Synthetic Projective Geometry

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 wellbeaten 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.

Excerpt:

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 apencil of rays instead of a bundle of rays, as later writers seem inclined to do. For a pointconsidered as made up of all the lines and planes through it he has ventured to use theterm point system, as being the natural dualization of the usual term plane system. He hasalso rejected the term foci of an involution, and has not used the customary terms forclassifying involutions-hyperbolic involution, elliptic involution and parabolicinvolution.

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 thepart of the student. Thus, the third example at the end of the first chapter will be found tobe very fruitful in interesting results.

A correspondence is there indicated between lines inspace and circles through a fixed point in space. If the student will trace a few of theconsequences of that correspondence, and determine what configurations of circlescorrespond to intersecting lines, to lines in a plane, to lines of a plane pencil, to linescutting three skew lines, etc., he will have acquired no little practice in picturing tohimself figures in space.

The writer has not followed the usual practice of inserting historical notes at the foot ofthe page, and has tried instead, in the last chapter, to give a consecutive account of thehistory of pure geometry, or, at least, of as much of it as the student will be able toappreciate 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 biographicalfootnotes, arranged in no sort of sequence. The writer, moreover, feels that the propertime to learn the history of a subject is after the student has some general ideas of thesubject 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 ofthe importance of work of this sort, but he does not believe it is possible at the presenttime 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 ordinaryelementary 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 nocall for a knowledge of trigonometry or analytical geometry. Naturally the student who isequipped with these subjects as well as with the calculus will be a little more mature, andmay be expected to follow the course all the more easily. The author has had nodifficulty, however, in presenting it to students in the freshman class at the University ofCalifornia.

The subject of synthetic projective geometry is, in the opinion of the writer, destinedshortly to force its way down into the secondary schools; and if this little book helps toaccelerate the movement, he will feel amply repaid for the task of working the materialsinto a form available for such schools as well as for the lower classes in the university.