PREFACE TO THE REVISED EDITION THE main purpose of the revision of this book has been to emphasize still further and to elaborate in greater detail the principal aim of the original edition, viz., to introduce the student systematically to original geometric work. To make the teaching of geometry both disciplinary and informational; to give to the student mental training instead of teaching him mere facts; to develop his power instead of making him memorize, these are the fundamental aims of this book. The means employed for this purpose are similar to those used in the first edition. Still greater emphasis, however, has been placed upon the general methods which may be used for the solution of original exercises. The grading and the selection of exercises have been carefully revised. All originals that appeared unfit or too difficult have been eliminated or replaced by simpler and better ones. Topics of fundamental importance, e.g. the methods of demonstrating the equality of lines, are represented in greater detail and illustrated by a greater number of exercises than in the first edition. In addition to these fundamental tendencies, a number of minor improvements have been introduced, among which may be mentioned: Improved presentation of the regular propositions. Many proofs have been simplified, a more pedagogic sequence of the propositions of Book I has been adopted, Books VI and VII have been considerably simplified, and a number of difficult theorems of minor importance have been omitted or placed in the appendix. As Simplification of the so-called "incommensurable case." this is a claim that is made by most text-books, it may be received with some degree of skepticism, but a repeated trial of this new method will reveal its simplicity. For the more conservative teacher, however, who dislikes fundamental changes, the time-honored method is given in the appendix. The introduction of many applied problems. These problems have been selected and arranged so as to increase the interest of the student, without sacrificing in the least the disciplinary value of the subject. Many such problems are given in the appendix. The arrangement of the propositions and the terminology are in accord with the best modern usage. Thus, statements and reasons have been separated and placed in parallel vertical columns; the term "congruent" and the corresponding symbol are introduced and applied consistently, etc. Many of the diagrams have been improved. The construction lines are drawn completely for most problems, graphical modes are employed for pointing out important facts, and many diagrams have been otherwise improved. Thanks are due to Dr. J. Kahn and Mr. W. S. Schlauch for assistance in reading the proof and for helpful suggestions. August 1, 1913. A. S. PLANE GEOMETRY INTRODUCTION DEFINITIONS 1. A physical body, such as a block of wood or iron, occu pies a definite portion of space. The portion of space occu pied by a physical body is called a geometric solid or a solid. A B 2. DEF. A solid is a limited portion of space. It has three dimensions, length, breadth, and thickness. 3. DEF. Surfaces are the boundaries of solids; as ABED or BEFC. They have two dimensions, length and breadth. The boundary between a window pane and the air is a surface. Obviously such a boundary has no thickness. 4. DEF. Lines are the boundaries of surfaces, as AB, ad. (Figure of § 1.) Lines have but one dimension, length. Thus, the annexed black line AB is not a geometric line, for it has breadth. B A true geometric line, however, is represented by the boundary between the black and the white. 5. DEF. lines. Points are the boundaries or the extremities of They are without dimensions, having position only. Surfaces may be conceived as existing independent of the solids whose boundaries they form. In like manner, lines and points may exist independently in space. |