WOOSTER WOODRUFF BEMAN PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN AND DAVID EUGENE SMITH PROFESSOR OF MATHEMATICS IN THE MICHIGAN STATE NORMAL SCHOOL BOSTON, U.S.A., AND LONDON 1896 PREFACE. 1. THE present work covers the elements of plane and solid geometry as offered in the best American high schools. It does not contain any discussion of conics, nor does it introduce the distinctive features of modern geometry. It does seek, however, to invest the ancient geometry with something of the spirit of modern mathematics. 2. The sequence of propositions in Books I to III is in the main that laid down by the Association for the Improvement of Geometrical Teaching, of England. The algebraic treatment of proportion is not concealed. The one-to-one correspondence between geometry and algebra which Professor Newcomb has termed the Law of Homology, is easily and naturally introduced in Book II. This law is distinctly postulated as the foundation of the treatment of proportion, and the student is informed that the purely geometric method is too difficult for the beginner. After proportion the sequence has been determined by the comparison of leading foreign and American text-books. It has been assumed that the student who has entered into the spirit of the subject will be so able to generalize that a number of the later propositions may be treated as corollaries. 3. Modern geometry offers many simple features that are usable. Among them are the notions of symmetry, of positive and negative, of continuity, of reciprocity and of similarity. These have been introduced on the pedagogical principle that the new should be brought in only where it is needed. 4. The terms and symbols are in no sense innovations. None have been used that are not generally recognized in the |