Bibliography

Bell, E., Men of Mathematics, New York: Simon and Schuster, 1937. Birkhoff, G., "A Set of Postulates for Plane Geometry, Based on Scale and Protractor,"

Annals of Math., 33 (1932),329-345. Bonola, R., Non-Euclidean Geometry, New York: Dover, 1955. Borsuk, K., and W. Szmielew, Foundations of Geometry, Amsterdam: North-Holland,

1960. Coexter, H., Introduction to Geometry, New York: Wiley, 1961. Fraleigh, J., A First Course in Abstract Algebra, 2nd Ed., Reading, Mass.:

Addison-Wesley, 1976. Greenberg, M., Euclidean and Non-Euclidean Geometry, 2nd Ed., San Francisco:

Freeman, 1980. Halstead, G., Girolamo Saccheri's Euclides Vindicatus, Chicago: Open Court, 1920. Heath, T., Euclid, The Thirteen Books of the Elements, 2nd Ed., New York:

Dover, 1956. Herstein, I., Topics in Algebra, 2nd Ed., Lexington, Mass.: Xerox, 1975. Hilbert, D., The Foundations of Geometry, 2nd Ed., tr. E. Townsend, Chicago: Open

Court, 1921. --, Grundlagen der Geometrie, 8th Ed., Stuttgart: Teubner, 1956. Kennedy, H., "The Origins of Modern Axiomatics: Pasch to Peano," Amer. Math.

Monthly, 79 (1972), 133-136. Kleven, D., "Morley's Theorem and a Converse," Amer. Math. Monthly, 85 (1978),

100-105. Martin, G., The Foundations of Geometry and the Non-Euclidean Plane, New York:

Intext, 1975. Millman, R., "Kleinian Transformation Geometry," Amer. Math. Monthly, 84 (1977),

338-349. --, "The Upper Half Plane Model for Hyperbolic Geometry," Amer. Math.

Monthly, 87 (1980), 48-53. --, and G. Parker, Elements of Differential Geometry, Englewood Cliffs:

Prentice-Hall, 1977. Moise, E., Elementary Geometry from an Advanced Standpoint, 2nd Ed., Reading, Mass.:

Addison-Wesley, 1974.

344

Bibliography 345

Moulton, F. R., "A Simple Non-desarguesian Plane Geometry." Transactions of A.M.S., 3 (1902),192-195.

Parker, G., "A Metric Model of Plane Euclidean Geometry," Amer. Math. Monthly, 87 (1980), 567-572.

Pasch, M., Vorlesungen aber neuere Geometrie, Leibzig: Teubner, 1882. Prenowitz, W., and M. Jordan, Basic Concepts of Geometry, New York: Blaisdell,

1965. Reyes, V., "Sur la theoreme relative au carre de l'hypotenuse et Ie cinquieme

postulate d'Euclide," Mathesis (2) 7 (1897),96. Smith, D., A Source Book in Mathematics, New York: McGraw-Hill, 1929. Spivak, M., A Comprehensive Introduction to Differential Geometry, vol. II, Boston:

Publish or Perish, 1970.

Index

PART I: The Notation ~r(C) (circle) 140 [) (Poincare Disk) 290 8 (Euclidean Plane) 16 .tt' (Hyperbolic Plane) 17 J(.AI) (isometry group) 323 .Y{ (Klein Plane) 288 vi! (Moulton Plane) 90 9 (Projective Plane) 24 9 c (pointed pencil) 299 9, (parallel pencil) 299 9 AB (asymptotic pencil) 299 fJl (Riemann Sphere) 19 fJl (set of polygonal regions) 237 ff (Taxicab Plane) 32

o (unit disk) 286 H (upper half plane) 17 P (set of projective points) 24 R 2 (real coordinate plane) 16 S2 (unit sphere) 18

!i'D (P-lines) 289 !i'E (Euclidean lines) 16 !.tH (hyperbolic lines) 17 !i'K (K-lines) 286 !i'M (Moulton lines) 90 !i'p (Projective lines) 24 !t'R (great circles) 18

