An Elementary Course in Synthetic Projective Geometry | Page 2

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.
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