# Book:Oswald Veblen/Projective Geometry/Volume 1

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## Oswald Veblen and John Wesley Young:

## Oswald Veblen and John Wesley Young: *Projective Geometry, Volume $\text { 1 }$*

Published $\text {1910}$, **Ginn and Company**

### Subject Matter

### 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 $\text {I}$. theorems of alignment and the principle of duality
- 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 $\text {II}$. projection, section, perspectivity, elementary configurations
- 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 $\text {III}$. projectivities of the primitive geometric forms of one, two, and three dimensions
- 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 $\text {IV}$. harmonic constructions and the fundamental theorem of projective geometry
- 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 $\text {V}$. conic sections
- 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 $\text {VI}$. algebra of points and one-dimensional coördinate systems
- 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 $\text {VII}$. coördinate systems in two- and three-dimensional forms
- 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 $\text {VIII}$. projectivities in one-dimensional forms
- 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 $\text {IX}$. geometric constructions. invariants
- 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 $\text {X}$. projective transformations of two-dimensional forms
- 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 $\text {XI}$. families of lines
- 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