CHAPTER I. 
PROJECTIVE TRANSFORMATIONS IN ONE-DIMEN- 
SIONAL SPACE. 
. General Properties of One-Dimensional Projective Transformations. 
. Types and Normal Forms of Projective Transformations. 
One-Parameter Groups of Projective Transformations. 
. Two- and Three-Parameter Groups of Projective Transformations. 
. Transformations of Pencils of Lines and Planes. 
. Real Projective Transformations. 
. Theory of Projection. 
. Geometric Theory of Projective Transformations. 
. Geometric Theory of Transformations of Pencils of Lines and Planes. 
Exercises. 
SRL IAM SR UR SR SR LN 
oman OurrwWn re 
1. The present chapter is devoted to an exposition of the 
theory of projective transformations in space of one dimension. 
The theory applies equally well to all three one-dimensional 
primary forms of projective geometry, viz., a range of points 
on a line, a pencil of lines through a point, and a pencil of 
planes through a line. The principal facts of one-dimensional 
projective transformations are set forth and on these are built 
a comprehensive theory of their continuous groups. The 
theory in one-dimension is sufficiently complete to serve as a 
foundation and model on which to build a consistent theory 
of collineations in two, three and higher dimensions. 
In $1 we shall define analytically a projective transforma- 
tion in one dimension. In $$1 to 5 are developed the conse- 
quences of this analytic definition, and in $6 is considered the 
special case when the variables and coefficients in the equation 
are all real quantities. A geometrical theory of one-dimen- 
sional projective transformations is developed in $$7 to 9, and 
the two theories, analytic and geometric, are shown to be in 
perfect harmony. Each method will be seen to have its spe- 
cial points of advantage. The chapter closes with a classified 
list of exercises illustrating both methods. 
(1) 
