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The Teaching of Geometry 40 THE TEACHING OF GEOMETRY BY DAVID EUGENE SMITH GINN AND COMPANY BOSTON " NEW YORK " CHICAGO " LONDON ' ' ' fli''/,T'/. / HARVARD COLLEGE LIBRARY 3y txcfcangefrom l^5!^"LINCOL.VEGt LIBRARY COPYRIGHT, 1911, BY DAVID EUGENE SMITH ALL RIGHTS BBSBBYBD ^1L6 GINN AND COMPANY " PRO-PRIETORS " BOSTON " U.S.A. PREFACE of A book upon the teaching geometry may be planned be ill divers ways. It may 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 fbr the purpose of setting forth the work as it now is, or with the tempting but dangerous idea of prophecy. of It may appeal to the iconoclast by its spirit destruc-tion, or to the disciples of laissez faire by its spirit of cbnserving 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 joitit domain of geometry and education, a writer may well afford before he and to pause sets his pen to paper, 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 iwrell-defined place in the curriculum. There are not wanting many earnest teachers who would change both the and radical fashion. matter the place in a very 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 iii iv PREFACE 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 revolu-tion or the extreme of stagnation. It is for this largerclass,the great body of progressive teachers, that this book is written. It stands for vitaliz-ing geometry in every legitimateway ; 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 desirabilityof establishinga 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 peru-sal they will come to admire the subject more and more, and to teach it with greater interest. It offers no pana-cea, 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 grad-ual 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 11. 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 HI 227 . XVII. The Leading Propositions of Book IV 252 . XVIIL 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 It is commonly said at the present time that the opening of the twentieth century is a period of unusual advancement in all that has to do with the school. It would be pleasant to feel that we are living in such an doubtful if the historian of educa-tion age, but it is future will find this to be the case, or that biographers will rank the leaders of our generation relatively as high as many who have passed away, or that any great movements of the present will be found that measure the world up to certain ones that now recognizes as epoch-making. Every generation since the invention of printing has been a period of agitation in educational matters, but out of all the noise and self-assertion, out of all the pretense of the chronic revolutionist, out of all the sham that leads to dogmatism, so little is remem-bered that we are apt to feel that the past had no prob-lems and was content simply to accept its inheritance. In one sense it is not a misfortune thus to be blinded by the dust of present agitation and to be deafened by the noisy clamor of the agitator, since it stirs us to action at finding ourselves in the midst of the skirmish ; but in another sense it is detrimental to Our progress, 1 2 THE TEACHING OF GEOMETRY since we thereby tend to lose the idea of perspective, and the coin comes to appear to our vision as large as the moon. In consideringa question like the teachingof geome-try, we at once find ourselves in the midst of a skirmish of this nature. If we jointhoughtlesslyin the noise,we may easilypersuade ourselves that we are waging a mighty battle,fightingfor some stupendous principle, doing deeds of great valor and of personal sacrifice. If, on the other hand, we stand aloof and think of the present movement as merely a chronic effervescence, fostered by the professionaleducator at the expense of the practicalteacher, we are equallyshortsighted.Sir Conan Doyle expressedthis sentiment most delightfully in these words : such The dead are good company that one may come to think too little of the living. It is a real and pressing danger with many of us that we should never find our own thoughts and our own souls,but be ever obsessed by the dead. In every generation it behooves the open-minded, earnest, progressiveteacher to seek for the best in the way of improvement, to endeavor to sift the few grains of gold out of the common dust, to weigh the values of proposed reforms, and to put forth his efforts to know and to use the best that the science of education has to offer. This has been the attitude of mind of the real leaders in the school life of the past, and it will be that of the leaders of the future. With these remarks to guide us, it is now proposed to take up the issues of the present day in the teaching of geometry, in order that we may consider them calmly and dispassionately,and may see where the opportunities for improvement lie. CERTAIN QUESTIONS NOW AT ISSUE 8 At the present time, in the educational circles of the United States,questionsof the followingtype are caus-ing the chief discussion among teachers of geometry : 1. Shall geometiy continue to be taught as an appli-cation of logic,or shall it be treated solelywith refer-ence to its applications? 2. If the latter is the purpose in view, shall the propositionsof geometry be limited to those that offer the an opportunityfor.real application,thus contracting whole subjectto very narrow dimensions ? 3. Shall a subjectcalled geometry be extended over several years, as 18 the case in Europe,^ or shall the name be appliedonly to serious demonstrative geome-try ^ as given in the second year of the four-yearhigh- school course in the United States at present? 4. Shall geometry be taught by itself,or shall it be either mixed with algebra (say a day of one subject followed by a day of the other)or fused with it in the form of a combined mathematics ? 5. Shall a textbook be used in which the basal propo-sitions are proved in full,the exercises furnishingthe opportunityfor originalwork and being looked upon as the most importantfeature,or shall one be employed in which the pupilis expectedto invent the proofsfor the basal propositionsas well as for the exercises ? " 6. Shall the terminologyand the spiritof a modified Euclid and Legendre prevailin the future as they have lAnd really,though not nominally, in the United States,where the first concepts are found in the kindergarten, and where an excellent course in mensuration is given in any of our better class of arithmetics. That we are wise in not attempting serious demonstrative geometry much earlier seems to be generally conceded. *The third stage of geometry as defined in the recent circular (No. 711) of the British Board' of Education, London, 1909.
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