sharpness of vision to try to focus his eyes on two things at once. Those
who prefer the usual method of procedure can, of course, develop the two
sets of theorems side by side; the author has not found this the better
plan in actual teaching.

As regards nomenclature, the author has followed the lead of the earlier
writers in English, and has called the system of lines in a plane which
all pass through a point a _pencil of rays_ instead of a _bundle of rays_,
as later writers seem inclined to do. For a point considered as made up of
all the lines and planes through it he has ventured to use the term _point
system_, as being the natural dualization of the usual term _plane
system_. He has also rejected the term _foci of an involution_, and has
not used the customary terms for classifying involutions—_hyperbolic
involution_, _elliptic involution_ and _parabolic involution_. He has
found that all these terms are very confusing to the student, who
inevitably tries to connect them in some way with the conic sections.

Enough examples have been provided to give the student a clear grasp of
the theory. Many are of sufficient generality to serve as a basis for
individual investigation on the part of the student. Thus, the third
example at the end of the first chapter will be found to be very fruitful
in interesting results. A correspondence is there indicated between lines
in space and circles through a fixed point in space. If the student will
trace a few of the consequences of that correspondence, and determine what
configurations of circles correspond to intersecting lines, to lines in a
plane, to lines of a plane pencil, to lines cutting three skew lines,
etc., he will have acquired no little practice in picturing to himself
figures in space.

The writer has not followed the usual practice of inserting historical