346

d D (Poincare distance) 290 dE (Euclidean distance) 25,41 d H (hyperbolic distance) 26 d K (Klein distance) 288 d M (Moulton distance) 91 dN (twisted distance) 63 d p (projective distance) 35 d R (spherical distance) 35 d T (taxicab distance) 26

mD (Poincare angle measure) 290 mE (Euclidean angle measure) 86 m H (hyperbolic angle measure) 88 mK (Klein angle measure) 288 mM (Moulton angle measure) 94

d(P, Q) (distance between points) 25

d(P, /) (distance from point to line) 135,205

d(l, I') (distance between lines) 205 int (interior) 75, 76, 78, 143, 237 La (vertical Euclidean line) 16 L m •b (nonvertical Euclidean line) 16 LAB (Euclidean line) 40 aL (type I hyperbolic line) 17 cLr (type II hyperbolic line) 17

Index

Mm. b (Moulton line) 90 TBA (Euclidean tangent) 87 fBA (Euclidean unit tangent) 331 r(P,/) (critical number) 177

8(!::,ABC) (defect) 194 ro (hyperbolic rotation) 329 'lJA (half-turn) 314 II(t) (critical function) 181 PI (reflection) 293 'TA (Euclidean translation) 337 </>A (value of isometry on A) 275 </>0 (Euclidean rotation) 337

...... AB (line) 20 AB (ray) 50 AB (segment) 49 AB (distance) 44 A-B-C (between) 44 A-B-C-D (between) 47 LABC (angle) 54

PART II: The Models

abstract geometry 16 angle addition 110 angle construction 107 angle measure 86, 89(A6), 107, III betweenness 47, 48(AI) circles 140 collineation 272, 283(AI,A2,A3) critical function I 83(A3) cycles 305(AI2) distance 25, 33(AI), 41 EPP I 84(A5) fixed points of isometries 298(A2) half planes 65, 68(A2), 69(A3) incidence geometry 20, 22(A5),

23(AI4),43(BI) isometry 343(A8)

!::,ABC (triangle) 56 £ABC (triangular region) 237 D ABCD (quadrilateral) 80 0ABCD (Saccheri quadrilateral)

167 []JABCD (Lambert quadrilateral)

175 LJAB ... E (polygon) 236 ~ AB ... E (polygonal region) 238 (A, B) (dot product) 39 II A II (length) 39 XJ.. (X perp) 64 - (similar) 223 - (equivalence of rays) 186 - r? (equivalent mod a pencil) 301 ..1 (perpendicular) 98 == (equivalent by finite

decomposition) 259 II (parallel) 22 I (asymptotic) 186

isometry group 340 Linear Pair Theorem 109 lines 16, 40, 64 metric geometry 28 parallel lines 22(AIO), 166(A7) PSA 66 Pythagorean Theorem 121(A6) rays 50 rulers 33(A4), 34(A9), 38(AI), 41,

43(A5) SAS 119 segments 50 triangle construction I 55(A5) triangle inequality 42 trigonometric functions 121(A7)

347

The Hyperbolic Plane .Yt' = {!HI, 2' H' d H' m H }

abstract geometry 17 angle addition 114 alternate interior angles I 66(A5) angle bisector IOI(AII,AI2),202(A8)

348

angle construction 113 angle measure 88, 89(AI,A2,A3), 112 area 257, 259(AIO,AII,BI) asymptotic rays I 93(B3) asymptotic triangle I 93(A8, A9) betweenness 45, 48(A2) circles 141, 148(A2) circumscribed circle 235(A5) collineation 273, 283(AIO) concurrence of perpendicular bisectors

219(AI9) critical function 179, 183(A4), 202(A3) cycles 305(B I) distance 26, 33(A2), 212(BI) distance scale 21 I (A8) equidistant sets I 84(A7,A8,A9) Euclidean tangent to a ray 87, III FLT 327, 343(A7,A9) half planes 67, 69(A4) HPP I 84(A6) incidence geometry 21, 22(A8) isometry 211(AIO)

isometry group 342 line of enclosure 202(A 7) lines 22(AI,A2,A6,A7,A9) metric geometry 30 midpoints 53(AII) Mirror Axiom 335 parallel displacement 312(A3) parallel lines 22(AII), 162,

Index

I 66(A6, A 10) parametrization 30, 34(A8) perpendicular lines 99, IOI(B3), 162 PSA 67 Pythagorean Theorem lOO(A9) quadrilateral 8 I (A4) reflection 298(A4) ruler 31,33, 34(A6,All) Saccheri quadrilateral I 74(AI,A2) SAS 335 segment congruence 53(A9) segment construction 52 segments 49 triangle congruence 121(A2)

