H t I tACH GEOMETRY SMITH BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF 1891 ^ ^a^inoa -|]vni^ 3777 Cornell University Library 3 1924 031 435 625 olin.anx The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031435625 THE TEACHING OF GEOMETRY BY DAVID EUGENE SMITH GINN AND COMPANY BOSTON ■ NEW YOKK • CHICAGO ■ LONDON GOPrRIGHT, 1911, BY DAVID BtJGENE SMITH ALL EIGHTS RESERVED 911.6 GINN AND COMPANY ■ PRO- PRIETORS • 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 destruc- tion, 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 afe 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 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 larger class, the great body of progressive teachers, that this book is written. It stands for vitaliz- ing 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 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 CHAPTEK 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 op 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 op 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 age, but it is doubtful if the future historian of educa- tion 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 up to certain ones that the world 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 considering a question like the teaching of geome- try, we at once find ourselves in the midst of a skirmish of this nature. If we join thoughtlessly in the noise, we may easily persuade ourselves that we are waging a mighty battle, fighting for 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 clironic effervescence, fostered by the professional educator at the expense of the practical teacher, we are equally shortsighted. Sir Conan Doyle expressed this sentiment most delightfully in these words : The dead are such 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, progressive teacher 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 NOAV AT ISSUE 3 At the present time, in the educational circles of the United States, questions of the following type are caus- ing the chief discussion among teachers of geometry : 1. Shall geometry continue to be taught as an appli- cation of logic, or shall it be treated solely with refer- ence to its applications ? 2. If the latter is the purpose in view, shall the propositions of geometry be limited to those that offer an opportunity for real application, thus contracting the whole subject to very narrow dimensions ? 3. Shall a subject called geometry be extended over several years, as is the case in Europe,^ or shall the name be applied only to serious demonstrative geome- try^ as given in the second year of the four-year high- 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 furnishing the opportunity for original work and being looked upon as the most important feature, or shall one be employed in which the pupil is expected to invent the proofs for the basal propositions as well as for the exercises ? 6. Shall the terminology and the spirit of a modified Euclid and Legendre prevail in the future as they have lAnd really, though not nominally, in the United States, where the first concepts ate 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. 2 The third stage of geometry as defined in the recent circular (No. 711) of the British Board of Education, London, 1909. 4 THE TEACHIISTG OF GEOMETRY in the past, or shall there be a revolution in the use of terms and in the general statements of the propositions ? 7. Shall geometry be made a strong elective subject, to be taken only by those whose minds are capable of serious work? Shall it be a required subject, diluted to the comprehension of the weakest minds? Or is it now, by proper teaching, as suitable for all pupils as is any other required subject in the school curriculum ? And in any case, will the various distinct types of high schools now arising call for distinct types of geometry ? This brief list might easily be amplified, but it is suffi- ciently extended to set forth the trend of thought at the present time, and to show that the questions before the teachers of geometry are neither particularly novel nor particularly serious. These questions and others of simi- lar nature are really side issues of two larger questions of far greater significance: (1) Are the reasons for teach- ing demonstrative geometry such that it should be a required subject, or at least a subject that is strongly recommended to all, whatever the type of high school? (2) If so, how can it be made interesting ? The present work is written with these two larger questions in mind, although it considers from time to time the minor ones already mentioned, together with others of a similar nature. It recognizes that the recent growth in popular education has brought into the high school a less carefully selected type of mind than was formerly the case, and that for this type a different kind of mathematical training will naturally be developed. It proceeds upon the theory, however, that for the normal mind, — for the boy or girl who is preparing to win out in the long run, — geometry will continue to be taught as demonstrative geometry, as a vigorous thought-compelling CERTAIN QUESTIONS NOW AT ISSUE 5 subject, and along the general lines that the experience of the world has shown to be the best. Soft mathe- matics is not interesting to this normal mind, and a sham treatment will never appeal to the pupil; and this book is written for teachers who believe in this principle, who believe in geometry for the sake of geometry, and who earnestly seek to make the subject so interesting that pupils will wish to study it whether it is required or elective. The work stands for the great basal proposi- tions that have come down to us, as logically arranged and as scientifically proved as the powers of the pupils in the American high school will permit; and it seeks to tell the story of these propositions and to show their possible and .their probable applications in such a way as to furnish teachers with a fund of interesting material with which to supplement the book work of their classes. After all, the problem of teaching any subject comes down to this: Get a subject worth teaching and then make every minute of it interesting. Pupils do not object to work if they like a subject, but they do object to aimless and uninteresting tasks. Geometry is particularly fortunate in that the feeling of accom- plishment comes with every proposition proved; and, given a class of fair intelligence, a teacher must be lacking in knowledge and enthusiasm who cannot foster an interest that will make geometry stand forth as the subject that briags the most pleasure, and that seems the most profitable of all that are studied in the first years of the high school. Continually to advance, continually to attempt to make mathematics fascinating, always to conserve the best of the old and to sift out and use the best of the new, to believe that "mankind is better served by 6 THE TEACHING OF GEOMETRY nature's quiet and progressive changes than by earth- quakes," ^ to believe that geometry as geometry is so val- uable and so interesting that the normal mind may rightly demand it, — this is to ally ourselves with progress. Continually to destroy, continually to follow strange gods, always to decry the best of the old, and to have no well-considered aim in the teaching of a subject, — this is to join the forces of reaction, to waste our time, to be recreant to our trust, to blind ourselves to the failures of the past, and to confess our weakness as teachers. It is with the desire to aid in the progressive movement, to assist those who believe that real geometry should be recommended to all, and to show that geometry is both attractive and valuable that this book is written* ^ The closing words of a sensible review of the British Board of Education circular (No. 711), on "The Teaching of Geometry" (London, 1909), by H. S. Hall in the ScJiool Wmid, 1909, p. 222. CHAPTER II WHY GEOMETRY IS STUDIED With geometry, as with other subjects, it is easier to set forth what are not the reasons for studying it than to proceed positively and enumerate the advantages. Although such a negative course is not satisfying to the mind as a finality, it possesses definite advantages in the beginning of such a discussion as this. Whenever false prophets arise, and with an attitude of pained superiority proclaim unworthy aims in human life, it is well to show the fallacy of their position before proceeding to a constructive philosophy. Taking for a moment this negative course, let us inquire as to what are not the reasons for studying geometry, or, to be more emphatic, as to what are not the worthy reasons. In view of a periodic activity in favor of the utilities of geometry, it is well to understand, in the first place, that geometry is not studied, and never has been stud- ied, because of its positive utility ua commercial life or- even in the workshop. In America we commonly allow at least a year to plane geometry and a half year to solid geometry; but all of the facts that a skilled mechanic or an engineer would ever need could be taught in a few lessons. All the rest is either obvious or is commercially and technically useless. We prove, for example, that the angles opposite the equal sides of a triangle are equal, a fact that is probably quite as obvious as the postulate that but one line can be drawn 7 8 THE TEACHmG OF GEOMETRY through a given point parallel to a given line. We then prove, sometimes by the unsatisfactory process of reductio ad absurdum, the converse of this proposition, — a fact that is as obvious as most other facts that come to our consciousness, at least after the preceding proposition has been proved. And these two theorems are perfectly fair types of upwards of one hundred sixty or seventy propositions comprising Euclid's books on plane geom- etry. They are generally not useful in daily life, and they were never intended to be so. There is an oft- repeated but not .well-authenticated story of Euclid that illustrates the feeling of the founders of geometry as well as of its most worthy teachers. A Greek writer, Stobaeus, relates the story in these words : Some one who had begun to read geometry witli Euclid, when he had learned the first theorem, asked, " But what shall I get by learning these things ? " Euclid called his slave and said, " Give him three obols, since he must make gain out of what he learns." Whether true or not, the story expresses the senti- ment that runs through Euclid's work, and not improb- ably we have here a bit of real biography, — practically all of the personal Euclid that has come down to us from the world's first great textbook maker. It is well that we read the story occasionally, and also such words as the following, recently uttered^ by Sir Conan Doyle, — words bearing the same lesson, although upon a dif- ferent theme: In the present utilitarian age one frequently hears the ques- tion asked, " What is the use of it aU ? " as if every noble deed was not its own justification. As if every action which makes for 1 In an address in London, June 15, 1909, at a dinner to Sir Ernest Shackelton. WHY GEOMETRY IS STUDIED 9 self-denial, for hardihood, and for endurance was not in itself a most precious lesson to mankind. That people can be found to ask such a question shows how far materialism has gone, and how needful it is that we insist upon the value of all that is nobler and higher in life. An American statesman and jurist, speaking upon a similar occasion,^ gave utterance to the same sentiments in these words : AVhen the time comes that knowledge will not be sought for its own sake, and men will not press forward simply in a desire of achievement, without hope of gain, to extend the limits of human knowledge and information, then, indeed, will the race enter upon its decadence. There have not been wanting, however, in every age, those whose zeal is in inverse proportion to their expe- rience, who were possessed with the idea that it is the duty of the schools to make geometry practical. We have them to-day, and the world had them yesterday, and the future shall see them as active as ever. These people do good to the world, and their labors should always be welcome, for out of the myriad of suggestions that they make a few have value, and these are helpful both to the mathematician and the artisan. Not infrequently they have contributed material that serves to make geometry somewhat more interesting, but it must be confessed that most of their work is merely the threshing of old straw, like the work of those who follow the will-o'-the-wisp of the circle squarers. The medieval astrologers wished to make geometry more practical, and so they carried to a considerable length the study of the star polygon, a figure that they could use in their profession. The cathedral builders, as their 1 Governor Hughes, now Justice Hughes, of New York, at the Peary testimonial on February 8, 1910, at New York City. 10 THE TEACHING OF GEOMETRY art progressed, found that architectural drawings were more exact if made with a siagle opening of the com- passes, and it is probable that their influence led to the development of this phase of geometry in the Middle Ages as a practical application of the science. Later, and about the beginning of the sixteenth century, the revival of art, and particularly the great development of paint- ing, led to the practical application of geometry to the study of perspective and of those curves ^ that occur most frequently in the graphic arts. The sixteenth and seventeenth centuries witnessed the publication of a large number of treatises on practical geometry, usually relat- ing to the measuring of distances and partly answering the purposes of our present trigonometry. Such were the well-known treatises of Belli (1569), Cataneo (1567), and Bartoli (1589).2 The period of two centuries from about 1600 to about 1800 was quite as much given to experiments in the creation of a practical geometry as is the present time, and it was no doubt as much by way of protest against this false idea of the subject as a desire to improve upon Euclid that led the great French mathematician, Legendre, to publish his geometry in 1794, — a work that soon replaced Euclid in the schools of America. It thus appears that the effort to make geometry prac- tical is by no means new. Euclid knew of it, the Mid- dle Ages contributed to it, that period vaguely styled the Renaissance joined in the movement, and the first three centuries of printing contributed a large literature to the 1 The first work upon this subject, and indeed the first printed treatise on curves in general, was written hy the famous artist of Ntirnberg, Albrecht Dlirer. 2 Several of these writers are mentioned in Chapter IV. WHY GEOMETRY IS STUDIED 11 subject. Out of all this effort some genuine good remains, but relatively not very mueh.^ And so it will be with the present movement ; it will serve its greatest purpose in making teachers think and read, and in adding to their interest and enthusiasm and to the interest of their pupils ; but it will not greatly change geometry, because no serious person ever believed that geometry was taught chiefly for practical purposes, or was made more inter- esting or valuable tlirough such a pretense. Changes in sequence, in deflnitions, and in proofs will come little by little ; but that there will be any such radical change in these matters in the immediate future, as some writers have anticipated, is not probable.^ A recent writer of much acumen^ has summed up this thought in these words : Not one tenth of the graduates of our high schools ever enter professions in which their algebra and geometry are applied to concrete realities ; not one day in three hundred sixty-five is a high-school graduate called upon to "apply," as it is called, an algebraic or a geometrical proposition. . . . Why, then, do we teach these Subjects, if this alone is the sense of the word " prac- tical " 1 . . . To me the solution of this paradox consists in boldly confronting the dilemma, and in saying that our conception of the' t)ractical utility of those studies must be readjusted, and that we have frankly to face the truth that the " practical " ends we seek are in a sense ideal practical ends, yet such as have, after all, an eminently utilitarian value in the intellectual sphere. 1 If any reader chances upon George Birkbeok's English transla^ tion of Charles Dupin's " Mathematics Practically Applied," Halifax, 1854, he will find that Dupin gave more good applications of geometry than all of our American advocates of practical geometry combined. 2 See, for example, Henrici's " Congruent Figures," London, 1879, and the review of Borel's "Elements of Mathematics," by Professor Sisam in the Bulletin of the American Mathematical Society, July, 1910, a matter discussed later in this work. 'T. J. McCormack, "Why do we study Mathematics: a Philo- sophical and Historical Retrospect," p. 9, Cedar Rapids, Iowa, 1910. 12 THE TEACHING OF GEOMETRY He quotes from C. S. Jackson, a progressive contem- porary teacher of mechanics in England, who speaks of pupils confusing millimeters and centimeters in some simple computation, and who adds : There is the enemy ! The real enemy we have to fight against, whatever we teach, is carelessness, inaccuracy, forgetfulness, and slovenliness. That battle has been fought and won with diverse weapons. It has, for instance, been fought with Latin grammar before now, and won. I say that because we must be very care- ful to guard against the notion that there is any one panacea for this sort of thing. It borders on quackery to say that elementary physics will cure everything. And of course the same thing maybe said for mathematics. Nevertheless it is doubtful if we have any other subject that does so much to bring to the front this danger of carelessness, of slovenly reasoning, of inaccuracy, and of forgetfulness as this science of geometry, which has been so polished and perfected as the centuries have gone on. There have been those who did not proclaim the utili- tarian value of geometry, but who fell into as serious an error, namely, the advocating of geometry as a means of training the memory. In times not so very far past, and to some extent to-day, the memorizing of proofs has been justified on this ground. This error has, however, been fully exposed by our modern psychologists. They have shown that the person who memorizes the propositions of Euclid by number is no more capable of memorizing other facts than he was before, and that the learning of proofs verbatim is of no assistance whatever in retaining matter that is helpful in other lines of work. Geometry, therefore, as a training of the memory is of no more value than any other subject in the curriculum. If geometry is not studied chiefly because it is prac- tical, or because it trains the memory, what reasons can WHY GEOMETRY IS STUDIED 13 be adduced for its presence in the courses of study of every civilized country ? Is it not, after all, a mere fetish, and are not those virulent writers correct who see noth- ing good in the subject save only its utilities ? ^ Of this tjrpe one of the most entertaining is William J. Locke,^ whose words upon the subject are well worth reading : ... I earned my living at school slavery, teaching to children the most useless, the most disastrous, the most soul-cramping branch of knowledge wherewith pedagogues in their insensate folly have crippled the minds and blasted the lives of thousands of their fellow creatures — elementary mathematics. There is no more reason for any human being on God's earth to be acquainted with the binomial theorem or the solution of triangles, unless he is a professional scientist, — when he can begin to specialize in mathematics at the same age as the lawyer begins to specialize in law or the surgeon in anatomy, — than for him to be expert in Choctaw, the Cabala, or the Book of Mormon. I look back with feelings of shame and degradation to the days when, for a crust of bread, I prostituted my intelligence to wasting the precious hours of impressionable childhood, which could have been filled with so many beautiful and meaningful things, over this utterly futile and inhuman subject. It trains the mind, — it teaches boys to think, they say. It does n't. In reality it is a cutand-dried sub- ject, easy to fit into a school curriculum. Its sacrosanctity saves educationalists an enormous amount of trouble, and its chief use is to enable mindless young men from the universities to make a dishonest living by teaching it to others, who in their turn may teach it to a future generation. To be fair we must face just such attacks, and we must recognize that they set forth the feelings of many 1 Of the fair and candid arguments against the culture value of mathematics, one of the best of the recent ones is that by G. F. Swain, in the Atti del IV Congresso Internazionale dei Matematici, Rome, 1909, Vol. Ill, p. 361. The literature of this school is quite exten- sive, but Perry's "England's Neglect of Science," London, 1900, and "Discussion on the Teaching of Mathematics," London, 1901, are typical. 2 In his novel, " The Morals of Marcus Ordeyne." 14 THE TEACHING OF GEOMETRY honest people. One is tempted to inquire if Mr. Locke could have written in such an incisive style if he had not, as was the case, graduated with honors in mathematics at one of the great universities. But he might reply that if his mind had not been warped by mathematics, he would have written more temperately, so the honors in the argument would be even. Much more to the point is the fact that Mr. Locke taught mathematics in the schools of England, and that these schools do not seem to the rest of the world to furnish a good type of the teaching of elementary mathematics. No country goes to England for its model in this particular branch of education, although the work is rapidly changing there, and Mr. Locke pictures a local condition in teaching rather than a general condition in mathematics. Few visitors to the schools of England would care to teach mathematics as they see it taught there, in spite of their recognition of the thoroughness of the work and the earnestness of many of the teachers. It is also of interest to note that the greatest protests against formal mathematics have come from England, as witness the utterances of such men as Sir William Hamilton and Professors Perry, Minchin, Henrici, and Alfred Lodge. It may therefore be questioned whether these scholars are not uncon- sciously protesting against the English methods and curriculum rather than against the subject itseK. "When Professor Minchin says that he had been through the six books of Euclid without really understanding an angle, it is Euclid's text and his own teacher that are at fault, and not geometry. Before considering directly the question as to why geometry should be taught, let us turn for a moment to the other subjects in the secondary curriculum. Why, WHY GEOMETRY IS STUDIED 15 for example, do we study literature ? " It does not lower the price of bread," as Malherbe remarked in speaking of the commentary of Bachet on the great work of Diophantus. Is it for the purpose of making authors ? Not one person out of ten thousand who study literature ever writes for publication. And why do we allow pupils to waste their time in physical education ? It uses valu- able hours, it wastes money, and it is dangerous to life and limb. Would it not be better to set pupils at sawing wood ? And why do we study music ? To give pleas- ure by our performances ? How many who attempt to play the piano or to sing give much pleasure to any but themselves, and possibly their parents ? The study of grammar does not make an accurate writer, nor the study of rhetoric an orator, nor the study of meter a poet, nor the study of pedagogy a teacher. The study of geography in the school does not make travel particularly easier, nor does the study of biology tend to populate the earth. So we might pass in review the various subjects that we study and ought to study, and in no case would we find utility the moving cause, and in every case would we find it difficult to state the one great reason for the pursuit of the subject in question, — and so it is with geometry. What positive reasons can now be adduced for the study of a subject that occupies upwards of a year in the school course, and that is, perhaps unwisely, required of all pupils ? Probably the primary reason, if we do not attempt to deceive ourselves, is pleasure. We study music because music gives us pleasure, not necessarily our own music, but good music, whether ours, or, as is more probable, that of others. We study literature because we derive pleasure from books ; the better the 16 THE TEACHmG OF GEOMETRY book the more subtle and lasting the pleasure. We study art because we receive pleasure from the great works of the masters, and probably we appreciate them the more because we have dabbled a little in pigments or in clay. We do not expect to be composers, or poets, or sculptors, but we wish to appreciate music and letters and the fine arts, and to derive pleasure from them and to be uplifted by them. At any rate, these are the nobler reasons for their study. So it is with geometry. We study it because we derive pleasure from contact with a great and an ancient body of learning that has occupied the attention of master minds during the thousands of years in which it has been perfected, and we are uplifted by it. To deny that our pupils derive this pleasure from the study is to confess ourselves poor teachers, for most pupils do have positive enjoyment in the pursuit of geometry, in spite of the tradition that leads them to proclaim a general dislike for all study. This enjoyment is partly that of the game, — the playing of a game that can always be won, but that cannot be won too easily. It is partly that of the sesthetic, the pleasure of symmetry of form, the delight of fitting things together. But probably it lies chiefly in the men- tal uplift that geometry brings, the contact with abso- lute truth, and the approach that one makes to the Infinite. We are not quite sure of any one thing in biology ; our knowledge of geology is relatively very slight, and the economic laws of society are uncertain to every one except some individual who attempts to set them forth; but before the world was fashioned the square on the hypotenuse was equal to the sum of the squares on the other two sides of a right triangle, and it will be so after this world is dead ; and the inhabitant of WHY GEOMETRY IS STUDIED 17 Mars, if he exists, probably knows its truth as we know it. The uplift of this contact with absolute truth, with truth eternal, gives pleasure to humanity to a greater or less^flegree, depending upon the mental equipment of the particular individual ; but it probably gives an appreciable amount of pleasure to every student of geometry who has a teacher worthy of the name. First, then, and fore- most as a reason for studying geometry has always stood, and will always stand, the pleasure and the mental uplift that comes from contact with such a great body of human learning, and particularly with the exact truth that it contains. The teacher who is imbued with this feeling is on the road to success, whatever method of presenta- tion he may use ; the one who is not imbued with it is on the road to failure, however logical his presentation or however large his supply of practical applications. Subordinate to these reasons for studying geometry are many others, exactly as with all other subjects of the curriculum. Geometry, for example, offers the best devel- oped application of logic that we have, or are likely to have, in the school course. This does not mean that it always exemplifies perfect logic, for it does not ; but to the J)upil who is not ready for logic, per se, it offers an example of close reasoning such as his other subjects do not offer. We may say, and possibly with truth, that one who studies geometry will not reason more clearly on a financial proposition than one who does not ; but in spite of the results of the very meager experiments of the psychologists, it is probable that the man who has had some drill in syllogisms, and who has learned to select the essentials and to neglect the nonessentials in reach- ing his conclusions, has acquired habits in reasoning that will help him in every line of work. As part of this 18 THE TEACHIIvTG OF GEOMETRY equipment there is also a terseness of statement and a clearness in arrangement of points in an argument that has been the subject of comment by many writers. Upon this same topic an English writer, in one of the sanest of recent monographs upon the subjeet,^ has expressed his views in the following words : The statement that a given individual has received a sound geometrical training implies that he has segregated from the whole of his sense impressions a certain set of these impressions, that he has then eliminated from their consideration all irrelevant impressions (in other words, acquired a subjective command of these impressions), that he has developed on the basis of these impressions an ordered and continuous system of logical deduc- tion, and finally that he is capable of expressing the nature of these impressions and his deductions therefrom in terms simple and free from ambiguity. Now the slightest consideration will convince any one not already conversant with the idea, that the same sequence of mental processes underlies the whole career of any individual in any walk of life if only he is not concerned entirely with manual labor ; consequently a full training in the performance of such sequences must be regarded as forming an essential part of any education worthy of the name. Moreover, the full appreciation of such processes has a higher value than is contained in the mental training involved, great though this be, for it induces an appreciation of intellectual unity and beauty which plays for the mind that part which the appreciation of schemes of shape and color plays for the artistic faculties ; or, again, that part which the appreciation of a body of religious doctrine plays for the ethical aspirations. Now geometry is not the sole possible basis for inculcating this appreciation. Logic is an alter- native for adults, provided that the individual is possessed of sufficient wide, though rough, experience on which to base his reasoning. Geometry is, however, highly desirable in that the objective bases are so simple and precise that they can be grasped at an early age, that the amount of training for the imagination is very large, that the deductive processes are not beyond the scope of 1 G. W. I/. Carson, " The Functions of Geometry as a Subject of Education," p. 3, Tonbridge, 1910. WHY GEOMETRY IS STUDIED 19 ordinary boys, and finally that it affords a better basis for exer- cise in the art of simple and exact expression than any other possible subject of a school course. Are these results really secured by teachers, however, or are they merely imagined by the pedagogue as a justi- fication for his existence ? Do teachers have any such appreciation of geometry as has been suggested, and eyen if they have it, do they impart it to their pupils ? In reply it may be said, probably with perfect safety, that teachers of geometry appreciate their subject and lead their pupils to appreciate it to quite as great a degree as obtains in any other branch of education. What teacher appreciates fully the beauties of " In Memoriam," or of " Hamlet," or of " Paradise Lost," and what one inspires his pupils with all the nobility of these world classics ? What teacher sees in biology all the grandeur of the evolution of the race, or imparts to his pupils the noble lessons of lif,e that the study of this subject should sug- gest ? What teacher of Latin brings his pupils to read the ancient letters with full appreciation of the dignity of style and the nobility of thought that they contain ? And what teacher of French succeeds in bringing a pupil to carry on a conversation, to read a French magazine, to see the history imbedded in the words that are used, to realize the charm and power of, the language, or to appreciate to the full a single classic ? In other words, none of us fully appreciates his subject, and none of us can hope to bring his pupils to the ideal attitude toward any part of it. But it is probable that the teacher of geometry succeeds relatively better than the teacher of other subjects, because the science has reached a rela- tively higher state of perfection. The body of truth in geometry has been more clearly marked out, it has been 20 THE TEACHING OF GEOMETRY more successfully fitted together, its lesson is more patent, and the experience of centuries has brought it into a shape that is more usable in the school. While, there- fore, we have all kinds of teaching in all kinds of sub- jects, the very nature of the case leads to the belief that the class in geometry receives quite as much from the teacher and the subject as the class in any other branch in the school curriculum. But is this not mere conjecture ? What are the results of scientific investigation of the teaching of geometry ? Unfortunately there is little hope from the results of such an inquiry, either here or in other fields. We cannot first weigh a pupil in an intellectual or moral balance, then feed him geometry, and then weigh him again, and then set back his clock of time and begin all over again virith the same individual. There is no " before taking " and " after taking " of a subject that extends over a year or two of a pupil's life. We can weigh utilities roughly, we can estimate the pleasure of a subject relatively, but we cannot say that geometry is worth so many dollars, and history so many, and so on through the curriculum. The best we can do is to ask ourselves what the various subjects, with teachers of fairly equal merit, have done for us, and to inquire what has been the experience of other persons. Such an investigation results in showing that, with few exceptions, people who have studied geometry received as much of pleasure, of mspiration, of satisfaction, of what they call training from geometry as from any other subject of study, — given teachers of equal merit, — and that they would not willingly give up the something which geometry brought to them. If this were not the feeling, and if humanity believed that geometry is what Mr. Locke's words would seem to WHY GEOMETRY IS STUDIED 21 indicate, it would long ago have banished it from the schools, since upon this ground rather thaa upon the ground of utility the subject has always stood. These seem to be the great reasons for the study of geometry, and to search for others would tend to weaken the argument. At first sight they may not seem to justify the expenditure of time that geometry demands, and they may seem unduly to neglect the argument that geometry is a stepping-stone to higher mathematics. Each of these points, however, has been neglected pur- posely. A pupil has a number of school years at his disposal ; to what shall they be devoted ? To literature ? What claim has letters that is such as to justify the exclusion of geometry? To music, or natural science, or language ? These are all valuable, and all should be studied by one seeking a liberal education ; but for the same reason geometry should have its place. What sub- ject, in fine, can supply exactly what geometry does ? And if none, then how can the pupil's time be better expended than in the study of this science ? ^ As to the second point, that a claim should be set forth that geom- etry is a sine qua non to higher mathematics, this belief is considerably exaggerated because there are relatively few who proceed from geometry to a higher branch of mathematics. This argument would justify its status as an elective rather than as a required subject. Let us then stand upon the ground already marked out, holding that the pleasure, the culture, the mental poise, the habits of exact reasoning that geometry brings, 1 It may well be, however, that the growing curriculum may jus- tify some reduction in the time formerly assigned to geometry, and any reasonable proposition of this nature should be fairly met by teachers of mathematics. 22 THE TEACHING OF GEOMETRY and the general experience of mankind upon the subject are sufficient to justify us in demanding for it a reason- able amount of time in the framing of a curriculum. Let us be fair in our appreciation of all other branches, but let us urge that every student may have an oppor- tunity to know of real geometry, say for a single year, thereafter pursuing it or not, according as we succeed in. making its value apparent, or fail ia our attempt to pre- sent worthily an ancient and noble science to the mind confided to our instruction. The shortsightedness of a narrow education, of an education that teaches only machines to a prospective mechanic, and agriculture to a prospective farmer, and cooking and dressmaking to the girl, and that would exclude all mathematics that is not utilitarian in the narrow sense, cannot endure. The community has found out that such schemes may be well fitted to give the children a good time in school, but lead them to a bad time afterward. Life is hard work, and if they have never learned in school to give their concentrated attention to that which does not appeal to them and which does not interest them immediately, they have missed the most valuable lesson of their school years. The little practical information they could have learned at any time; the energy of attention and concentration can no longer be learned if the early years are wasted. However narrow and commercial the standpoint which is chosen may be, it can always be found that it is the general education which pays best, and the more the period of cultural work can be expanded the more efficient will be the services of the school for the prac- tical services of the nation.