It is for this larger class, the great body of progressive teachers, that this book is written. It stands for Vitaliz ing geometry in every legitimate way; for improving the subject matter in such manner as not to destroy the pupil's interest; for so teaching geometry as to make it appeal to pupils as strongly as any other subject in the curriculum; but for the recognition of geometry for geometry's sake and not for the sake of a fancied utility that hardly exists. Expressing full appreciation of the desirability of establishing a motive for all studies, so as to have the work proceed with interest and vigor, it does not hesitate to express doubt as to certain motives that have been exploited, nor to stand for such a genuine, thought-compelling development of the science as is in harmony with the' mental powers of the pupils in the American high school. For this class of teachers the author hopes that the book will prove of service, and that through its peru sal they will come to admire the subject more and more, and to teach it with greater interest. It offers no pana cea, it champions no single method, but it seeks to set forth plainly the reasons for teaching a geometry of the kind that we have inherited, and for hoping for a grad ual but definite improvement in the science and in the methods of its presentation.