The Klein Plane .ff={D, ~K' dK , mK }

angle measure 291(BI) distance 291(BI)

The Max Plane {R2, ~E' ds' mEl angle measure 89(B2) circles 148(AI), 149(A6,A8) distance 34(B5) equidistant lines I 75(B3)

The Missing Strip Plane {ff,~, d'}

circles I 49(B II) Crossbar Theorem 79(A20) EPP I 66(B2)

angle measure 94 angle sum 95(AIO) circles I 49(B6) distance 91 EPP 166(BI)

lines 286, 291(A6) parallel lines 29 I (A7)

metric geometry 34(B5) segment construction 53(A8) triangle inequality 43(B3)

incidence geometry 75(A4) lines 73 PSA 74

incidence geometry 90 lines 95(AI) metric geometry 92 perpendicular lines I OI (B7) protractor geometry 94

Index

PSA 92 SAS 121(BI)

angle measure 291(B2) distance 291(B2)

triangle inequality 96(B2)

lines 289 parallel lines 29 I (A9)

The Projective Plane Ef' = {IF», .!t' P' d p }

betweenness 48(B I ) distance 3S(BlO) lines 24(B9)

distance 3S(B9) lines 18, 22(A3,A4)

parallel lines 24(B I 0) plane separation 63(B9) points 8(A6), 24(B8)

parallel lines 22(AI2)

The Taxicab Plane Y={1R 2, .!t'E' dr, mel

angle measure 89(AS) PSA 69(AS). betweenness 48(A3,AlO) Pythagorean Theorem 138(AS) circles 140, 148(AS) rulers, 33, 34(AS,AlO) 38(A3) collineation 283(AII) SAS 117, 132(BI) distance 26 segment congruence S3(A 7) equidistant lines 17S(B2) segment construction S3(AIO) metric geometry 32 SSS 12S(B2) perpendicular bisectors 139(B2) triangle congruence 101(AI6) perpendicular distance 138(A4) triangle construction ISS(A6) Pons asinorum 121(B2) triangle inequality 43(B2)

The Twisted Plane {IR 2, .!t'E' d N }

circles 149(B7) distance 63(B 10)

PSA 63(BIO)

PART III: The Terminology

AAA Congruence Theorem 197 Absolute geometry (see Neutral

geometry) Abstract geometry IS

Allor None Theorem 183 Altitude 13S

concurrence of 230 foot 13S

349

350

Angle 54 acute 96 Addition Axiom 85 Addition Theorem 100 alternate interior 160 base 120, 167 bisector 100

concurrence of 228 comparison 126 complementary 96 congruence 100 Construction Axiom 85 Construction Theorem 100 corresponding 160 exterior 126, 194 measure 83

degree 84 grade 84 radian 84

obtuse 96 remote interior 126 right 96 Subtraction Theorem 100 sum for hyperbolic geometry 196 supplementary 96 trisector 10 1

Area function 239 Euclidean 250

uniqueness 251 hyperbolic 256

special form in .Yf 257 uniqueness 268

special 265 Aristotle's Theorem 207 ASA Congruence Theorem 122

equivalent to SAS 123 Asymptotic lines 192

convergence of 208 Asymptotic rays 185, 186

strictly 185 Asymptotic triangle 192

congruence 192 doubly 202 trebly 304

Beltrami, E. 286 Betweenness 44

of numbers 45 Biangle (see Open triangle)

Bijection 11 Binary relation 4 Birkhoff, G. 25, 157 Bolyai, J. 159 Bolyai's Theorem 260, 270

for triangles 266 for special area 267

Index

Boundary of polygonal region 238

Carroll, L. 159 Cartesian Plane 17 Cauchy-Schwarz Inequality 42 Cayley, A. 286 Centroid 229 Change of scale 209 Chord 140 Circle 35, 140

chord 140 determined by three points 142 diameter 140 radius 140 secant of 144 tangent to 144

Circumcenter 230 Circumcircle 218,235 Classification Theorem 317

of double reflections 309 of finite isometry groups 326 of involutions 314 of triple reflections 316

Closed triangle (see Asymptotic triangle)