^ Of course no one should construe these remarks as opposing in the slightest degree the laudable efforts that are constantly being put forth to make geometry more 1 Professor Munsterberg, in the Metropolitan Magazine for July, 1910. AVHY GEOMETRY IS STUDIED 23 interesting and to vitalize it by establishing as strong motives as possible for its study. Let the home, the workshop, physics, art, play, — all contribute their quota of motive to geometry as to all mathematics and all other branches. But let us never forget that geometry has a raison d'etre beyond all this, and that these applications are sought primarily for the sake of geometry, and that geometry is not taught primarily for the sake of these applications. When we consider how often geometry is attacked by those who profess to be its friends, and how teachers who have been trauied in mathematics occasionally seem to make of the subject little besides a mongrel course in drawing and measuring, all the time insisting that they are progressive while the champions of real geometry are reactionary, it is well to read some of the opinions of the masters. The following quotations may be given occa- sionally in geometry classes as showing the esteem in which the subject has been held in various ages, and at any rate they should serve to inspire the teacher to greater love for his subject. The enemies of geometry, those who know it only imperfectly, look upon the theoretical problems, which constitute the most difficult part of the subject, as mental games which consume time and energy that might better be employed in other ways. Such a belief is false, and it would block the progress of science if it were credible. But aside from the fact that the speculative prob- lems, which at first sight seem barren, can often be applied to use- ful purposes, they always stand as among the best means to develop and to express all the forces of the human intelligence. — Abbe Bossut. The sailor whom an exact observation of longitude saves from shipwreck owes his life to a theory developed two thousand years ago by men who had in mind merely the speculations of abstract geometry. — Condorcet. 24 THE TEACHING OF GEOMETRY If mathematical heights are hard to climb, the fundamental principles lie at every threshold, and this fact allows them to be comprehended by that common sense which Descartes declared was " apportioned equally among all men.'' — Collet. It may seem strange that geometry is unable to define the terms which it uses most frequently, since it defines neither movement, nor number, nor space, — the three things with which it is chiefly concerned. But we shall not be surprised if we stop to consider that this admirable science concerns only the most sim- ple things, and the very quality that renders these things worthy of study renders them incapable of being defined. Thus the very lack of definition is rather an evidence of perfection than a defect, since it comes not from the obscurity of the terms, but from the fact that they are so very well known. — Pascal. God eternally geometrizes. — Plato. God is a circle of which the center is everywhere and the cir- cumference nowhere. — Rabelais. Without mathematics no one can fathom the depths of philos- ophy. Without philosophy no one can fathom the depths of mathematics. Without the two no one can fathom the depths of anything. — Bokdas-Demoulin. We may look upon geometry as a practical logic, for the truths which it studies, being the most simple and most clearly under- .stood of all truths, are on this 'account the most susceptible of ready application in reasoning. — D'Alembert. The advance and the perfecting of mathematics are closely joined to the prosperity of the nation Napoleox. Hold nothing as certain save what can be demonstrated. — Newton. To measure is to know. — Kepler. The method of making no mistake is sought bv every one. The logicians profess to show the way, but the geometers alone ever reach it, and aside from their science there is no genuine demonstration Pascal. The taste for exactness, the impossibility of contenting one's self with vague notions or of leaning upon mere hypotheses, the necessity for perceiving clearly the connection between certain propositions and the object in view, — these are the most precious fruits of the study of mathematics Lacroix. WHY GEOMETRY IS STUDIED 25 Bibliography. Smith, The Teaching of Elementary Mathemair ica, p. 234, New York, 1900 ; Henrici, Presidential Address before the British Association, Nature, Vol. XXVIII, p. 497 ; Hill, Edu- cational Value of Mathematics, Educational RerU'ie, Vol. IX, p. 349 ; Young, The Teaching of Mathematics, p. 9, New York, 1907. The closing quotations are from Rebifere, Math^matiques et Mathematicians, Paris, 1893. CHAPTER III A BRIEF HISTORY OF GEOMETRY The geometry of very ancient peoples was largely the mensuration of simple areas and solids, such as is taught to children in elementary arithmetic to-day. They early learned how to find the area of a rectangle, and in the oldest mathematical records that have come down to us there is some discussion of the area of triangles and the volume of solids. The earliest documents that we have relating to geom- etry come to us from Babylon and Egypt. Those from Babylon are written on small clay tablets, some of them about the size of the hand, these tablets afterwards having been baked in the sun. They show that the Babylonians of that period knew something of land measures, and per- haps had advanced far enough to compute the area of a trapezoid. For the mensuration of the circle they later used, as did the early Hebrews, the value tt = 3. A tab- let in the British Museum shows that they also used such geometric forms as triangles and circular segments in astrology or as talismans. The Egyptians must have had a fair knowledge of practical geometry long before the date of any mathe- matical treatise that has come down to us, for the building of the pyramids, between 3000 and 2400 B.C., required the application of several geometric principles. Some knowledge of surveying must also have been necessary 26 A BRIEF HISTORY OF GEOMETRY 27 to carry out the extensive plans for irrigation that were executed under Amenemhat III, about 2200 B.C. The first definite knowledge that we have of Egyp- tian mathematics comes to us from a manuscript copied on papyrus, a kind of paper used about the Mediterranean in early times. This copy was made by one Aah-mesu (The Moon-born), commonly called Ahmes, who prob- ably flourished about 1700 B.C. The original from which he copied, written about 2300 B.C., has been lost, but the papyrus of Ahmes, written nearly four thousand years ago, is still preserved, and is now in the British Museum. In this manuscript, which is devoted chiefly to fractions and to a crude algebra, is found some work on mensuration. Among the curious rules are the incorrect ones that the area of an isosceles triangle equals half the product of the base and one of the equal sides ; and that the area of a trapezoid having bases 5, 6', and the nonparallel sides each equal to a, is ^a(b + J'). One noteworthy advance appears, however. Ahmes gives a rule for finding the area of a circle, substantially as follows : Multiply the square on the radius by (^-)^ which is equivalent to taking for tt the value 3.1605. This papyrus also con- tains some treatment of the mensuration of solids, par- ticularly with reference to the capacity of granaries. There is also some slight mention of similar figures, and an extensive treatment of unit fractions, — fractions that were quite universal among the ancients. In the line of algebra it contains a brief treatment of the equation of the first degree with one unknown, and of progressions. ^ 1 It was published in German translation by A. Eisenlohr, "Ein mathematisohes Handbuch der alten Aegypter," Leipzig, 1877, and in facsimile by the British Museum, under the title, " The Rhind Papyrus," in 1898. 28 THE TEACHING OF GEOMETRY Herodotus tells ns that Sesostris, king of Egypt,^ divided the land among his people and marked out the boundaries after the overflow of the Nile, so that survej'- ing must have been well known in his day. Indeed, the Tiarpedonaptce, or rope stretchers, acquired their name because they stretched cords, in which were knots, so as to make the right triangle 3, 4, 5, when they wished to erect a perpendicular. This is a plan occasionally used by surveyors to-day, and it shows that the practical application of the Pythagorean Theorem was known long before Pythagoras gave what seems to have been the first general proof of the proposition. From Egypt, and possibly from Babylon, geometry passed to the shores of Asia JNIinor and Greece. The scientific study of the subject begins with Thales, one of the Seven Wise Men of the Grecian civilization. Born at Miletus, not far from Smyrna and Ephesus, about 640 B.C., he died at Athens in 548 B.C. He spent his early manhood as a merchant, accumulating the wealth that enabled him to spend his later years in study. He visited Egypt, and is said to have learned such elements of geometry as were known there. He founded a school of mathematics and philosophy at ISIiletus, known from the country as the Ionic School. How elementary the knowledge of geometry then was may be understood from the fact that tradition attributes only about four propositions to Thales, — (1) that vertical angles are equal, (2) that equal angles lie opposite the equal sides of an isosceles triangle, (3) that a triangle is determined by two angles and the included side, (4) that a diameter bisects the circle, and possibly the propositions about the 1 Generally known as Rameses II. He reigned in Egypt about 1350 B.C. A BRIEF HISTORY OF GEOMETRY 29 angle-sum of a triangle for special cases, and the angle inscribed in a semicircle.^ The greatest pupil of Thales, and one of the most remarkable men of antiquity, was Pythagoras. Born probably on the island of Samos, just off the coast of Asia Minor, about the year 580 B.C., Pythagoras set forth as a young man to travel. He went to Miletus and studied under Thales, probably spent several years in Egypt, very likely went to Babylon, and possibly went even to India, since tradition asserts this and the nature of his work in mathematics suggests it. In later life he went to a Greek colony in southern Italy, and at Cro- tona, in the southeastern part of the peninsula, he founded a school and established a secret society to propagate his doctrines. In geometry he is said to have been the first to demonstrate the proposition that the square on the hjrpotenuse is equal to the sum of the squares upon the other two sides of a right triangle. The proposition was known in India and Egypt before his time, at any rate for special cases, but he seems to have been the first to prove it. To him or to his school seems also to have been due the construction of the regular pentagon and of the five regular polyhedrons. The construction of the regular pentagon requires the dividing of a line into extreme and mean ratio, and this problem is commonly assigned to the Pythagoreans, although it played an important part in Plato's school. Pythagoras is also said to have known that six equilateral triangles, three 1 Two excellent works on Thales and liis successors, and indeed the best in English, are the following : G. J. Allman, " Greek Geometry from Thales to Euclid," Dublin, 1889 ; J. Gow, "A History of Greek Mathematics," Cambridge, 1884. On all mathematical subjects the best general history is that of M. Cantor, " Geschichte der Mathe- matik," 4 vols, Leipzig, 1880-1908. 30 THE TEACHING OF GEOMETRY regular hexagons, or four squares, can be placed about a point so as just to fill the 360°, but that no other reg- ular polygons can be so placed. To his school is also due the proof for the general case that the sum of the angles of a triangle equals two right angles, the first knowledge of the size of each angle of a regular polygon, and the construction of at least one star-poly- gon, the star-pen- tagon, which be- came the badge of his fraternity. The brotherhood founded by Py- thagoras proved so offensive to the government that it was dispersed before the death of the master. Pythagoras fled to Megapontum, a sea- port lying to the north of Crotona, and there he died about 501 B.c.i For two centuries after Pythagoras geometry passed tlirough a period of discovery of propositions. The state 1 Another good work on Greek geometry, with considerable mate- rial on Pythagoras, is by C. A. Bretschneider, " Die Geometrie und die Geometer vor Eukleides," Leipzig, 1870. Fanciful Portkait of Pythagoras Calandri's Arithmetic, 1491 A BRIEF HISTORY OF GEOMETRY 31 of the science may be seen from the fact that CEnopides of Chios, who flourished about 465 B.C., and who had studied in Egypt, was celebrated because he showed how to let fall a perpendicular to a line, and how to make an angle equal to a given angle. A few years later, about 440 B.C., Hippocrates of Chios wrote the first Greek textbook on mathematics. He knew that the areas of circles are proportional to the squares on their radii, but was ignorant of the fact that equal central angles or equal inscribed angles intercept equal arcs. Antiphon and Bryson, two Greek scholars, flourished about 430 B.C. The former attempted to find the area of a circle by doubling the number of sides of a regular inscribed polygon, and the latter by doing the same for both inscribed and circumscribed polygons. They thus approximately exhausted the area between the polygon and the circle, and hence this method is known as the method of exhaustions. About 420 B.C. Hippias of Elis invented a certain curve called the quadratrix, by means of which he could square the circle and trisect any angle. This curve cannot be constructed by the unmarked straight^ edge and the compasses, and when we say that it is impossible to square the circle or to trisect any angle, we mean that it is impossible by the help of these two instruments alone. ~- — -otttti?. During this period the great philosophic school of Plato (429-348 B.C.) flourished at Athens, and to this school is due the first systematic attempt to create exact definitions, axioms, and postulates, and to distinguish be- tween elementary and higher geometry. It was at this time that elementary geometry became limited to the use of the compasses and the unmarked straightedge, 32 THE TEACHING OF GEOMETRY which took from this domain the possibiHty of construct- ing a square equivalent to a given circle (" squaring the circle "), of trisecting any given angle, and of construct- ing a cube that should have twice the volume of a given cube ("duplicating the cube"), these being the three famous problems of antiquity. Plato and his school interested themselves with the so-called Pythagorean numbers, that is, with numbers that would represent the three sides of a right triangle and hence fulfill the condition that a^-\-h^= and < in the works of Harriot (1560- 1621) ; and x in a publication by Oughtred (1574-1660). 1 Smith and Karpinski, "The Hindu-Arabic Numerals," Boston, 1911. 2 For a sketch of his life see Smith and Karpinski, loo. cit. 38 THE TEACHING OF GEOMETRY The most noteworthy advance in geometry in modem times was made by the great French philosopher Des- cartes, who pubUshed a small work entitled " La Geo- m^trie " in 1637. From this springs the modern analytic geometry, a subject that has revolutionized the methods of all mathematics. Most of the subsequent discoveries in mathematics have been in higher branches. To the great Swiss mathematician Euler (1707-1783) is due, however, on^ proposition that has found its way into elementary geometry, the one showing the relation between the number of edges, vertices, and faces of a polyhedron. There has of late arisen a modern elementary geom- etry devoted chiefly to special points and lines relating to the triangle and the circle, and many interesting prop- ositions have been discovered. The subject is so extensive that it cannot find any place in our crowded curriculum, and must necessarily be left to the specialist.^ Some idea of the nature of the work may be obtaiaed from a men- tion of a few propositions : The medians of a triangle are concurrent in the cen- troid, or center of gravity of the triangle. The bisectors of the various interior and exterior angles of a triangle are concurrent by threes in the incenter or in one of the three excenters of the triangle. The common chord of two intersecting circles is a special case of their radical axis, and tangents to the circles from any point on the radical axis are equal. 1 Those who care for a brief description of this phase of the sub- ject may consult J. Casey, " A Sequel to Euclid," Dublin, fifth edi- tion, 1888 ; W. J. M'Clelland, "A Treatise on the Geometry of the Circle," New York, 1891 ; M. Simon, " Uber die Entwicklung der Elementar-Geometrie im XIX. Jahrhundert," Leipzig, 1906, A BRIEF HISTORY OP GEOMETRY 39 If is the orthocenter of the triangle ABC^ and X, F, Z are the fe'et of the perpendiculars from A, B, C respectively, and P, Q, E are the mid-points of a, b, c respectively, and L, M, N are the mid-points of OA, OB, 00 respectively; then the points L, M, N; F, Q, B\ X, F, Z all lie on a circle, the " nine points circle." In the teaching of geometry it adds a human interest to the subject to mention occasionally some of the his- torical facts connected with it. For this reason this brief sketch will be supplemented by many notes upon the various important propositions as they occur in the sev- eral books described in the later chapters of this work. CHAPTER IV DEVELOPMENT OF THE TEACHING OF GEOMETRY We know little of the teaching of geometry in very ancient times, but we can infer its nature from the teaching that is still seen in the native schools of the East. Here a man, learned in any science, will have a group of voluntary students sitting about him, and to them he will expound the truth. Such schools may still be seen in India, Persia, and China, the master sitting on a mat placed on the ground or on the floor of a veranda, and the pupils reading aloud or listening to his words of exposition. In Egypt geometry seems to have been in early times mere mensuration, confined largely to the priestly caste. It was taught to novices who gave promise of success in this subject, and not to others, the idea of general culture, of training in logic, of the cultivation of exact expression, and of commg in contact with truth being wholly wanting. In Greece it was taught in the schools of philosophy, often as a general preparation for philosophic study. Thus Thales introduced it into his Ionic school, Pythag- oras made it very prominent iii his great school at Crotona m southern Italy (Magna Grsecia), and Plato placed above the door of his Academia the words, " Let no one ignorant of geometry enter here," — a kind of entrance examination for his school of philosophy. In 40 DEVELOPMENT OF GEOMETRY 41 these gatherings of students it is probable that geometry was taught in much the way already mentioned for the schools of the East, a small group of students being instructed by a master. Printing was unknown, papyrus was dear, parchment was only in process of invention. Paper such as we know had not yet appeared, so that instruction was largely oral, and geometric figures were drawn by a pointed stick on a board covered with fine sand, or on a tablet of wax. But with these crude materials there went an abun- dance of time, so that a number of great results were accomplished in spite of the difficulties attending the study of the subject. It is .said that Hippocrates of Chios (ca. 440 B.C.) wrote the first elementary textbook on mathematics and invented the method of geometric re- duction, the replacing of a proposition to be proved by another which, when proved, allows the first one to be demonstrated. A little later Eudoxus of Cnidus (j;a. 375 B.C.), a pupil of Plato's, used the reductio ad ah- surdum, and Plato is said to have invented the method of proof by analysis, an elaboration of the plan used by Hippocrates. Thus these early philosophers taught their pupils not facts alone, but methods of proof, giving them power as well as knowledge. Furthermore, they taught them how to discuss their problems, investigating the conditions under which they are capable of solution. This feature of the work they called the diorismus, and it seems to have started with Leon, a follower of Plato. Between the time of Plato (ca. 400 B.C.) and Euclid (ea. 300 B.C.) several attempts were made to arrange the accumulated material of elementary geometry in a text- book. Plato had laid the foundations for the science, in the form of axioms, postulates, and definitions, and he 42 THE TEACHING OF GEOMETRY had limited the instruments to the straightedge and the compasses. Aristotle (ca. 350 B.C.) had paid special at- tention to the history of the subject, thus finding out what had already been accomplished, and had also made much of the applications of geometry. The world was therefore ready for a good teacher who should gather the material and arrange it scientifically. After several attempts to find the man for such a task, he was dis- covered in Euclid, and to his work the next chapter is devoted. After Euclid, Archimedes (ca. 250 B.C.) made his great contributions. He was not a teacher like his illustrious predecessor, but he was a great discoverer. He has left us, ■ however, a statement of his methods of investiga- tion which is helpful to those who teach. These methods were largely experimental, even extending to the weigh- ing of geometric forms to discover certain relations, the proof being given later. Here was born, perhaps, what has been called the laboratory method of the present. Of the other Greek teachers we have ■ but little in- formation as to methods of imparting instruction. It is not until the Middle Ages that there is much known in this line. Whatever of geometry was taught seems to have been imparted by word of mouth in the way of expounding Euclid, and this was done in the ancient fashion. The early Church leaders usually paid no attention to geometry, but as time progressed the quadrivium, or four sciences of arithmetic, music, geometry, and astron- omy, came to rank with the trivium (grammar, rhet- oric, dialectics), the two making up the "seven liberal arts." All that there was of geometry in the first thou- sand years of Christianity, however, at least in the great DEVELOPMENT OF GEOMETRY 43 majority of Church schools, was summed up in a few definitions and rules of mensuration. Gerbert, who became Pope Sylvester II in 999 a.d., gave a new impetus to geometry by discovering a manuscript of the old Roman surveyors and a copy of the geometry of Boethius, who paraphrased Euclid about 500 a.d. He thereupon wrote a brief geometry, and his elevation to the papal chair tended to bring the study of mathe- matics again into prominence. Geometry now began to have some place in the Church schools, naturally the only schools of high rank in the Middle Ages. The study of the subject, however, seems to have been merely a matter of memorizing. Geometry received another impetus in the book written by Leonardo of Pisa in 1220, the " Practica Geometriae." Euclid was also translated into Latin about this time (strangely enough, as already stated, from the Arabic instead of the Greek), and thus the treasury of elemen- tary geometry was opened to scholars in Europe. From now on, until the invention of printing (ca. 1450), numer- ous writers on geometry appear, but, so far as we know, the method of instruction remained much as it had always been. The universities began to appear about the thir- teenth century, and Sacrobosco, a well-known medieval mathematician, taught mathematics about 1250 in the University of Paris. In 1336 this university decreed that mathematics should be required for a degree. In the thirteenth century Oxford required six books of Euclid for one who was to teach, but this amount of work seems to have been merely nominal, for in 1450 only two books were actually read. The universities of Prague (founded in 1350) and Vienna (statutes of 1389) required most of plane geometry for the teacher's 44 THE TEACHING OF GEOMETRY license, although Vienna demanded but one book for the bachelor's degree. So, in general, the universities of the thirteenth, fourteenth, and fifteenth centuries required less for the degree of master of arts than we now require from a pupil in our American high schools. On the other hand, the university students were younger than now, and were really doing only high-school work. The invention of printing made possible the study of geometry in a new fashion. It now became possible for any one to study from a book, whereas before this time instruction was chiefly by word of moutli, consisting of an explanation of Euclid. The first Euclid was printed in 1482, at Venice, and new editions and variations of this text came out frequently in the next century. Practical geometries became very popular, and the re- action against the idea of mental disciplme tlireatened to abolish the old style of text. It was argued that geometry was unmteresting, that it was not sufficient in itself, that boys needed to see the practical uses of the subject, that only those propositions that were capable of application should be retained, that there must be a fusion between the demands of culture and the demands of business, and that every man who stood for mathe- matical ideals represented an obsolete type. Such writers as Finaeus (1556), Bartoli (1589), BelU (1569), and Cataneo (1567), in the sixteenth century, and Capra (1673), GargioUi (1655), and many others in the seven- teenth century, either directly or inferentially, took this attitude towards the subject, — exactly the attitude that is being taken at the present time by a number of teachers in the United States. As is always the case, to such an extreme did this movement lead that there was a reaction that brought the Euclid type of book DEVELOPMENT OF GEOMETRY 45 again to the front, and it has maintained its prominence even to the present. The study of geometry in the high schools is rela- tively recent. The Gymnasium (classical school prepar- atory to the university) at Niirnberg, founded in 1526, and the Cathedral school at Wiirttemberg (as shown by the curriculum of 1556) seem to have had no geometry before 1600, although the' Gymnasium at Strassburg included some of this branch of mathematics in 1578, and an elective course in geometry was offered at Zwickau, in Saxony, in 1521. In the seventeenth cen- tury geometry is found in a considerable number of secondary schools, as at Coburg (1605), Kurfalz (1615, elective), Erfurt (1643), Gotha (1605), Giessen (1605), and numerous other places in Germany, although it appeared but rarely in the secondary schools of France before the eighteenth century. In Germany the Real- sehulen — schools with more science and less classics than are found in the Gymnasium — came into being in the eighteenth century, and considerable effort was made to construct a course in geometry that should be more practical than that of the modified Euclid. At the open- ing of the nineteenth century the Prussian schools were reorganized, and from that time on geometry has had a firm position in the secondary schools of all Germany. In the eighteenth century some excellent textbooks on geometry appeared in France, among the best being that of Legendre (1794), which influenced in such a marked degree the geometries of America. Soon after the open- ing of the nineteenth century the lycees of France became strong institutions, and geometry, chiefly based on Legendre, was well taught in the mathematical divi- sions. A worthy rival of Legendre's geometry was the 46 THE TEACHING OF GEOMETRY work of Lacroix, who called attention continually to the analogy between the theorems of plane and solid geometry, and even went so far as to suggest treating the related propositions together in certain cases. In England the preparatory schools, such as Rugby, Harrow, and Eton, did not commonly teach geometry until quite recently, leaving this work for the universi- ties. In Christ's Hospital, London, however, geometry was taught as early as 1681, from a work written by several teachers of prominence. The highest class at Harrow studied " Euclid and vulgar fractions " one period a week in 1829, but geometry was not seriously studied before 1837. In the Edinburgh Academy as early as 1835, and in Rugby by 1839, plane geometry was completed. Not until 1844 did Harvard require any plane geom- etry for entrance. In 1855 Yale required only two books of Euclid. It was therefore from 1850 to 1875 that plane geometry took a definite place in the Amer- ican high school. Solid geometry has not been gener- ally required for entrance to any eastern college, although in the West this is not the case. The East teaches plane geometry more thoroughly, but allows a pupil to enter college or to go into business with no solid geometry. Given a year to the subject, it is possible to do little more than cover plane geometry ; with a year and a half the solid geometry ought easily to be covered also. Bibliography. Stamper, A History of the Teaching of Elemen- tary Geometry, New York, 1909, with a, very full bibliography of the subject; Cajori, The Teaching of Mathematics in the United States, Washington, 1890 ; Cantor, Geschichte der Mathe- raatik, Yol. IV, p. 321, Leipzig, 1908; Schotten, Inhalt imd Methode des plauimetrischen Unterrichts, Leipzig, 1890. CHAPTER V EUCLID It is fitting that a chapter in a book upon the teach- ing of this subject should be devoted to the life and labors of the greatest of all textbook writers, Euclid, — a man whose name has been, for more than two thousand years, a synonym for elementary plane geometry wherever the subject has been studied. And yet when an effort is made to pick up the scattered fragments of his biogra- phy, we are surprised to find how little is known of one whose fame is so universal. Although more editions of his work have been printed than of any other book save the Bible,i we do not know when he was born, or in what city, or even in what country, nor do we know his race, his parentage, or the time of his death. We should not feel that we knew much of the life of a man who lived when the Magna Charta was wrested from King John, if our first and only source of in- formation was a paragraph in the works of some his- torian of to-day ; and yet this is about the situation in respect to Euclid. Proclus of Alexandria, philosopher, teacher, and mathematician, lived from 410 to 485 a.d., and wrote a commentary on the works of Euclid. In his writings, which seem to set forth in amplified form his lectures to the students in the Neoplatonist School 1 Riccardi, Saggio di una tibliografla Euclidea, Part I, p. 3, Bo- logna, 1887. Riccardi lists well towards two thousand editions. 47 48 THE TEACHING OF GEOMETRY of Alexandria, Proclus makes this statement, and of Euclid's life we have little else: Not much younger than these ^ is Euclid, who put together the " Elements," collecting many of the theorems of Eudoxus, perfecting many of those of Thesetetus, and also demonstrating with perfect certainty what his predecessors had but insufficiently proved. He flourished in the time of the first Ptolemy, for Archimedes, who closely followed this ruler,^- speaks of Euclid. Furthermore it is related that Ptolemy one time demanded of him if there was in geometry no shorter way than that of the " Elements," to whom he replied that there was no royal road to geometry.'^ He was therefore younger than the pupils of Plato, but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says.* Thus we have in a few lines, from one who lived per- liaps seven or eight hundred years after Euclid, nearly all that is known of the most famous teacher of geom- etry that ever lived. Nevertheless, even this little tells us about when he flourished, for Hermotimus and Phi- lippus were pupils of Plato, who died in 347 B.C., whereas Archimedes was bom about 287 B.C. and was writing about 250 B.C. Furthermore, since Ptolemy I reigned from 306 to 283 B.C., Euclid must have been teaching about 300 B.C., and this is the date that is generally assigned to him. Euclid probably studied at Athens, for until he him- self assisted in transferring the center of mathematical 1 Hermotimus of Colophon and Philippus of Mende. 2 Literally, " Who closely followed the first," i.e. the first Ptolemy. 3 Mensechmus is said to have replied to a similar question of Alex- ander the Great : " King, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all." *This is also shown in a letter from Archimedes to Eratosthenes, recently discovered by Heiberg. EUCLID 49 culture to Alexandria, it had long been in the Grecian capital, indeed since the time of Pythagoras. Moreover, numerous attempts had been made at Athens to do ex- actly what Euclid succeeded in doing, — to construct a logical sequence of propositions ; in other words, to write a textbook on plane geometry. It was at Athens, there- fore, that he could best have received the inspiration to compose his " Elements." ^ After finishing his education at Athens it is quite probable that he, like other savants of the period, was called to Alexandria by Ptolemy Soter, the king, to assist in establishing the great school which made that city the center of the world's learning for several centuries. In this school he taught, and here he wrote the " Elements " and numerous other, works, perhaps ten in, all. Although the Greek writers who may have known something of the life of Euclid have little to say of him, the Arab writers, who could have known nothing save from Greek sources, have allowed their imaginations the usual latitude in speaking of him and of his labors. Thus Al-Qifti, who wrote in the thirteenth century, has this to say in his biographical treatise " Ta'rikh al- Hukama": Euclid, son of jSTauorates, grandson of Zenarchus, called the author of geometry, a Greek by nationality, domiciled at Damascus, bom at Tyre, most learned in the science of geometry, published a most excellent and most useful work entitled " The Foundation or Elements of Geometry,'' a subject in which no more general treatise existed before among the Greeks ; nay, there was no one even of later date who did not walk in his footsteps and frankly profess his doctrine. 1 On this phase of the subject, and indeed upon Euclid and his propositions and works in general, consult T. L. Heath, " The Thirteen Books of Euclid's Elements," 3 vols., Cambridge, 1908, a masterly treatise of which frequent use has been made in preparing this work. 50 THE TEACHIN^G OF GEOMETRY This is rather a specimen of the Arab tendency to manufacture history than a serious contribution to the biograpliy of Euchd, of whose personal history we have only the information given by Proclus. EfCLII) Fi-oiii a]i old print Euclid's works at once took high rank, and they are mentioned by various classical authors. Cicero knew of them, and Capella (m. 470 A.i).), Cassiodorius (en. 515 A.D. ), and Boethius (ca. 4S0-5"2-!: a.d.) were all more ELX'LID 51 or less familiar with the " Elements." With the advance of the Dark Ages, however, learning was held in less and less esteem, so that Euclid was finally forgotten, and manuscripts of his works were either destroyed or buried in some remote cloister. The Arabs, however, whose civilization assumed prominence from about 750 a.d. to about 1500, translated the most important treatises of the Greeks, and Euclid's " Elements " among the rest. One of these Arabic editions an English monk of the twelfth century, one Athelhard (^thelhard) of Bath, found and translated into Latin (ca. 1120 a.d.). A little later Grherard of Cremona (1114-1187) made a new translation from the Arabic, differing m essential fea- tures from that of Athelhard, and about 1260 Johannes Campanus made still a third translation, also from Arabic into Latm.^ There .is reason to believe that Athelhard, Campanus, and Gherard may all have had access to an earlier Latin translation, suice all are quite alike in some particulars while diverging noticeably in others. Indeed, there is an old English verse that relates : The clerk Euclide on this wyse hit fonde Thys craft of gemetry yn Egypte londe . . . Thys craft com into England, as y yow say, Yn tyme of good Kyng Adelstone's day. If this be true, Euclid was known in England as early as 924-940 a.d. Without going into particulars further, it suffices to say that the modern knowledge of Euclid came first through the Arabic into the Latin, and the first printed 1 A contemporary copy of this translation is now in the library of George A. Plimpton, Esq., of New York. See the author's "Kara Arithmetica," p. 433, Boston, 1909. 52 THE TEACHING OF GEOMETRY edition of the "Elements" (Venice, 1482) was the Campanus translation. Greek manuscripts now began to appear, and at the present time several are known. There is a manuscript of the ninth century in the Bod- leian library at Oxford, one of the tenth century in the Vatican, another of the tenth century in Florence, one of the eleventh century at Bologna, and two of the twelfth century at Paris. There are also fragments con- taining bits of Euclid in Greek, and going back as far as the second and third century A.D. The iirst modern translation from the Greek into the Latin was made by Zamberti (or Zamberto),i and was printed at Venice in 1513. The first translation into English was made by Sir Henry Billingsley and was printed in 1570, sixteen years before he became Lord Mayor of London. Proclus, in his commentary upon Euclid's work, remarks : In the whole of geometry there are certain leading theorems, bearing to those which follow the relation of a principle, all-per- vading, and furnishing proofs of many properties. Such theorems are called by the name of elements, and their function may be compared to that of the letters of the alphabet in relation to language, letters being indeed called by the same name in Greek \crTOi.)