Preface 

ix  
1 Euclid's Elements 

1  (16) 

1.1 Geometry before Euclid 


1  (1) 

1.2 The logical structure of Euclid's Elements 


2  (1) 

1.3 The historical importance of Euclid's Elements 


3  (2) 

1.4 A look at Book I of the Elements 


5  (3) 

1.5 A critique of Euclid's Elements 


8  (3) 

1.6 Final observations about the Elements 


11  (6) 
2 Axiomatic Systems and Incidence Geometry 

17  (14) 

2.1 Undefined and defined terms 


17  (1) 


18  (1) 


18  (1) 


19  (1) 

2.5 An example of an axiomatic system 


19  (6) 

2.6 The parallel postulates 


25  (2) 

2.7 Axiomatic systems and the real world 


27  (4) 
3 Theorems, Proofs, and Logic 

31  (12) 

3.1 The place of proof in mathematics 


31  (1) 

3.2 Mathematical language 


32  (2) 


34  (3) 


37  (2) 


39  (1) 

3.6 The theorems of incidence geometry 


40  (3) 
4 Set Notation and the Real Numbers 

43  (9) 

4.1 Some elementary set theory 


43  (2) 

4.2 Properties of the real numbers 


45  (3) 


48  (1) 

4.4 The foundations of mathematics 


49  (3) 
5 The Axioms of Plane Geometry 

52  (42) 

5.1 Systems of axioms for geometry 


53  (3) 


56  (1) 

5.3 Existence and incidence 


56  (1) 


57  (6) 


63  (4) 


67  (3) 

5.7 Betweenness and the Crossbar Theorem 


70  (14) 


84  (5) 

5.9 The parallel postulates 


89  (1) 


90  (4) 
6 Neutral Geometry 

94  (41) 

6.1 Geometry without the parallel postulate 


94  (1) 

6.2 AngleSideAngle and its consequences 


95  (2) 

6.3 The Exterior Angle Theorem 


97  (5) 

6.4 Three inequalities for triangles 


102  (5) 

6.5 The Alternate Interior Angles Theorem 


107  (3) 

6.6 The SaccheriLegendre Theorem 


110  (3) 


113  (3) 

6.8 Statements equivalent to the Euclidean Parallel Postulate 


116  (7) 

6.9 Rectangles and defect 


123  (8) 

6.10 The Universal Hyperbolic Theorem 


131  (4) 
7 Euclidean Geometry 

135  (26) 

7.1 Geometry with the parallel postulate 


135  (2) 

7.2 Basic theorems of Euclidean geometry 


137  (2) 

7.3 The Parallel Projection Theorem 


139  (2) 


141  (2) 

7.5 The Pythagorean Theorem 


143  (2) 


145  (2) 

7.7 Exploring the Euclidean geometry of the triangle 


147  (14) 
8 Hyperbolic Geometry 

161  (33) 

8.1 The discovery of hyperbolic geometry 


161  (2) 

8.2 Basic theorems of hyperbolic geometry 


163  (5) 

8.3 Common perpendiculars 


168  (3) 

8.4 Limiting parallel rays and asymptotically parallel lines 


171  (10) 

8.5 Properties of the critical function 


181  (4) 

8.6 The defect of a triangle 


185  (4) 

8.7 Is the real world hyperbolic? 


189  (5) 
9 Area 

194  (31) 

9.1 The Neutral Area Postulate 


195  (3) 

9.2 Area in Euclidean geometry 


198  (8) 

9.3 Dissection theory in neutral geometry 


206  (7) 

9.4 Dissection theory in Euclidean geometry 


213  (3) 

9.5 Area and defect in hyperbolic geometry 


216  (9) 
10 Circles 

225  (39) 


226  (1) 


227  (4) 

10.3 Circles and triangles 


231  (7) 

10.4 Circles in Euclidean geometry 


238  (6) 


244  (3) 

10.6 Circumference and area of Euclidean circles 


247  (8) 

10.7 Exploring Euclidean circles 


255  (9) 
11 Constructions 

264  (20) 

11.1 Compass and straightedge constructions 


265  (2) 

11.2 Neutral constructions 


267  (3) 

11.3 Euclidean constructions 


270  (2) 

11.4 Construction of regular polygons 


272  (4) 


276  (3) 

11.6 Three impossible constructions 


279  (5) 
12 Transformations 

284  (43) 

12.1 The transformational perspective 


285  (1) 

12.2 Properties of isometrics 


286  (6) 

12.3 Rotations, translations, and glide reflections 


292  (8) 

12.4 Classification of Euclidean motions 


300  (3) 

12.5 Classification of hyperbolic motions 


303  (1) 

12.6 A transformational approach to the foundations 


304  (6) 

12.7 Euclidean inversions in circles 


310  (17) 
13 Models 

327  (18) 

13.1 The significance of models for hyperbolic geometry 


327  (2) 

13.2 The Cartesian model for Euclidean geometry 


329  (2) 

13.3 The Poincaré disk model for hyperbolic geometry 


331  (5) 

13.4 Other models for hyperbolic geometry 


336  (5) 

13.5 Models for elliptic geometry 


341  (4) 
14 Polygonal Models and the Geometry of Space 

345  (47) 


346  (11) 

14.2 Approximate models for the hyperbolic plane 


357  (6) 


363  (6) 

14.4 The geometry of the universe 


369  (6) 


375  (1) 


375  (9) 


384  (8) 
APPENDICES 



392  (6) 


392  (2) 


394  (1) 


394  (1) 


394  (4) 


398  (10) 


398  (2) 


400  (1) 


401  (1) 


402  (3) 


405  (3) 

C The Postulates Used in this Book 


408  (3) 


408  (1) 

C.2 The postulates of neutral geometry 


408  (1) 

C.3 The parallel postulates 


409  (1) 


409  (1) 

C.5 The reflection postulate 


410  (1) 

C.6 Logical relationships 


410  (1) 


411  (1) 

E Hints for Selected Exercises 


412  (9) 
Bibliography 

421  (4) 
Index 

425  