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 I. THE HISTORICAL DEVELOPMENT OF NON-EUCLIDEAN GEOMETRY SECTION PAGE
  1.1 Euclid
  1.2 Saccheri and Lambert
  1.3 Gauss, Wächter, Schweikart, Taurinus
  1.4 Lobatschewsky
  1.5 Bolyai
  1.6 Riemann
  1.7 Klein
 II. REAL PROJECTIVE GEOMETRY: FOUNDATIONS
  2.1 Definitions and axioms
  2.2 Models
  2.3 The principle of duality
  2.4 Harmonic sets
  2.5 Sense
  2.6 Triangular and tetrahedral regions
  2.7 Ordered correspondences
  2.8 One-dimensional projectivities
  2.9 Involutions
 III. REAL PROJECTIVE GEOMETRY: POLARITIES, CONICS AND QUADRICS
  3.1 Two-dimensional projectivities
  3.2 Polarities in the plane
  3.3 Conies
  3.4 Projectivities on a conic
  3.5 The fixed points of a collineation
  3.6 Cones and reguli
  3.7 Three-dimensional projectivities
  3.8 Polarities in space
 IV. HOMOGENEOUS COORDINATES
  4.1 The von Staudt-Hessenberg calculus of points
  4.2 One-dimensional projectivities
  4.3 Coordinates in one and two dimensions
  4.4 Collineations and coordinate transformations
  4.5 Polarities
  4.6 Coordinates in three dimensions
  4.7 Three-dimensional projectivities
  4.8 Line coordinates for the generators of a quadric
  4.9 Complex projective geometry
 V. ELLIPTIC GEOMETRY IN ONE DIMENSION
  5.1 Elliptic geometry in general
  5.2 Models
  5.3 Reflections and translations
  5.4 Congruence
  5.5 Continuous translation
  5.6 The length of a segment
  5.7 Distance in terms of cross ratio
  5.8 Alternative treatment using the complex line
 VI. ELLIPTIC GEOMETRY IN TWO DIMENSIONS
  6.1 Spherical and elliptic geometry
  6.2 Reflection
  6.3 Rotations and angles Ill
  6.4 Congruence
  6.5 Circles
  6.6 Composition of rotations
  6.7 Formulae for distance and angle
  6.8 Rotations and quaternions
  6.9 Alternative treatment using the complex plane
 VII. ELLIPTIC GEOMETRY IN THREE DIMENSIONS
  7.1 Congruent transformations
  7.2 Clifford parallels
  7.3 The Stephanos-Cartan representation of rotations by points
  7.4 Right translations and left translations
  7.5 Right parallels and left parallels
  7.6 Study's representation of lines by pairs of points
  7.7 Clifford translations and quaternions
  7.8 Study's coordinates for a line
  7.9 Complex space
 VIII. DESCRIPTIVE GEOMETRY
  8.1 Klein's projective model for hyperbolic geometry
  8.2 Geometry in a convex region
  8.3 Veblen's axioms of order
  8.4 Order in a pencil
  8.5 The geometry of lines and planes through a fixed point
  8.6 Generalized bundles and pencils
  8.7 Ideal points and lines
  8.8 Verifying the projective axioms
  8.9 Parallelism
 IX. EUCLIDEAN AND HYPERBOLIC GEOMETRY
  9.1 The introduction of congruence
  9.2 Perpendicular lines and planes
  9.3 Improper bundles and pencils
  9.4 The absolute polarity
  9.5 The Euclidean case
  9.6 The hyperbolic case
  9.7 The Absolute
  9.8 The geometry of a bundle
 X. HYPERBOLIC GEOMETRY IN TWO DIMENSIONS
  10.1 Ideal elements
  10.2 Angle-bisectors
  10.3 Congruent transformations
  10.4 Some famous constructions
  10.5 An alternative expression for distance
  10.6 The angle of parallelism
  10.7 Distance and angle in terms of poles and polars
  10.8 Canonical coordinates
  10.9 Euclidean geometry as a limiting case
 XI. CIRCLES AND TRIANGLES
  11.1 Various definitions for a circle
  11.2 The circle as a special conic
  11.3 Spheres
  11.4 The in- and ex-circles of a triangle
  11.5 The circum-circles and centroids
  11.6 The polar triangle and the orthocentre
 XII. THE USE OF A GENERAL TRIANGLE OF REFERENCE
  12.1 Formulae for distance and angle
  12.2 The general circle
  12.3 Tangential equations
  12.4 Circum-circles and centroids
  12.5 In- and ex-circles
  12.6 The orthocentre
  12.7 Elliptic trigonometry
  12.8 The radii
  12.9 Hyperbolic trigonometry
 XIII. AREA
  13.1 Equivalent regions
  13.2 The choice of a unit
  13.3 The area of a triangle in elliptic geometry
  13.4 Area in hyperbolic geometry
  13.5 The extension to three dimensions
  13.6 The differential of distance
  13.7 Arcs and areas of circles
  13.8 Two surfaces which can be developed on the Euclidean plane
 XIV. EUCLIDEAN MODELS
  14.1 The meaning of "elliptic" and "hyperbolic"
  14.2 Beltrami's model
  14.3 The differential of distance
  14.4 Gnomonic projection
  14.5 Development on surfaces of constant curvature
  14.6 Klein's conformai model of the elliptic plane
  14.7 Klein's conformai model of the hyperbolic plane
  14.8 Poincaré's model of the hyperbolic plane
  14.9 Conformai models of non-Euclidean space
 XV. CONCLUDING REMARKS
  15.1 HjelmsleVs mid-line
  15.2 The Napier chain
  15.3 The Engel chain
  15.4 Normalized canonical coordinates
  15.5 Curvature
  15.6 Quadratic forms
  15.7 The volume of a tetrahedron
  15.8 A brief historical survey of construction problems
  15.9 Inversive distance and the angle of parallelism
 APPENDIX: ANGLES AND ARCS IN THE HYPERBOLIC PLANE
 BIBLIOGRAPHY
 INDEX