(aa., stoicheia].^ This characterizes the work of Euclid, a collection of the basic propositions of geometry, and chiefly of plane geometry, arranged in logical sequence, the proof of each depending upon some preceding proposition, defi- nition, or assumption (axiom or postulate). The number 1 A beautiful vellum manuscript of this translation is in the library of George A. Plimpton, Esq., of New York. See the author's " Rara Arithmetica," p. 481, Boston, 1909. 2 Heath, loc. cit., Vol. I, p. 114. EUCLID 53 of the propositions of plane geometry iiicluded in the " Elements " is not entirely certain, owing to some dis- agreement in the manuscripts, but it was between one hundred sixty and one hundred seventy-five. It is possible to reduce this number by about thirty or forty, because Euclid included a certain amount of geo- metric algebra; but beyond this we cannot safely go in the way of elimination, since from the very nature of the " Elements " these propositions are basic. The efforts at revising Euclid have been generally confined, there- fore, to rearranging his material, to rendering more mod- ern his phraseology, and to making a book that is more usable with beginners if not more logical in its presen- tation of the subject. While there has been an improve- ment upon Euclid in the art of bookmaking, and in minor matters of phraseology and sequence, the educa- tional gain has not been commensurate with the effort put forth. With a little modification of Euclid's semi- algebraic Book II and of his treatment of proportion, with some scattering of the definitions and the inclusion of well-graded exercises at proper places, and with atten- tion to the modern science of bookmaking, the " Ele- ments " would answer quite as well for a textbook to- day as most of our modern substitutes, and much better than some of them. It would, moreover, have the advan- tage of being a classic, — somewhat the same advantage that comes from reading Homer in the original instead of from Pope's metrical translation. This is not a plea for a return to the Euclid text, but for a recognition of the excellence of Euclid's work. The distinctive feature of Euclid's " Elements," com- pared with the modern American textbook, is perhaps this : Euclid begins a book with what seems to him the 54 THE TEACHING OF GEOMETRY easiest proposition, be it theorem or problem ; upon this he builds another ; upon these a third, and so on, con- cerning himself but little with the classification of prop- ositions. Furthermore, he arranges his propositions so as to construct his figures before using them. We, on the other hand, make some little attempt to classify our propositions within each book, and we make no attempt to construct our figures before using them, or at least to prove that the constructions are correct. Indeed, we go so far as to study the properties of figures that we cannot construct, as when we ask for the size of the angle of a regular heptagon. Thus Euclid begins Book I by a problem, to construct an equilateral triangle on a given line. His object is to follow this by problems on drawing a straight line equal to a given straight line, and cutting off from the greater of two straight lines a line equal to the less. He now introduces a theorem, which might equally well have been his first proposition, namely, the case of the congruence of two triangles, hav- ing given two sides and the included angle. By means of his third and fourth propositions he is now able to prove the pons asinorum,that the angles at the base of an isosceles triangle are equal. We, on the other hand, seek to group our propositions where this can conveniently be done, putting the congruent propositions together, those about inequalities by themselves, and the propositions about parallels in one set. The results of the two arrangements are not radically different, and the effect of either upon the pupil's mind does not seem particularly better than that of the other. Teachers who have used both plans quite commonly feel that, apart from Books H and V, Euclid is nearly as easily understood as our modern texts, if presented in as satisfactory dress. EUCLID 55 The topics treated and the number of propositions in the plane geometry of the " Elements " are as follows : Book I. Rectilinear figures . . 48 Book II. Geometric algebra . l-i Book III. Circles .... . 37 Book IV. Problems about circles . . 16 Book V. Proportion . . .25 Book VI. Applications of proportion . 33 173 Of these we now omit Euclid's Book II, because we have an algebraic symbolism that was unknown in his time, although he would not have used it in geometry even had it been known. Thus his first proposition in Book II is as follows : If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments. This amounts to saying that ii x=p + q-i-r-\ , then ax = ap + aq + ar + ■ ■ ■ . We also materially simplify Euclid's Book V. He, for example, proves that "If four magnitudes be proportional, they will also be pro- portional alternately." This he proves generally for any kind of magnitude, while we merely prove it for num- bers having a common measure. We say that we may substitute for the older form of proportion, namely. a:b = o:d, a _c b^d' From this we have ad = be. the fractional form Whence a _b c d 56 THE TEACHING OF GEOMETRY In this work we assume that we may multiply equals by h and d. But suppose h and d are cubes, of which, indeed, we do not even know the approximate numerical measure ; what shall we do ? To Euclid the multipli- cation by a cube or a polygon or a sphere would have been entirely meaningless, as it always is from the standpoint of pure geometry. Hence it is that our treat- ment of proportion has no serious standing in geometry as compared with Euclid's, and our only justification for it lies in the fact that it is easier. Euclid's treatment is much more rigorous than ours, but it is adapted to the comprehension of only advanced students, while ours is merely a confession, and it should be a frank confes- sion, of the weakness of our pupils, and possibly, at times, of ourselves. If we should take Euclid's Books II and V for granted, or as sufficiently evident from our study of algebra, we should have remaining only one hundred thirty-four prop- ositions, most of which may be designated as basal propo- sitions of plane geometry. Revise Euclid as we will, we shall not be able to eliminate any large number of his fundamental truths, while we might do much worse than to adopt these one hundred thirty-four propositions in toto as the bases, and indeed as the definition, of elemen- tary plane geometry. Bibliography. Heath, The Thirteen Books of Euclid's Elements, 3 vols., Cambridge, 1908 ; Frankland, The First Book of Euclid, Cambridge, 1906 ; Smith, Dictionary of Greek and Roman Biog- raphy, article Eukleides ; Simon, Euclid und die sechs plani- metrisohen Bticher, Leipzig, 1901 ; Gow, Plistory of Greek Mathe- matics, Cambridge, 1884, and any of the standard histories of mathematics. Both Heath and Simon give extensive bibliogra- phies. The latest standard Greek and Latin texts are Heiberg's, published by Teubner of Leipzig. CHAPTER VI EFFORTS AT IMPROVING EUCLID From time to time an effort is made by some teacher, or association of teachers, animated by a serious desire to improve the instruction in geometry, to prepare a new syllabus that shall mark out some "royal road," and it therefore becomes those who are mterested in teaching to consider with care the results of similar efforts in recent years. There are many questions which such an attempt suggests : What is the real purpose of the move- ment ? What will the teaching world say of the result ? Shall a reckless, ill-considered radicalism dominate the effort, bringing in a distasteful terminology and symbol- ism merely for its novelty, insisting upon an ultra- logical treatment that is beyond the powers of the learner, rearrangmg the subject matter to fit some narrow notion of the projectors, seeking to emasculate mathematics by looking only to the applications, riding some little hobby in the way of some particular class of exercises, and cut- ting the number of propositions to a minimum that will satisfy the mere demands of the artisan ? Such are some of the questions that naturally arise in the mind of every one who wishes well for the ancient science of geometry. It is not proposed in this chapter to attempt to answer these questions, but rather to assist in understanding the problem by considering the results of similar attempts. 57 58 THE TEACHING OF GEOMETRY If it shall be found that syllabi have been prepared under circumstances quite as favorable as those that obtain at present, and if these syllabi have had little or no real influence, then it becomes our duty to see if new plans may be worked out so as to be more successful than their predecessors. If the older attempts have led to some good, it is well to know what is the nature of this good, to the end that new efforts may also result in something of benefit to the schools. It is proposed in this chapter to call attention to four important syllabi, setting forth briefly their distinguish- ing features and drawing some conclusions that may be helpful in other efforts of this nature. In England two noteworthy attempts have been made within a century, looking to a more satisfactory sequence and selection of propositions than is found in Euclid. Each began with a list of propositions arranged in proper sequence, and each was thereafter elaborated into a text- book. Neither accomplished fully the purpose intended, but each was instrumental in provoking healthy discus- sion and in improving the texts from which geometry is studied. The first of these attempts was made by Professor Augustus de Morgan, under the auspices of the Society for the Diffusion of Useful Knowledge, and it resulted in a textbook, including " plane, solid, and spherical " geometry, in six books. According to De Morgan's plan, plane geometry consisted of three books, the number of jDropositions being as follows : Book I. Rectilinear figures 60 Book II. Ratio, proportion, applications .... 69 Book III. The circle . . . . 65 Total for plane geometry . 194 EFFORTS AT IMPKOVIJs'G EUCLID 59 Of the 194 propositions De Morgan selected 11-i with their corollaries as necessary for a beginner who is teaching himself. In solid geometry the plan was as follows : Book IV. Lines in different planes, solids con- tained by planes . . 52 Book V. Cylinder, cone, sphere . . 25 Book VI. Figures on a sphere . . 42 Total for solid geometry . ... . . 119 Of these 119 propositions De Morgan selected 76 with their corollaries as necessary for a beginner, thus making 190 necessary propositions out of 305 desirable ones, besides the corollaries in plane and solid geometry. In other words, of the desirable propositions he considered that about two thirds are absolutely necessary. It is interesting to note, however, that he summed up the results of his labors by saying : It will be found that the course just laid down, excepting the sixth book of it only, is not of much greater extent, nor very dif- ferent in point of matter from that of Euclid, whose " Elements " have at all times been justly esteemed a model not only of easy and progressive instruction in geometry, bvit of accuracy and perspicuity in reasoning. De Morgan's effort, essentially that of a syllabus-maker rather than a textbook writer, although it was published under the patronage of a prominent society with which were associated the names of men like Henry Hallam, Rowland Hill, Lord John Russell, and George Peacock, had no apparent influence on geometry either in England or abroad. Nevertheless the syllabus was in many respects excellent ; it rearranged the matter, it classified the propo- sitions, it improved some of the terminology, and it re- duced the number of essential propositions ; it had the assistance of De Morgan's enthusiasm and of the society 60 THE TEACHING OF GEOMETRY with which he was so prominently connected, and it was circulated with considerable generosity throughout the English-speaking world ; but in spite of all this it is to-day practically unknown. A second noteworthy attempt in England was made about a quarter of a century ago by a society that was organized practically for this very purpose, the Associa- tion for the Improvement of Geometrical Teaching. This society was composed of many of the most progressive teachers in England, and it included in its membership men of high standing in mathematics in the universities. As a result of their labors a syllabus was prepared, which was elaborated into a textbook, and in 1889 a revised syllabus was issued. As to the arrangement of matter, the syllabus departs from Euclid chiefly by separating the problems from the theorems, as is the case in our American textbooks, and in improving the phraseology. The course is preceded by some simple exercises in the use of the compasses and ruler, a valuable plan that is followed by many of the best teachers everywhere. Considerable attention is paid to logical processes before beginning the work, such terms as " contrapositive " and " obverse," and such rules as the " rule of conversion " and the " rule of identity " being introduced before any propositions are considered. The arrangement of the work and the number of propositions in plane geometry are as follows : Book I. The straight line . . . . . 51 Book II. Equality of areas . . .19 Book III. The circle . . .42 Book IV. Ratio and proportion . . 32 Book V. Proportion . . . ... 24 Total for plane geometry . . . . . 168 EFFORTS AT IMPROVING EUCLID 61 Here, then, is the result of several years of labor by a somewhat radical organization, fostered by excellent mathematicians, and carried on in a country where ele- mentary geometry is held in highest esteem, and where Euclid was thought unsuited to the needs of the begin- ner. The number of propositions remains substantially the same as in Euclid, and the introduction of some unusable logic tends to counterbalance the improvement in sequence of the propositions. The report provoked thought ; it shook the Euclid stronghold ; it was prob- ably mstrumental in bringing about the present upheaval in geometry in England, but as a working syllabus it has not appealed to the world as the great improvement upon Euclid's "Elements" that was hoped by many of its early advocates. The same association published later, and republished in 1905, a "Report on the Teaching of (reometry," in which it returned to Euclid, modifying the " Elements " by omitting certain propositions, changing the order and proof of others, and introducing a few new theorems. It seems to reduce the propositions to be proved in plane geometry to about one hundred fifteen, and it recommends the omission of the incommensurable case. This number is, however, somewhat misleading, for Euclid frequently puts in one proposition what we in America, for educational reasons, find it better to treat in two, or even tliree, propositions. This report, there- fore, reaches about the same conclusion as to the geo- metric facts to be mastered as is reached by our later textbook writers in America. It is not extreme, and it stands for good mathematics. In the United States the influence of our early wars with England, and the sympathy of France at that time, 62 THE TEACHING OF GEOMETRY turned the attention of our scholars of a century ago from Cambridge to Paris as a mathematical center. The influx of French mathematics brought with it such works as Legendre's geometry (1794) and Bourdon's algebra, and made known the texts of Lacroix, Bertrand, and Bezout. Legendre's geometry was the result of the efforts of a great mathematician at syllabus-making, a natural thing in a country that had early broken away from Euclid. Legendre changed the Greek sequence, sought to select only propositions that are necessary to a good understanding of the subject, and added a good course in solid geometry. His arrangement, with the number of propositions as given in the Davies transla- tion, is as follows : Book I. Rectilinear figures . . 81 Book II. Ratio and proportion . . ... 14 Book III. The circle . 48 Book IV. Proportions of figures and areas . 51 Book V. Polygons and circles . . 17 Total for plane geometry . . 161 Legendre made, therefore, practically no reduction in the number of Euclid's propositions, and his improve- ment on Euclid consisted chiefly in his separation of problems and theorems, and in a less rigorous treatment of proportion which boys and girls could comprehend. D'Alembert had demanded that the sequence of propo- sitions should be determined by the order in which they had been discovered, but Legendre wisely ignored such an extreme and gave the world a very usable book. The principal effect of Legendre's geometry in Amer- ica was to make every textbook writer his own syllabus- maker, and to put solid geometry on a more satisfactory footing. The minute we depart from a standard text EFFORTS AT IMPROVING EUCLID 63 like Euclid's, and have no recognized examining body, every one is free to set up his own standard, always within the somewhat uncertain boundary prescribed by public opinion and by the colleges. The efforts of the past few years at syllabus-making have been merely attempts to define this boundary more clearly. Of these attempts two are especially worthy of con- sideration as having been very carefully planned and having brought forth such definite results as to appeal to a large number of teachers. Other syllabi have been made and are familiar to many teachers, but in point of clearness of purpose, conciseness of expression, and form of publication they have not been such as to compare with the two in question. The first of these is the Harvard syllabus, which is placed in the hands of students for reference when try- ing the entrance examinations of that university, a plan not followed elsewhere. It sets forth the basal proposi- tions that should form the essential part of the -student's preparation, and that are necessary and sufficient for prov- ing any " original proposition " (to take the common expression) that may be set on the examination. The propositions are arranged by books as follows: Book I. Angles, triangles, parallels . . 25 Book II. The circle, angle measure . 18 Book III. Similar polygons . . 10 Book IV. Area of polygons .... 8 Book V. Polygons and circle measure . . . 11 Constructions .... .21 Ratio and proportion . . 6 Total for plane geometry . . 99 The total for solid geometry is 79 propositions, or 178 for both plane and solid geometry. Thi^ is perhaps the 64 THE TEACHING OP GEOMETRY most successful attempt that has been made at reaching a minimum number of propositions. It might well be further reduced, since it includes the proposition about two adjacent angles formed by one line meeting another, and the one about the circle as the limit of the inscribed and circumscribed regular polygons. The first of these leads a beginner to doubt the value of geometry, and the second is beyond the powers of the majority of stu- dents. As compared with the syllabus reported by a Wisconsin committee in 1904, for example, here are 99 propositions against 132. On the other hand, a commit- tee appointed by the Central Association of Science and Mathematics Teachers reported in 1909 a syllabus with what seems at first sight to be a list of only 59 propo- sitions in plane geometry. This number is fictitious, however, for the reason that numerous converses are indicated with the propositions, and are not included in the count, and directions are given to include " related theorems " and " problems dealing with the length and area of a circle," so that in some cases one proposition is evidently intended to cover several others. This syllabus is therefore lacking in definite- ness, so that the Harvard list stands out as perhaps the best of its type. The second noteworthy recent attempt in America is that made by a committee of the Association of Mathe- matical Teachers in New England. This committee was organized in 1904. It held sixteen meetings and carried on a great deal of correspondence. As a result, it pre- pared a syllabus arranged by topics, the propositions of solid geometry being grouped immediately after the corresponding ones of plane geometry. For example, the nine propositions on congruence in a plane are followed EFFORTS AT IMPROVING EUCLID 65 by nine on congruence in space. As a result, the follow- ing summarizes the work in plane geometry : Congruence in a plane . . ... . 9 Equivalence . . ... . . 3 Parallels and perpendiculars . . 9 Symmetry .... . . 20 Angles .... 15 Tangents . . . . 4 Similar figures . .... 18 Inequalities ..... . 8 Lengths and areas . . 17 Loci . . . 2 Concurrent lines ... 5 Total for plane geometry . . . 110 Not SO conventional in arrangement as the Harvard syllabus, and with a few propositions that are evidently not basal to the same extent as the rest, the list is never- theless a very satisfactory one, and the parallelism shown between plane and solid geometry is suggestive to both student and teacher. On the whole, however, the Harvard selection of basal propositions is perhaps as satisfactory as any that has been made, even though it appears to lack a " factor of safety," and it is probable that any further reduction would be unwise. What, now, has been the effect of all these efforts ? What teacher or school would be content to follow any one of these syllabi exactly ? What textbook writer would feel it safe to limit his regular propositions to those in any one syllabus ? These questions suggest their own answers, and the effect of all this effort seems at first thought to have been so slight as to be entirely out of proportion to the end in view. This depends, however, on what this end is conceived to be. If the 66 THE TEACHING OF GEOMETRY purpose has been to cut out a very large number of the propositions that are found in Euclid's plane geometry, the effort has not been successful. We may reduce this number to about one hundred thirty, but in general, whatever a syllabus may give as a minimum, teachers will favor a larger number than is suggested by the Harvard list, for the purpose of exercise in the read- ing of mathematics if for no other reason. The French geometer, Lacroix, who wrote more than a century ago, proposed to limit the propositions to those needed to prove other important ones, and those needed in prac- tical mathematics. If to this we should add those that are used in treating a considerable range of exercises, we should have a list of about one hundred thirty. But this is not the real purpose of these syllabi, or at most it seems like a relatively unimportant one. The purpose that has been attained is to stop the indefinite increase in the number of propositions that would fol- low from the recent developments in the geometry of the triangle and circle, and of similar modern topics, if some such counter-movement as this did not take place. If the result is, as it probably will be, to let the basal propositions of Euclid remain about as they always have been, as the standards for beginners, the syllabi will have accomplished a worthy achievement. If, ia addi- tion, they furnish an irreducible minimum of proposi- tions to which a student may have access if he desires it, on an examination, as was intended in the case of the Harvard and the New England Association syllabi, the achievement may possibly be still more worthy. In preparing a syllabus, therefore, no one should hope to bring the teaching world at once to agree to any great reduction in the number of basal propositions, nor to EFFORTS AT IMPROVmG EUCLID 67 agree to any radical change of terminology, symbolism, or sequence. Rather should it be the purpose to show- that we have enough topics in geometry at present, and that the number of propositions is really greater than is absolutely necessary, so that teachers shall not be led to introduce any considerable number of proposi- tions out of the large amount of new material that has recently been accumulating. Such a syllabus will always accomplish a good purpose, for at least it will provoke thought and arouse interest, but any other kind is bound to be ephemeral. 1 Besides the evolutionary attempts at rearrangiag and reducing in number the propositions of Euclid, there have been very many revolutionary efforts to change his treatment of geometry entirely. The great French math- ematician, D'Alembert, for example, in the eighteenth century, wished to divide geometry into three branches : (1) that dealing with straight lines and circles, appar- ently not limited to a plane ; (2) that dealing with sur- faces; and (3) that dealing with solids. So Meray in France and De Paolis ^ in Italy have attempted to fuse plane and solid geometry, but have not produced a sys- tem that has been particularly successful. More recently Bourlet, Grevy, Borel, and others in France have produced several works on the elements of mathematics that may lead to something of A^alue. They place intuition to the front, favor as much appUed mathematics as is reasonable, to all of which American teachers would generally agree, 1 The author is a member of a committee that has for more than a year been considering a syllabus in geometry. This committee will probably report sometime during the year 1911. At the present writing it seems disposed to recommend about the usual list of basal propositions. 2 " Elementi di Geometria," Milan, 1884. 68 THE TEACHING OF GEOMETRY but they claim that the basis of elementary geometry in the future must be the "investigation of the group of motions." It is, of course, possible that certain of the notions of the higher mathematical thought of the nuie- teenth century may be so simplified as to be within the comprehension of the tyro in geometry, and we should be ready to receive all efforts of this kind with open mind. These writers have not however produced the ideal work, and it may seriously be questioned whether a work based upon their ideas will prove to be educationally any more sound and usable than the labors of such excellent writers as Henrici and Treutlein, and H. Miiller, and Schlegel a few years ago in Germany, and of Veronese in Italy. All such efforts, however, should be welcomed and tried out, although so far as at present appears there is nothing in sight to replace a well-arranged, vitalized, simplified textbook based upon the labors of Euclid and Legendre. The most broad-minded of the great mathematicians who have recently given attention to secondary prob- lems is Professor Klein of Gottingen. He has had the good sense to look at something besides the mere ques- tion of good mathematics. 1 Thus he insists upon the psychologic point of view, to the end that the geometry shall be adapted to the mental development of the pupil, — a thing that is apparently ignored by j\Ieray (at least for the average pupil), and, it is to be feared, by the other recent French writers. He then demands a careful selection of the subject matter, which in our American schools would mean the elimination of propositions that are not basal, that is, that are not used for most of the 1 See his " Elementarmathematik vom holieren Standpunkt aiis," Part II, Leipzig, 1909. EFFORTS AT IMPROA'ING EUCLID 69 exercises that one naturally meets in elementary geom- etry and in applied work. He further insists upon a reasonable correlation with practical work, to which e\'ery teacher will agree so long as the work is really or even potentially practical. And finally he asks that we look with favor upon the union of plane and solid geometry, and of algebra and geometry. He does not make any plea for extreme fusion, but presumably he asks that to which every one of open mind would agree, namely, that whenever the opportunity offers in teachmg plane geometry, to open the vision to a generalization in space, or to the measurement of well-known solids, or to the use of the algebra that the pupil has learned, the opportunity should be seized. CHAPTER VII THE TEXTBOOK IN GEOMETRY In considering the nature of the textbook in geometry we need to bear in mind the fact that the subject is being taught to-day in America to a class of pupils that is not composed like the classes found in other countries or in earlier generations. In general, in other countries, geom- etry is not taught to mixed classes of boys and girls. Furthermore, it is generally taught to a more select group of pupils than in a country where the high school and college are so popular with people in all the walks of life. In America it is not alone the boy who is in- terested ui education in general, or in mathematics .in particular, who studies geometry, and who joins with others of like tastes in this pursuit, but it is often the boy and the girl who are not compelled to go out and work, and who fill the years of youth with a not over- strenuous school life. It is therefore clear that we can- not hold the interest of such pupils by the study of Euclid alone. Geometry must, for them, be less formal than it was half a century ago. We cannot expect to make our classes enthusiastic merely over a logical sequence of proved propositions. It becomes necessary to make the work more concrete, and to give a much larger number of simple exercises in order to create the interest that comes from independent work, from a feeling of conquest, and from a desire to do something 70 THE TEXTBOOK IN GEOMETRY 71 original. If we would " cast a glamor over the multipli- cation table," as an admirer of Macaulay has said that the latter could do, we must have the facilities for so doing. It therefore becomes necessary in weighing the merits of a textbook to consider : (1) if the number of proved propositions is reduced to a safe minimum ; (2) if there is reasonable opportunity to apply the theory, the actual applications coming best, however, from the teacher as an outside interest ; (3) if there is an abundance of material in the way of simple exercises, since such, mate- rial is not so readily given by the teacher as the seem- ingly local applications of the propositions to outdoor measurements ; (4) if the book gives a reasonable amount of introductory work in the use of simple and inexpen- sive instruments, not at that time emphasizing the formal side of the subject ; (5) if there is afforded some oppor- tunity to see the recreative side of the subject, and to know a little of the story of geometry as it has devel- oped from ancient to modern times. But this does not mean that there is to be a geometric cataclysm. It means that we must have the same safe, conservative evolution in geometry that we have in other subjects. Geometry is not going to degenerate into mere measuring, nor is the ancient sequence going to become a mere hodge-podge without system and with no incen- tive to strenuous effort. It is now about fifteen hundred years since Proclus laid down what he considered the essential features of a good textbook, and in all of our efforts at reform we cannot improve very much upon his statement. " It is essential," he says, " that such a treat- ise should be rid of everything superfluous, for the super- fluous is an obstacle to the acquisition of knowledge ; it should select everythmg that embraces the subject and 72 THE TEACHING OF GEOMETRY brings it to a focus, for this is of the highest service to science ; it must have great regard both to clearness and to conciseness, for their opposites trouble our understand- ing ; it must aim to generalize its theorems, for the divi- sion of knowledge into small elements renders it difficult of comprehension." It being prefaced that we must make the book more concrete in its applications, either directly or by suggest- ing seemingly practical outdoor work ; that we must in- crease the number of simple exercises calling for original work; that we must reasonably reduce the number of proved propositions ; and that we must not allow the good of the ancient geometry to depart, let us consider in detail some of the features of a good, practical, com- mon-sense textbook. The early textbooks in geometry contained only the propositions, with the proofs in full, preceded by lists of definitions and assumptions (axioms and postulates). There were no exercises, and the proofs were given in essay form. Then came treatises with exercises, these exercises being grouped at the end of the work or at the close of the respective books. The next step was to the unit page, arranged in steps to aid the eye, one proposi- tion to a page whenever this was possible. Some effort was made in this direction in France about two hundred years ago, but with no success. The arrangement has so much to commend it, however, the proof bemg so much more easily followed by the eye than was the case in the old-style works, that it has of late been revived. In this respect the Wentworth geometry was a pioneer in Amer- ica, and so successful was the effort that this type of page has been adopted, as far as the various writers were able to adopt it, in all successful geometries that have THE TEXTBOOK IN GEOMETRY 73 appeared of late years in this country. As a result, the American textbooks on this subject are more helpful and pleasing to the eye than those found elsewhere. The latest improvements in textbook-making have removed most of the blemishes of arrangement that re- mained, scattering the exercises through the book, grad- ing them with greater care, and making them more modern in character. But the best of the latest works do more than this. They reduce the number of proved theorems and increase the number of exercises, and they simplify the proofs whenever possible and eliminate the most difficult of the exercises of twenty-five years ago, It would be possible to carry this change too far by put- ting in only half as many, or a quarter as many, regular propositions, but it should not be the object to see how the work can be cut down, but to see how it can be improved. What should be the basis of selection of propositions and exercises ? Evidently the selection must include the great basal propositions that are needed in mensuration and in later mathematics, together with others that are necessary to prove them. Euclid's one hundred seventy- three propositions of plane geometry were really upwards of one hundred eighty, because he several times com- bined two or more in one. These we may reduce to about one hundred thirty with perfect safety, or less than one a day for a school year, but to reduce still further is undesirable as well as unnecessary. It would not be difficult to dispense with a few more ; indeed, we might dispense with thirty more if we should set about it, although we must never forget that a goodly number in addition to those needed for the logical sequence are necessary for the wide range of exercises that are offered. 