Book:Oswald Veblen/Projective Geometry/Volume 1

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

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

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 $\text {I}$. 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