# Book:Oswald Veblen/Projective Geometry/Volume 1

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

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

### 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
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
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