74 THE TEACHING OF GEOMETRY But let it be clear that if we teach 100 instead of 130, our results are liable to be about \^^ as satisfactory. We may theorize on pedagogy as we please, but geometry will pay us about in proportion to what we give. And as to the exercises, what is the basis of selection ? In general, let it be said that any exercise that pretends to be real should be so, and that words taken from science or measurements do not necessarily make the problem genuine. To take a proposition and apply it in a man- ner that the world never sanctions is to indulge in deceit. On the other hand, wholly to neglect the common appli- cations of geometry to handwork of various kinds is to miss one of our great opportunities to make the subject vital to the pupil, to arouse new interest, and to give a meaning to it that is otherwise wanting. It should always be remembered that mental discipline, whatever the phrase may mean, can as readily be obtained from a genu- ine application of a theorem as from a mere geometric puzzle. On the other hand, it is evident that not more than 25 per cent of propositions have any genuine appli- cations outside of geometry, and that if we are to attempt any applications at all, these must be sought mainly in the field of pure geometry. In the exercises, therefore, we seek to-day a sane and a balanced book, giving equal weight to theory and to practice, to the demands of the - artisan and to those of the mathematician, to the applica- tions of concrete science and to those of pure geometry, thus making a fusion of pure and applied mathematics, with the latter as prominent as the supply of genuine problems permits. The old is not all bad and the new is not all good, and a textbook is a success in so far as it selects boldly the good that is in the old and rejects with equal boldness the bad that is in the new. THE TEXTBOOK IX GEOMETRY 75 Lest the nature of the exercises of geometry may be misunderstood, it is well that we consider for a moment what constitutes a genuuie application of the subject. It is the ephemeral fashion just at present in America to call these genuiae applications by the name of "real problems." The name is an unfortunate importation, but that is not a matter of serious moment. The impor- tant thing is that we should know what makes a prob- lem "real" to the pupil of geometry, especially as the whole thing is coming rapidly into disrepute through the mistaken zeal of some of its supporters. A real problem is a problem that the average citizen may sometime be called upon to solve ; that, if so called upon, he will solve in the manner indicated ; and that is expressed in terms that are familiar to the pupil. This definition, which seems fairly to state the condi- tions under which a problem can be called " real " in the schoolroom, involves three points : (1) people must be liable to meet such a problem; (2) in that case they will solve it in the way suggested by the book ; (3) it must be clothed in language familiar to the pupil. For example, let the problem be to find the dimensions of a rectangular field, the data being the area of the field and the area of a road four rods wide that is cut from three sides of the field. As a real problem this is ridiculous, since no one would ever meet such a case outside the puzzle department of a schoolroom. Again, if by any stretch of a vigorous imagination any human being should care to find the area of a piece of glass, bounded by the arcs of circles, in a Gothic window in York Minster, it is fairly certain that he would not go about it in the way suggested in some of the earnest attempts that have been made by several successful teachers to add interest to 76 THE TEACHING OF GEOMETRY geometry. And for the third point, a problem is not real to a pupil simply because it relates to moments of inertia or the tensile strength of a steel bar. Indeed, it is unreal precisely because it does talk of these things at a time when they are unfamiliar, and properly so, to the pupil. It must not be thought that puzzle problems, and unreal problems generally, have no value. All that is insisted upon is that such problems as the above are not " real," and that about 90 per cent of problems that go by this name are equally lacking in the elements that make for reality in this sense of the word. For the other 10 per cent of such problems we should be thankful, and we should endeavor to add to the number. As for the great mass, however, they are no better than those that have stood the test of generations, and by their pretense they are distinctly worse. It is proper, however, to consider whether a teacher is not justified in relating his work to those geometric forms that are found in art, let us say in floor patterns, in domes of buildings, in oilcloth designs, and the like, for the purpose of arousing interest, if for no other reason. The answer is apparent to any teacher : It is certainly justi- fiable to arouse the pupil's interest in his subject, and to call his attention to the fact that geometric design plays an important part in art ; but we must see to it that our efforts accomplish this purpose. To make a course in geometry one on oilcloth design would be absurd, and nothing more unprofitable or depressing could be imag- ined in connection with this subject. Of course no one would advocate such an extreme, but it sometimes seems as if we are getting painfully near it in certain schools. A pupil has a passing interest in geometric design. He should learn to use the instruments of geometry, and THE TEXTBOOK IN GEOMETRY 77 he learns this most easily by drawing a few such pat- terns. But to keep him week after week on questions relating to such designs of however great variety, and especially to keep him upon designs relating to only one or two types, is neither sound educational policy nor even common sense. That this enthusiastic teacher or that one succeeds by such a plan is of no significance ; it is the enthusiasm that succeeds, not the plan. The experience of the world is that pupils of geometry like to use the subject practically, but that they are more interested in the pure theory than in any fictitious applications, and this is why pure geometry has endured, while the great mass of applied geometry that was brought forward some three hundred years ago has long since been forgotten. The question of the real applications of the subject is considered in subsequent chapters. In Chapter VI we considered the question of the number of regular propositions to be expected in the text, and we have just considered the nature of the exer- cises which should follow those propositions. It is well to turn our attention next to the nature of the proofs of the basal theorems. Shall they appear in full? Shall they be merely suggested demonstrations ? Shall they be only a series of questions that lead to the proof? Shall the proofs be omitted entirely ? Or shall there be some combination of these plans ? The natural temptation in the nervous atmosphere of America is to listen to the voice of the mob and to pro- ceed at once to lynch Euclid and every one who stands for that for which the " Elements " has stood these two thousand years. This is what some who wish to be con- sidered as educators tend to do ; in the language of the mob, to "smash things"; to call reactionary that which 78 THE TEACHING OF GEOMETRY does not conform to their ephemeral views. It is so easy to be an iconoclast, to think that eui bono is a conclusive argument, to say so glibly that Raphael was not a great painter, — to do anything but construct. A few years ago every one must take up with the heuristic method developed in Germany half a century back and contain- ing much that was commendable. A little later one who did not believe that the Culture Epoch Theory was vital in education was looked upon with pity by a consider- able number of serious educators. A little later the man who did not thiak that the principle of Concentration in education was a regula aurea was thought to be hopeless. A little later it may have been that Correlation was the saving factor, to be looked upon in geometry teaching as a guiding beacon, even as the fusion of all mathematics is the temporary view of a few enthusiasts to-day.^ And just now it is vocational traming that is the catch phrase, and to many this phrase seems to sound the funeral knell of the standard textbook in geometry. But does it do so ? Does this present cry of the pedagogical circle really mean that we are no longer to have geom- etry for geometry's sake ? Does it mean that a panacea has been found for the ills of memorizing without under- standing a proof in the class of a teacher who is so ineffi- cient as to allow this kind of work to go on ? Does it mean that a teacher who does not see the human side of I For some classes of schools and under certain circumstances courses in combined jnathematics are very desirable. All that is here insisted upon is that any general fusion all along the line would result in weak, insipid, and uninteresting mathematics. A beginning, inspi- rational course in combined mathematics has a good reason for being in many high schools in spite of its manifest disadvantages, and such a course may be developed to cover all of the required mathematics given in certain schools. THE TEXTBOOK IN GEOMETRY 79 geometry, who does not know the real uses of geometry, and who has no faculty of making pupils enthusiastic over geometry, — that this teacher is to succeed with some scrappy, weak, pretending apology for a real work on the subject ? No one believes in stupid teaching, in memorizing a textbook, in having a book that does all the work for a pupil, or in any of the other ills of inefficient instruction. On the other hand, no fair-minded person can condemn a type of book that has stood for generations until some- thing besides the mere transient experiments of the moment has been suggested to replace it. Let us, for example, consider the question of having the basal prop- ositions proved in full, a feature that is so easy to con- demn as leading to memorizing. The argument in favor of a book with every basal proposition proved in full, or with most of them so proved, the rest having only suggestions for the proof, is that the pupil has before him standard forms exhibit- ing the best, most succinct, most clearly stated demon- strations that geometry contains. The demonstrations stand for the same thing that the type problems stand for in algebra, and are generally given in full in the same way. The argument against the plan is that it takes away the pupil's originality by doing all the work for him, allowing him to merely memorize the work. Now if all there is to geometry were in the basal propositions, this argument might hold, just as it would hold in algebra in case there were only those exercises that are solved in full. But just as this is not the case in algebra, the solved exercises standing as types or as bases for the pupil's real work, so the demonstrated proposition forms a relatively small part of geometry, standing as a type, 80 THE TEACHING OF GEOMETRY a basis for the more important part of the work. More- over, a pupil who uses a syllabus is exposed to a danger that should be considered, namely, that of dishonesty. Any textbook in geometry will furnish the proofs of most of the propositions in a syllabus, whatever changes there may be in the sequence, and it is not a healthy con- dition of mind that is induced by getting the proofs surreptitiously. Unless a teacher has more time for the course than is usually allowed, he cannot develop the new work as much as is necessary with only a syllabus, and the result is that a pupil gets more of his work from other books and has less time for exercises. The ques- tion therefore comes to this: Is it better to use a book containing standard forms of proof for the basal propo- sitions, and have time for solving a large number of original exefcises and for seeking the applications of geometry ? Or is it better to use a book that requires more time on the basal propositions, with the danger of dishonesty, and allows less time for solving originals ? To these questions the great majority of teachers answer in favor of the textbook with most of the basal proposi- tions fully demonstrated. In general, therefore, it is a good rule to use the proofs of the basal propositions as models, and to get the original work from the exercises. Unless we preserve these model proofs, or unless we supply them with a syllabus, the habit of correct, succinct self-expression, which is one of the chief assets of geom- etry, will tend to become atrophied. So important is this habit that " no system of education in which its perform- ance is neglected can hope or profess to evolve men and women who are competent in the full sense of the word. So long as teachers of geometry neglect the possibilities of the subject in this respect, so long will the time devoted THE TEXTBOOK IN GEOMETRY 81 to it be in large part wasted, and so long will their pupils continue to imbibe the vicious idea that it is much more important to be able to do a thing than to say how it can be done." ^ It is here that the chief danger of syllabus-teaching lies, and it is because of this patent fact that a syllabus without a carefully selected set of model proofs, or with- out the unnecessary expenditure of time by the class, is a dangerous kind of textbook. What shall then be said of those books that merely suggest the proofs, or that give a series of questions that lead to the demonstrations ? There is a certain plausi- bility about such a plan at first sight. But it is easily seen to have only a fictitious claim to educational value. In the first place, it is merely an attempt on the part of the book to take the place of the teacher and to "develop " every lesson by the heuristic method. The questions are so framed as to admit, in most cases, of only a single answer, so that this answer might just as well be given instead of the question. The pupil has therefore a proof requiring no more effort than is the case in the standard form of textbook, but not given in the clear language of a careful writer. Furthermore, the pupil is losing here, as when he uses only a syllabus, one of the very things that he should be acquu'ing, namely, the habit of reading mathematics. If he met only syllabi without proofs, or " suggestive " geometries, or books that endeavored to question every proof out of him, he would be in a sorry plight when he tried to read higher mathematics, or even other elementary treatises. It is for reasons such as these that the heuristic textbook has never succeeded for any great length of time or in any wide territory. 1 Carson, loc. cit., p. 15. 82 THE TEACHING OF GEOMETRY And finally, upon this point, shall the demonstrations be omitted entirely, leaving only the list of propositions, — ia other words, a pure syllabus ? This has been suffi- ciently answered above. But there is a modification of the pure syllabus that has much to commend itself to teachers of exceptional strength and with more confidence in themselves than is usually found. This is an arrange- ment that begins like the ordinary textbook and, after the pupil has acquired the form of proof, gradually merges into a syllabus, so that there is no temptation to go surreptitiously to other books for help. Such a book, if worked out with skill, would appeal to an enthusiastic teacher, and would accomplish the results claimed for the cruder forms of manual already described. It would not be in general as safe a book as the standard form, but with the right teacher it would bring good results. In conclusion, there are two types of textbook that have any hope of success. The first is the one with all or a large part of the basal propositions demonstrated in full, and with these propositions not unduly reduced in number. Such a book should give a large number of simple exercises scattered through the work, with a rel- atively small number of difficult ones. It should be mod- ern in its spirit, with figures systematically lettered, with each page a unit as far as possible, and with every proof a model of clearness of statement and neatness of form. Above all, it should not yield to the demand of a few who are always looking merely for something to change, nor should it in a reactionary spirit return to the old essay form of proof, which hinders the pupil at this stage. The second type is the semisyllabus, otherwise with all the spirit of the first type. In both there should be an honest fusion of pure and applied geometry, with no THE TEXTBOOK IN GEOMETRY 83 exercises that pretend, to be practical without being so, with no forced applications that lead the pupil to meas- ure things in a way that would appeal to no practical man, with no merely narrow range of applications, and with no array of difficult terms from physics and engi- neering that submerge all thought of mathematics in the slough of despond of an unknown technical vocabulary. Outdoor exercises, even if somewhat primitive, may be introduced, but it should be perfectly understood that such exercises are given for the purpose of increasing the interest in geometry, and they should be abandoned if they fail of this purpose. Bibliography. For a list of standard textbooks issued prior to the present generation, consult the bibliography in Stamper, His- tory of the Teaching of Geometry, New York, 1908. CHAPTER VIII THE RELATION OF ALGEBRA TO GEOMETRY From the standpoint of theory there is or need be no relation whatever between algebra and geometry. Alge- bra was originally the science of the equation, as its name ^ indicates. This means that it was the science of finding the value of an unknown quantity in a statement of equality. Later it came to mean much more than this, and Newton spoke of it as universal arithmetic, and wrote an algebra with this title. At present the term is applied to the elements of a science in which numbers are represented by letters and in which certain functions are studied, functions which it is not necessary to specify at this time. The work relates chiefly to functions involv- ing the idea of number. In geometry, on the other hand, the work relates chiefly to form. Indeed, in pure geom- etry number plays practically no part, while in pure algebra form plays practically no part. In 1637 the great French philosopher, Descartes, wish- ing to picture certain algebraic functions, wrote a work of about a hundred pages, entitled " La Geometric," and in this he showed a correspondence between the num- bers of algebra (which may be expressed by letters) and the concepts of geometry. This was the first great step in the analytic geometry that finally gave us the graph ^ Al-jabr waH-muqabalah : "restoration and equation" is a fairly good translation of the Arabic. 84 THE RELATION OF ALGEBRA TO GEOMETRY 85 ill algebra. Since then there have been brought out from time to time other analogies between algebra and geom- etry, always to the advantage of each science. This has led to a desire on the part of some teachers to unite algebra and geometry into one science, having simply a class in mathematics without these special names. It is well to consider the advantages and the disad- vantages of such a plan, and to decide as to the rational attitude to be taken by teachers concerning the question at issue. On the side of advantages it is claimed that there is economy of time and of energy. If a pupil is studying formulas, let the formulas of geometry be stud- ied ; if he is taking up ratio and proportion, let him do so for algebra and geometry at the same time ; if he is solving qu.adratics, let him apply them at once to certain propositions concerning secants ; and if he is proving that (a -t- by equals a^ + 'iah + b% let him do so by algebra and by geometry simultaneously. It is claimed that not only is there economy in this arrangement, but that the pupil sees mathematics as a whole, and thus acquires more of a mastery than comes by our present " tandem arrangement." ^ On the side of disadvantages it may be asked if the same arguments would not lead us to teach Latin and Greek together, or Latin and French, or all three simul- taneously ? If pupils should decline nouns in all three languages at the same time, learn to count in all at the same time, and begin to translate in all simultaneously, would there not be an economy of time and effort, and would there not be developed a much broader view of language ? Now the fusionist of algebra and geometry does not like this argument, and he says that the cases are not parallel, and he tries to tell why they are not. 86 THE TEACHING OF GEOMETRY He demands that his opponent abandon argument by analogy and advance some positive reason why algebra and geometry should not be fused. Then his opponent says that it is not for him to advance any reason for what already exists, the teaching of the two separately ; that he has only to refute the fusionist's arguments, and that he has done so. He asserts that algebra and geom- etry are as distinct as chemistry and biology ; that they have a few common points, but not enough to require teaching them together. He claims that to begin Latin and Greek at the same time has always proved to be confusing, and that the same is true of algebra and geometry. He grants that unified knowledge is desirable, but he argues that when the fine arts of music and color work fuse, and when the natural sciences of chemistry and physics are taught in the same class, and when we follow the declension of a German noun by that of a French noun and a Latin noun, and when we teach draw- ing and penmanship together, then it is well to talk of mixing algebra and geometry. It is well, before deciding such a question for our- selves (for evidently we cannot decide it for the world), to consider what has been the result of experience. Alge- bra and geometry were always taught together in early times, as were trigonometry and astronomy. The Ahmes papyrus contains both primitive algebra and primitive geometry. Euclid's " Elements " contains not only pure geometry, but also a geometric algebra and the theory of numbers. The early works of the Hindus often fused geometry and arithmetic, or geometry and algebra. Even the first great printed compendium of mathematics, the "Stima" of Paciuolo (1494) contained all of the branches of mathematics. Much of this later attempt was not, THE RELATION OF ALGEBRA TO GEOMETRY 87 however, an example of perfect fusion, but rather of as- signmg one set of chapters to algebra, another to geome- try, and another to arithmetic. So fusion, more or less perfect, has been tried over long periods, and abandoned as each subject grew more complete in itself, with its own language and its peculiar symbols. But it is asserted that fusion is being carried on suc- cessfully to-day by more than one enthusiastic teacher, and that this proves the contention that the plan is a good one. Books are cited to show that the arrangement is feasible, and classes are indicated where the work is progressing along this hne. What, then, is the conclusion ? That is a question for the teacher to settle, but it is one upon which a writer on the teaching of mathematics should not fear to express his candid opinion. It is a fact that the Greek and Latin fusion is a fair analogy. There are reasons for it, but there are many more against it, the chief one being the confusion of beginning two languages at once, and the learning simul- taneously of two vocabularies that must be kept sepa- rate. It is also a fact that algebra and geometry are fully as distinct as physics and chemistry, or chemistry and biology. Life may be electricity, and a brief cessa- tion of oxidization in the lungs brings death, but these facts are no reasons for fusing the sciences of physics, biology, and chemistry. Algebra is primarily a theory of certain elementary functions, a generalized arithmetic, while geometry is primarily a theory of form with a highly refined logic to be used in its mastery. They have a few things in common, as many other subjects have, but they have very many more features that are peculiar to the one or the other. The experience of the world has led 88 THE TEACHING OF GEOMETRY it away from a simultaneous treatment, and the contrary experience of a few enthusiastic teachers of to-day proves only their own powers to succeed with any method. It is easy to teach logarithms in the seventh school year, but it is not good policy to do so under present condi- tions. So the experience of the world is against the plan of strict fusion, and no arguments have as yet been advanced that are likely to change the world's view. No one has written a book combining algebra and geometry in this fashion that has helped the cause of fusion a particle ; on the contrary, every such work that has appeared has damaged that cause by showing how unsci- entific a result has come from the labor of an enthusiastic supporter of the movement. But there is one feature that has not been considered above, and that is a serious handicap to any effort at combining the two sciences in the high school, and this is the question of relative difficulty. It is sometimes said, in a doctrinaire fashion, that geometry is easier than algebra, since form is easier to grasp than function, and that therefore geometry should precede algebra. But every teacher of mathematics knows better than this. He knows that the simplest form is easier to grasp than the simplest function, but nevertheless that plane geom- etry, as we understand the term to-day, is much more difficult than elementary algebra for a pupil of fourteen. The child studies form in the kindergarten before he stud- ies number, and this is sound educational policy. He studies form, in mensuration, throughout his course in arithmetic, and this, too, is good educational policy. This kind of geometry very properly precedes algebra. But the demonstrations of geometry, the study by pupils of fourteen years of a geometry that was written for THE RELATION OF ALGEBRA TO GEOMETRY 89 college students and always studied by them until about fifty years ago, — that is by no means as easy as the study of a simple algebraic symbolism and its applica- tion to easy equations. If geometry is to be taught for the same reasons as at present, it cannot advantageously be taught earlier than now without much simplification, and it cannot successfully be fused with algebra save by some teacher who is willing to sacrifice an undue amount of energy to no really worthy purpose. When great mathematicians like Professor Klein speak of the fusion of all mathematics, they speak from the standpoint of advanced students, not for the teacher of elementary geometry. ' It is therefore probable that simple mensuration will continue, as a part of arithmetic, to precede algebra, as at present ; and that algebra into or through quadratics will precede geometry,^ drawing upon the mensuration of arithmetic as may be needed ; and that geometry will follow this part of algebra, using its principles as far as possible to assist in the demonstrations and to express and manipulate its formulas. Plane geometry, or else a year of plane and solid geometry, will probably, in this country, be followed by algebra, completmg quadratics and studying progressions; and by solid geometry, or a supplementary course in plane and solid geometry, this work being elective in many, if not all, schools.^ It is also probable that a general review of mathematics, where the fusion idea may be carried out, will prove to be a feature of the last year of the high school, and one 1 Or be carried along at the same time as a distinct topic. 2 With a single year for required geometry it would be better from every point of view to cut the plane geometry enough to admit a fair course in solid geometry. 90 THE TEACHING OF GEOMETRY that will grow in popularity as time goes on. Such a plan will keep algebra and geometry separate, but it will allow each to use all of the other that has preceded it, and will encourage every effort in this direction. It will accomplish all that a more complete fusion really hopes to accomplish, and it will give encouragement to all who seek to modernize the spirit of each of these great branches of mathematics. There is, however, a chance for fusion in two classes of school, neither of which is as yet well developed in this country. The first is the technical high school that is at present coming into some prominence. It is not prob- able even here that the best results can be secured by eliminating all mathematics save only what is applicable in the shop, but if this view should prevail for a time, there would be so little left of either algebra or geometry that each could readily be jouaed to the other. The ac- tual amount of algebra needed by a foreman in a machine shop can be taught m about four lessons, and the geom- etry or mensuration that he needs can be taught in eight lessons at the most. The necessary trigonometry may take eight more, so that it is entirely feasible to unite these three subjects. The boy who takes such a course would know as much about mathematics as a child who had read ten pages in a primer would know about literature, but he would have enough for his immediate needs, even though he had no appreciation of mathematics as a science. If any one asks if this is not all that the school should give him, it might be well to ask if the school should give only the ability to read, without the knowledge of any good literature ; if it should give only the ability to smg, without the knowledge of good music ; if it should give only the ability to speak, without any training in the use THE RELATION OF ALGEBRA TO GEOMETRY 91 of good language ; and if it should give a knowledge of home geography, without any intimation that the world is round, — an atom in the unfathomable universe about us. The second opportunity for fusion is possibly (for it is by no means certain) to be found in a type of school in which the only required courses are the initial ones. These schools have some strong advocates, it being claimed that every pupil should be introduced to the large branches of knowledge and then allowed to elect the ones in which he finds himself the most interested. Whether or not this is sound educational policy need not be discussed at this time; but if such a plan were developed, it might be well to offer a somewhat super- ficial (in the sense of abridged) course that should em- body a little of algebra, a little of geometry, and a little of trigonometry. This would unconsciously become a bait for students, and the result would probably be some good teaching in the class in question. It is to be hoped that we may have some strong, well-considered text- books upon this phase of the work. As to the fusion of trigonometry and plane geometry little need be said, because the subject is in the doctri- naire stage. Trigonometry naturally follows the chapter on similar triangles, but to put it there means, in our crowded curriculum, to eliminate something from geom- etry. Which, then, is better, — to give up the latter por- tion of geometry, or part of it at least, or to give up trigonometry ? Some advocates have entered a plea for two or three lessons in trigonometry at this point, and this is a feature that any teacher may introduce as a bit of interest, as is suggested in Chapter XVI, just a-s he may give a popular talk to his class upon the fourth dimension or the non-Euclidean geometry. The lasting 92 THE TEACHING- OF GEOMETRY impression upon the pupil will be exactly the same as that of four lessons in Sanskrit while he is studying Latin. He might remember each with pleasure, Latin being related, as it is, to Sanskrit, and trigonometry being an outcome of the theory of similar triangles. But that either of these departures from the regular sequence is of any serious mathematical or linguistic significance no one would feel like asserting. Each is allowable on the score of interest, but neither will add to the pupil's power in any essential feature. Each of these subjects is better taught by itself, each using the other as far as possible and being followed by a review that shall make use of all. It is not improbable that we may in due time have high schools that give less extended courses in algebra and geometry, adding brief practical courses in trigonometry and the elements of the calculus ; but even in such schools it is likely to be found that geometry is best taught by itself, making use of all the mathematics that has preceded it. It will of course be understood that the fusion of al- gebra and geometry as here understood has nothing to do with the question of teaching the two subjects simul- taneously, say two days in the week for one and three days for the other. This plan has many advocates, al- though on the whole it has not been well received in this country. But what is meant here is the actual fusing of algebra and geometry day after day, — a plan that has as yet met with only a sporadic success, but which may be developed for beginning classes in due time. CHAPTER IX THE INTRODUCTIOIf TO GEOMETRY There are two difficult crises in the geometry course, both for the pupil and for the teacher. These crises are met at the beginning of the subject and at the beginning of solid geometry. Once a class has fairly got into Book I, if the interest in the subject can be maintained, there are only the incidental difficulties of logical advance through- out the plane geometry. When the pupil who has been seeing figures in one plane for a year attempts to visual- ize solids from a flat drawing, the second difficult place is reached. Teachers going over solid geometry from year to year often forget this difficulty, but most of them can easily place themselves in the pupil's position by look- ing at the working drawings of any artisan, — usually sim- ple cases in the so-called descriptive geometry. They will then realize how difficult it is to visualize a solid from an unfamiliar kind of picture. The trouble is usually avoided by the help of a couple of pieces of heavy card- board or box board, and a few knitting needles with which to represent lines in space. If these are judiciously used in class for a few days, until the figures are under- stood, the second crisis is easily passed. The continued use of such material, however, or the daily use of either models or photographs, weakens the pupil, even as a child is weakened by being kept too long in a perambu- lator. Such devices have their place ; they are useful 94 THE TEACHING OF GEOMETRY when needed, but they are pernicious when unnecessary. Just as the mechanic must be able to make and to vis- ualize his working drawings, so the student of solid geometry must be able to get on with pencil and paper, representing his solid figures in the flat. But the introduction to plane geometry is not so easily disposed of. The pupil at that time is entering a field that is entirely unfamiliar. He is only fourteen or fifteen years of age, and his thoughts are distinctly not on geom- etry. Of logic he knows little and cares less. He is not interested in a subject of which he knows nothing, not even the meanmg of its name. He asks, naturally and properly, what it all signifies, what possible use there is for studying geometry, and why he should have to prove what seems to him evident without proof. To pass him successfully through this stage has taxed the ingenuity of every real teacher from the time of Euclid to the present ; and just as Euclid remarked to King Ptolemy, his patron, that there is no royal road to geometry, so we may affirm that there is no royal road to the teachi:ig of geometry. Nevertheless the experience of teachers counts for a great deal, and this experience has shown that, aside from the matter of technic in handling the class, certain sugges- tions are of value, and a few of these will now be set forth. First, as to why geometry is studied, it is manifestly impossible successfully to explain to a boy of fourteen or fifteen the larger reasons for studying anything what- ever. When we confess ourselves honestly we find that these reasons, whether in mathematics, the natural sci- ences, handwork, letters, the vocations, or the fine arts, are none too clear in our own minds, in spite of any pre- tentious language that we may use. It is therefore most satisfactory to anticipate the question at once, and to set THE INTRODUCTION TO GEOMETRY 95 the pupils, for a few days, at using the compasses and ruler in the drawing of geometric designs and of the most common figures that they will use. This serves several purposes : it excites their interest, it guards against the slovenly figures that so often lead them to erroneous conclusions, it has a genuine value for the future artisan, and it shows that geometry is something besides mere theory. Whether the textbook provides for it or not, the teacher will find a few days of such work well spent, it being a simple matter to supplement the book in this respect. There was a time when some form of mechanical drawing was generally taught in the schools, but this has given place to more genuine art work, leaving it to the teacher of geometry to impart such knowledge of drawing as is a necessary prelimi- nary to the regular study of the subject. Such work in drawing should go so far, and only so far, as to arouse an interest in geometric form without becoming wearisome, and to familiarize the pupil with the use of the instruments. He should be counseled about making fine lines, about being careful in setting the point of his compasses on the exact center that he wishes to use, and about representing a point by a very fine dot, or, preferably at first, by two crossed lines. Unless these details are carefully considered, the pupil will soon find that the lines of his drawings do not fit together, and that the result is not pleasing to the eye. The figures here given are good ones upon which to begin, the dotted construction lines being erased after the work is completed. They may be constructed with the compasses and ruler alone, or the draftsman's T-square, triangle, and protractor may be used, although these latter instruments are not necessary. We should 96 THE TEACHING OF GEOMETRY constantly remember that there is a danger in the slav- ish use of instruments and of such helps as squared paper. Just as Euclid rode roughshod over the growing intellects of boys and girls, so may instruments ride roughshod over their growing perceptions by interfering with natural and healthy intui- tions, and making them the subject of laborious measurement.^ The pupil who cannot see the equality of vertical angles intuitively better than by the use of the protractor is abnormal. Nevertheless it is the pupil's interest that is at stake, together with his ability to use the instru- ments of daily life. If, therefore, he can readily be 1 Carson, loo. cit., p. 13. THE INTRODUCTION TO GEOMETRY 97 supplied with draftsmen's materials, and is not com- pelled to use them in a foolish manner, so much the better. They will not hurt his geometry if the teacher does not interfere, and they will help his practical draw- ing ; but for obvious reasons we cannot demand that the pupil purchase what is not really essential to his study of the subject. The most valuable single instrument of the three just mentioned is the protractor, and since a paper one costs only a few cents and is often helpful m the drawing of figures, it should be recommended to pupils. There is also another line of work that often arouses a good deal of interest, namely, the simple field meas- ures that can easily be made about the school grounds. Guarding against the ever-present danger of doing too much of such work, of doing work that has no interest for the pupil, of requiring it done in a way that seems unreal to a class, and of neglecting the essence of geom- etry by a line of work that involves no new principles, — such outdoor exercises in measurement have a posi- tive value, and a plentiful supply of suggestions in this line is given in the subsequent chapters. The object is chiefly to furnish a motive for geometry, and for many pupils this is quite unnecessary. For some, however, and particularly for the energetic, restless boy, such work has been successfully offered by various teachers as an alternative to some of the book work. Because of this value a considerable amount of such work will be suggested for teachers who may care to use it, the textbook being manifestly not the place for occasional topics of this nature. For the purposes of an introduction only a tape line need be purchased. Wooden pins and a plumb line can 98 THE TEACHING OF GEOMETRY easily be made. Even before he comes to the proposi- tions in mensuration in geometry the pupil knows, from his arithmetic, how to find ordinary areas and volumes, and he may therefore be set at work to find the area of the school ground, or of a field, or of a city block. The following are among the simple exercises for a beginner : 1. Drive stakes at two corners, A and S, of the school grounds, putting a cross on top of each; or make the crosses on the sidewalk, so as to get two points between which to measure. Measure from ^ to 5 by holding the tape taut and level, dropping perpendiculars when nec- essary by means of the plumb line, as shown in the figure. Check the work by measuring from £ back to A in the same way. Pupils will find that their work should always be checked, and they will be sur- prised to see how the results will vary in such a simple measurement as this, unless very great care is taken. If they learn the lesson of accuracy thus early, they will have gained much. 2. Take two stakes, X, Y, in a field, preferably two or three hundred feet apart, always marked on top with crosses so as to have exact points from which to work. Let it then be required to stake out or " range " the line from X to F by plac- , , ^ ing stakes at specified ^ P Q B Y distances. One boy stands at ¥ and another at X, each with a plumb line. A third one takes a plumb line and stands at P, the observer at X motioning to him to move his plumb line to the right or the left until it is exactly in line with X and Y. A stake is then driven at P, and the pupil at X moves on to the stake P. Then THE INTRODUCTION TO GEOMETRY 99 Q is located in the same way, and then J?, and so on. The work is checked by ranging back from Y to X. In some of tlie simple exercises suggested later it is neces- sary to range a line so that this work is useful in making measurements. The geometric principle involved is that two points determine a straight line. 3. To test a perpendicular or to draw one line perpendicular to another in a field, we may take a stout cord twelve feet long, having a knot at the end of every foot. If this is laid along four feet, the ends of this part being fixed, and it is stretched as here shown, so that the next vertex is five feet from one of these ends and three feet from the other end, a right angle will be formed. A right angle can also be run by making a simple instru- ment, such as is described in Chapter XV. Still another plan of drawing a line per- pendicular to another line AB, from a point P, consists in swinging a tape from P, cutting AB at X and I', and then bisecting XY by doubling the tape. This fixes the foot of the perpendicular. 