stomary 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
notes at the foot of the page, and has tried instead, in the last chapter,
to give a consecutive account of the history of pure geometry, or, at
least, of as much of it as the student will be able to appreciate who has
mastered the course as given in the preceding chapters. One is not apt to
get a very wide view of the history of a subject by reading a hundred
biographical footnotes, arranged in no sort of sequence. The writer,
moreover, feels that the proper time to learn the history of a subject is
after the student has some general ideas of the subject itself.

The course is not intended to furnish an illustration of how a subject may
be developed, from the smallest possible number of fundamental
assumptions. The author is aware of the importance of work of this sort,
but he does not believe it is possible at the present time to write a book
along such lines which shall be of much use for elementary students. For
the purposes of this course the student should have a thorough grounding
in ordinary elementary geometry so far as to include the study of the
circle and of similar triangles. No solid geometry is needed beyond the
little used in the proof of Desargues' theorem (25), and, except in
certain metrical developments of the general theory, there will be no call
for a knowledge of trigonometry or analytical geometry. Naturally the
student who is equipped with t