The Project Gutenberg EBook of An Elementary Course in Synthetic Projective Geometry by Lehmer, Derrick Norman

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Title: An Elementary Course in Synthetic Projective Geometry

Author: Lehmer, Derrick Norman

Release Date: November 4, 2005 [Ebook #17001]

Language: English

***START OF THE PROJECT GUTENBERG EBOOK AN ELEMENTARY COURSE IN SYNTHETIC PROJECTIVE GEOMETRY***

An Elementary Course in Synthetic Projective Geometry

by Lehmer, Derrick Norman

Edition 1, (November 4, 2005)

[pg iii]

Preface

The following course is intended to give, in as simple a way as possible, the essentials of synthetic projective geometry. While, in the main, the theory is developed along the well-beaten track laid out by the great masters of the subject, it is believed that there has been a slight smoothing of the road in some places. Especially will this be observed in the chapter on Involution. The author has never felt satisfied with the usual treatment of that subject by means of circles and anharmonic ratios. A purely projective notion ought not to be based on metrical foundations. Metrical developments should be made there, as elsewhere in the theory, by the introduction of infinitely distant elements.

The author has departed from the century-old custom of writing in parallel columns each theorem and its dual. He has not found that it conduces to sharpness of vision to try to focus his eyes on two things at once. Those who prefer the usual method of procedure can, of course, develop the two sets of theorems side by side; the author has not found this the better plan in actual teaching.

As regards nomenclature, the author has followed the lead of the earlier writers in English, and has called the system of lines in a plane which all pass through a point a pencil of rays instead of a bundle of rays, as later writers seem inclined to do. For a point considered [pg iv] as made up of all the lines and planes through it he has ventured to use the term point system, as being the natural dualization of the usual term plane system. He has also rejected the term foci of an involution, and has not used the customary terms for classifying involutions—hyperbolic involution, elliptic involution and parabolic involution. He has found that all these terms are very confusing to the student, who inevitably tries to connect them in some way with the conic sections.

Enough examples have been provided to give the student a clear grasp of the theory. Many are of sufficient generality to serve as a basis for individual investigation on the part of the student. Thus, the third example at the end of the first chapter will be found to be very fruitful in interesting results. A correspondence is there indicated between lines in space and circles through a fixed point in space. If the student will trace a few of the consequences of that correspondence, and determine what configurations of circles correspond to intersecting lines, to lines in a plane, to lines of a plane pencil, to lines cutting three skew lines, etc., he will have acquired no little practice in picturing to himself figures in space.

The writer has not followed the usual practice of inserting historical notes at the foot of the page, and has tried instead, in the last chapter, to give a consecutive account of the history of pure geometry, or, at least, of as much of it as the student will be able to appreciate who has mastered the course as given in the preceding chapters. One is not apt to get a very wide view of the history of a subject by reading a hundred [pg v] biographical footnotes, arranged in no sort of sequence. The writer, moreover, feels that the proper time to learn the history of a subject is after the student has some general ideas of the subject itself.

The course is not intended to furnish an illustration of how a subject may be developed, from the smallest possible number of fundamental assumptions. The author is aware of the importance of work of this sort, but he does not believe it is possible at the present time to write a book along such lines which shall be of much use for elementary students. For the purposes of this course the student should have a thorough grounding in ordinary elementary geometry so far as to include the study of the circle and of similar triangles. No solid geometry is needed beyond the little used in the proof of Desargues' theorem (25), and, except in certain metrical developments of the general theory, there will be no call for a knowledge of trigonometry or analytical geometry. Naturally the student who is equipped with these subjects as well as with the calculus will be a little more mature, and may be expected to follow the course all the more easily. The author has had no difficulty, however, in presenting it to students in the freshman class at the University of California.

The subject of synthetic projective geometry is, in the opinion of the writer, destined shortly to force its way down into the secondary schools; and if this little book helps to accelerate the movement, he will feel amply repaid for the task of working the materials into a form available for such schools as well as for the lower classes in the university.

[pg vi]

The material for the course has been drawn from many sources. The author is chiefly indebted to the classical works of Reye, Cremona, Steiner, Poncelet, and Von Staudt. Acknowledgments and thanks are also due to Professor Walter C. Eells, of the U.S. Naval Academy at Annapolis, for his searching examination and keen criticism of the manuscript; also to Professor Herbert Ellsworth Slaught, of The University of Chicago, for his many valuable suggestions, and to Professor B. M. Woods and Dr. H. N. Wright, of the University of California, who have tried out the methods of presentation, in their own classes.