4. It is now possible to find the area of a field of irregu- lar shape by dividing it into triangles and trapezoids, as shown in the figure. Pupils know from their work in arithmetic how to find the area of a triangle or a trapezoid, so that the area of the field is easily found. The work may be checked by comparing XX 100 THE TEACHING OF GEOMETRY the results of different groups of pupils, or by drawing another diagonal and dividing the field into other tri- angles and trapezoids. These are about as many types of field work as there is any advantage in undertaking for the purpose of secur- ing the interest of pupils as a preliminary to the work in geometry. Whether any of it is necessary, and for what pupils it is necessary, and how much it should trespass upon the time of scientific geometry are matters that can be decided only by the teacher of a particular class. A second difficulty of the pupil is seen in his attitude of mind towards proofs in general. He does not see why vertical angles should be proved equal when he knows that they are so by looking at the figure. This difficulty should also be anticipated by giving him some opportu- nity to know the weakness of his judg- ^x ,^ 5 and if x = y, then a + x>b + y, a — x>h — y, ax>hy, etc. The reason for thus combining six axioms in one in the case of inequalities is apparent. They are rarely used in geometry, and if a teacher is in doubt as to the pupil's knowledge, he can easily inquire in the few cases that arise, whereas it would consume a great deal of time to do this for the many equations that are met. The axiom is stated in such a way as to exclude multiplying or dividing by negative numbers, this case not being needed. 7. If unequals are added to unequals in the same order, the sums are unequal in the same order; if unequals are subtracted from equals, the remainders are unequal in the reverse order. These are the only cases in which unequals are necessarily combined with unequals, or operate upon equals in geometry, and the axiom is easily explained to the class by the use of numbers. 8. Quantities that are equal to the same quantity or to equal quantities are equal to each other. In this axiom the word " quantity " is used, in the common manner of the present time, to include number and all geometric mag- nitudes (length, area, volume). 124 THE TEACHING OF GEOMETRY 9. A quantity may he substituted for its equal in an equation or in an inequality. This axiom is tacitly assumed by all writers, and is very useful in the proofs of geom- etry. It is really the basis of several other axioms, and if we were seeking the " irreducible minimum," it would replace them. Since, however, we are seeking only a reasonably abridged list of convenient assumptions that beginners will understand and use, this axiom has much to commend it. If we consider the equations (1) a = x and (2) b = x,we see that for x in equation (1) we may substitute b from equation (2) and have a = b; in other words, that Axiom 8 is included in Axiom 9. Further- more, if (1) a=b and (2) x = y, then since a -|- a; is the same as a + x, we may, by substituting, say that a + x = a + x = b+x = b + y. In other words, Axiom 1 is included in Axiom 9. Thus an axiom that includes others has a legitimate place, because a beginner would be too much confused by seeing its entire scope, and because he will make frequent use of it in his mathe- matical work. 10. If tJie first of three quantifies is greater than the second, and the second is greater than the third, then the first is greater than the third. This axiom is needed sev- eral times in geometry. The case in which a>b and b = c, therefore a> c, is provided for in Axiom 9. 11. The whole is greater than any of its parts and is equal to the sum of all its parts. The latter part of this axiom is really only the definition of "whole," and it would be legitimate to state a definition accordingly and refer to it where the word is employed. Where, how- ever, we wish to speak of a polygon, for example, and wish to say that the area is equal to the combined areas of the triangles composing it, it is more satisfactory to THE AXIOMS AND POSTULATES 125 have this axiom to which to refer. It will be noticed that two related axioms are here combined in one, for a reason similar to the one stated under Axiom 5. In the case of the postulates we are met by a problem similar to the one confronting us in connection with the axioms, — the problem of the " irreducible minimum " as related to the question of teaching. Manifestly Euclid used postulates that he did not state, and proved some statements that he might have postulated. ^ The postulates given by Euclid under the name a'iTriixara (aitemata) were requests made by the teacher to his pupil that certain things be conceded. They were five in number, as follows : 1. Let the following he conceded : to draw a straight line from any point to any point. Strictly speaking, Euclid might have been required to postulate that points and straight lines exist, but he evi- dently considered this statement sufficient. Aristotle had, however, already called attention to the fact that a mere definition was .sufficient only to show what a con- cept is, and that this must be followed by a proof that the thing exists. We might, for example, define a; as a line that bisects an angle without meeting the vertex, but this would not show that an x exists, and indeed it does not exist. Euclid evidently intended the postulate to assert that this line joining two points is unique, which is only another way of saying that two points determine a straight line, and really includes the idea 1 Tor example, he moves figures without deformation, but states no postulate on the subject ; and he proves that one side of a triangle is less than the sum of the other two sides, when he might have postu- lated that a straight line is the shortest path between two points. Indeed, his followers were laughed at for proving a fact so obvious as this one concerning the triangle. 126 THE TEACHING OF GEOMETRY that two straight lines cannot inclose space. For pur- poses of instruction, the postulate would be clearer if it read, One straight line, and only one, can he drawn through two given points. 2. To produce a finite straight line continuously in a straight line. In this postulate Euclid practically assumes that a straight line can be produced only in a straight line ; in other words, that two different straight lines cannot have a commoia segment. Several attempts have been made to prove this fact, but without any marked success. 3. To describe a circle with any center and radius. 4. That all right angles are equal to one another. While this postulate asserts the essential truth that a right angle is a determinate magnitude so that it really serves as an inva- riable standard by which other (acute and obtuse) angles may be measured, much more than this is implied, as will easily be seen from the following consideration- If the statement is to be proved, it can only be proved by the method of applying one pair of right angles to another and so arguing their equality. But this method would not be valid unless on the assumption of the invariability of figures, which would have to be asserted as an antecedent postu- late. Euclid preferred to assert as a postulate, directly, the fact that all right angles are equal ; and hence his postulate must be taken as equivalent to the principle of invariability of figures, or its equivalent, the homogeneity of space.^ It is better educational policy, however, to assert this fact more definitely, and to state the additional assump- tion that figures may be moved about in space without deformation. The fourth, of Euclid's postulates is often given as an axiom, following the idea of the Greek philosopher Geminus (who flourished in the first century B.C.), but this is becavise Euclid's distinction between IT. L. Heath, "Euclid," Vol. I, p. 200. THE AXIOMS AND POSTULATES 127 axiom and postulate is not always understood. Proclus (410-485 A.D.) endeavored to prove the postulate, and a later and more scientific effort was made by the Ital- ian geometrician Saccheri (1667-1733). It is very com- monly taken as a postulate that all straight angles are equal, this being more evident to the senses, and the equality of right angles is deduced as a corollary. This method of procedure has the sanction of many of our best modern scholars. 5. That, if a straight line falling on ttvo straight lines make the interior angle on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. This famous postulate, long since abandoned in teach- ing the beginner in geometry, is a remarkable evidence of the clear vision of Euclid. For two thousand years math- ematicians sought to prove it, only to demonstrate the wisdom of its author in placing it among the assump- tions. ^ Every proof adduced contains some assumption that practically conceals the postulate itself. Thus the great English mathematician John Wallis (1616-1703) gave a proof based upon the assumption that "given a figure, another figure is possible which is similar to the given one, and of any size whatever." Legendre (1752- 1833) did substantially the same at one time, and offered several other proofs, each depending upon some equally unprovable assumption. The definite jDroof that the postulate cannot be demonstrated is due to the Italian Beltrami (1868). 1 For a r6sum6 of the best known attempts to prove this postulate, see Heath, "Euclid," Vol. I, p. 202 ; W. B. Prankland, "Theories of Parallelism," Cambridge, 1910. 128 THE TEACHING OF GEOMETRY Of the alternative forms of the postulate, that of Proclus is generally considered the best suited to begin- ners. As stated by Playfair (1795), this is, " Through a given point only one parallel can be drawn to a given straight line"; and as stated by Proclus, "If a straight line intersect one of two parallels, it will intersect the other also." Playf air's form is now the common "pos- tulate of parallels," and is the one that seems destined to endure. Posidonius and Geminus, both Stoics of the iirst cen- tury B.C., gave as their alternative, " There exist straight lines everywhere equidistant from one another." One of Legendre's alternatives is, "There exists a triangle in which the sum of the three angles is equal to two right angles." One of the latest attempts to suggest a substitute is that of the Italian Ingrami (1904), " Two parallel straight lines intercept, on every transversal which passes tlirough the mid-pomt of a segment included between them, another segment the mid-point of which is the mid-point of the first." Of course it is entirely possible to assume that through a point more than one line can be drawn parallel to a given straight line, in which case another type of geom- etry can be built up, equally rigorous with Euclid's. This was done at the close of the first quarter of the nineteenth century by Lobachevsky (1793-1856) and Bolyai (1802-1860), resulting in the first of several "non-Euclidean" geometries. ^ 1 For the early history of this movement see Engel and Staokel, "Die Theorie der Parallellinien von EuWid bis auf Gauss," Leipzig, 1895 ; Bonola, Sulla teoria delle parallels e sulle geometrie non- euolidee, in his " Question! riguardanti la geometria elementare," 1900 ; Karagiannides, " Die niohteuklidisohe Geometrie vom Alter- thum bis zur Gegenwart," Berlin, 1893. THE AXIOMS AND POSTULATES 129 Taking the problem to be that of statmg a reasonably small number of geometric assumptions that may form a basis to supplement the general axioms, that shall cover the most important matters to which the student must refer, and that shall be so simple as easily to be under- stood by a beginner, the following are recommended : 1. One straight line, and only one, can he drawn through two given joints. This should also be stated for conven- ience in the form, Two points determine a straight line. From it may also be drawn this corollary. Two straight lines can intersect in only one point, since two points would determine a straight line. Such obvious restate- ments of or corollaries to a postulate are to be com- mended, since a beginner is often discouraged by having to prove what is so obvious that a demonstration fails to commend itself to his mind. 2. A straight line may he produced to any required length. This, like Postulate 1, requires the use of a straightedge for drawing the physical figure. The required length is attained by using the compasses to measure the distance. The straightedge and the compasses are the only two drawing instruments recognized in. elementary geometry. ^ While this involves more than Euclid's postulate, it is a better working assumption for beginners. 3. A straight line is the shortest path hetween two points. This is easily proved by the method of Euclid ^ for the case where the paths are broken lines, but it is needed as a postulate for the case of curve paths. It is a better statement than the common one that a straight line is the shortest distance between two points; for distance is 1 This limitation upon elementary geometry was placed by Plato (died 347 e.g.), as already stated. 2 Book I, Proposition 20. 130 THE TEACHING OF GEOMETRY measured on a line, but it is not itself a line. Further- more, there are scientific objections to using the word " distance " any more than is necessary. 4. A circle may he described with any given point as a center and any given line as a radius. This involves the use of the second of the two geometric instruments, the compasses. 5. Any figure may be moved from one place to another without altering the size or shape. This is the postulate of the homogeneity of space, and asserts that space is such that we may move a figure as we please without deformation of any kind. It is the basis of all cases of superposition. 6. All straight angles are equal. It is possible to prove this, and therefore, from the standpoint of strict logic, it is unnecessary as a postulate. On the other hand, it is poor educational policy for a beginner to attempt to prove a thing that is so obvious. The attempt leads to a loss of interest in the subject, the proposition being (to state a paradox) hard because it is so easy. It is, of course, possible to postulate that straight angles are equal, and to draw the conclusion that their halves (right angles) are equal ; or to proceed in the opposite direction, and postulate that all right angles are equal, and draw the conclusion that their doubles (straight angles) are equal. Of the two the former has the advantage, since it is probably more obvious that all straight angles are equal. It is well to state the following definite corol- laries to this postulate : (1) All right angles are equal ; (2) From a point in a line only one perpendicular can be drawn to the line, since two perpendiculars would make the whole (right angle) equal to its part; (3) Equal' angles have equal complements, equal supplements, and equal THE AXIOMS AND POSTULATES 131 conjugates ; (4) The greater of two angles has the less com- plement, the less supplemerit, and the less conjugate. All of these four might appear as propositions, but, as already stated, they are so obvious as to be more harmful than useful to beginners when given in such form. The postulate of parallels may properly appear in con- nection with that topic in Book I, and it is accordingly treated in Chapter XIV. There is also another assumption that some writers are now trying to formulate in a simple fashion. We take, for example, a line segment AB, and describe cir- cles with A and B respectively as centers, and with a radius AB. We say that the circles will intersect as at C and D. But how do we know that they intersect ? We assume it, just as we assume that an indefinite straight line drawn from a point inclosed by a circle will, if pro- duced far enough, cut the circle twice. Of course a pupil would not think of this if his attention was not called to it, and the harm outweighs the good in doing this with one who is beginning the study of geometry. With axioms and with postulates, therefore, the con- clusion is the same : from the standpoint of scientific geometry there is an irreducible mmimum of assump- tions, but from the standpoint of practical teaching this list should give place to a working set of axioms and postulates that meet the needs of the beginner. Bibliography. Smith, Teaching of Elementary Mathematics, New York, 1900 ; Young, The Teaching of Mathematics, New York, 1901 ; Moore, On the Foundations of Mathematics, Bulletin of the American Mathematical Society, 1903, p. 402 ; Betz, Intuition and Logic in Geometry, The Mathematics Teacher, Vol. II, p. 3 ; Hilbert, The Foundations of Geometry, Chicago, 1902 ; Veblen, A System of Axioms for Geometry, Transactions of the American Mathematical Society, 1904, p. 343. CHAPTER XII THE DEFINITIONS OF GEOMETRY When we consider the nature of geometry it is evident that more attention must be paid to accuracy of defini- tions than is the case in most of the other sciences. The essence of all geometry worthy of serious study is not the knowledge of some fact, but the proof of that fact; and this proof is always based upon preceding proofs, assumptions (axioms or postulates), or definitions. If we are to prove that one line is perpendicular to another, it is essential that we have an exact definition of " per- pendicular," else we shall not know when we have reached the conclusion of the proof. The essential features of a definition are that the term defined shall be described in terms that are simpler than, or at least better known than, the thing itself ; that this shall be done in such a way as to limit the term to the thing defined; and that the description shall not be redundant. It would not be a good definition to say that a right angle is one fourth of a perigon and one half of a straight angle, because the concept " perigon" is not so simple, and the term "perigon" is not so well known, as the term and the concept " right angle," and because the definition is redundant, containing more than is necessary. It is evident that satisfactory definitions are not always possible ; for since the number of terms is limited, there must be at least one that is at least as simple as any 132 THE DEFINITIONS OF GEOMETRY 133 other, and this cannot be described in terms simpler than itself. Such, for example, is the term " angle." We can easily explain the meaning of this word, and we can make the concept clear, but this must be done by a cer- tain amount of circumlocution and explanation, not by a concise and perfect definition. Unless a beginner in geometry knows what an angle is before he reads the definition in a textbook, he will not know from the defi- nition. This fact of the impossibility of defining some of the fundamental concepts will be evident when we come to consider certain attempts that have been made in this direction. It should also be understood in this connection that a definition makes no assertion as to the existence of the thing defined. If we say that a tangent to a circle is an unlimited straight line that touches the circle in one point, and only one, we do not assert that it is possible to have such a line ; that is a matter for proof. Not in all cases, however, can this proof be given, as in the existence of the simplest concepts. We cannot, for exam- ple, prove that a point or a straight line exists after we have defined these concepts. We therefore tacitly or explicitly assume (postulate) the existence of these fundamentals of geometry. On the other hand, we can prove that a tangent exists, and this may properly be considered a legitimate proposition or corollary of ele- mentary geometry. In relation to geometric proof it is necessary to bear in mind, therefore, that we are per- mitted to define any term we please ; for example, "a seven-edged polyhedron" or Leibnitz's "ten-faced regular polyhedron," neither of which exists ; but, strictly speak- ing, we have no right to make use of a definition in a proof until we have shown or postulated that the thing defined 134 THE TEACHING OF GEOMETRY has an existence. This is one of the strong features of Euclid's textbook. Not being able to prove that a jDoint, a straight line, and a circle exists, he practically postu- lates these facts ; but he uses no other definition in a proof without showing that the thing defined exists, and tliis is his reason for mingling his problems with his theorems. At the present time we confessedly sacrifice his logic in this respect for the reason that we teach geometry to pupils who are too young to appreciate that logic. It was pointed out by Aristotle, long before Euclid, that it is not a satisfactory procedure to define a thmg by means of terms that are strictly not prior to it, as when we attempt to define somethuag by means of its opposite. Thus to define a curve as "a line, no part of which is straight," would be a bad definition unless " straight " had already been explicitly defined ; and to define " bad " as " not good " is unsatisfactory for the reason that " bad " and " good " are concepts that are evolved simultaneously. But all this is only a detail under the general principle that a definition must employ terms that are better understood than the one defined. It should be understood that some definitions are much more important than others, considered from the point of view of the logic of geometry. Those that enter into geometric proofs are basal ; those that form part of the conversational language of geometry are not. Euclid gave twenty-three definitions in Book I, and did not make use of even all of these terms. Other terms, those not employed in his proofs, he assumed to be known, just as he assumed a knowledge of any other words in his language. Such procedure would not be satisfactory under modern conditions, but it is of great importance THE DEFINITIONS OF GEOMETRY 135 that the teacher should recognize that certain definitions are basal, while others are merely informational. It is now proposed to consider the basal definitions of geometry, first, that the teacher may know what ones are to be emphasized and learned; and second, that he may know that the idea that the standard definitions can easily be improved is incorrect. It is hoped that the result will be the bringing into prominence of the basal concepts, and the discouraging of attempts to change in unimportant respects the definitions in the textbook used by the pupil. In order to have a systematic basis for work, the defi- nitions of two books of Euclid will first be considered.^ 1. Point. A point is that which has no part. This was incorrectly translated by Capella in the fifth century, " Punctum est cuius pars nihil est " (a point is that of which a part is nothing), which is as much as to say that the point itself is nothing. It generally appears, however, as in the Campanus edition,^ "Functus est cuius pars non est," which is substantially Euclid's word- ing. Aristotle tells of the definitions of point, line, and surface that prevailed in his time, saying that they all defined the prior by means of the posterior.^ Thus a point was defined as " an extremity of a line," a line as " the extremity of a surface," and a surface as "the extremity of iFree use has been made of "W". B. Frankland, "The First Book of Euclid's ' Elements,' " Cambridge, 1905 ; T. L. Heath, " The Thir- teen Books of Euclid's 'Elements,' " Cambridge, 1908 ; H. Schotten, "Inhalt und Methode des planimetrischen Unterrichts," Leipzig, 1893; M. Simon, "Euclid und die sechs planimetrischen Bticher," Leipzig, 1901. 2 For a facsimile of a thirteenth-century MS. containing this defi- nition, see the author's " Rara Arithmetica," Plate IV, Boston, 1909. 8 Our slang expression " The cart before the horse " is suggestive of this procedure. 136 THE TEACHING OF GEOMETRY a solid," — definitions still in use and not without their value. For it must not be assumed that scientific priority is necessarily priority in fact ; a child knows of " solid " before he knows of "point," so that it may be a very good way to explain, if not to define, by beginning with solid, passing thence to surface, thence to line, and thence to point. The first definition of point of which Proclus could learn is attributed by him to the Pythagoreans, namely, " a monad having position," the early form of our pres- ent popular definition of a point as "position without magnitude." Plato defined it as "the beginning of a line," thus presupposing the definition of "line"; and, strangely enough, he anticipated by two thousand years Cavalieri, the Italian geometer, by speaking of points as "indivisible lines." To Aristotle, who protested against Plato's defi- nitions, is due the definition of a point as " something indivisible but having position." Euclid's definition is essentially that of Aristotle, and is followed by most modern textbook writers, except as to its omission of the reference to position. It has been criticized as being negative, "which has no part"; but it is generally admitted that a negative definition is admissible in the case of the most elementary concepts. For example, "blind" must be defined in terms of a negation. At present not much attention is given to the defini- tion of " point," since the term is not used as the basis of a proof, but every effort is made to have the con- cept clear. It is the custom to start from a small solid, conceive it to decrease in size, and think of the point as the limit to which it is approaching, using these terms in their usual sense without further explanation. THE DEFINITIONS OF GEOMETRY 137 2. Line. A line is breadthless length. This is usually modified in modern textbooks by saying that " a line is that which has length without breadth or thickness," a statement that is better understood by beginners. Euclid's definition is thought to be due to Plato, and is only one of many definitions that have been suggested. The Pythagoreans having spoken of the pomt as a monad naturally were led to speak of the line as dyadic, or related to two. Proclus speaks of another definition, "magnitude in one dimension," and he gives an excel- lent illustration of line as "the edge of a shadow," thus making it real but not material. Aristotle speaks of a line as a magnitude " divisible in one way only," as con- trasted with a surface which is divisible in two ways, and with a solid which is divisible m three ways. Proclus also gives another definition as the "flux of a point," which is sometimes rendered as the path of a moving point. Aristotle had suggested the idea when he wrote, " They say that a line by its motion produces a surface, and a point by its motion a line." Euclid did not deem it necessary to attempt a classi- fication of lines, contenting himself with defining only a straight line and a circle, and these are really the only lines needed in elementary geometry. His commenta- tors, however, made the attempt. For example. Heron (first century A.D.) probably followed his definition of line by this classification : r straight Lines .j r Circular circumferences L Not straight I Spiral shaped L Curved (generally) Proclus relates that both Plato and Aristotle divided lines into " straight," " circular," and " a mixture of the 138 THE TEACHING OF GEOMETRY two," a statement which is not quite exact, but which shows the origin of a classification not infrequently found in recent textbooks. Geminus {ca. 50 B.C.) is said by Proclus to have given two classifications, of which one will suffice for our purposes : r Composite (broken line forming an angle) I r Forming a figure, or determinate. (Circle, ellipse, cissoid.) Not forming a iigure, or indeterminate and extending without a limit. (Straight line, parabola, hyperbola, conchoid.) Lines ^ Incomposite Of course his view of the cissoid, the curve represented by the equation 'if {a-\-oe) = (a — a;)', is not the modern view. 3. Tlie extremities of a line are points. This is not a definition in the sense of its two predecessors. A modern writer would put it as a note under the defini- tion of line. Euclid did not wish to define a point as the extremity of a line, for Aristotle had asserted that this was not scientific ; so he defined point and line, and then added this statement to show the relation of one to the other. Aristotle had improved upon this by stating that the " division " of a line, as well as an extremity, is a point, as is also the intersection of two lines. These statements, if they had been made by Euclid, would have avoided the objection made by Proclus, that some Imes have no extremities, as, for example, a circle, and also a straight line extending infinitely in both directions. 4. Straight Line. A straight line is that which lies evenly with respect to the points on itself. This is the least satisfactory of all of the definitions of Euclid, and em- phasizes the fact that the straight line is the most diffi- cult to define of the elementary concepts of geometry. THE DEFINITIONS OF GEOMETRY 139 What is meant by " lies evenly " ? Who would know what a straight line is, from this definition, if he did not know in advance ? The ancients suggested many definitions of straight line, and it is well to consider a few in order to ap- preciate the difficulties involved. Plato spoke of it as "that of which the middle covers the ends," meaning that if looked at endways, the middle would make it impossible to see the remote end. This is often modified to read that " a straight line when looked at endways appears as a point," — an idea that involves the postulate that our line of sight is straight. Archimedes made the statement that " of all the lines which have the same extremities, the straight line is the least," and this has been modified by later writers into the statement that " a straight line is the shortest distance between two points." This is open to two objections as a definition: (1) a line is not distance, but distance is the length of a line, — it is measured on a line ; (2) it is merely stating a property of a straight line to say that " a straight line is the shortest path between two points," — a proper pos- tulate but not a good definition. Equally objectionable is one of the definitions suggested by both Heron and Proclus, that "a straight line is a line that is stretched to its uttermost"; for even then it is reasonable to think of it as a catenary, although Proclus doubtless had in mind the Archimedes statement. He also stated that " a straight line is a line such that if any part of it is in a plane, the whole of it is in the plane," — a definition that runs in a circle, since plane is defined by means of straight line. Proclus also defines it as "a uniform line, capable of sliding along itself," but this is also true of a circle. 140 THE TEACHING OF GEOMETRY Of the various definitions two of the best go back to Heron, about the beginning of our era. Proclus gives one of them in this form, "That line which, when its ends remain fixed, itself remains fixed." Heron proposed to add, " when it is, as it were, turned round in the same plane." This has been modified into "that which does not change its position when it is turned about its ex- tremities as poles," and appears m substantially this form in the works of Leibnitz and Gauss. The defini- tion of a straight line as "such a line as, with another straight Ime, does not inclose space," is only a modifica- tion of this one.. The other definition of Heron states that in a straight line " all its parts fit on all in all ways," and this in its modern form is perhaps the most satisfactory of all. In this modern form it may be stated, " A line such that any part, placed with its ends on any other part, must lie wholly in the line, is called a straight line," in which the force of the word " must " should be noted. This whole historical discussion goes to show how futile it is to attempt to define a straight line. What is needed is that we should explain what is meant by a straight line, that we should illustrate it, and that pupils should then read the definition understandingly. 5. SuEFACB. A surface is that which has length and breadth. This is substantially the common definition of our modern textbooks. As with line, so with surface, the definition is not entirely satisfactory, and the chief consideration is that the meaning of the term should be made clear by explanations and illustrations. The shadow cast on a table top is a good illustration, since all idea of thickness is wanting. It adds to the understanding of the concept to introduce Aristotle's statement that a sur- face is generated by a moving line, modified by saying THE DEFINITIONS OF GEOMETRY 141 that it may be so generated, since the line might shcle along its own trace, or, as is commonly said in mathe- matics, along itself. 6. The extremities of a surface are lines. This is open to the same explanation and objection as definition 3, and is not usually given in modern textbooks. Proclus calls attention to the fact that the statement is hardly true for a complete spherical surface. 7. Plane. A -plane surface is a surface which lies evenly with the straight lines on itself. Euclid here fol- lows his definition of straight line, with a result that is equally unsatisfactory. For teaching purposes the trans- lation from the Greek is not clear to a beginner, since " lies evenly " is a term not simpler than the one defined. As with the definition of a straight line, so with that of a plane, numerous efforts at improvement have been made. Proclus, following a hint of Heron's, defines it as "the surface which is stretched to the utmost," and also, this time influenced by Archimedes's assumption concerning a straight line, as "the least surface among all those which have the same extremities." Heron gave one of the best definitions, "A surface all the parts of which have the property of fitting on [each other]." The definition that has met with the widest acceptance, how- ever, is a modification of one due to Proclus, " A surface such that a straight line fits on all parts of it." Proclus elsewhere says, "[A plane surface is] such that the straight line fits on it all ways," and Heron gives it in this form, " [A plane surface is] such that, if a straight line pass through two points on it, the line coincides with it at every spot, all ways." In modern form this appears as follows : " A surface such that a straight line joining any two of its points lies wholly in the surface is called a 142 THE TEACHING OF GEOMETRY plane," and for teaching purposes we have no better defi- nition. It is often known as Simson's definition, having , been given by Robert Simson in 1756. The French mathematician, Fourier, proposed to define a plane as formed by the aggregate of all the straight lines which, passing through one point on a straight line ill space, are perpendicular to that line. This is clear, but it is not so usable for beginners as Simson's defini- tion. It appears as a theorem in many recent geometries. The German mathematician, Crelle, defined a plane as a surface containing all the straight lines (throughout their whole length) passing through a fixed point and also intersectmg a straight line in space, but of course this intersected straight line must not pass through the fixed point. Crelle's definition is occasionally seen in modern textbooks, but it is not so clear to the pupil as Simson's. Of the various ultrascientiflc definitions of a plane that have been suggested of late it is hardly of use to speak in a book concerned primarily with practical teaching. No one of them is adapted to the needs and the com- prehension of the beginner, and it seems that we are not likely to improve upon the so-called Simson form. 8. Plane Angle. A plane angle is the inclination to each other of two lines in a plane which meet each other and do not lie in a straight line. This definition, it will be noticed, includes curvilinear angles, and the expression " and do not lie in a straight line " states that the lines must not be continuous one with the other, that is, that zero and straight angles are excluded. Since Euclid does not use the curvilinear angle, and it is only the recti- linear angle with which we are concerned, we will pass to the next definition and consider this one in connection therewith. THE DEFINITIONS OF GEOMETRY 143 9. Rectilinear Angle. When the lines containing the angle are straight, the angle is cabled rectilinear. This definition, taken with the preceding one, has always been a subject of criticism. In the first place it expressly excludes the straight angle, and, indeed, the angles of Euclid are always less than 180°, contrary to our mod- ern concept. In the second place it defines angle by means of the word " inclination," which is itself as diffi- cult to define as angle. To remedy these defects many substitutes have been proposed. ApoUonius defined angle as " a contracting of a surface or a solid at one point under a broken line or surface." Another of the Greeks defined it as " a quantity, namely, a distance be- tween the lines or surfaces containing it." Schotteni says that the definitions of angle generally fall into three groups: a. An angle is the difference of direction between two lines that meet. This is no better than Euclid's, since " difference' of direction " is as difficult to define as " inclination." b. An angle is the amount of turning necessary to bring one side to the position of the other side. c. An angle is the portion of the plane included be- tween its sides. Of these, h is given by way of explanation in most modern textbooks. Indeed, we cannot do better than simply to define an angle as the opening between two lines which meet, and then explain what is meant by size, through the bringing in of the idea of rotation. This is a simple presentation, it is easily understood, and it is sufficiently accurate for the real purpose in 1 Loc. cit., Vol. II, p. 94. 