1. Prologue: Euclid’s Elements 1.1 Geometry before Euclid 1.2 The logical structure of Euclid’s Elements 1.3 The historical significance of Euclid’s Elements 1.4 A look at Book I of the Elements 1.5 A critique of Euclid’s Elements 1.6 Final observations about the Elements   2. Axiomatic Systems and Incidence Geometry 2.1 The structure of an axiomatic system 2.2 An example: Incidence geometry 2.3 The parallel postulates in incidence geometry 2.4 Axiomatic systems and the real world 2.5 Theorems, proofs, and logic 2.6 Some theorems from incidence geometry   3. Axioms for Plane Geometry 3.1 The undefined terms and two fundamental axioms 3.2 Distance and the Ruler Postulate 3.3 Plane separation 3.4 Angle measure and the Protractor Postulate 3.5 The Crossbar Theorem and the Linear Pair Theorem 3.6 The Side-Angle-Side Postulate 3.7 The parallel postulates and models   4. Neutral Geometry 4.1 The Exterior Angle Theorem and perpendiculars 4.2 Triangle congruence conditions 4.3 Three inequalities for triangles 4.4 The Alternate Interior Angles Theorem 4.5 The Saccheri-Legendre Theorem 4.6 Quadrilaterals 4.7 Statements equivalent to the Euclidean Parallel Postulate 4.8 Rectangles and defect 4.9 The Universal Hyperbolic Theorem   5. Euclidean Geometry 5.1 Basic theorems of Euclidean geometry 5.2 The Parallel Projection Theorem 5.3 Similar triangles 5.4 The Pythagorean Theorem 5.5 Trigonometry 5.6 Exploring the Euclidean geometry of the triangle   6. Hyperbolic Geometry 6.1 The discovery of hyperbolic geometry 6.2 Basic theorems of hyperbolic geometry 6.3 Common perpendiculars 6.4 Limiting parallel rays and asymptotically parallel lines 6.5 Properties of the critical function 6.6 The defect of a triangle 6.7 Is the real world hyperbolic?   7. Area 7.1 The Neutral Area Postulate 7.2 Area in Euclidean geometry 7.3 Dissection theory in neutral geometry 7.4 Dissection theory in Euclidean geometry 7.5 Area and defect in hyperbolic geometry   8. Circles 8.1 Basic definitions 8.2 Circles and lines 8.3 Circles and triangles 8.4 Circles in Euclidean geometry 8.5 Circular continuity 8.6 Circumference and area of Euclidean circles 8.7 Exploring Euclidean circles   9. Constructions 9.1 Compass and straightedge constructions 9.2 Neutral constructions 9.3 Euclidean constructions 9.4 Construction of regular polygons 9.5 Area constructions 9.6 Three impossible constructions   10. Transformations 10.1 The transformational perspective 10.2 Properties of isometries 10.3 Rotations, translations, and glide reflections 10.4 Classification of Euclidean motions 10.5 Classification of hyperbolic motions 10.6 Similarity transformations in Euclidean geometry 10.7 A transformational approach to the foundations 10.8 Euclidean inversions in circles   11. Models 11.1 The significance of models for hyperbolic geometry 11.2 The Cartesian model for Euclidean geometry 11.3 The Poincaré disk model for hyperbolic geometry 11.4 Other models for hyperbolic geometry 11.5 Models for elliptic geometry 11.6 Regular Tessellations   12. Polygonal Models and the Geometry of Space 12.1 Curved surfaces 12.2 Approximate models for the hyperbolic plane 12.3 Geometric surfaces 12.4 The geometry of the universe 12.5 Conclusion 12.6 Further study 12.7 Templates   APPENDICES A. Euclid’s Book I A.1 Definitions A.2 Postulates A.3 Common Notions A.4 Propositions   B. Systems of Axioms for Geometry B.1 Filling in Euclid’s gaps B.2 Hilbert’s axioms B.3 Birkhoff’s axioms B.4 MacLane’s axioms B.5 SMSG axioms B.6 UCSMP axioms   C. The Postulates Used in this Book C.1 The undefined terms C.2 Neutral postulates C.3 Parallel postulates C.4 Area postulates C.5 The reflection postulate C.6 Logical relationships   D. Set Notation and the Real Numbers D.1 Some elementary set theory D.2 Properties of the real numbers D.3 Functions   E. The van Hiele Model   F. Hints for Selected Exercises   Bibliography Index

Earlier presentation of neutral geometry has been achieved by shifting some topics to appendices and covering others more efficiently. For example, material on set theory and the real numbers was moved to an appendix because much of it is review for most students. The review of proof writing has been incorporated into the chapter on Axiomatic Systems. Description of different axiom systems for elementary geometry has been moved to its own appendix (previously found at the beginning of the chapter on The Axioms of Plane Geometry). Instructors can now cover this material at any time they choose during the course. New topics/content include: More exercises in elementary Euclidean geometry in Chapter 5 A new section on ideal triangles and the classification of parallels in Chapter 6 Reworking of Chapter 7 (Area) to include a relatively elementary proof of the Euclidean case of the dissection theorem, giving instructors the option of omitting all the non-Euclidean material in that chapter A section on similarity transformations in Euclidean geometry has been added to the chapter on Transformations (Chapter 10) Technology material facilitates the use of the open-source software Geogebra. A new Appendix B (Systems of Axioms for Geometry) explores in depth the range of options available in the choice of axioms and explains the rationale for the choices made in this book.