For this class of teachers the author hopes that the book will prove of service, and that through its perusal they will come to admire the subject more and more, and to teach it with greater interest. It offers no panacea, it champions no single method, but it seeks to set forth plainly the reasons for teaching a geometry of the kind that we have inherited, and for hoping for a gradual but definite improvement in the science and in the methods of its presentation. DAVID EUGENE SMITH CONTENTS CHAPTER PAGE I. CERTAIN QUESTIONS NOW AT ISSUE 1 II. WHY GEOMETRY IS STUDIED 7 III. A BRIEF HISTORY OF GEOMETRY 26 IV. DEVELOPMENT OF THE TEACHING OF GEOMETRY 40 V. EUCLID 47 VI. EFFORTS AT IMPROVING EUCLID 57 VII. THE TEXTBOOK IN GEOMETRY 70 VIII. THE RELATION OF ALGEBRA TO GEOMETRY 84 IX. THE INTRODUCTION TO GEOMETRY 93 X. THE CONDUCT OF A CLASS IN GEOMETRY 108 XI. THE AXIOMS AND POSTULATES 116 XII. THE DEFINITIONS OF GEOMETRY 132 XIII. HOW TO ATTACK THE EXERCISES 160 XIV. BOOK I AND ITS PROPOSITIONS 165 XV. THE LEADING PROPOSITIONS OF BOOK II 201 XVI. THE LEADING PROPOSITIONS OF BOOK III 227 XVII. THE LEADING PROPOSITIONS OF BOOK IV 252 XVIII. THE LEADING PROPOSITIONS OF BOOK V 269 XIX. THE LEADING PROPOSITIONS OF BOOK VI 289 XX. THE LEADING PROPOSITIONS OF BOOK VII 303 XXI. THE LEADING PROPOSITIONS OF BOOK VIII 321 INDEX 335 THE TEACHING OF GEOMETRY CHAPTER I CERTAIN QUESTIONS NOW AT ISSUE