BY HERBERT E. HAWKES, Ph.D. PROFESSOR OF MATHEMATICS IN COLUMBIA UNIVERSITY WILLIAM A. LUBY, M.A. HEAD OF THE DEPARTMENT OF MATHEMATICS IN THE JUNIOR COLLEGE OF KANSAS CITY AND FRANK C. TOUTON, Ph.D. PUBLIC INSTRUCTION, MADISON, WISCONSIN H35 COPYRIGHT, 1920, BY AND FRANK C. TOUTON ALL RIGHTS RESERVED 321.3 CAJORI The Athen zum Press PREFACE Although the study of geometry is important from an informational point of view, it is generally recognized that a genuine mastery of the subject means real achievement in the solution of original exercises. The chief aim of the authors in preparing this text has been to give such assistance to students as will stimulate insight and develop power to solve exercises of gradually increasing difficulty. The content and organization of the first book of geometry are determining factors in the student's progress in the subject. The first ten or twenty theorems determine whether a student will grasp quickly the general trend of the subject or remain bewildered for an indefinite period. The simplicity and directness of the early theorems and the order in which they are presented are elements of the highest importance in an effective introduction. These elements were given the most careful consideration in the preparation of this text. If the first theorem is used in proving the second and the second in proving the third, and so on, the student will soon see a reason for mastering the content of each theorem. But if the initial theorem is used for the first time in the fifteenth theorem, and the second is used next in the eighth, the first month's work will give the impression that geometry deals with unrelated facts which lead nowhere. Such an arrangement of the theorems of geometry will hamper even the strongest students, and will make progress almost impossible for the less capable. 918336 In this text no time is wasted on side issues or on discussive explanations, but the student is brought by a direct route to the theorems on angle sums of polygons, the theorems on parallel lines, and those on parallelograms. Before reaching Theorem 13 he meets theorems on angle sums, the one topic in Book I which has varied numerical applications. Exercises based on angle sums are presented on page 35, at which point the student begins his really independent work upon numerical exercises. Having acquired some ability in solving numerical exercises he is prepared to begin the devising of general proofs involving congruent triangles. The most important single method of elementary geometry is the use of congruent triangles. The best experience of teachers has shown that practice with this method should begin early and last long. Superposition is used here only when unavoidable, and the more difficult topics, such as inequalities and indirect methods, are deferred until near the end of the book. Attention is invited to certain general features of the text. For example, axioms, postulates, and definitions are given when needed, not before. The usual group of theorems on proportion are distributed, each one being presented at the proper point. An important feature is the placing of a few exercises involving suitable numerical and general applications of a theorem immediately after it. Many exercises are presented which are designed to correlate geometry with algebra, and to compel the student to use his knowledge of fractions, equations, and radicals. A large number of pertinent and stimulating queries are given which demand only direct answers. Occasionally queries are used to develop a specific topic such as loci, but usually they are designed to broaden and perfect the student's knowledge of details supplementary to the theorems and exercises. The parallelcolumn arrangement of the demonstrations has been chosen in the belief that it is an aid to clear thinking, and that it places unusual and proper emphasis on the necessity of stating a reason for each assertion. Frequently, to encourage independent thinking, reasons have been omitted and " Why? has been inserted. This is done sparingly at first, but later on with an increasing frequency graduated to the student's growing knowledge of the subject. Certain special features have received careful treatment. Emphasis on right triangles having one angle 30°, 45°, or 60°, which begins with Theorem 24 of Book I, is continued over into other books, especially Book III. This is justified by the experience of every teacher of trigonometry and by the specific recommendation of the Society for the Promotion of Engineering Education. In fact, no theorem has been omitted which any important educational body considers necessary. In Book II particular attention is called to the emphasis placed on constructions and loci and their interrelations. An appreciation of these relations makes it clear why real constructional work should not come earlier in a course on demonstrative geometry. If it does, the results must be assumed without proof. It seems far more desirable to assume, in the few cases needed, the existence of certain lines and figures, and to defer the actual construction of them until it is possible to prove that the method used is correct. Full advantage has been taken of the unusual opportunity offered by Book V for useful and necessary constructions. The presentation of the matter of that book is one of the unique features of the text. The graphical methods of solution given on pages 218 and 219 satisfy the most rigorous criteria for practical problems. A further illustration of the |