An Elementary Course in Synthetic Projective Geometry | Page 2

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.
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