PREFACE THE present text, the outgrowth of over twenty years' experience with classes in geometry, carries the work from the simple elements necessary in the beginning class of the high school to the most advanced requirements of university preparation. For many years we have followed in our own classes the plan here presented, with such modifications and improvements as experience has suggested. It is generally conceded that much of the pupil's difficulty in demonstration arises from his failure to grasp thoroughly and keep vividly in mind as separate and distinct statements, first, the exact data of the proposition, and, second, the precise fact to be established. To remove this stumbling-block, we have stated the hypothesis and conclusion separately for every theorem and corollary demonstrated. Through the "open" arrangement of the printed matter, we have sought to make each successive step stand out clearly; and by so adjusting diagrams and text that in the course of any single demonstration it is unnecessary to turn the page, we have endeavored to avoid waste of effort on the part of the pupil. All the original exercises should be mastered with only such help as is given by the book itself. If a proposition has been found difficult for the average pupil, it has been broken up into a series of exercises in such sequence that the difficulties are presented one at a time and in natural order, the truth of the main proposition being established by means of these graded exercises. It has been our purpose to eliminate discouraging elements, to refresh the memory of the student before he begins inventive work, to arouse his interest and to inspire his confidence in his ability to discover hidden truths. We desire to express our acknowledgments for valuable counsel and suggestions to Professor F. N. Cole of Columbia University, to Professor Irving Stringham of the University of California, to Professor R. E. Gaines of Richmond College, Virginia, and to J. A. C. Chandler, LL.D., formerly Dean of Richmond Academy, Virginia. SAN FRANCISCO, WALTER N. BUSH. PLAN AND SCOPE THERE are few students who fail to respond to the stimulus of original work in Geometry. The energy expended and enthusiasm displayed in the solution of exercises is in sharp contrast with their apathy toward the study of theorems in the text. That the student may arrive by the shortest path to the point where the real development of his mental power begins, where his interest in the independent solution of original exercises becomes an active part of his school life, it is of the first importance that the theorems of the text be so classified and demonstrated as to offer his attempts to master them the least resistance. It is of equal importance that these exercises should be so carefully graded as to stimulate and not discourage the pupil, and that some method of systematically attacking and solving problems should be devised and presented for his assistance. Nothing is presented in the pages that follow that has not stood the actual test of class-room experience for many years. In accordance with these essentials of a text for use in Geometry classes, we call special attention to the following features of this work: First. The classification of Definitions and Axioms. Second. The arrangement into groups of Theorems relating to the same topic. For example, theorems concerning isosceles triangles in the "Isosceles Triangle Group"; congruent tri angles in the "Congruent Triangle Group"; comparison of areas in the "Areal Ratio Group." |