GEOMETRY BY WILLIAM J. MILNE, PH.D., LL.D. PRESIDENT OF NEW YORK STATE NORMAL COLLEGE, ALBANY, N.Y. NEW YORK: CINCINNATI .:. CHICAGO HARVARD MILNE'S MATHEMATICS MILNE'S ELEMENTS OF ARITHMETIC MILNE'S STANDARD ARITHMETIC MILNE'S MENTAL ARITHMETIC MILNE'S ELEMENTS OF ALGEBRA MILNE'S GRAMMAR SCHOOL ALGEBRA MILNE'S HIGH SCHOOL ALGEBRA MILNE'S PLANE AND SOLID GEOMETRY MILNE'S PLANE GEOMETRY-SEPARATE COPYRIGHT, 1899, BY MILNE'S GEOM. E-P 10 PREFACE It is generally conceded that geometry is the most interesting of all the mathematical sciences, yet many students have failed to find either pleasure or profit in studying it. The most serious hindrance to the proper understanding of the subject has been the failure on the part of the student to grasp the geometrical concept which he has been endeavoring to establish by a process of reasoning. Many attempts have been made by thorough teachers to remedy the difficulty, but there is a very general agreement that the most successful method has been by exercises in "Inventional" Geometry. Students who have been fortunate enough to have the subject presented in that way have usually understood it, and, better still, they have enjoyed it. While Inventional Geometry has been full of interest to the student, it has often failed to develop that knowledge of the science which is necessary to thorough mastery, because it has not been progressive, and, what is more to be deplored, it has failed to give that acquaintance with the forms of rigid deductive reasoning which is one of the chief objects sought in the study of the science. The student has often been led by this objective method of study to rely upon his visual recognition of the relations of lines and angles in a drawing rather than upon the demonstration based upon definitions, axioms, and propositions that have been proved. In this book the effort is made to introduce the student to geometry through the employment of inventional steps, but the somewhat fragmentary and unsatisfactory result of such teaching is supplemented by demonstrations, in consecutive order, of the fundamental propositions of the science. The desirability of training students to form proper inferences from the study of accurately drawn figures has been recognized by the author; such a method awakens keen interest in the subject and develops right habits of investigation, but there is necessity also for the accuracy of statement and the logical training of the older methods to assure the pleasure and profit that belong with both. Every theorem has been introduced by questions designed to lead the student to discover the geometrical concept clearly and fully before a demonstration is attempted. They are not intended to lead to a demonstration, but rather to a correct and definite idea of what is to be proved. |