The following course is intended to give, in as simplea way as possible, the essentials of synthetic projectivegeometry. While, in the main, the theory is developedalong the well-beaten track laid out by the great mastersof the subject, it is believed that there has been a slightsmoothing of the road in some places. Especially willthis be observed in the chapter on Involution. Theauthor has never felt satisfied with the usual treatmentof that subject by means of circles and anharmonicratios. A purely projective notion ought not to be basedon metrical foundations. Metrical developments shouldbe made there, as elsewhere in the theory, by theintroduction of infinitely distant elements.
The author has departed from the century-old customof writing in parallel columns each theorem and itsdual. He has not found that it conduces to sharpnessof 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 actualteaching.
As regards nomenclature, the author has followedthe lead of the earlier writers in English, and has calledthe system of lines in a plane which all pass through apoint a pencil of rays instead of a bundle of rays, as laterwriters seem inclined to do. For a point considered[pg iv]as made up of all the lines and planes through it hehas ventured to use the term point system, as beingthe natural dualization of the usual term plane system.He has also rejected the term foci of an involution, andhas not used the customary