The Teaching of Geometry

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THE TEACHING OF GEOMETRY CHAPTER I CERTAIN QUESTIONS NOW AT ISSUE

ANNOUNCEMENTS

BOOKS FOR TEACHERS

BY
DAVID EUGENE SMITH


GINN AND COMPANY

BOSTON · NEW YORK · CHICAGO · LONDON

COPYRIGHT, 1911, BY DAVID EUGENE SMITH
ALL RIGHTS RESERVED
911.6

The AthenÆum Press

GINN AND COMPANY · PROPRIETORS BOSTON · U.S.A.


PREFACE

A book upon the teaching of geometry may be planned in divers ways. It may be written to exploit a new theory of geometry, or a new method of presenting the science as we already have it. On the other hand, it may be ultraconservative, making a plea for the ancient teaching and the ancient geometry. It may be prepared for the purpose of setting forth the work as it now is, or with the tempting but dangerous idea of prophecy. It may appeal to the iconoclast by its spirit of destruction, or to the disciples of laissez faire by its spirit of conserving what the past has bequeathed. It may be written for the few who always lead, or think they lead, or for the many who are ranked by the few as followers. And in view of these varied pathways into the joint domain of geometry and education, a writer may well afford to pause before he sets his pen to paper, and to decide with care the route that he will take.

At present in America we have a fairly well-defined body of matter in geometry, and this occupies a fairly well-defined place in the curriculum. There are not wanting many earnest teachers who would change both the matter and the place in a very radical fashion. There are not wanting others, also many in number, who are content with things as they find them. But by far the largest part of the teaching body is of a mind to welcome the natural and gradual evolution of geometry toward better things, contributing to this evolution as much as it can, glad to know the best that others have to offer, receptive of ideas that make for better teaching, but out of sympathy with either the extreme of revolution or the extreme of stagnation.

It is for this larger class, the great body of progressive teachers, that this book is written. It stands for vitalizing geometry in every legitimate way; for improving the subject matter in such manner as not to destroy the pupil's interest; for so teaching geometry as to make it appeal to pupils as strongly as any other subject in the curriculum; but for the recognition of geometry for geometry's sake and not for the sake of a fancied utility that hardly exists. Expressing full appreciation of the desirability of establishing a motive for all studies, so as to have the work proceed with interest and vigor, it does not hesitate to express doubt as to certain motives that have been exploited, nor to stand for such a genuine, thought-compelling development of the science as is in harmony with the mental powers of the pupils in the American high school.

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


                                                                                                                                                                                                                                                                                                           

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