Book:Oswald Veblen/Projective Geometry/Volume 1

Subject Matter

 * Projective Geometry

Contents

 * PREFACE


 * INTRODUCTION
 * 1. Undefined elements and unproved propositions
 * 2. Consistency, categoricalness, independence. Example of a mathematical science
 * 3. Ideal elements in geometry
 * 4. Consistency of the notion of points, lines, and plane at infinity
 * 5. Projective and metric geometry


 * CHAPTER $\text {I}$.
 * 6. The assumption of alignment
 * 7. The plane
 * 8. The first assumption of alignment
 * 9. The three-space
 * 10. The remaining assumptions of extension for a space of three dimensions
 * 11. The principle of duality
 * 12. The theorems of alignment for a space of $n$ dimensions


 * CHAPTER $\text {II}$.
 * 13. Projection, section, perspectivity
 * 14. The complete $n$-point, etc.
 * 15. Configurations
 * 16. The Desargues configuration
 * 17. Perspective tetrahedra
 * 18. The quadrangle-quadrilateral configuration
 * 19. The fundamental theorem on quadrangular sets
 * 20. Additional remarks concerning the Desargues configuation


 * CHAPTER $\text {III}$.
 * 21. The nine primitive geometric forms
 * 22. Perspectivity and projectivity
 * 23. The projectivity of one-dimensional primitive forms
 * 24. General theory of correspondence. Symbolic treatment
 * 25. The notion of a group
 * 26. Groups of correspondences. Invariant elements and figures
 * 27. Group properties of projectivities
 * 28. Projective transformations of two-dimensional forms
 * 29. Projective collineations of three-dimensional forms


 * CHAPTER $\text {IV}$.
 * 30. The projectivity of quadrangular sets
 * 31. Harmonic sets
 * 32. Nets of rationality on a line
 * 33. Nets of rationality in the plane
 * 34. Nets of rationality in space
 * 35. The fundamental theorem of projectivity
 * 36. The configuration of Pappus. Mutually inscribed and circumscribed triangles
 * 37. Construction of projectivities on one-dimensional forms
 * 38. Involutions
 * 39. Axis and center of homology
 * 40. Types of collineations in the plane


 * CHAPTER $\text {V}$.
 * 41. Definitions. Pascal's and Brianchon's theorems
 * 42. Tangents. Points of contact
 * 43. The tangents to a point conic form a line conic
 * 44. The polar system of a conic
 * 45. Degenerate conics
 * 46. Desargues's theorem on conics
 * 47. Pencils and ranges of conics. Order of contact


 * CHAPTER $\text {VI}$.
 * 48. Addition of points
 * 49. Multiplication of points
 * 50. The commutative law for multiplication
 * 51. The inverse operations
 * 52. The abstract concept of a number system. Isomorphism
 * 53. Nonhomogeneous coordinates
 * 54. The analytic expression for a projectivity in a one-dimensional primitive form
 * 55. Von Staudt's algebra of throws
 * 56. The cross ratio
 * 57. Coördinates in a net of rationality on a line
 * 58. Homogeneous coördinates on a line
 * 59. Projective correspondence between the points of two different lines


 * CHAPTER $\text {VII}$.
 * 60. Nonhomogeneous coördinates in a plane
 * 61. Simultaneous point and line coördinates
 * 62. Condition that a point be on a line
 * 63. Homogeneous coördinates in the plane
 * 64. The line on two points. The point on two lines
 * 65. Pencils of points and lines. Projectivity
 * 66. The equation of a conic
 * 67. Linear transformations in a plane
 * 68. Collineations between two different planes
 * 69. Nonhomogeneous coördinates in space
 * 70. Homogeneous coördinates in space
 * 71. Linear transformations in space
 * 72. Finite spaces


 * CHAPTER $\text {VIII}$.
 * 73. Characteristic throw and cross ratio
 * 74. Projective projectivities
 * 75. Groups of projectivities on a line
 * 76. Projective transformations between conics
 * 77. Projectivities on a conic
 * 78. Involutions
 * 79. Involutions associated with a given projectivity
 * 80. Harmonic transformations
 * 81. Scale on a conic
 * 82. Parametric representation of a conic


 * CHAPTER $\text {IX}$.
 * 83. The degree of a geometric problem
 * 84. The intersection of a given line with a given conic
 * 85. Improper elements. Proposition $\mathrm K_2$
 * 86. Problems of the second degree
 * 87. Invariants of linear and quadratic binary forms
 * 88. Proposition $\mathrm K_n$
 * 89. Taylor's theorem. Polar forms
 * 90. Invariants and covariants of binary forms
 * 91. Ternary and quaternary forms and their invariants
 * 92. Proof of Proposition $\mathrm K_n$


 * CHAPTER $\text {X}$.
 * 93. Correlations between two-dimensional forms
 * 94. Analytic representation of a correlation between two planes
 * 95. General projective group. Representations by matrices
 * 96. Double points and double lines of a collineation in a plane
 * 97. Double pairs of a correlation
 * 98. Fundamental conic of a polarity in a plane
 * 99. Poles and polars with respect to a conic. Tangents
 * 100. Various definitions of conics
 * 101. Pairs of conics
 * 102. Problems of the third and fourth degree


 * CHAPTER $\text {XI}$.
 * 103. The regulus
 * 104. The polar system of a regulus
 * 105. Projective conics
 * 106. Linear dependence on lines
 * 107. The linear congruence
 * 108. The linear complex
 * 109. The Plücker line coördinates
 * 110. Linear families of lines
 * 111. Interpretation of line coördinates as point coördinates in $\mathrm S_5$


 * INDEX