Collinear points 20 Collineation 272 Comparison:

angle 126 segment 126

Complementary angle 96 Composition of functions II Concurrence 214

of altitudes 230 of angle bisectors 228 of medians 229 of perpendicular bisectors 215,230

Congruence: of angles 100 of asymptotic triangles 192 of quadrilaterals 168 of Saccheri quadrilaterals 168

Index

of segments 51 of triangles 116

conjugate fractional linear transformation 341

Consecutive vertices 236 Continuous function 145 Convex set 58 Convex polygonal region 239

defect of 254 Coordinate system (see Ruler) Corresponding angles 160 Cosh(t) 29 Cosine function 104 Critical function 181

decreasing 195 nonincreasing 181 suIjectivity of 199

Critical number 177 Crossbar Theorem 77

converse 79(AI2) Crossbar interior 79 Cycle 302

center of 302 degenerate 302 determined by three points 302

Dedekind 157 Defect 194

Addi tion Theorem 196 of convex polygonal region 254 of polygonal region 256 of triangle 194 of triangular region 252

Descartes, R. 17 Diameter of circle 140 Distance:

between lines 205 between a point and a line 135, 205 between points 25 scale 210

change of 209 Divergently parallel lines 203

distance between 205 divergence of 207

Dodgson, C. 159 Double reflection 306

classification 309 Doubly asymptotic triangle 202

351

Endpoints of segment 50 uniqueness 50

EPP (see Euclidean parallel property) Equiangular triangle 121 Equiangular quadrilateral 175 Equidistant sets 172 Equilateral triangle 120 Equivalence relation 5 Equivalent by finite decomposition

259 Equivalent rays 186 Equivalent with respect to a pencil 301 Erlangen Program 286, 295 Euclid 120, 158

Fifth Postulate of 158, 163 equivalence to EPP 165

Euclidean geometry 183 angle sum 171

Euclidean parallel property (EPP) 165 equivalent to Euclid's Fifth Postulate

165 equivalent forms 213 and Pythagorean Theorem 226 and rectangles 214 and Saccheri quadrilaterals 214

Euclidean Plane 16 (see also Part II of this index)

Euler line 232 Euler points 232 Even isometry 313 Exterior angle 126

of open triangle 194 Exterior Angle Theorem 126

for closed triangle 195 Exterior of circle 143 Exterior of cycle 304 External Tangent Theorem 148

Family of parallel lines 247 Fixed point of isometry 292 Fractional linear transformation (FL T)

327 associated matrix 327 conjugate 341

Function 8 bijective 11 domain of 8 image of 8 injective 10

352

Function (cont.) range of 8 swjective 9

Gauss, C. 159 Geometry:

absolute 1I8 abstract 15 Euclidean 183 hyperbolic 183 incidence 20 induced 24,275,285 metric 28 neutral 118 Pasch 69 protractor 84 scissors 62

Giordano's Theorem 173 Glide 315 Great circle 18 Group 321

cyclic 326 dihedral 326 finite 326 isometry 323 isomorphism 324 order of element 324

HA Theorem 137 Half planes determined by a line 59 uniqueness 60

edge 62 uniqueness 61

Half-tum 314 Hilbert, D. 24, 156

axioms of geometry 156 Hinge Theorem 130 Hjelmsev's Lemma and Theorem 320 HL Theorem 136 HPP (see Hyperbolic parallel property) Hyperbolic functions 29 Hyperbolic geometry 183 Hyperbolic parallel property 183 Hyperbolic Plane 17

(see also Part II of this index) Hypotenuse 133

uniqueness 133

Improper integral 102 Incenter 228

Incidence geometry 20 Induced angle measure 284 Induced distance function 284 Induced geometry:

by a bijection 275, 285 on a subset 24

Injection II Inscribed circle 235

Index

Inside of a polygonal region 238 Intermediate Value Theorem 67, 145 Interior:

of angle 76 of circle 143 of convex polygon 237 of cycle 304 of ray 75 of segment 75 of triangle 78

Internal bisector 125 Invariant set 303 Inverse cosine 103 Inverse function 12 Inversion, special 329 Involution 314

classification of 314 Isometry 210, 275

as a collineation 280 classification 317 even 313 odd 313 preserves angle measure 279