D. N. LEHMER

Berkeley, California

Contents

Preface

Contents

CHAPTER I - ONE-TO-ONE CORRESPONDENCE

1. Definition of one-to-one correspondence

2. Consequences of one-to-one correspondence

3. Applications in mathematics

4. One-to-one correspondence and enumeration

5. Correspondence between a part and the whole

6. Infinitely distant point

7. Axial pencil; fundamental forms

8. Perspective position

9. Projective relation

10. Infinity-to-one correspondence

11. Infinitudes of different orders

12. Points in a plane

13. Lines through a point

14. Planes through a point

15. Lines in a plane

16. Plane system and point system

17. Planes in space

18. Points of space

19. Space system

20. Lines in space

21. Correspondence between points and numbers

22. Elements at infinity

PROBLEMS

CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE CORRESPONDENCE WITH EACH OTHER

23. Seven fundamental forms

24. Projective properties

25. Desargues's theorem

26. Fundamental theorem concerning two complete quadrangles

27. Importance of the theorem

28. Restatement of the theorem

29. Four harmonic points

30. Harmonic conjugates

31. Importance of the notion of four harmonic points

32. Projective invariance of four harmonic points

33. Four harmonic lines

34. Four harmonic planes

35. Summary of results

36. Definition of projectivity

37. Correspondence between harmonic conjugates

38. Separation of harmonic conjugates

39. Harmonic conjugate of the point at infinity

40. Projective theorems and metrical theorems. Linear construction

41. Parallels and mid-points

42. Division of segment into equal parts

43. Numerical relations

44. Algebraic formula connecting four harmonic points

45. Further formulae

46. Anharmonic ratio

PROBLEMS

CHAPTER III - COMBINATION OF TWO PROJECTIVELY RELATED FUNDAMENTAL FORMS

47. Superposed fundamental forms. Self-corresponding elements

48. Special case

49. Fundamental theorem. Postulate of continuity

50. Extension of theorem to pencils of rays and planes

51. Projective point-rows having a self-corresponding point in common

52. Point-rows in perspective position

53. Pencils in perspective position

54. Axial pencils in perspective position

55. Point-row of the second order

56. Degeneration of locus

57. Pencils of rays of the second order

58. Degenerate case

59. Cone of the second order

PROBLEMS

CHAPTER IV - POINT-ROWS OF THE SECOND ORDER

60. Point-row of the second order defined

61. Tangent line

62. Determination of the locus

63. Restatement of the problem

64. Solution of the fundamental problem

65. Different constructions for the figure

66. Lines joining four points of the locus to a fifth

67. Restatement of the theorem

68. Further important theorem

69. Pascal's theorem

70. Permutation of points in Pascal's theorem

71. Harmonic points on a point-row of the second order

72. Determination of the locus

73. Circles and conics as point-rows of the second order

74. Conic through five points

75. Tangent to a conic

76. Inscribed quadrangle

77. Inscribed triangle

78. Degenerate conic

PROBLEMS

CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER

79. Pencil of rays of the second order defined

80. Tangents to a circle

81. Tangents to a conic

82. Generating point-rows lines of the system

83. Determination of the pencil

84. Brianchon's theorem

85. Permutations of lines in Brianchon's theorem

86. Construction of the penvil by Brianchon's theorem

87. Point of contact of a tangent to a conic

88. Circumscribed quadrilateral

89. Circumscribed triangle

90. Use of Brianchon's theorem

91. Harmonic tangents

92. Projectivity and perspectivity

93. Degenerate case

94. Law of duality

PROBLEMS

CHAPTER VI - POLES AND POLARS

95. Inscribed and circumscribed quadrilaterals

96. Definition of the polar line of a point

97. Further defining properties

98. Definition of the pole of a line

99. Fundamental theorem of poles and polars

100. Conjugate points and lines

101. Construction of the polar line of a given point

102. Self-polar triangle

103. Pole and polar projectively related

104. Duality

105. Self-dual theorems

106. Other correspondences

PROBLEMS

CHAPTER VII - METRICAL PROPERTIES OF THE CONIC SECTIONS

107. Diameters. Center

108. Various theorems

109. Conjugate diameters

110. Classification of conics

111. Asymptotes

112. Various theorems

113. Theorems concerning asymptotes

114. Asymptotes and conjugate diameters

115. Segments cut off on a chord by hyperbola and its asymptotes

116. Application of the theorem

117. Triangle formed by the two asymptotes and a tangent

118. Equation of hyperbola referred to the asymptotes

119. Equation of parabola

120. Equation of central conics referred to conjugate diameters

PROBLEMS

CHAPTER VIII - INVOLUTION

121. Fundamental theorem

122. Linear construction

123. Definition of involution of points on a line

124. Double-points in an involution

125. Desargues's theorem concerning conics through four points

126. Degenerate conics of the system

127. Conics through four points touching a given line

128. Double correspondence

129. Steiner's construction

130. Application of Steiner's construction to double correspondence

131. Involution of points on a point-row of the second order.

132. Involution of rays

133. Double rays

134. Conic through a fixed point touching four lines

135. Double correspondence

136. Pencils of rays of the second order in involution

137. Theorem concerning pencils of the second order in involution

138. Involution of rays determined by a conic

139. Statement of theorem

140. Dual of the theorem

PROBLEMS

CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS

141. Introduction of infinite point; center of involution

142. Fundamental metrical theorem

143. Existence of double points

144. Existence of double rays

145. Construction of an involution by means of circles

146. Circular points

147. Pairs