144 THE TEACHING OF GEOMETRY mind, namely, the grasping of the concept. We should frankly acknowledge that the concept of angle is such a simple one that a satisfactory definition is impossible, and we should therefore confine our attention to having the concept understood. 10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. We at present separate these definitions and simplify the language. 11. An obtuse angle is an angle greater than a right angle. 12. An acute angle is an angle less than a right angle. The question sometimes asked as to whether an angle of 200° is obtuse, and whether a negative angle, say — 90°, is acute, is answered by saying that Euclid did not conceive of angles equal to or greater than 180° and had no notion of negative quantities. Generally to- day we define an obtuse angle as " greater than one and less than two right angles." An acute angle is defined as "an angle less than a right angle," and is considered as positive under the general understanding that all geometric magnitudes are positive unless the contrary is stated. 13. A boundary is that which is an extremity/ of any- thing. The definition is not exactly satisfactory, for a circle is the boundary of the space inclosed, but we hardly consider it as the extremity of that space. Euclid wishes the definition before No. 14. 14. A figure is that which is contained by any boundary or boundaries. The definition is not satisfactory, since it excludes the unlimited straight line, the angle, an THE DEFINITIONS OF GEOMETRY 145 assemblage of points, and other combinations of lines and points which we should now consider as figures. 15. A circle is a plane figure contained hy one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another. 16. And the point is called the center of the circle. Some commentators add after " one line," definition 15, the words " which is called the circumference," but these are not in the oldest manuscripts. The Greek idea of a circle was usually that of part of a plane which is bounded by a line called in modern times the circumference, although Aristotle used "circle" as synonymous with " the bounding line." With the growth of modern math- ematics, however, and particularly as a result of the development of analytic geometry, the word " circle " has come to mean the bounding line, as it did with Aris- totle, a century before Euclid's time. This has grown out of the equations of the various curves, 0^+^^=?^ representing the circle-Zme, a^ + 6V = a'^b" representing the ellipse-Zme, and so on. It is natural, therefore, that circle, ellipse, parabola, and hyperbola should all be looked upon as lines. Since this is the modern use of " circle " in English, it has naturally found its way into elementary geometry, in order that students should not have to form an entirely different idea of circle on be- ginning analytic geometry. The general body of Ameri- can teachers, therefore, at present favors using " circle " to mean the bounding line and " circumference " to mean the length of that line. This requires redefining " area of a circle," and this is done by saying that it is the area of the plane space inclosed. The matter is not of greatest consequence, but teachers will probably prefer to join in the modern American usage of the term. 146 THE TEACHING OF GEOMETRY 17. Diameter. A diameter of the circle is any straight line drawn through the center and terminated in both direc- tions hy the circumference of the circle, and such a straight line also bisects the circle. The word " diameter" is from two Greek words meaning a " through measurer," and it was also used by EucUd for the diagonal of a square, and more generally for the diagonal of any parallelo- gram. The word " diagonal " is a later term and means the "through angle." It will be noticed that Euclid adds to the usual definition the statement that a diameter bisects the circle. He does this apparently to justify his definition (18), of a semicircle (a half circle). Thales is said to have been the first to prove that a diameter bisects the circle, this being one of three or four propositions definitely attributed to him, and it is sometimes given as a proposition to be proved. As a proposition, however, it is unsatisfactory, since the proof of what is so evident usually instills more doubt than certainty in the miads of beginners. 18. Semiciecle. a semicircle is the figure contained by the diameter mid the circumference cut off by it. And the center of the semicircle is the same as that of the circle. Proclus remarked that the semicircle is the only plane figure that has its center on its perimeter. Some writers object to defining a circle as a line and then speaking of the area of a circle, showing minds that have at least one characteristic of that of Proclus. The modern defini- tion of semicircle is " half of a circle," that is, an arc of 180°, although the term is commonly used to mean both the arc and the segment. 19. Rectilineae Figures. Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained THE DEFINITIONS OF GEOMETRY 147 hy four, and multilateral those contained by more than four, straight lines. 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene tri- angle that which has its three sides unequal. 21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled tri- angle that which has its three angles acute. These three definitions may properly be considered together. " Rectilinear " is from the Latin translation of the Greek euthygrammos, and means "right-lined," or " straight-lined." Euclid's idea of such a figure is that of the space inclosed, while the modern idea is tending to become that of the inclosing lines. In elementary geom- etry, however, the Euclidean idea is still held. " Trilat- eral " is from the Latin translation of the Greek tripleuros (three-sided). In elementary geometry the word "tri- angle " is more commonly used, although " quadrilateral " is more common than "quadrangle." The use of these two different forms is eccentric and is merely a matter of fashion. Thus we speak of a pentagon but not of a tetragon or a trigon, although both words are correct in form. The word " multilateral " (many-sided) is a trans- lation of the Greek polypleuros. Fashion has changed this to "polygonal" (many-angled), the word "multi- lateral " rarely being seen. Of the triangles, "equilateral" means " equal-sided"; "isosceles" is from the Greek isoskeles, meaning "with equal legs," and " scalene " from skalenos, possibly from skazo (to limp), or from skolios (crooked). Euclid's lim- itation of isosceles to a triangle with two, and only two. 148 THE TEACHING OF GEOMETRY equal sides would not now be accepted. We are at pres- ent more given to generalizing than he was, and when we have proved a proposition relating to the isosceles triangle, we wish to say that we have thereby proved it for the equilateral triangle. We therefore say that an isosceles triangle has two sides equal, leaving it possible that all three sides should be equal. The expression " equal legs " is now being discarded on the score of inel- egance. In place of "right-angled triangle " modern writers speak of " right triangle," and so for the obtuse and acute triangles. The terms are briefer and are as readily understood. It may add a little interest to the subject to know that Plutarch tells us that the ancients thought that " the power of the triangle is expressive of the nature of Pluto, Bacchus, and Mars." He also states that the Pythagoreans called " the equilateral triangle the head-bom Minerva and Tritogeneia (born of Triton) because it may be equally divided by the perpendicular lines drawn from each of its angles." 22. Of quadrilateral figures a square is that which is both equilateral and right-angled ; an oblong that which is right- angled but not equilateral; a rhombus that which is equi- lateral and not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another, but is neither equilateral nor right-angled. And let all quadrilat- erals other than these be called trapezia. In this definition Euclid also specializes in a manner not now generally approved. Thus we are more apt to-day to omit the oblong and rhomboid as unnecessary, and to define " rhombus " in such a manner as to include a square. We use "paral- lelogram " to cover " rhomboid," " rhombus," " oblong," and "square." For "oblong" we use "rectangle," let- ting it include square. Euclid's definition of " square " THE DEFINITIONS OF GEOMETRY 149 illustrates his freedom in stating more attributes than are necessary, in order to make sure that the concept is clear ; for he might have said that it " is that which is equilateral and has one right angle." We may profit by his method, sacrificing logic to educational necessity. Euclid does not use "oblong," "rhombus," "rhomboid," and "trapezium" (^plural, " trapezia ") in his proofs, so that he might well have omitted the definitions, as we often do. 23. Parallels. Parallel straight lines are straight lines which, being in the same plane and being produced in- dejinitely in both directions, do not meet one another in either direction. This definition of parallels, simplified in its language, is the one commonly used to-day. Other definitions have been suggested, but none has been so generally used. Proclus states that Posidonius gave the definition based upon the lines always being at the same distance apart. Geminus has the same idea in his defi- nition. There are, as Schotten has pointed out, three general types of definitions of parallels, namely: a. They have no point in common. This may be ex- pressed by saying that (1) they do not intersect, (2) they meet at infinity. b. They are equidistant from one another. c. They have the same direction. Of these, the first is Euclid's, the idea of the point at infinity being suggested by Kepler (1604). The sec- ond part of this definition is, of course, unusable for beginners. Dr. (now Sir Thomas) Heath says, " It seems best, therefore, to leave to higher geometry the concep- tion of infinitely distant points on a line and of two straight lines meeting at infinity, like imaginary points of intersection, and, for the purposes of elementary geom- etry, to rely on the plain distinction between 'parallel' 150 THE TEACHING OF GEOMETRY and 'cutting,' which average human intelligence can readily grasp." The direction definition seems to have originated with Leibnitz. It is open to the serious objection that " direc- tion " is not easy of definition, and that it is used very loosely. If two people on different meridians travel due north, do they travel in the same direction ? on paral- lel lines ? The definition is as objectionable , as that of angle as the "difference of direction" of two intersect- ing lines. From these definitions of the first book of Euclid we see (1) what a small number Euclid considered as basal ; (2) what a change has taken place in the generalization of concepts ; (3) how the language has varied. Never- theless we are not to be commended if we adhere to Euclid's small number, because geometry is now taught to pupils whose vocabulary is limited. It is necessary to define more terms, and to scatter the definitions through the work for use as they are needed, instead of massing them at the beginning, as in a dictionary. The most important lesson to be learned from Euclid's definitions is that only the basal ones, relatively few in number, need to be learned, and these because they are used as the foundations upon which proofs are built. It should also be noticed that Euclid explains nothing in these defini- tions ; they are hard statements of fact, massed at the beginning of his treatise. Not always as statements, and not at all in their arrangement, are they suited to the needs of our boys and girls at present. Having considered Euclid's definitions of Book I, it is proper to turn to some of those terms that have been added from time to time to his list, and are now usually incorporated in American textbooks. It will be seen that THE DEFINITIONS OF GEOMETRY 151 most of these were assumed by Euclid to be known by his mature re^aders. Tliey need to be defined for young people, but most of them are not basal, that is, they are not used in the proofs of propositions. Some of these terms, such as magnitudes, curve line, broken line, curvi- linear figure, bisector, adjacent angles, reflex angles, oblique angles and lines, and vertical angles, need merely a word of explanation so that they may be used intelligently. If they were numerous enough to make it worth the while, they could be classified in our textbooks as of minor importance, but such a course would cause more trouble than it is worth. Other terms have come into use in modern times that are not common expressions with which students are familiar. Such a term is " straight angle," a concept not used by Euclid, but one that adds so materially to the interest and value of geometry as now to be generally recognized. There is also the word " perigon," meaning the whole angular space about a point. This was excluded by the Greeks because their idea of angle required it to be less than a straight angle. The word means " around angle," and is the best one that has been coined for the purpose. " Flat angle " and " whole angle " are among the names suggested for these two modern concepts. The terms "complement," "supplement," and "conju- gate," meaning the difference between a given angle and a right angle, straight angle, and perigon respectively, have also entered our vocabulary and need defining. There are also certain terms expressing relationship which Euclid does not define, and which have been so changed in recent times as to require careful definition at present. Chief among these are the words " equal," " con- gruent," and " equivalent." Euclid used the single word 152 THE TEACHING OF GEOMETRY "equal" for all three concepts, although some of liis recent editors have changed it to " identically equal" in the case of congruence. In modern speech we use the word "equal" commonly to mean "like-valued," "having the same measure," as when we say the circumference of a circle " equals " a straight line whose length is 2 irr, although it could not coincide with it. Of late, there- fore, in Europe and America, and wherever European influence reaches, the word " congruent " is coming into use to mean "identically equal" in the sense of super- posable. We therefore speak of congruent triangles and congruent parallelograms as being those that are superposable. It is a little unfortunate that " equal " has come to be so loosely used in orduiary conversation that we can- not keep it to mean " congruent "; but our language will not permit it, and we are forced to use the newer word. Whenever it can be used without misunderstanding, however, it should be retained, as in the case of " equal straight lines," " equal angles," and " equal arcs of the same circle." The mathematical and educational world will never consent to use " congruent straight lines,"- or " congruent angles," for the reason that the terms are unnecessarily long, no misunderstanding being possible when " equal " is used. The word " equivalent "' was introduced by Legendre at the close of the eighteenth century to indicate equal-' ity of length, or of area, or of volume. Euclid had said, " Parallelograms which are on the same base and in the same parallels are equal to one another," while Legendre and his followers would modify the wording somewhat and introduce " equivalent " for " equal." This usage has been retained. Congruent polygons are therefore THE DEFINITIONS OP GEOMETRY 153 necessarily equivalent, but equivalent polygons are not in general congruent. Ctagruent polygons have mutually equal sides and mutually equal angles, while equivalent polygons have no equality save that of area. In general, as already stated, these and other terms should be defined just before they are used instead of at the beginning of geometry. The reason for this, from the educational standpoint and considering the present position of geometry in the curriculum, is apparent. We shall now consider the definitions of Euclid's Book III, which is usually taken as Book II in America. 1. Equal Circles. Hqual circles are those the diam- eters of which are equal, or the radii of which are equal. Manifestly this is a theorem, for it asserts that if the radii of two circles are equal, the circles may be made to coincide. In some textbooks a, proof is given by super- position, and the proof is legitimate, but Euclid usually avoided superposition if possible. Nevertheless he might as well have proved this as that two triangles are con- gruent if two sides and the included angle of the one are respectively equal to the corresponding parts of the other, and he might as well have postulated the latter as to have substantially postulated this fact. For in reality this definition is a postulate, and it was so con- sidered by the great Italian mathematician Tartaglia (ca. 1500-ca. 1557). The plan usually followed in Amer- ica to-day is to consider this as one of many unproved propositions, too evident, indeed, for proof, accepted by intuition. The result is a loss in the logic of Euclid, but the method is thought to be better adapted to the mind of the youthful learner. It is interesting to note in this connection that the Greeks had no word for "radius," and were therefore compelled to use some such phrase as 154 THE TEACHING OF GEOMETRY "the straight line from the center," or, briefly, "the from the center," as if " from the center " were one word. 2. Tangent. A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle. Teachers who prefer to use " circumference " instead of "circle" for the line should notice how often such phrases as " cut the circle " and " intersecting circle " are used, — phrases that signify nothing unless " circle " is taken to mean the line. So Aristotle uses an expres- sion meaning that the locus of a certain point is a circle, and he speaks of a circle as passing through " all the angles." Our word " touch " is from the Latin tangere, from which comes "tangent," and also "tag," an old touching game. 3. Tangent Circles. Circles are said to touch one another which, meeting one another, do not cut one another. The definition has not been looked upon. as entirely satisfactory, even aside from its unfortunate phraseology. It is not certain, for instance, whether Euclid meant that the circles could not cut at some other point than that of tangency. Furthermore, no distinction is made be- tween external and internal contact, although both forms are used in the propositions. Modern textbook makers find it convenient to define tangent circles as those that are tangent to the same straight line at the same point, and to define external and internal tangency by refer- ence to their position with respect to the line, although this may be characterized as open to about the same objection as Euclid's. 4. Distance. In a circle straight lines are said to be equally distant from the center, when the perpendiculars drawn to them from the center are equal. THE DEFINITIONS OF GEOMETRY 155 It is now customary to define " distance " from a point to a line as the length of the perpendicular from the point to the line, and to do this in Book I. In higher math- ematics it is found that distance is not a satisfactory term to use, but the objections to it have no particular significance in elementary geometry. 5. Greater Distance. And that straight line is said to be at a greater distance on which the greater perpendicular falls. Such a definition is not thought essential at the present time. 6. Segment. A segment of a circle is the figure con- tained hy a straight line and the circumference of a circle. The word " segment " is from the Latin root sect, meaning " cut." So we have " sector " (a cutter), " sec- tion " (a cut), " intersect," and so on. The word is not limited to a circle ; we have long spoken of a spherical segment, and it is common to-day to speak of a line seg- ment, to which some would apply a new name "sect." There is little confusion in the matter, however, for the context shows what kind of a segment is to be under- stood, so that the word "sect" is rather pedantic than important. It will be noticed that Euclid here uses " circumference " to mean " arc." 7. Angle op a Segment. An angle of a segment is that contained hy a straight line and a circumference of a circle. This term has entirely dropped out of geometry, and few teachers would know what it meant if they should hear it used. Proclus called such angles " mixed." 8. Angle in a Segment. An angle in a segment is the angle which., when a point is taken on the circumfer- ence of the segment and straight lines are joined from it to 156 THE TEACHING OF GEOMETRY the extremities of the straight line which is the base of the segment, is contained hy the straight lines so joined. Such an involved definition would not be usable to-day. Moreover, the words "circumference of the segment" would not be used. 9. And when the straight lines containing the angle cut off a circumference^ the angle is said to stand upon that circumference. 10. Sector. A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained hy the straight lines containing the angle and the circumference out off hy them. There is no reason for such an extended definition, our modern phraseology being both more exact (as seen in the above use of "circumference" for "arc") and more intelligible. The Greek word for " sector " is " knife " (tomeus'), "sector" being the Latin translation. A sector is supposed to resemble a shoemaker's knife, and hence the significance of the term. Euclid followed this by a defi- nition of similar sectors, a term now generally abandoned as unnecessary. It will be noticed that Euclid did not use or define the word " polygon." He uses " rectilinear figure " instead. Polygon may be defined to be a bounding line, as a circle is now defined, or as the space inclosed by a broken line, or as a figure formed by a broken line, thus including both the limited plane and its boundary. It is not of any great consequence geometrically which of these ideas is adopted, so that the usual definition of a portion of a plane bounded by a broken line may be taken as suffi- cient for elementary purposes. It is proper to call atten- tion, however, to the fact that we may have cross polygons of various types, and that the line that "bounds" the THE DEFINITIONS OF GEOMETRY 157 polygon must be continuous, as the definition states. That is, in the second of these figures the shaded portion is not considered a polygon. Such special cases are not liable to arise, but if questions relating to them are suggested, the teacher should be prepared to answer them. If sug- gested to a class, a note of this kind should come out only inci- dentally as a bit of interest, and should not occupy much time nor be unduh'^ emphasized. ^ It may also be mentioned to a class at some convenient time that the old idea of a polygon was that of a convex figure, and that the modern idea, which is met in higher mathematics, leads to a modification of earlier concepts. For example, here is a ^j be g quadrilateral with one ^ ~'/f y\ /^ of its diagonals, BB, out- \\n// \j^^^/ si'Je the figure. Further- \ ; / /\ 1 /\ more, if we consider a \ ( / Xl/ quadrilateral as a figure W /\ formed by four intersect- ^ ing lines, AC, OF, BE, and EA, it is apparent that this general quadrilateral has six vertices. A, B, C, I), E, F, and tliree diagonals, AD, BE, and CE. Such broader ideas of geometry form the basis of what is called modern elementary geometry. The other definitions of plane geometry need not be discussed, since all that have any historical interest have been considered. On the whole it may be said that our definitions to-day are n6t in general so carefully consid- ered • as those of Euclid, who weighed each word with 158 THE TEACHING OF GEOMETRY greatest skill, but they are more teachable to beginners, and are, on the whole, more satisfactory from the educa- tional standpoint. The greatest lesson to be learned from this discussion is that the number of basal definitions to be learned for subsequent use is very small. Since teachers are occasionally disturbed over the form in which definitions are stated, it is well to say a few words upon this subject. There are several standard types that may be used. (1) We may use the diction- ary form, putting the word defined first, thus: ^^ Might triangle. A triangle that has one of its angles a right angle." This is scientifically correct, but it is not a com- plete sentence, and hence it is not easily repeated when it has to be quoted as an authority. (2) We may put the word defined at the end, thus : " A triangle that has one of its angles a right angle is called a right triangle." This is more satisfactory. (3) We may combine (1) and (2), thus : "Right triangle. A triangle that has one of its angles a right angle is called a right triangle." This is still better, for it has the catchword at the beginning of the paragraph. There is occasionally some mental agitation over the trivial things of a definition, such as the use of the words "is called." It would not be a very serious matter if they were omitted, but it is better to have them there. The reason is that they mark the statement at once as a definition. For example, suppose we say that " a trian- gle that has one of its angles a right angle is a right triangle." We have also the fact that " a triangle whose base is the diameter of a semicircle and whose vertex lies on the semicircle is a right triangle." The style of statement is the same, and we have nothing in the phrase- ology to show that the first is a definition and the second THE DEFINITIONS OF GEOMETRY 159 a theorem.. This may happen with most of the definitions, and hence the most careful writers have not consented to omit the distinctive words in question. Apropos of the definitions of geometry, the great French philosopher and mathematician, Pascal, set forth certain rules relating to this subject, as also to the axioms employed, and these may properly sum up this chapter. 1. Do not attempt to define terms so well known in themselves that there are no simpler terms by which to express them. 2. Admit no obscure or equivocal terms without defining them. 3. Use in the definitions only terms that are perfectly understood or are there explained. 4. Omit no necessary principles without general agree- ment, however clear and evident they may be. 5. Set forth in the axioms only those things that are in themselves perfectly evident. 6. Do not attempt to demonstrate anything that is so evident in itself that there is nothing more simple by which to prove it. 7. Prove whatever is in the least obscure, using in the demonstration only axioms that are perfectly evident in themselves, or propositions already demonstrated or allowed. 8. In case of any uncertainty arising from a term em- ployed, always substitute mentally the definition for the term itself. Bibliography. Heath, Euclid, as cited; Frankland, The First Book of Euclid, as cited ; Smith, Teaching of Elementary Mathe- matics, p. 257, New York, 1900 ; Young, Teaching of Mathematics, p. 189, New York, 1907 ; Veblen, On Definitions, in the Monist, 1903, p. 303. CHAPTER XIII HOW TO ATTACK THE EXERCISES The old geometry, say of a century ago, usually con- sisted, as has been stated, of a series of theorems fully proved and of problems fully solved. During the nuie- teenth century exercises were gradually mtroduced, thus developing geometry from a science in which one learned by seeing things done, into one in which he gained power by actually doing things. Of the nature of these exer- cises (" originals," " riders "), and of their gradual change in the past few years, mention has been made in Chapter VII. It now remains to consider the methods of attacking these exercises. It is evident that there is no single method, and this is a fortunate fact, since if it were not so, the attack would be too mechanical to be interesting. There is no one rule for solving every problem nor even for seeing how to begin. On the other hand, a pupil is saved some time by having his attention called to a few rather definite lines of attack, and he will undoubtedly fare the better by not wasting his energies over attempts that are in advance doomed to failure. There are two general questions to be considered : first, as to the discovery of new truths, and second, as to the proof. With the first the pupil will have little to do, not having as yet arrived at this stage in his progress. A bright student may take a little interest in seeing what 160 HOW TO ATTACK THE EXERCISES 161 he can find out that is new (at least to him), and if so, he may be told that many new propositions have been dis- covered by the accurate drawing of figures ; that some have been found by actually weighing pieces of sheet metal of certain sizes ; and that still others have made themselves known tlarough paper folding. In all of these cases, however, the supposed proposition must be proved before it can be accepted. As to the proof, the pupil usually wanders about more or less until he strikes the right line, and then he follows this to the conclusion. He should not be blamed for doing this, for he is pursuing the method that the world followed in the earliest times, and one that has always been common and always will be. This is the synthetic method, the building up of the proof from propositions previously proved. If the proposition is a theorem, it is usuallj'' not difficult to recall propositions that may lead to the demonstration, and to select the ones that are really needed. If it is a problem, it is usually easy to look ahead and see what is necessary for the solution and to select the preceding propositions accordingly. But pupils should be told that if they do not rather easily find the necessary propositions for the construc- tion or the proof, they should not delay in resorting to another and more systematic method. This is known as the method of analysis, and it is applicable both to theo- rems and to problems. It has several forms, but it is of little service to a pupil to have these differentiated, and it suffices that he be given the essential feature of all these forms, a feature that goes back to Plato and his school in the fifth century B.C. For a theorem, the method of analysis consists ui reasoning as follows : " I can prove this proposition if I 162 THE TEACHING OF GEOMETRY can prove this thing ; I can prove this thing if I can prove that ; I can prove that if I can prove a third thing," and so the reasoning runs until the pupil comes to the point where he is able to add, " but I can prove that." This does not prove the proposition, but it enables him to reverse the process, begmning with the thing he can prove and going back, step by step, to the thing that he is to prove. Analysis is, therefore, his method of dis- covery of the way in which he may arrange his synthetic proof. Pupils often wonder how any one ever came to know how to arrange the proofs of geometry, and this answers the question. Some one guessed that a statement was true ; he applied analysis and found that he could prove it ; he then applied synthesis and did prove it. For a problem, the method of analysis is much the same as in the case of a theorem. Two things are in- volved, however, instead of one, for here we must make the construction and then prove that this construction is correct. The pupil, therefore, first supposes the problem solved, and sees what results follow. He then reverses the process and sees if he can attain these results and thus effect the required construction. If so, he states the proc- ess and gives the resulting proof. For example : In a triangle ABC, to draw PQ parallel to the base AB, cutting the sides in P and Q, so that PQ shall equal AP + BQ. Ansilysis. Assume the problem solved. Then AP must equal some part of PQ as PX, and BQ must equal QX. But if AP= PX, what must /.PXA equal ? •/ Pq is II to AB, what does /^PXA equal ? Then why must Z BAX =ZXAP? Similarly, what about Z QBX and Z XBA ? Construction. Now reverse the process. What may we do to zi 4 and Bin order to fix X? Then how shall PQ be drawn? Now give the proof . HOW TO ATTACK THE EXERCISES 163 The third, general method of attack applies chiefly to problems where some point is to be determined. This is the method of the intersection of loci. Thus, to locate an electric light at a point eighteen feet from the point of intersection of two streets and equidistant from them, evi- dently one locus is a circle with a radius eighteen feet and the .center at the ver- tex of the angle made by the streets, and the other locus is the bisector of the angle. The method is also occasionally applicable to theorems. For example, to prove that the perpendicular bisec- tors of the sides of a triangle are concurrent. Here the locus of points equidistant from A and B is PP', and the locus of points equidistant from £ and C is QQ'. These can easily be shown to intersect, as at 0. Then 0, being equidistant from A, B, and C, is also on the perpendicular bisector of AC. Therefore these bisectors are concurrent in 0. These are the chief methods of attack, and are all that should be given to an average class for practical use. Besides the methods of attack, there are a few general directions that should be given to pupils. 1. In attacking either a theorem or a problem, take the most general figure possible. Thus, if a proposition relates to a quadrilateral, take one with unequal sides and unequal angles rather than a square or even a rectangle. The simpler figures often deceive a pupil into feeling that he has a proof, when in reality he has one only for a special case. 164 THE TEACHING OF GEOMETKY 2. Set forth very exactly the thing that is given, using letters relating to the figure that has been drawn. Then set forth with the same exactness the thing that is to be proved. The neglect to do this is the cause of a large per cent of the failures. The knowing of exactly what we have to do and exactly what we have with which to do it is half the battle. 3. If the proposition seems hazy, the difficulty is prob- ably with the wording. In this case try substituting the definition for the name of the thing defined. Thus in- stead of thinking too long about proving that the median to the base of an isosceles triangle is perpendicular to the base, draw the figure and think that there is given AC=BC, / AD = BD, / and that there is to be proved that / Z CDA = Z BBC. ^ Here we have replaced " median," " isosceles," and " per- pendicular " by statements that express the same idea in simpler language. Bibliography. Petersen, Methods and Theories for the Solution of Geometric Problems of Construction, Copenhagen, 1879, a curious piece of English and an extreme view of the subject, but well worth consulting; Alexandroff, Problfemes de g6om6trie 616mentaire, Paris, 1899, with a German translation in 1903 ; Loomis, Original Investigation ; or, How to attack an Exercise in Geometry, Boston, 1901 ; Sauvage, Les Lieux gdom^triques en g6om6trie 616mentaire, Paris, 1893 ; Hadamard, Legons de g^om^trie, p. 261, Paris, 1898 ; Duhamel, Des M6thodes dans les sciences de raisonnement, 3'= 6d., Paris, 1885 ; Henrici and Treut- lein, Lehrbuch der Elementar-Georaetrie, Leipzig, 3. Aufl., 1897 ; Henrici, Congruent Figures, London, 1879. CHAPTER XIV BOOK I AND ITS PROPOSITIONS Having considered tlie nature of the geometry tliat we have inherited, and some of tlie opportunities for improving upon the methods of presenting it, the next question tliat arises is the all-important one of the sub- ject matter, What shall geometry be in detail ? Shall it be the text or the sequence of Euclid ? Few teachers have any such idea at the present time. Shall it be a mere dabbling with forms that are seen in mechanics or architecture, with no serious logical sequence ? This is an equally dangerous extreme. Shall it be an enthely new style of geometry based upon groups of motions ? This may sometime be developed, but as yet it exists in the future if it exists at all, since the recent eif orts in this respect are generally quite as ill suited to a young pupil as is Euclid's " Elements " itself. No one can deny the truth of M. Bourlet's recent assertion that " Industry, daughter of the science of the nineteenth century, reigns to-day the mistress of the world ; she has transformed all ancient methods, and she has absorbed m herself almost all human activity." ^ Neither can one deny the justice of his comparison of Euclid with a noble piece of Gothic architecture and of his assertion that as modern life demands another type of building, so it demands another type of geometry. ' Address at Brussels, August, 1910. 