren
WilliamRobertGreen VirginiaNoonburg RuthVanderpool
CharlesR.Hampton
2010MathematicsSubjectClassification.Primary51-01.
Foradditionalinformationandupdatesonthisbook,visit
www.ams.org/bookpages/text-44
LibraryofCongressCataloging-in-PublicationData
Names:Carroll,MaureenT.,1966–author.|Rykken,Elyn,1967–author.
Title:Geometry:Thelineandthecircle/MaureenT.Carroll,ElynRykken.
Description:Providence,RhodeIsland:MAAPress,animprintoftheAmericanMathematicalSociety,
[2018]|Series:AMS/MAAtextbooks;volume44|Designedforanupper-levelcollegegeometrycourse.
|Includesbibliographicalreferencesandindexes.
Identifiers:LCCN2018034790|ISBN9781470448431(alk.paper)
Subjects:LCSH:Geometry–Textbooks.|Geometry–Studyandteaching(Higher)
Classification:LCCQA445.C29852018|DDC516–dc23
LCrecordavailableathttps://lccn.loc.gov/2018034790
Copyingandreprinting. Individualreadersofthispublication,andnonprofitlibrariesactingforthem,
arepermittedtomakefairuseofthematerial,suchastocopyselectpagesforuseinteachingorresearch.
Permissionisgrantedtoquotebriefpassagesfromthispublicationinreviews,providedthecustomaryac-
knowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublicationispermit-
tedonlyunderlicensefromtheAmericanMathematicalSociety.Requestsforpermissiontoreuseportions
ofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. Formoreinformation,please
visitwww.ams.org/publications/pubpermissions.
Sendreq[e-mail].
©2018bytheauthors.Allrightsreserved.
PrintedintheUnitedStatesofAmerica.
⃝1Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttps://www.ams.org/
10987654321 232221201918
Toourparentsfortheirlove,supportandencouragement

Contents
NotetotheInstructor xi
Outlineofthebook xii
Designingacourseusingthistext xiii
NotetotheReader xvii
Acknowledgments xix
1 TheLineandtheCircle 1
1.1 Introduction 1
1.2 Whichcamefirst? 2
1.3 Whatisastraightline,anyways? 4
2 Euclid’sElements: DefinitionsandAxioms 7
2.1 TheElements 7
2.2 Definitions 9
2.3 Postulatesandcommonnotions 12
3 BookIofEuclid’sElements: NeutralGeometry 17
3.1 PropositionsI.1throughI.8 18
3.2 PropositionsI.9throughI.15 32
3.3 PropositionsI.16throughI.28andI.31 41
4 SphericalGeometry 59
4.1 Whatisastraightline,anyways? -Part2 60
4.2 TrianglesinSphericalgeometry 65
4.3 Euclid’saxiomsviewedinSphericalgeometry 68
4.4 Neutralgeometryonthesphere 71
4.5 AreainSphericalgeometry 77
4.6 Trigonometryforsphericaltriangles 82
4.7 Uniquelysphericalconstructions 88
5 TaxicabGeometry 93
5.1 Points,lines,angles,distancesandcircles 94
5.2 Euclid’spostulatesinTaxicabgeometry 100
5.3 CongruenceschemesinTaxicabgeometry 101
5.4 TherestofNeutralgeometry 102
6 HilbertandGödel 105
6.1 Axiomaticsystems 106
6.2 AFourPointgeometry 110
vii
viii Contents
6.3 Hilbert’saxiomsforEuclideanplanegeometry 113
6.4 SphericalandTaxicabgeometries 120
6.5 Gödelandconsistency 123
7 BookI:Non-NeutralGeometry 127
7.1 Parallellines 128
7.2 PropositionsI.32andI.33 132
7.3 Area 135
7.4 PropositionsI.34throughI.41 138
7.5 PropositionsI.42throughI.46 145
7.6 ThePythagoreanTheorem 151
8 BookII:GeometricAlgebra 159
8.1 PropositionII.1throughII.10 160
8.2 PropositionsII.11throughII.14 165
8.3 Quadratureonthesphere 171
9 BookVI:Similarity 175
9.1 BookV:Ratioandproportion 176
9.2 Similarity 176
9.3 AgeneralizedPythagoreanTheorem 184
10 BookIII:Circles 189
10.1 Definitions 190
10.2 Tangency 193
10.3 Arcs,chordsandangles 201
10.4 AreaPropositions: III.35throughIII.37 210
10.5 Thecircumferenceofacircle&𝜋 216
11 BookIV:Circles&Polygons 221
11.1 Definitions 222
11.2 Circles&triangles 223
11.3 Circles&squares 236
11.4 Circles&pentagons 241
11.5 Constructingregularpolygons 246
11.6 Theareaofacircle&𝜋 251
12 ModelsfortheHyperbolicPlane 261
12.1 Historicaloverview 262
12.2 Modelsofthehyperbolicplane 266
12.3 Arclength&distanceinthehalf-plane
model 277
13 AxiomaticHyperbolicGeometry 289
13.1 Parallellines 291
13.2 Omegatriangles 302
13.3 Saccheriquadrilaterals 307
13.4 Hyperbolicarea 320
Contents ix
14 FiniteGeometries 331
14.1 FourPointgeometry-Part2 332
14.2 Fano’splane 335
14.3 Projectivegeometry 339
14.4 Affineplanes 355
14.5 Transformingafineintoprojective 366
14.6 Openprobleminfinitegeometry 370
15 Isometries 373
15.1 Rigidmotionsorisometries 374
15.2 Reflections 378
15.3 IsometriesoftheEuclideanplane 382
15.4 InversionsintheEuclideanplane 399
15.5 Isometriesofthehyperbolicplane 406
16 Constructibility 415
16.1 Fourfamousproblemsofantiquity 416
16.2 Constructiblenumbers 419
16.3 Fourcounterexamples 429
16.4 Thelimitsofgeometry 435
AppendixA Euclid’sDefinitionsandAxioms 437
A.1 Definitions 437
A.2 Postulates 438
A.3 Commonnotions 438
AppendixB Euclid’sPropositions 439
B.1 BookI 439
B.2 BookII 442
B.3 BookIII 443
B.4 BookIV 444
B.5 BookVI 445
AppendixC VisualGuidetoEuclid’sPropositions 447
C.1 BookI 447
C.2 BookII 453
C.3 BookIII 454
C.4 BookIV 456
AppendixD Euclid’sProofs 457
D.1 BookI 457
AppendixE Hilbert’sAxiomsforPlaneEuclideanGeometry 461
Credits,PermissionsandAcknowledgements 463
Bibliography 465
NotationIndex 471
Index 473