An Elementary Course in Synthetic Projective Geometry

by Lehmer, Derrick Norman

Edition 1, (November 4, 2005)

PREFACE

The following course is intended to give, in as simple a way as possible, the essentials of synthetic projective geometry. While, in the main, the theory is developed along the well-beaten track laid out by the great masters of the subject, it is believed that there has been a slight smoothing of the road in some places. Especially will this be observed in the chapter on Involution. The author has never felt satisfied with the usual treatment of that subject by means of circles and anharmonic ratios. A purely projective notion ought not to be based on metrical foundations. Metrical developments should be made there, as elsewhere in the theory, by the introduction of infinitely distant elements.

The author has departed from the century-old custom of writing in parallel columns each theorem and its dual. He has not found that it conduces to 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.