hese 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. R