165 166 THE TEACHING OF GEOMETRY But what does this mean ? That geometry is to exist merely as it touches industry, or that bad architecture is to replace the good ? By no means. A building should to-day have steam heat and elevators and electric lights, but it should be constructed of just as enduring materials as the Parthenon, and it should have lines as pleasing as those of a Gothic facade. Architecture should still be artistic and construction should still be substantial, else a building can never endure. So geometry must still exemplify good logic and must still bring to the pupil a feeling of exaltation, or it will perish and become a mere relic in the museum of human culture. What, then, shall the propositions of geometry be, and in what manner shall they answer to the challenge of the industrial epoch in which we live ? In reply, they must be better adapted to young minds and to all young minds than Euclid ever intended his own propositions to be. Furthermore, they must have a richness of application to pure geometry, in the way of carefully chosen exer- cises, that Euclid never attempted. And finally, they must have application to this same life of industry of which we have spoken, whenever this can really be found, but there must be no sham and pretense about it, else the very honesty that permeated the ancient geometry will seem to the pupil to be wanting in the whole subject.^ Until some geometry on a radically different basis shall appear, and of this there is no very hopeful sign at pres- ent, the propositions will be the essential ones of Euclid, excluding those that may be considered merely intuitive, and excluding all that are too difficult for the pupil who 1 For a recent discussion of this general subject, see Professor Hobson on " The Tendencies of Modern Mathematics," in the Educa^ tional Beview, New York, 1910, Vol. XL, p. 524. BOOK I AND ITS PROPOSITIONS 167 to-day takes up their study. The number will be limited in a reasonable way, and every genuine type of applica- tion will be placed before the teacher to be used as necessity requires. But a fair amount of logic will be retained, and the effort to make of geometry an empty bauble of a listless mind will be rejected by every worthy teacher. What the propositions should be is a matter upon which opinions may justly differ; but in this chapter there is set forth a reasonable list for Book I, arranged in a workable sequence, and this list may fairly be taken as typical of what the American school will prob- ably use for many years to come. With the list is given a set of typical applications, and some of the general m- f ormation that will add to the interest in the work and that should form part of the equipment of the teacher. An ancient treatise was usually written on a kind of paper called papyrus, made from the pith of a large reed formerly common in Egypt, but now growing luxuriantly only above Khartum in Upper Egypt, and near Syracuse m Sicily ; or else it was written on parchment, so called from Pergamos in Asia Minor, where skins were first prepared in parchment form ; or occasionally they were written on ordinary leather. In any case they were gen- erally written on long strips of the material used, and these were rolled up and tied. Hence we have such an expression as " keeping the roll " in school, and such a word as " volume," which has in it the same root as " involve " (to roll in), and " evolve " (to roll out). Sev- eral of these rolls were often necessary for a single treat- ise, in which case each was tied, and all were kept together in a receptacle resembling a pail, or in a compartment on a shelf. The Greeks called each of the separate parts of a treatise hihlion (/3t/3\toi'), a word meaning " book." 168 THE TEACI-imG OF GEOMETRY Hence we have the books of the Bible, the books of Homer, and the books of Euchd. From the same root, indeed, comes Bible, bibliophile (booklover), bibliography (list of books), and kindred words. Thus the books of geometry are the large chapters of th6 subject, " chapter " being from the Latin caput (head), a' section under a new heading. There have been efforts to change " books " to "chapters," but they have not succeeded, and there is no reason why they should succeed, for the term is clear and has the sanction of long usage. Theoeem. If two lines intersect, the vertical angles are equal. This was Euclid's Proposition 15, being put so late because he based the proof upon his Proposition 13, now thought to be best taken without proof, namely, " If a straight line set upon a straight line makes angles, it will make either two right angles or angles equal to two right angles." It is found to be better pedagogy to assume that this follows from the definition of straight angle, with reference, if necessary, to the meaning of the sum of two angles. This proposition on vertical angles is prob- ably the best one with which to begin geometry, since it is not so evident as to seem to need no proof, although some prefer to rank it as semiobvious,' while the proof is so simple as easily to be understood. Eudemus, a Greek who wrote not long before Euclid, attributed the discovery of this proposition to Thales of Miletus (ca. 640-548 B.C.), one of the Seven Wise Men of Greece, of whom Proclus wrote : " Thales it was who visited Egypt and first transferred to Hellenic soil this theory of geometry. He himself, indeed, discovered much, but still more did he introduce to his successors the prin- ciples of the science." BOOK I AND ITS PROPOSITIONS 169 The proposition is the only basal one relating to the. intersection of two lines, and hence there are no others with which it is necessarily grouped. This is the reason for placing it by itself, followed by the congruence theorems. There are many familiar illustrations of this theorem. Indeed, any two crossed lines, as in a jjair of shears or the legs of a camp stool, bring it to mind. The word " straight " is here omitted before " lines " in accordance with the modern convention that the word " line " un- modified means a straight line. Of course in cases of special emphasis the adjective should be used. Theorem. Two triangles are congruent if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other. This is Euclid's Proposition 4, his first tliree proposi- tions being problems of construction. This would there- fore have been his first proposition if he had placed his problems later, as we do to-day. The words "congruent" and " equal " are not used as in Euclid, for reasons already set forth on page 151. There have been many attempts to rearrange the propositions of Book I, putting in sepa- rate groups those concerning angles, those concerning triangles, and those concerning parallels, but they have all failed, and for the cogent reason that such a scheme destroys the logical sequence. This proposition may properly follow the one on vertical angles simply because the latter is easier and does not involve superposition. As far as possible, Euclid and all other good geome- ters avoid the proof by superposition. As a practical test superposition is valuable, but as a theoretical one it is open to numerous objections. As Peletier pointed out in his (1557) edition of Euchd, if the superposition of 170 THE TEACHING OF GEOMETRY lines and figures could freely be assumed as a method of demonstration, geometry would be full of such proofs. There would be no reason, for example, why an angle should not be constructed equal to a given angle by superposing the given angle on another part of the plane. Indeed, it is possible that we might then assume to bisect an angle by imagining the plane folded like a piece of paper. Heath (1908) has pointed out a subtle defect in Euclid's proof, in that it is said that because two lines are equal, they can be made to coincide. Euclid says, practically, that if two lines can be made to coincide, they are equal, but he does not say that if two straight lines are equal, they can be made to coincide. For the purposes of elementary geometry the matter is hardly worth bringing to the attention of a pupil, but it shows that even Euclid did not cover every point. Applications of this proposition are easily found, but they are all very much alike. There are dozens of meas- urements that can be made by simply constructing a triangle that shall be congruent to another triangle. It seems hardly worth the while at this time to do more than mention one typical case,i leaving it to teachers who may find it desirable to suggest others to their pupils. "Wishing to measure the distance across a river, some boys sighted from ^ to a point P. They then turned and measured AB at right angles to AP. They placed a stake at 0, halfway from A to B, and drew a perpendicular to AB at B. They placed a stake at C, on this perpendicular, and in line with and P. They then found the width by measuring BC. Prove that they were right. 1 A more extended list of applications is given later in this work. BOOK I AND ITS PROPOSITIONS 171 This involves the ranging of a line, and the running of a line at right angles to a given line, both of which have been described in Chapter IX. It is also fairly accurate to run a line at any angle to a given line by sighting along two pins stuck in a protractor. Theorem. Two triangles are congruent if two angles and the included side of the one are equal respectively to two angles and the included side of the other. Euclid combines this with his Proposition 26 : If two triangles have the two angles equal to two angles re- spectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the re- maining sides, and the remaining angle to the remaining angle. He proves this cumbersome statement without super- position, desiring to avoid this method, as already stated, whenever possible. The proof by superposition is old, however, for Al-Nairizi ^ gives it and ascribes it to some earlier author whose name he did not know. Proclus tells us that " Eudemus in his geometrical history refers this theorem to Thales. For he says that in the method by which they say that Thales proved the distance of ships in the sea, it was necessary to make use of this theorem." How Thales did this is purely a matter of conjecture, but he might have stood on the top of a tower rising from the level shore, or of such headlands as abound near Miletus, and by some simple instrument sighted to the ship. Then, turning, he might have sighted along the shore to a point having the same angle of declina- tion, and then have measured the distance from the tower 1 Abu'l-'Abbas al-Fadl ibn Hatim al-NairizI, so called from his birthplace, Nairiz, was a well-known Arab writer. He died about 922 A.D. He wrote a commentary on Euclid. 17i THE TKACIIING OF GEOMETRY to tlii.s jxjiiit. Tlii« seems more reasonable than any of the various plans suggested, and it is found in so many practical geometries of the first century of printing that it seems to have long been a common expedient. The stone astrolabe from Mesopotamia, now preserved in the Brit- ish Museum, shows that such instruments for the meas- uring of angles are very old, and for the purposes of Sixteenth-Centuky Mensuration Belli's " Del Misurar con la Vista," Venice, 1569 Thales even a pair of large compasses would have an- swered very well. An illustration of the method is seen in Belli's work of 1569, as here shown. At the top of tlie picture a man is getting the angle by means of the visor of his cap ; at the bottom of the picture a man is using a ruler screwed to a staff, i The story goes that 1 This illustration, taken from a book in the author's library, appeared in a valuable monograph by W. E. Stark, " Measuring In- struments of Long Ago," published in School Science and Mathematics, Vol. X, pp. 48, 126. With others of the same nature it is here re- produced by the courtesy of Principal Stark and of tlie editors of the journal in which it appeared. BOOK I AND ITS PROPOSITIONS 173 one of Napoleon's engineers won the imperial favor by quickly measuring the width of a stream that blocked the progress of the army, using this very method. This proposition is the reciprocal or dual of the pre- ceding one. The relation between the two may be seen from the following arrangement : Two triangles are congruent Two triangles are congruent if two sides and the included if two angles and the included angle of the one are equal re- side of the one are equal re- spectively to two sides and the speotively to two angles and the included angle of the other. included side of the other. In general, to every proposition involving points and lines there is a reciprocal proposition involving lines and points respectively that is often true, — indeed, that is always true in a certain line of propositions. This rela- tion is known as the Principle of Reciprocity or of Dual- ity; Instead of points and lines we have here angles (suggested by the vertex points) and lines. It is inter- esting to a class to have attention called to such rela- tions, but it is not of sufficient importance in elementary geometry to justify more than a reference here and there. There are other dual features that are seen in geometry besides those given above. Theorem. In an isosceles triangle the angles opposite the equal sides are equal. This is Euclid's Proposition 5, the second of his theo- rems, but he adds, "and if the equal straight lines be produced further, the angles under the base will be equal to one another." Since, however, he does not use this second part, its genuineness is doubted. He would not admit the common proof of to-day of supposing the ver- tical angle bisected, because the problem about bisecting an angle does not precede this proposition, and therefore 174 THE TEACHIXG OF GEOMETRY his proof is much more involved than ours. He riiakes CX=CY, and proves AXBC and YAC congruent,^ and also A XBA and YAB congruent. Then from ZYAC he takes Z YAB, leaving Z BA C, and so on the other side, leaving Z CBA, these therefore being equal. This proposition has long been called the pons asinorum, or bridge of asses, but no one knows where or when the name arose. It is usually stated that it came from the fact that fools could not cross this bridge, and it is a fact that in the Middle Ages this was often the hmit of the student's progress in geometry. It has how- ever been suggested that the name came from Euclid's figure, which resembles the simplest type of a wooden truss bridge. The name is applied by the French to the Pythagorean Theorem. Proclus attributes the discovery of this proposition to Thales. He also says that Pappus (third century a.d.), a Greek commentator on Euclid, proved the proposition as follows : Let ABC he the triangle, with AB =AC. Conceive of this as two triangles; then AB =AC, AC=AB, and Z ^ is common; hence the ^ABC and ACB are congruent, and /.B oi the one equals Z C of the other. This is a better plan than that followed by some text- book writers of imagining A ABC taken up and laid down on itself. Even to lay it down on its " trace " is more objectionable than the plan of Pappus. 1 In speaking of two congruent triangles it is somewhat easier to follow the congruence if the two are read in the same order, even though the relatively unimportant counterclockwise reading is neg- lected. No one should be a slave to such a formalism, but should fol- low the plan when convenient. BOOK I AND ITS PROPOSITIONS 175 Theoeem. If two angles of a triangle are equal, the sides opposite the equal angles are equal, and the triangle is isosceles. The statement is, of course, tautological, the last five words being unnecessary from the mathematical stand- point, but of value at this stage of the student's progress as emphasizing the nature of the triangle. Euclid stated the proposition thus, " If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another." He did not define "subtend," supposing such words to be already under- stood. This is the first case of a converse proposition in geometry. Heath distinguishes the logical from the geometric converse. The logical converse of Euclid I, 5, would be that '■'■some triangles with two angles equal are isosceles," while the geometric converse is the propo- sition as stated. Proclus called attention to two forms of converse (and in the course of the work, but not at this time, the teacher may have to do the same) : (1) the complete converse, in which that which is given in one becomes that which is to be proved m the other, and vice versa, as m this and the preceding proposition ; (2) the partial converse, in which two (or even more) things may be given, and a certain thmg is to be proved, the converse being that one (or more) of the preceding things is now given, together with what was to be proved, and the other given thing is now to be proved. Symbol- ically, if it is given that a = b and c = d, to prove that x = y, the partial converse would have given a = 5 and x = y, to prove that c= d. Several proofs for the proposition have been sug- gested, but a careful examination of all of them shows that the one given below is, all things considered, the best one for pupils beginning geometry and f oUowmg the 176 THE TEACIimG OF GEOMETKY sequence laid down in this chapter. It has the sanction of some of the most eminent mathematicians, and while not as satisfactory in some respects as the reductio ad ah- surduni, mentioned below, it is more satisfactory in most particulars. The proof is as follows : c C A' B' Given the triangle ABC, with the angle A equal to the angle B. To prore that AC = BC. Proof. Suppose the second triangle A'B'C to be an exact re- production of the given triangle ABC. Turn the triangle A'B'C over and place it upon ABC so that B' shall fall on A and A' shall fall on B. Then B'A' will coincide with AB. Since ZA' = ZB', Given and ZA=ZA', Hyp. .•.ZA=ZB'. .-. B'C ■willlie along AC. Similarly, A'C will lie along BC. Therefore C will fall on both A C and BC, and hence at their intersection. . n//-<'_^/^ But B'C was made equal to BC. .■.AC = BC. Q.E.D. If the proposition should be postponed until after the one on the sum of the angles of a triangle, the proof would be simpler, but it is advantageous to couple it with its immediate predecessor. This simpler proof consists BOOK I AND ITS PROPOSITIONS 177 in bisecting the v-ertical angle, and then proving the two triangles congruent. Among the other proofs is that of the reductio ad absurdum, which the student might now meet, but which may better be postponed. The phrase reductio ad absurdum seems lUiely to continue ia spite of the efforts to find another one that is simpler. Such a proof is also called an indirect proof, but this term is not altogether satisfactory. Probably both names should be used, the Latin to explain the nature of the English. The Latin name is merely a translation of one of several Greek names used by Aristotle, a second being in Eng- lish " proof by the impossible," and a third being " proof leading to the impossible." If teachers desire to intro- duce this form of proof here, it must be borne in mind that only one supposition can be made if such a proof is to be valid, for if two are made, then an absurd con- clusion simply shows that either or both must be false, but we do not know which is false, or if only one is false. Theoeem. Two triangles are congruent if the three sides of the one are equal respectively to the three sides of the other. It would be desirable to place this after the fourth proposition mentioned in this list if it could be done, so as to get the triangles in a group, but we need the fourth one for proving this, so that the arrangement cannot be made, at least with this method of proof. This proposition is a " partial converse " of the- second propo- sition in this list; for if the triangles are ABC and A'B'C, with sides a, b, c and o', b% c', then the second proposition asserts that iih = V, c = c', and /.A —A A', thien a = a' and the triangles are congruent, while this proposition asserts that ii a = a',b = b', and c = c', then ZA = ZA' and the triangles are congruent. The proposition was known at least as early as Aris- totle's time. Euclid proved it by inserting a prelimmary 178 THE TEACHING OF GEOMETRY proposition to the effect tliat it is impossible to have on the same base AB and the same side of it two different triangles ABC and ABC', with AC = AC', and BC = BC'. The proof ordinarily given to-day, wherein the two tri- angles are constructed on opposite sides of the base, is due to Philo of Byzantium, who lived after Euclid's time but before the Christian era, and it is also given by Pro- clus. There are really three cases, if one wishes to be overparticular, corresponding to the three pairs of equal sides. But if we are allowed to take the longest side for the common base, only one case need be considered. •Of the applications of the proposition one of the most important relates to making a figure rigid by means of diagonals. For example, how many diagonals must be drawn in order to make a quadrilateral rigid ? to make a pentagon rigid ? a hexagon ? a polygon of n sides. In particular, the following questions may be asked of a class : 1. Three iron rods are hinged at the ex- tremities, as shown in this figure. Is the figure rigid ? Why ? 2. Four iron rods are hinged, as shown in this figure. Is the figure rigid ? If not, where would you put in the fifth rod to make it rigid ? Prove that this would accomplish the result. Another interesting application relates to the most ancient form of leveling instrument known to us. This kuid of level is pic- tured on very ancient monuments, and it is still used in many parts of the world; Pupils in manual training may make such an in- strument, and indeed one is easily made out of cardboard. BOOK I AXI) ITS PROPOSITION'S 179 If the plumb line passes through the mid-pomt of the base, the two triangles are congruent and the plumb line is then perpendicular to the base. In other words, the base • „'. ; t \f Comefi MtSUimtoTe OUi^ta, Veiyendicolmi ,iy Base, de '^"'-^^° " Honti^ StttSmente msurare le Supttficie con dif • ^karTsl'impKBi^f)Tolti cUclMcmi ^'/^ > / * / ■ 'i Eakly Methods ok Leveling Pomodoro's "La geoiuetria prattioa," Rome, 1624 is level. With such simple primitive instruments, easily made Ijy pupils, a good deal of practical mathematical work can be performed. The uiteresting old illustration here given shows how tliis form of level was used three hundred years ago. Teachers who seek for geometric figures in practical mechanics will find this proposition illustrated in the ordinary lioisting apparatus of the kind here shown. From the stud)" of such forms and of simple roof and bridge trusses, a numljer of the usual properties of the isosceles triangle may be derived. 180 THE TEACHING OF GEOMETRY Theokem. The sum of two lines drawn from a given point to the extremities of a given line is greater than the sum of two other lines similarly drawn, but included hy them. It should be noted that the words "the extremities of " are necessary, for it is possible to draw from a cer- tain point within a certam triangle two lines to the base such that their sum is greater than the sum of the other two sides. c Thus, in the right triangle ABC draw any line CX from C to the hase. Make XY=AC, and CP = PY. Then it is easily shown that PB + PA' > CB + CA. It is interesting to a class to have a teacher point out that, in this figure, ^P + P-B < .-1C+ Cfi, and AP' + P'B S^\ we may infer (although we have not /^^^-^'"''^P^^^^ proved) that " a straight line (AB) is ^^ ' '^^~^^^ the shortest path between two points." A ' B Theokem. Only one perpendicular can be drawn to a given line from a given external point. Theoeem. Two lines drawn from a point in a perpen- dicular to a given line, cutting off on the given line equal segments from the foot of the perpendicular, are equal and make equal angles with the perpendicular. Theorem. Of two lines drawn from the same point in a perpendicular to a given line, cutting off on the line un- equal segments from the foot of the perpendicular, the more remote is the greater. Theorem. The perpendicular is the shortest lins that can be drawn to a straight line from a given external point. BOOK I AXD ITS PROPOSITIONS 181 These four propositions, while known to the ancients and incidentally used, are not explicitly stated by Euclid. The reason seems to be that he interspersed his problems with his theorems, and in his Propositions 11 and 12, which treat of drawing a perpendicular to a line, the essential features of these theorems are proved. Further mention will be made of them when we come to consider the problems in question. Many textbook writers put the second and third of the four before the first, forgetting that the first is assumed in the other two, and hence should precede them. Theorem. Two right triangles are congruent if the hypotenuse and a side of the one are equal respectively to the hypotenuse and a side of the other. Theorem. Tivo right triangles are congruent if the hypotenuse and an adjacent angle of the one are equal re- spectively to the hypotenuse and an adjacent angle of the other. As stated in the notes on the third proposition in this sequence, Euclid's cumbersome Proposition 26 covers several cases, and these two among them. Of course this present proposition could more easily be proved after the one concerning the sum of the angles of a triangle, but the proof is so simple that it is better to leave the propo- sition here in connection with others concerning triangles. Theorem. Two lines in the same plane perpendicular to the same line cannot meet, however far they are produced. This proposition is not in Euclid, and it is introduced for educational rather than for mathematical reasons. Euclid introduced the subject by the proposition that, if alternate angles are equal, the lines are parallel. It is, however, simpler to begin with this proposition, and there is some advantage in stating it in such a way as to 182 THE TEACHING OF GEOMETRY prove that parallels exist before they are defined. The proposition is properly followed by the definition of parallels and by the postulate that has been discussed on page 127. A good application of this proposition is the one con- cerning a method of drawing parallel lines by the use of a carpenter's square. Here two lines are drawn perpen- dicular to the edge of a board or a ruler, and these are parallel. Theorem. If a line is perpendicular to one of two parallel lines, it is perpendicular to the other also. This, like the preceding proposition, is a special case under a later theorem. It simplifies the treatment of parallels, however, and the beginner finds it easier to approach the difficulties gradually, through these two cases of perpendiculars. It should be noticed that this is an example of a partial converse, as explained on page 175. The preceding proposition may be stated thus : If a is ± to a; and 5 is ± to x, then a is II to 6. This propo- sition may be stated thus : If a is _L to a; and a is II to b, then & is _L to x. This is, therefore, a partial converse. These two propositions having been proved, the usual definitions of the angles made by a transversal of two parallels may be given. It is unfortunate that we have no name for each of the two groups of four equal angles, and the name of " transverse angles " has been suggested. This would simplify the statements of certain other prop- ositions ; thus : " If two parallel lines are cut by a trans- versal, the transverse angles are equal," and this includes two propositions as usually given. There is not as yet, however, any general sanction for the term. Theoeem. If two parallel lines are cut hy a trans- versal, the alternate-interior angles are equal. BOOK I AND ITS PROPOSITIONS 183 Euclid gave this as half of his Proposition 29. Indeed, he gives only four theorems on parallels, as against five propositions and several corollaries in most of our Amer^ ican textbooks. The reason for increasmg the number is that each proposition may be less involved. Thus, instead of having one proposition for both exterior and interior angles, modern authors usually have one for the exterior and one for the interior, so as to make the difficult sub- ject of parallels easier for beginners. Theorem. When two straight lines in the same plane are cut hy a transversal, if the alternate-interior angles are equal, the two straight lines are parallel. This is the converse of the preceding theorem, and is half of Euclid I, 28, his theorem being divided for the reason above stated. There are several typical pairs of equal or supplemental angles that would lead to parallel lines, of which Euclid uses only part, leaving the other cases to be inferred. This accounts for the number of corollaries in this connection in later textbooks. Surveyors make use of this proposition when they wish, without using a transit instrument, to run one line parallel to another. For example, suppose two boys are laying out a tennis court and they wish to run a line through P parallel to AB. Take a 60-foot tape and swing it around P until the other end rests on AB, as at M. Put a stake at 0, 30 feet from P and Af. Then take any convenient point JV on AB, and measure ON. Sup- pose it equals 20 feet. Then sight from N through 0, and put a stake at Q just 20 feet from 0. Then P and Q deterinine the parallel, according to the proposition just mentioned. 184 THE TEACHING OF GEOMETRY Theorem. If two parallel lines are cut hy a transversal, the exterior-interior angles are equal. This is also a part of Euclid I, 29. It is usually fol- lowed by several corollaries, covering the minor and ob- vious cases omitted by the older writers. While it would be possible to dispense with these corollaries, they are helpful for definite reference in later propositions. Theorem. The sum of the three angles of a triangle is equal to two right angles. Euclid stated this as follows : " In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles." This states more than is necessary for the basal fact of the prop- osition, which is the constancy of the sum of the angles. The theorem is one of the three most important propo- sitions in plane geometry, the other two being the so- called Pythagorean Theorem, and a proposition relating to the proportionality of the sides of two triangles. These three form the foundation of trigonometry and of the mensuration of plane figures. The history of the proposition is extensive. Eutocius (ca. 510 A.D.), in his commentary on ApoUonius, says that Geminus (first century B.C.) testified that "the ancients investigated the theorem of the two right angles in each individual species of triangle, first in the equilateral, again in the isosceles, and afterwards in the scalene triangle." This, indeed, was the ancient plan, to proceed from the particular to the general. It is the natural order, it is the world's order, and it is well to follow it in all cases of difficulty in the classroom. Proclus (410-485 a.d.) tells us that Eudemus, who lived just before Euclid (or probably about 325 B.C.), BOOK I AND ITS PROPOSITIONS 185 affirmed that the theorem was due to the Pythagoreans, although this does not necessarily mean to the actual pupils of Pythagoras. The proof as he gives it consists in showing that a=a\ h=h', and a'+c + h'= two right angles. Since the proposition about the exterior angle of a tri- angle is attributed to Philip- pus of Mende (ca. 380 B.C.), the figure given by Eudemus is probably the one used by the Pythagoreans. There is also some reason for believing that Thales (ca. 600 B.C.) knew the theorem, for Diogenes Laertius (ca. 200 A.D.) quotes Pamphilius (first century a.d.) as saying that "he, having learned geometry from the Egyptians, was the first to inscribe a right triangle in a circle, and sacrificed an ox." The proof of this propo- sition requires the knowledge that the sum of the angles, at least in a right triangle, is two right angles. The propo- sition is frequently referred to by Aristotle. There have been numerous attempts to prove the proposition without the use of parallel lines. Of these a German one, first given by Thibaut in the early part of the eighteenth century, is among the most interesting. This, in simplified form, is as follows : Suppose an indefi- nite line XY to lie on A B. Let it swing about A, counter- clockwise, through Z.A, so as to lie on AC, as A'T'. Then let it swing about C, through Z. C, so as to lie on CB, as X" Y" Then let it swing about B, through ZB, so as to lie on BA, as X"'Y'". It now lies on AB, 186 THE TEACHING OF GEOMETRY but it is turned over, X'" being where Y was, and ]"" where A' was. In turning through A A, B, and C it has therefore turned through two right angles. One trouble with the proof is tliat the rotation has not been about the same point, so that it has never been looked upon as other than an interesting illustration. Proclus tried to prove the theorem by saying that, if we have two perpendiculars to the same line, and sup- pose them to revolve about their feet so as to make a triangle, then the amount taken from the right angles is added to the vertical angle of the triangle, and there- fore the sum of the angles continues to be two right angles. But, of course, to prove his statement requires a perpendicular to be drawn from the vertex to the base, and the theorem of parallels to be applied. Pupils will find it interesting to cut off the corners of a paper triangle and fit the angles together so as to ■ make a straight angle. This theorem furnishes an opportunity for many in- teresting exercises, and in particular for determining the third angle when two angles of a triangle are given, or the second acute angle of a right triangle when one acute angle is given. Of the simple outdoor applications of the proposition, one of the best is illustrated in ^^ this figure. To ascertain the height of a tree or of the school building, fold a piece of paper so as to make an angle of 45°. Then walk back from the tree until the top is seen at an angle of 45° with the ground (being therefore careful to have the base of the triangle level). Then the height AC will equal the base AB, since ABC is isosceles. A paper protractor may be used for the same purpose. BOOK I AND ITS PROPOSITIONS 187 Distances can easily be measured by constructing a large equilateral triangle of heavy pasteboard, and stand- ing pins at the vertices for the purpose of sighting. To measure PC, stand at some convenient point A and sight along APC and also along AB. Then walk along AB until a point B is reached from which B C makes with BA an angle of the triangle (60°). Then AC = AB, and since AP can be measured, we can find PC. Another simple method of measuring a distance AC across a stream is shown in this figure. Measure the angle CAX, either in degrees, with a pro- tractor, or by sighting along a piece of paper and marking down the angle. Then go along XA produced until a point B is reached from which BC makes with A an angle equal to half of angle CAX. Then it is easily shown that AB- AC. A navigator vises the same principle when he " doubles the angle on the bow " to find his distance from a light- house or other object. If he is sailing on the course j4.BC and notes a lighthouse L when he is at A, and takes the angle A, and if he notices '^'^ when the angle that the light- house makes with his course is just twice the angle noted at A, then BL = AB. He has AB from his log (an instru- ment that tells how far a ship goes in a given time), so he knows BL. He has " doubled the angle on the bow " to get this distance. 188 THE TEACHING OF GEOMETRY It would have been possible for Thales, if he knew this proposition, to have measured the distance of the ship at sea by some such device as this : Make a large isosceles triangle out of wood, and, standing at T, sight to the ship and along the shore on a line TA, using the vertical angle of the triangle. Then go along TA until a point P is reached, from which T and 5 can be seen along the sides of a base angle of the triangle. Then TP = TS. By measuring TB, BS can then be found. Theorem. The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side. If the postulate is assumed that a straight line is the shortest path between two points, then the first part of this theorem requires no further proof, and the second part follows at once from the axiom of inequalities. This seems the better plan for beginners, and the proposition may be considered as semiobvious. Euclid proved the first part, not having assumed the postulate. Proclus tells us that the Epicureans (the followers of Epicurus, the Greek philosopher, 342-270 B.C.) used to ridicule this theorem, saying that even an ass knew it, for if he wished to get food, he walked in a straight line and not along two sides of a triangle. Proclus replied that it was one thing to know the truth and another thing to prove it, meaning that the value of geometry lay in the proof rather than in the mere facts, a thing that all who seek to reform the teaching of geometry would do well to keep in mind. The theorem might simply appear as a corollary under the postulate if it were of any importance to reduce the number of propositions one, more. BOOK I AND ITS PROPOSITIONS 189 For example, making Then If the proposition is postponed until after tliose con- cerning the inequalities of angles and sides of a triangle, there are several good proofs. produce A C to X, CX = CB. AX = ZXBC. .