Isometry group 323 of 8 340 finite subgroups 326 of./f 342

Isosceles triangle 120

Klein, F. 286,294 Klein Plane 286

(see also Part II of this index) K-line 286

Lambert, J. 159 Lambert quadrilateral 175 Law of cosines 119 Least upper bound 177 Legs 133, 167 Length of segment 50 Line-Circle Theorem 147 Line of enclosure 202, 217

Index

Line segment 49 Linear pair 96 Linear Pair Theorem 97

converse 98 Lobachevsky, N. 159 Longest side 133

Max Plane 34 (see also Part II of this index)

Median 138, 229 concurrence 229

Metric geometry 28 Midpoint 53

existence and uniqueness 54 Mirror 296 Mirror Axiom 296

and SAS 296 Missing Strip Plane 73

(see also Part II of this index) Morley's Theorem 233 Moulton Plane 90

(see also Part II of this index)

Nassir-Eddin 159 Neutral geometry 1I8 Nine point circle 232 Noncollinear points 20

Odd isometry 313 Omar Khayam's Theorem 174 One-to-one (see Function, injective) Onto (see Function, surjective) Open Mouth Theorem 130

converse 132 Open triangle 185 Opposite sides of line 61 Opposite sides of quadrilateral 80 Orthocenter 230

Pappus 120 Parallel decomposition 248 Parallel displacement 306 Parallel lines 22 Parallelogram 172 Parametrization 30 Pasch, M. 69 Pasch geometry 69 Pasch's Postulate 70

equivalent to PSA 71

Pasch's Theorem 69 strong form 78

Passing point 63 Passing set 63 Peano's Axiom 74 Pencil 299

asymptotic 299 center of 299 parallel 299 pointed 299

Perpendicular bisector 99, 137

353

concurrence 215,230 Perpendicular circles 288 Perpendicular Distance Theorem 134 Perpendicular lines 98 Perpendicular to a line:

existence 99, 124 uniqueness 99, 127

Plane 18 Plane Separation Axiom (PSA) 59 Playfair, I. 159 Plethora of lines (see Pencil) P-line 289 Poincare, H. 18,24, 159,288 Poincare Disk 289

(see also Part II of this index) Poincare Plane (see Hyperbolic Plane) Polygon 236

convex 237 interior of 237

Polygon Inequality 169 Polygonal region 237

boundary of 238 defect of 256

Pons asinorum 120, 125 converse 122

Preserve angle measure 277 distance 275 lines 272 right angles 277

Proclus 158, 159 Projective Plane 8, 24

(see also Part II of this index) Protractor 83 Protractor geometry 84 PSA (see Plane Separation Axiom) Ptolemy 159 Pythagorean Theorem 121, 133, 137,

147, 225, 243

354

Pythagorean Theorem (conI.) equivalent to EPP 226 Euclid's proof 241

Quadrilateral 80 angles of 80 congruence of 168 convex 81 diagonal of 80 sides of 80 vertex of 80

Radius 140 Ray 50

in terms of a ruler 51 Rectangle 172

Reflection 293 double 306 special 334

Regular polygon 244 Remote interior angle 126 Riemann, G. 19 Riemann Sphere 19

(see also Part II of this index) Right triangle 133 Rotation 306

special 329 Ruler 27 Ruler Placement Theorem 36 Ruler Postulate 28

SAA Theorem 128 Saccheri 159, 167 Saccheri quadrilateral 167

congruence 168 Saccheri's Theorem 170 Same side of line 61 SAS Axiom 118 SAS Similarity Theorem 227 Scalene triangle 120 Scissors geometry 62 Secant of a circle 144 Sech(t) 29 Segment 49

comparison 126 congruence 51 length 50

Segment Addition Theorem 53 Segment Construction Theorem 52

Segment Subtraction Theorem 53 Similar triangles 223 Similarity theory from area 243 Simson line 235 Sine function 104 Sinh(t) 29 Size of polygonal region 245 Sloping Ladder Theorem 150 Special area function 165 Square 172 SSS Congruence Theorem 122

equivalence to SAS 154 SSS Similarity Theorem 225 Standard rulers 33 Stereographic projection 287 Subgroup 323

normal 326 Sub tangent 149 Supplementary angles 96 Surjection 11 Synthetic approach 155

