The present work is written with these two larger questions in mind, although it considers from time to time the minor ones already mentioned, together with others of a similar nature. It recognizes that the recent growth in popular education has brought into the high school a less carefully selected type of mind than was formerly the case, and that for this type a different kind of mathematical training will naturally be developed. It proceeds upon the theory, however, that for the normal mind,--for the boy or girl who is preparing to win out in the long run,--geometry will continue to be taught as demonstrative geometry, as a vigorous thought-compelling subject, and along the general lines that the experience of the world has shown to be the best. Soft mathematics is not interesting to this normal mind, and a sham treatment will never appeal to the pupil; and this book is written for teachers who believe in this principle, who believe in geometry for the sake of geometry, and who earnestly seek to make the subject so interesting that pupils will wish to study it whether it is required or elective. The work stands for the great basal propositions that have come down to us, as logically arranged and as scientifically proved as the powers of the pupils in the American high school will permit; and it seeks to tell the story of these propositions and to show their possible and their probable applications in such a way as to furnish teachers with a fund of interesting material with which to supplement the book work of their classes. After all, the problem of teaching any subject comes down to this: Get a subject worth teaching and then make every minute of it interesting. Pupils do not object to work if they like a subject, but they do object to aimless and uninteresting tasks. Geometry is particularly fortunate in that the feeling of accomplishment comes with every proposition proved; and, given a class of fair intelligence, a teacher must be lacking in knowledge and enthusiasm who cannot foster an interest that will make geometry stand forth as the subject that brings the most pleasure, and that seems the most profitable of all that are studied in the first years of the high school.