
The purpose of this book is to provide a first course in Projective Geometry for
undergraduate majors in mathematics and for prospective teachers of high school
geometry. For the former it will furnish an introduction to the important concept of
projective spaces; for the latter it will introduce a more general geometry from which,
by proper specialization, the familiar metric geometry is obtained. Since only the
real geometry of one and two dimensions is considered here, every theorem may be
illustrated by a diagram in the construction of which nothing more than a straight
edge is required.
Chapter 1 begins with a brief survey of the geometry of Euclid and his associates.
That part of this geometry which is concerned solely with the incidence of points and
lines is called projective. The projective plane is then obtained by properly modifying
the fundamental plane of Euclidean geometry. In Chapters 16 and 812, the reader
will find the basic propositions of plane projective geometry developed entirely by
synthetic methods.
Chapter 7 is concerned with an axiomatic approach. In the course of providing
models for the axioms, certain finite projective geometries are introduced. Since this
leads eventually to the geometry of points defined by number triples over the field of
real numbers of Chapter 15, the reading of the chapter may be postponed until that
time.
In Chapter 13 the procedure is reversed. Taking the projective plane as funda
mental, modifications are made to obtain the affine plane in which parallel lines
reappear. Additional modifications are made in Chapter 14 in order to define per
pendicular lines and thus permit a return to the metric plane. Of interest here is the
fact, observed perhaps for the first time by the reader, that so much of the metric
geometry with which he is familiar depends on parallelism rather than on perpendicu
larity. Also to be noted is the great variety of metric theorems which often follow
from a single projective theorem.
In Chapters 1517 the reader is introduced to analytic methods in projective geom
etry. In these chapters an acquaintance with matrix algebra is assumed. For those
who would wish a brief review, the Appendix will be found helpful.
The final chapter parallels Chapters 1314. Beginning with the set of all projective
transformations of the plane onto itself, the reader is led by successive steps to the
familiar rigid motions of Plane Analytic Geometry.
The author wishes to take this opportunity to express his gratitude to the staff
of the Schaum Publishing Company for their splendid cooperation. 