Tangent of circle 144

existence and uniqueness 145 of cycle 303

Tanh(t) 29 Taxicab Plane 32

(see also Part II of this index) Transformation 326

fractional linear 327 orthogonal 339 special orthogonal 337

Translation 306 Euclidean 337 special hyperbolic 328

Transversal 159 Trapezoid 240 Trebly asymptotic triangle 304 Triangle 56

asymptotic 192 closed 192 congruence 116

and isometries 295 defect of 194 equiangular 121 equilateral 120 isosceles 120 open 185

Index

Index

right 133 scalene 120 sides of 57 similar 223 vertices 56

uniqueness 56 Triangle Construction Theorem 151 Triangle Inequality 42, 130

general form 132 Triangular region 237

defect of 252 Triangulation 245

base 245, 246 refinement of 250 star 252

Trigonometric functions 114(A7) Twisted Plane 63

(see also Part II of this index) Two Circle Theorem 153

Uniqueness of angle measure in a neutral

geometry 280

355

of cycles through three points 302 of Euclidean area 251 of Euclidean geometry 281 of hyperbolic area 268 of hyperbolic geometry 282

Unit sphere 18

Vertex of angle 56

uniqueness 55 of polygon 236 of ray 51

uniqueness 51 of triangle 56

uniqueness 57 Vertical angles 96 Vertical Angle Theorem 100 Vitale, G. 159

Wallis, J. 159

Z Theorem 76

Undergraduate Texts in Mathematics

Apostol: Introduction to Analytic Number Theory. 1976. xii, 338 pages. 24 illus.

Childs: A Concrete Introduction to Higher Algebra. 1979. xiv, 338 pages. 8 illus.

Chung: Elementary Probability Theory with Stochastic Processes. 1975. xvi, 325 pages. 36 illus.

Croom: Basic Concepts of Algebraic Topology. 1978. x, 177 pages. 46 illus.

Fleming: Functions of Several Variables. Second edition. 1977. xi, 411 pages. 96 illus.

Foulds: Optimization Techniques: An Introduction. 1981. xii, 502 pages. 72 illus.

Franklin: Methods of Mathematical Economics. Linear and Nonlinear Programming, Fixed-Point Theorems. 1980. x, 297 pages. 38 illus.

Halmos: Finite-Dimensional Vector Spaces. Second edition. 1·974. viii, 200 pages.

Halmos: Naive Set Theory. 1974. vii, 104 pages.

100ss/Joseph: Elementary Stability and Bifurcation Theory. 1980. xv, 286 pages. Approx. 47 illus.

Kemeny/Snell: Finite Markov Chains. 1976. ix, 224 pages. 11 illus.

Lax/Burstein/Lax: Calculus with Applications and Computing, Volume 1. 1976. xi, 513 pages. 170 illus.

LeCuyer: College Mathematics with A Programming Language. 1978. xii, 420 pages. 144 illus.

MackilStrauss: Introduction to Optimal Control Theory. 1981. xiii, 168 pages. 68 illus.

Malitz: Introduction to Mathematical Logic: Set Theory - Computable Functions - Model Theory. 1979. xii, 198 pages. 2 illus.

Millman/Parker: Geometry: A Metric Approach with Models. 1981. viii, 355 pages. 259 iIlus.

PrenowitzlJantosciak: Join Geometrics: A Theory of Convex Set and Linear Geometry. 1979. xxii, 534 pages. 404 iIlus.

Priestley: Calculus: An Historical Approach. 1979. xvii, 448 pages. 335 illus.

Protter/Morrey: A First Course in Real Analysis. 1977. xii, 507 pages. 135 illus.

Ross: Elementary Analysis: The Theory of Calculus. 1980. viii, 264 pages. 34 illus.

Sigler: Algebra. 1976. xii, 419 pages. 27 illus.

Singer/Thorpe: Lecture Notes on Elementary Topology and Geometry. 1976. viii, 232 pages. 109 iIlus.

Smith: Linear Algebra. 1978. vii, 280 pages. 21 illus.

Thorpe: Elementary Topics in Differential Geometry. 1979. xvii. 253 pages. 126 illus.

Whybum/Duda: Dynamic Topology. 1979. xiv, 338 pages. 20 illus.

Wilson: Much Ado About Calculus: A Modem Treatment with Applicatior Prepared for Use with the Computer. 1979. xvii, 788 pages. 145 illus.