ELEMENTARY PLANE GEOMETRY INDUCTIVE AND DEDUCTIVE BY ALFRED BAKER, M.A., F.R.S.C. PROFESSOR OF MATHEMATICS, UNIVERSITY OF TORONTO BOSTON, U.S.A. The Athenæum Press 333 COPYRIGHT, 1903, BY GINN & COMPANY Entered according to the Act of Parliament of Canada, in the Office of the CAJORI се PREFACE. THE geometry of Euclid is deductive. Yet the processes of all sciences, other than pure mathematics, involve both induction and deduction. All the knowledge which we have of life, with its varied phenomena, is reached by induction and deduction. Any science, then, which permits the student, from a number of observations, to reach a general result, and again from such generalization to draw conclusions, must have distinct educational value. The present little book is an attempt to make the processes of elementary geometry both inductive and deductive. I feel that in making this attempt I am adapting the study of Geometry to immature minds. The mind of youth receives its knowledge in the form of isolated facts; it is for the educator to point out that isolated facts fall into groups and may be crystallized into general conclusions. Special opportunities present themselves in elementary geometry for following this method. Thus, if a number of triangles be accurately constructed with bases of 45 millimetres and angles at the bases 75° and 62°, by actual measurement the learner finds that all the sides opposite to the angles of 75° are equal, and likewise those opposite to the angles of 62°, and that the remaining angles of the triangles have the same magnitude. Analogous constructions and measurements being repeated in a number of cases, the learner, as a matter of inductive observation, feels himself justified in making the generalization expressed in the enunciation of Euclid I., 26. In the process the intellectual interest and curiosity of the pupil are excited, and in reaching the conclusion he feels almost as if he had made the discovery himself. If, subsequently, geometrical forms are presented to him where he can utilize his previous conclusion, he feels with keenness the value of his previous work. He has, in fact, been going through the process of induction and deduction,—the process through which every scientific discoverer goes-with, in miniature, the emotions of the investigator. 911324 It is claimed that deductive geometry inculcates accuracy of thought. Most admirably in this respect it does its work. It too often happens, however, that in the class-room triangles are alleged to be equal which are ridiculously unlike, and lines are proved to be equal which the eye tells us differ in length by several inches. In fact, in spite of accuracy of thought, the utmost contempt for physical accuracy is often inculcated. The whole spirit of the following pages is accuracy of construction. Only by exact drawing can results be attained, and the pupil will find that inaccuracy means failure. My object is to make the class-room in geometry a sort of workshop, where exactness in drawing lines of required length, in measuring lines that are drawn, in constructing angles of given magnitude, in measuring angles that are constructed, and generally in constructing all figures, is insisted on. The attitude of the pupil towards his geometrical figures should be that of the skilled mechanic towards an instrument or machine of precision which he is making, where inaccuracy in measurement would mean loss of time and of material, and would be considered evidence of stupidity. I do not suggest this book as a substitute for the ordinary works on deductive geometry used in the schools, but rather as an introduction to their study. Hence I have included the leading geometrical facts reached in such works, and have introduced them in what is more or less an accepted order. Teachers will find here about one year's work for a class of beginners. If the pupils pursue the subject of geometry no further, I humbly trust that the practical work they have done in connection with this course will have impressed the leading facts of elementary geometry indelibly on their minds; if on the other hand they take up the study of deductive geometry, I hope they will the better, from following this concrete course, appreciate the absolutely general and irrefragable character of methods purely deductive. UNIVERSITY OF TORONTO, May, 1903. A. B. CONTENTS. (At the close of Chapter VIII. the suggestion is made that Chap- |