PREFACE. 1x 
points into points, lines into lines, and planes into planes. 
It is, therefore, a self-dualistic transformation. 
A collineation may be regarded from two distinct points of 
view, viz., the analytic andthe synthetic. From the synthetic 
point of view the phenomena of a collineation appeal directly 
to the eye or to the space intuitions. On the other hand, from 
the analytic point of view the operation is seen through the 
medium of a linear substitution on the requisite number of 
variables. The two methods have long been in use side by 
side and each has its special advantages. Each also has its 
special votaries, and each will continue to have its advocates 
as long as human minds continue to be constructed on differ- 
ent patterns. To methesynthetic method isthe more attractive, 
for the reason that it enables one to get closer to the facts and 
to view them at first hand. In all applications of analysis 
to geometry a formula is only the vehicle which conveys the 
thought, not the thought itself. The inevitable tendency is 
to confuse the vehicle with the thought, to mistake the vessel 
for the contents, and to lay hold on the shadow rather than 
the substance of the thing sought. 
My interest in the collineation as an object of research dates 
from the time when it was my rare good fortune to be a stu- 
dent of Lie at Leipzig in 1887-’88. I followed with special in- 
terest his lectures on Modern Geometry and on Continuous 
Groups. The latter course was afterward published under the 
title Vorlesungen weber Conlinuerliche Gruppen. Almost every 
example used to illustrate the theory of continuous groups was 
a group of projective transformations. Lie’s method of ap- 
proach to the theory of projective groups was through the in- 
finitesimal transformation. I early became dissatisfied with 
the infinitesimal method because there seemed to meso wide a 
gap between the analytic processes and the geometric interpre- 
tation of the results. I was constantly asking myself the 
question, whether it was not possible to develop the theory 
of the projective group directly from the finite form of the 
equations of a linear transformation or from geometric con- 