-. ZXBA >ZX. .-. AX > AB. .■.AC+ CB >AB. The above proof is due to Euclid. Heron of Alexandria (first century A.D.) is said by Proclus to have given the following: Let . Then Similarly, CX bisect Z C, ZBXO ZACX. .ZBXOZXCB. .-. CB > XB. AOAX. Adding, AC + CB> AB. Theorem. If two sides of a triangle are unequal, the angles opposite these sides are unequal, and the angle oppo- site the greater side is the greater. Euclid stated this more briefly by saying, "In any tri- angle the greater side subtends the greater angle." This is not so satisfactory, for there may be no greater side. Theorem. // two angles of a triangle are unequal, the sides opposite these angles are unequal, and the side oppo- site the greater angle is the greater. Euclid also stated this more briefly, but less satis- factorily, thus, " In any triangle the greater angle is subtended by the greater side." Students should have their attention called to the fact that these two theorems 190 THE TEACHING OF GEOMETRY are reciprocal or dual theorems, the words " sides " and " angles " of the one corresponding to the words "angles" and " sides " respectively of the other. It may also be noticed that the proof of this proposition in- volves what is known as the Law of Converse ; for (1) ifJ = c, then^B = ZC; (2) if5>c, then ZB > ZC; (3) if6 Z C, then 6 cannot equal c without violating (1), and b cannot be less than c without violating (3), therefore b>c; similarly, if ZB I This proposition is // j a very simple test for i U :> y -\ a parallelogram. It is the principle involved in the case of the common folding parallel ruler, an instrument that has long been 192 THE TEACHING OF GEOMETRY tafiromenta d* ttrtr tatt par^lelie , a\ f7\ \ ( U ^ I , ■ ^ Pakallel Rhler op the Seven- teenth Centuky San Giovanni's " Seconda squara mobile," Vicenza, 1686 recognized as one of the valuable tools of practical geom- etry. It will be of some interest to teachers to see one of the early forms of this parallel ruler, as shown in the illustration.^ If such an instrument is not available in the school, one suitable for illustrative purposes can easily be made from cardboard. A somewhat more complicated form of this instrument may also be made by pupils in manual training, as is shown in this illustration from Bion's great treatise. The prin- ciple involved may be taken up in class, even ' ; ^ ;^>». *' ' " " " <» — ^„ .- ^y:r::^ ~ if the instrument is ' — »-r;---;r-^ -j not used. It is evident that, unless the work- manship is unusu- ally good, this form of parallel ruler is not as accurate as the com- mon one illustrated above. The principle is sometimes used in iron gates. Theorem. Two parallelograms are congruent if two sides and the included angle of the one are equal respec- tively to two sides and the included angle of the other. This proposition is discussed ui connection with the one that follows. 1 Stark, loc. cit. Parallel Eulek of the Eighteenth Century N. Bion's "Traite de la construction . . . des instrumens de math^raatique," The Hague, 1723 BOOK I AND ITS PROPOSITIONS 193 Theorem. If three or more parallels intercept equal segments on one transversal, they intercept equal segments on every transversal. These two propositions are not given in Euclid, although generally required by American syllabi of the present time. The last one is particularly useful in sub- sequent work. Neither one offers any difficulty, and neither has any interesting history. There are, how- ever, numerous in- teresting applica- tions to the last one. One that is used in mechani- cal drawing is here illustrated. If it is desired to divide a line^iJ into five equal parts, we may take a piece of ruled tracing paper and lay it over the given line so that line passes through A, and line 5 through B. We may then prick through the paper and thus determine the points on AB. Similarly, we may divide AB into any other number of equal parts. Among the applications of these propositions is an in- teresting one due to the Arab Al-Nairizi (««. 900 a.'d. ). The problem is to divide a line into any number of equal parts, and he begins with the case of trisecting AB. It may be given as a case of practical drawing even before the problems are reached, particularly if some prelimi- nary work with the compasses and straightedge has been given. 194 THE TEACHING OP GEOMETRY Make BQ and AQ' perpendicular to AB, and make BP = PQ = AP' = P'Q'. Then AXYZ is congruent to A YBP, and also to A XAP'. Therefore AX = XY=YB. In the same way we might continue to produce BQ until it is made up of ra — 1 lengths BP, and so for A Q', and by properly joining points we could divide AB into n equal parts. In particular, if we join P and P', we bisect the line AB. Theorem. If two sides of a quadrilateral are equal and parallel, then the other two sides are equal and parallel, and the figure is a parullelogram. This was Euclid's first proposition on parallelograms, and Proclus speaks of it as the connecting link between the theory of parallels and that of parallelograms. The ancients, writing for mature students, did not add the words " and the figure is a parallelogram," because that follows at once from the first part and from the defi- nition of "parallelogram," but it is helpful to younger students because it emphasizes the fact that here is a test for this kind of figure. Theorem. The diagonals of a parallelogram bisect each other. This proposition was not given in Euclid, but it is usually required in American syllabi. There is often given in connection with it the exercise in which it is proved that the diagonals of a rectangle are equal. When this is taken, it is well to state to the class that carpen- ters and builders find this one of the best checks in lay- ing out floors and other rectangles. It is frequently applied also in laying out tennis courts. If the class is doing any work in mensuration, such as finding the area of the school grounds, it is a good plan to check a few rectangles by this method. BOOK I AND ITS PROPOSITIONS 195 All interesting outdpor application of the theory of parallelograms is the following: Suppose you are on the side of this stream opposite to XY, and wish to measure the length of XV. Run a line AB along the bank. Then take a carpenter's square, or even a large book, and walk along AB until you reach P, a point from which you can just see A' and B along two sides of the square. Do the same for Y, thus fixing P and Q. Using the tape, bisect PQ at M. Then walk along YM produced until you reach a point F' that is ex- actly in line with M and Y, and also with P and A'. Then walk along XM produced until you reach a point A' that is exactly in line with M and A', and also with Q and 1'. Then measure Y'X' and you have the length of XY. For since FA" is ± to PQ, and XT is also J- to PQ, FA" is II to AF'. And since PM = MQ, therefore XM = MX' and Y'M = MY. Therefore Y'X'YX is a parallelogram. The properties of the parallelogram are often applied to proving figures of various kinds congruent, or to con- structing them so c c that they will be congruent. For example, if -"^ we draw A'B' equal and parallel to AB, B'C equal and par- -^ -^ allel to BC, and so on, it is easily proved that ABCD and A'B'CD' are congruent. This may be done by ordinary superposition, or by sliding ABCD along the dotted parallels. There are many applications of this principle of par- allel translation in practical construction work. The prin- ciple is more far-reaching than here intimated, however, and a few words as to its significance will now be in place. 196 THE TEACHING OF GEOMETRY The efforts usually made to improve the spirit of Euclid are trivial. They ordinarily relate to some com- monplace change of sequence, to some slight change ui language, or to some narrow line of applications. Such attempts require no particular thought and yield no very noticeable result. But there is a possibility, remote though it may be at present, that a geometry will be developed that will be as serious as Euclid's and as effective in the education of the thinking individual. If so, it seems probable that it will not be based upon the congruence of triangles, by which so many proposi- tions of Euclid are proved, but upon certain postulates of motion, of which one is involved in the above illus- tration, — the postulate of parallel translation. If to this we join the two postulates of rotation about an axis,' leading to axial symmetry ; and rotation about a point,^ leading to symmetry with respect to a center, we have a group of three motions upon which it is possible to base an extensive and rigid geometry.^ It will be through some such effort as this, rather than through the weak- ening of the Euclid-Legendre style of geometry, that any improvement is likely to come. At present, in America, the important work for teachers is to vitalize the geom- etry they have, — an effort in which there are great possibilities, — seeing to it that geometry is not reduced to mere froth, and recognizing the possibility of another geometry that may sometime replace it, — a geometry 1 Of which so much was made by Professor Olaus Henrici in his "Congruent Figures," London, 1879, — a book that every teacher of geometry should own. 2 Much is made of this in the excellent work by Henrici and Treu1> lein, "Lehrbuch der Geometrie," Leipzig, 1881. ^ M6ray did much f orthis movement in France, and the recent works of Bourlet and Borel have brought it to the front In that country. BOOK I AND ITS PROPOSITIONS 197 as rigid, as thought-compelling, as logical, and as truly educational. Theorem. The sum of the interior angles of a polygon is equal to two right angles, taken as many times less two as the figure has sides. This interesting generalization of the proposition about the sum of the angles of a triangle is given by Proclus. There are several proofs, but all are based upon the possi- bility of dissecting the polygon into triangles. The point from which lines are drawn to the vertices is usually taken at a vertei, so that there are w — 2 triangles. It may how- ever be taken within the figure, making n triangles, from the sum of the angles of which the four right angles about the point must be subtracted. The point may even be taken on one side, or outside the polygon, but the proof is not so simple. Teachers who desire .to do so may sug- gest to particularly good students the proving of the theorem for a concave polygon, or even for a cross poly- gon, although the latter requires negative angles. Some schools have transit instruments for the use of their classes in trigonometry. In such a case it is a good plan to measure the angles in some piece of land so as to verify the proposition, as well as show the care that must be taken in reading angles. In the absence of this exer- cise it is well to take any irregular polygon and measure the angles by the help of a protractor, and thus accom- plish the same results. Theorem. The sum of the exterior angles of a polygon, made by producing each of its sides in succession, is equal to fow right angles. This is also a proposition not given by the ancient writers. We have, however, no more valuable theorem for the purpose of showing the nature and significance 198 THE TEACHING OF GEOMETRY of the negative angle ; and teachers may arouse a great deal of interest in the negative quantity by showing to a class that when an interior angle becomes 180° the exterior angle becomes 0, and when the polygon becomes concave the exterior angle becomes negative, the theo- rem holding for all these cases. We have few better illustrations of the significance of the negative quantity, and few better opportunities to use the knowledge of this kind of quantity already acquired in algebra. In the hilly and mountainous parts of America, where irregular-shaped fields are more common than in the more level portions, a common test for a survey is that of find- ing the exterior angles when the transit instrument is set at the comers. In this field these angles are given, and it will be seen that the sum is 360°. In the absence of any outdoor work a protractor may be used to measure the exterior angles of a polygon drawn on paper. If there is an irregular piece of land near the school, the exterior angles can be fairly well measured by an ordinary paper protractor. The idea of locus is usually introduced at the end of Book I. It is too abstract to be introduced successfully any earlier, although authors repeat the attempt from time to time, unmindful of the fact that all experience is opposed to it. The loci propositions are not ancient. The Greeks used the word "locus" (in Greek, topos), however. Proclus, for example, says, " I call those locus theorems in which the same property is found to exist on the whole of some locus." Teachers should be careful to have the pupils recognize the necessity for proving BOOK I AND ITS PROPOSITIONS 199 two things with respect to any locus : (1) that any point on the supposed locus satisfies the condition ; (2) that any point outside the supposed locus does not satisfy the given condition. The first of these is called the " sufficient condition," and the second the " necessary condition." Thus m the case of the locus of points m a plane equidistant from two given pomts, it is sufficient that the point be on the perpendicular bisector of the line joining the given points, and this is the first part of the proof ; it is also necessary that it be on this line, i.e. it cannot be outside this line, and this is the second part of the proof. The proof of loci cases, therefore, involves a consideration of "the necessary and sufficient condition" that is so often spoken of in higher mathematics. This expression might well be incorporated into elementary geometry, and when it becomes better understood by teachers, it probably will be more often used. In teaching loci it is helpful to call attention to loci in space (meaning thereby the space of three dimensions), without stopping to prove the proposition involved. Indeed, it is desirable all through plane geometry to refer incidentally to solid geometry. In the mensuration of plane figures, which may be boundaries of solid figures, this is particularly true. It is a great defect in most school coiirses in geometry that they are entirely confined to two dimensions. Even if solid geom- etry in the nsual sense is not attempted, every occasion should be taken to liberate boys' minds from what becomes the tyranny of paper. Thus the questions : " What is the locus of a point equi- distant from two given points ; at a constant distance from a given straight line or from a given point ? " should be extended to space.^ i"W. N. Bruce, "Teaching of Geometry and Graphic Algebra in Secondary Schools," Board of Education circular (No. 711), p. 8, London, 1909. 200 THE TEACHING OF GEOMETRY The two loci problems usually given at this time, referring to a point equidistant from the extremities of a given line, and to a point equidistant from two inter- secting lines, both permit of an interesting extension to three dimensions without any formal proof. It is possible to give other loci at this poiat, but it is preferable merely to introduce the subject in Book I, reserving the further discussion until after the circle has been studied. It is well, in speaking of loci, to remember that it is entirely proper to speak of the " locus of a point " or the " locus of points." Thus the locus of a point so moving in a plane as constantly to be at a given distance from a fixed point in the plane is a circle. In analytic geom- etry we usually speak of the locus of a point, thinking of the point as being anywhere on the locus. Some teachers of elementary geometry, however, prefer to speak of the locus of points, or the locus of all points, thus tending to make the language of elementary geom- etry differ from that of analytic geometry. Since it is a trivial matter of phraseology, it is better to recognize both forms of expression and to let pupils use the two interchangeably. CHAPTER XV THE LEADING PROPOSITIONS OP BOOK II Having taken up all of the propositions usually given in Book I, it seems unnecessary to consider as specifi- cally all those in subsequent books. It is therefore proposed to select certain ones that have some special interest, either from the standpoint of mathematics or from that of history or application, and to discuss them as fully as the circumstances seem to warrant. Theoeems. In the same circle or in equal circles equal central angles intercept equal arcs; and of two unequal central angles the greater intercepts the greater arc, and conversely for both of these cases. Euclid made these the twenty -sixth and twenty-seventh propositions of his Book III, but he limited them as fol- lows : " In equal circles equal angles stand on equal cir- cumferences, whether they stand at the centers or at the circumferences, and conversely." He therefore included two of our present theorems in one, thus making the proposition doubly hard for a beginner. After these two propositions the Law of Converse, already mentioned on page 190, may properly be introduced. Theorems. In the same circle or in equal circles, if two arcs are equal, they are subtended by equal chords ; and if two arcs are unequal, the greater is subtended hy the greater chord, and conversely. Euclid dismisses all this with the simple theorem, " In equal circles equal circumferences are subtended by 201 202 THE TEACHING OF GEOMETRY equal straight lines." It will therefore be noticed that he has no special word for " chord " and none for " arc," and that the word " circumference," which some teachers are so anxious to retain, is used to mean both the whole circle and any arc. It cannot be doubted that later writers have greatly improved the language of geometry by the use of these modern terms. The word " arc " is the same, etymologically, as "arch," each being derived from the Latin arcus (a bow). "Chord" is from the Greek, meaning "the string of a musical instrument." "Subtend" is from the Latin sub (under), and tendere (to stretch). It should be noticed that Euclid speaks of "equal circles," while we speak of " the same circle or equal circles," confining our proofs to the latter, on the suppo- sition that this sufficiently covers the former. Theoeem. a line through the center of a circle perpen- dicular to a chord bisects the chord and the arcs subtended by it. This is an improvement on Euclid, III, 3 : " If in a circle a straight line through the center bisects a straight line not through the center, it also cuts it at right angles ; and if it cuts it at right angles, it also bisects it." It is a very important proposition, theoretically and practi- cally, for it enables us to find the center of a circle if we know any part of its arc. A civil engineer, for example, who wishes to find the center of the circle of which some curve (lilce that on a running track, on a railroad, or in a park) is an arc, takes two chords, say of one hundred feet each, and erects perpendicular bisectors. It is well to ask a class why, in practice, it is better to take these chords some distance apart. Engineers often check then- work by taking three chords, the perpendicular bisectors LEADING PROPOSITIONS OF BOOK II 203 of the three passing through a single point. Illustrations of this kind of work are given later in this chapter. Theoeem. In the same circle or in equal circles equal chords are equidistant from the center, and chords equidis- tant from the center are equal. This proposition is practically used by engineers in locating points on an arc of a circle that is too large to be described by a tape, or that cannot easily be reached from the center on account of obstructions. If part of the curve APB is known, take P as the mid-point. Then stretch the tape from A to B and draw PM perpendicular to it. Then swing the length AM about P, and PM about B, until they meet at L, and stretch the length AB along PL to Q. This fixes the point Q. In the same way fix the point C. Points on the curve can thus be fixed as near together as we wish. The chords AB, PQ, BC, and so on, are equal and are equally distant from the center. Theoeem. A line perpendicular to a radius at its extremity is tangent to the circle. The enunciation of this proposition by Euclid is very interesting. It is as follows : The straight line drawn at right angles to the diameter of a circle at its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed ; further, the angle of the semi- circle is greater and the remaining angle less than any acute rectilineal angle. The first assertion is practically that of tangency, — "will fall outside the circle." The second one states, substan- tially, that there is only one such tangent, or, as we say in modem mathematics, the tangent is unique. The third statement relates to the angle formed by the diameter 204 THE TEACHING OF GEOMETRY and the circumference, — a mixed angle, as Proclus called it, and a kind of angle no longer used in elemen- tary geometry. The fourth statement practically asserts that the angle between the tangent and circumference is less than any assignable quantity. This gives rise to a difficulty that seems to have puzzled many of Euclid's commentators, and that will interest a pupil: As the circle diminishes this angle apparently increases, while as the circle increases the angle decreases, and yet the angle is always stated to be zero. Vieta (1540-1603), who did much to improve the science of algebra, attempted to explain away the difficulty by adopting a notion of circle that was prevalent in his time. He said that a circle was a polygon of an infinite number of sides (which it cannot be, by definition), and that a tangent simply coincided with one of the sides, and therefore made no angle with it; and this view was also held by Galileo (1514-1642), the great physicist and mathema- tician who first stated the law of the pendulum. Thboeem. Parallel lines intercept equal arcs on a circle. The converse of this proposition has an interesting application in outdoor work. Suppose we wish to run a line through P parallel to a given line AB. With any convenient point as a center, and OP as a radius, describe a circle cutting AB in Xan.iY. Draw PA'. Then with K as a center and PX as a radius draw an arc cutting the circle in Q. Then run the line from P to Q. PQ is parallel to ABhj the converse of the above theorem, which is easily shown to be true for this figure. LEADING PROPOSITIONS OF BOOK II 205 Thbokem. If two circles are tangent to each other, the line of centers passes through the point of contact. There are many illustrations of this theorem in prac- tical work, as in the case of cogwheels. An interesting application to engineering is seen in the case of two par- allel streets or lines of track which are to be connected by a " reversed ; curve." "^ ^TsTX i^ I >A \, If the lines are I / ^^Ajf.,— -/fo' AB and CD, and iC Tv / j the connection is [ \ \, I to be made, as i .J^^rs^liJ I On shown, from £ to ' '^ u C, we may proceed as follows : Draw BC and bisect it at M. Erect PO, the perpendicular bisector of BM; and BO, perpendicular to AB. Then is one center of curvature. In the same way fix 0'. Then to check the work apply this theorem, M being in the line of centers 00'. The curves may now be drawn, and they will be tangent to ^B, to CD, and to each other. At this point in the American textbooks it is the custom to insert a brief treatment of measurement, ex- plaining what is meant by ratio, commensurable and incommensurable quantities, constant and variable, and limit, and introducing one or more propositions relating to limits. The object of this departure from the ancient sequence, which postponed this subject to the book on ratio and proportion, is to treat the circle jnore com- pletely in Book III. It must be confessed that the treat- ment is not as scientific as that of Euclid, as will be explained under Book III, but it is far better suited to the mind of a boy or girl. It begins by defining measurement in a practical way, as the finding of the number of times a quantity of any kind contains a known quantity of the same kind. Of 206 THE TEACHING OF GEOMETRY course this gives a number, but this number may be a surd, like V 2. In other words, the magnitude measured may be incommensurable with the unit of measure, a seeming paradox. With this difficulty, however, the pupil should not be called upon to contend at this stage in his progress. The whole subject of incommensurables might safely be postponed, although it may be treated in an elementary fashion at this time. The fact that the measure of the diagonal of a square, of which a side is unity, is V2, and that this measure is an incommensu- rable number, is not so paradoxical as it seems, the paradox being verbal rather than actual. It is then customary to define ratio as the quotient of the numerical measures of two quantities in terms of a common unit. This brings all ratios to the basis of numerical fractions,, and while it is not scientifically so satisfactory as the ancient concept which considered the terms as lines, surfaces, angles, or solids, it is more prac- tical, and it suffices for the needs of elementary pupils. " Commensurable," " iacommensurable," " constant," and "variable" are then defined, and these definitions are followed by a brief discussion of limit. It simplifies the treatment of this subject to state at once that there are two classes of limits, — those which the variable actually reaches, and those which it can only approach indefinitely near. We find the one as frequently as we find the other, although it is the latter that is referred to in geometry. For example, the superior limit of a chord is a diameter, and this limit the chord may reach. The inferior limit is zero, but we do not consider the chord as reaching this limit. It is also well to call the attention of pupils to the fact that a quantity may de- crease towards its limit as well as uacrease towards it. LEADING PROPOSITIONS OP BOOK II 207 Such further definitions as are needed in the theory of limits are now introduced. Among these is "area of a circle." It might occur to some pupil that since a circle is a line (as used in modern mathematics), it can have no area. This is, however, a mere quibble over words. It is not pretended that the line has area, but that " area of a circle " is merely a shortened form of the expression " area inclosed by a circle." The Principle of Limits is now usually given as fol- lows : " If, while approaching their respective lunits, two variables are always equal, their limits are equal." This was expressed by D'Alembert in the eighteenth century as " Magnitudes which are the limits of equal magnitudes are equal," or this in substance. It would easily be pos- sible to elaborate this theory, proving, for example, that if X approaches y as its limit, then ax approaches ay as its limit, and - approaches - as its limit, and so on. Very much of this theory, however, wearies a pupil so that the entire meaning of the subject is lost, and at best the treatment in elementary geometry is not rigorous. It is another case of having to sacrifice a strictly scientific treatment to the educational abilities of the pupil. Teach- ers wishing to find a scientific treatment of the subject should consult a good work on the calculus. Theorem. In the same circle or in equal circles two central angles have the same ratio as their intercepted arcs. This is usually proved first for the commensurable case and then for the incommensurable one. The latter is rarely understood by all of the class, and it may very properly be required only of those who show some apt- itude in geometry. It is better to have the others under- stand fully the commensurable case and see the nature 208 THE TEACHING OF GEOMETRY of its applications, possibly reading the incommensurable proof with the teacher, than to stumble about in the dark- ness of the incommensurable case and never reach the goal. In. Euclid there was no distinction between the two because his definition of ratio covered both ; but, as we shall see in Book III, this definition is too difficult for our pupils. Theon of Alexandria (fourth century A.D.), the father of the Hypatia who is the heroine of Kingsley's well-known novel, wrote a commentary on Euclid, and he adds that sectors also have the same ratio as the arcs, a fact very easily proved. In propositions of this type, referring to the same circle or to equal circles, it is not worth while to ask pupils to take up both cases, the proof for either being obviously a proof for the other. Many writers state this proposition so that it reads that " central angles are measured hy their intercepted arcs." This, of course, is not literally true, since we can measure anything only by something of the same kind. Thus we measure a volume by finding how many times it contains another volume which we take as a unit, and we measure a length by taking some other length as a unit; but we cannot measure a given length in quarts nor a given weight in feet, and it is equally impossible to measure an arc by an angle, and vice versa. Nevertheless it is often found convenient to define some brief expres- sion that has no meaning if taken literally, in such way that it shall acquire a meaning. Thus we define " area of a circle," even when we use " circle " to mean a line ; and so we may define the expression " central angles are measured by their intercepted arcs " to mean that central angles have the same numerical measure as these arcs. This is done by most writers, and is legitimate as ex- plaining an abbreviated expression. ' LEADING PROPOSITIONS OF BOOK II 209 Theorem. An inscribed angle is measured hy half the intercepted arc. In Euclid this proposition is combined with the pre- ceding one in his Book VI, Proposition 33. Such a procedure is not adapted to the needs of students to-day. Euclid gave in Book III, however, the proposition (No. 20) that a central angle is twice an inscribed angle stand- ing on the same arc. Since Euclid never considered an angle greater than 180°, his inscribed angle was neces- sarily less than a right angle. The first one who is known to have given the general case, taking the central angle as being also greater than 180°, was Heron of Alexan- dria, probably of the first century A.D.i In this he was followed^ by various later commentators, including Tar- taglia and Clavius in the sbcteenth century. One of the many interesting exercises that may be derived from this theorem is seen in the case of the "horizontal danger angle" ob- served by ships. If sonie dangerous rooks lie off the shore, and L and L' are two lighthouses, the angle A is deter- mined by observation, so that A -sn^x ^~~—-^j will lie on a circle inclosing the '^ dangerous area. Angle A is called the " horizontal danger angle." Ships passing in sight of the two lighthouses L and L' must keep out far enough so that the angle L'SL shall be less than angle A'. To this proposition there are several important corol- laries, including the following : 1. An angle inscribed in a semicircle is a right angle. This corollary is mentioned by Aristotle and is attributed 1 This is the latest opinion. He is usually assigned to the first century b.c. 210 THE TEACHING OF GEOMETRY to Thales, being one of the few propositions with which his name is connected. It enables us to describe a circle by letting the arms of a carpenter's square slide along two nails driven in a board, a pencil being held at the vertex. A more practical use for it is made by machinists to determine whether a casting is a true semicircle. Tak- ing a carpenter's square as here shown, if the vertex touches th curve at every point as the squai sUdes around, it is a true semicircl( By a similar method a circle ma be described by sliding a drafts- man's triangle so that two sides touch two tacks driven in a board. Another interesting application of this corollary may be seen by taking an ordinary paper protractor ACB, and fastening a plumb line at B. If the protractor is so held that the plumb line cuts the semicircle at C, then AC is level because it is perpendicular to the vertical line BC. Thus, if a class wishes to deter- mine the horizontal line AC, while sighting up a hiU in the direction AB, this is easily determined without a spirit level. It follows from this corollary, as the pupil has already found, that the mid-point of the hypotenuse of a right triangle is equidistant from the three vertices. This is useful in outdoor measuring, forming the basis of one of the best methods of letting fall a perpendicular from an external point to a line. LEADING PROPOSITIONS OF BOOK II 211 Suppose JTFto be the edge of a sidewalk, and P a point in the street from which we wish to lay a gas pipe perpendicular to the walk. From P swing a cord or tape, say 60 feet long, imtil it meets XY &t A. Then take M, the mid- point of PA, and swing MP about M, to meet XY &i, B. Then B is the foot of the perpendicular, since Z. PBA can be inscribed in a semi- circle. 2. Angles inscribed in the same segment are equal. By driving two nails in a board, at A and B, and taking an angle P made of rigid material (in particular, as already stated, a carpenter's square), a pencil placed at P will gen- erate an arc of a circle if the arms slide along A and B. This is an interesting exercise for pupils. Theorem. An angle formed hy two chords intersecting within the circle is measured hy half the sum of the intercepted arcs. Theorem. An angle formed hy a tangent and a chord drawn from the point of tangency is meas- ured hy half the intercepted arc. Theorem. An angle formed hy two secants, a secant and a tangent, or two tangents, drawn to a circle from an external point, is measured hy half the difference of the intercepted arcs. These three theorems are all special cases of the gen- eral proposition that the angle included between two lines that cut (or touch) a circle is measured by half the sum of the intercepted arcs. If the point passes from within the circle to the circle itself, one arc becomes zero and the angle becomes an inscribed angle. If the point passes outside the circle, the smaller arc becomes negative, having passed through zero. The point may even " go to 212 THE TEACHING OF GEOMETRY infinity," as is said in higher mathematics, the lines then becoming parallel, and the angle becoming zero, being measured by half the sum of one arc and a negative arc of the same absolute value. This is one of the best illustrations of the Principle of Continuity to be found in geometry. Problem. To let fall a perpendicular upon a given line from a given external point. This is the first problem that a student meets in most American geometries. The reason for treating the prob- lems by themselves instead of mingling them with the theorems has already been discussed.^ The student now has a sufficient body of theorems, by which he can prove that his constructions are correct, and the advantage of treating these constructions together is greater than that of following Euclid's plan of introducing them when- ever needed. Proclus tells us that "this problem was first investi- gated by CEnopides,^ who thought it useful for astron- omy." Proclus speaks of such a line as a gnomon, a common name for the perpendicular on a sundial, which casts the shadow by which the time of day is known. He also speaks of two kinds of perpendicu- lars, the plane and solid, the former being a line per- pendicular to a line, and the latter a liae perpendicular to a plane. It is interesting to notice that the solution tacitly assumes that a certain arc is going to cut the given line in two points, and only two. Strictly speaking, why may it not cut it in only one point, or even in three points ? We really assume that if a straight line is drawn through 1 See page 54. 2 A Greek philosopher and mathematician of the fifth century B.C. LEADING PEOPOSITIONS OF BOOK II 213 a point within a circle, this line must get out. of the circle on each of two sides of the given point, and in getting out it must cut the circle twice. Proclus noticed this assumption and endeavored to prove it. It is better, however, not to raise the question with beginners, since it seems to them like hair-sphtting to no purpose. The problem is of much value in surveying, and teach- ers would do well to ask a class to let fall a perpendic- ular to the edge of a sidewalk from a point 20 feet from the walk, using an ordinary 66-foot or 50-foot tape. Practically, the best plan is to swing 30 feet of the tape about the point and mark the two points of intersection with the edge of the walk. Then measure the distance between the points and take half of this distance, thus fixing the foot of the perpendicular. Problem:. At a given point in a line, to erect a per- pendicular to that line. This might be postponed until after the problem to bisect an angle, since it merely requires the bisection of a straight angle ; but considering the immaturity of the average pupil, it is better given independently. The usual case considers the point not at the extremity of the line, and the solution is essentially that of Euclid. In practice, however, as for example in A surveying, the point may be at the ex- tremity, and it may not be convenient to produce the line. Surveyors sometimes measure PB = 3 ft., and then take 9 ft. of tape, the ends being held at B and P, and the tape being stretched to A, so that PA = 4 ft. and AB = 5 ft. Then B P is a right angle by the Pythagorean Theorem. This theorem not having yet been proved, it cannot be used at this time. 214 THE TEACHING OF GEOMETRY A solution for the problem of erecting a perpendicular from the extremity of a line that cannot be produced, depending, however, on the problem of bisecting an angle, and therefore to be given after that problem, is attributed by Al-Nairizi (tenth century a.d.) to Heron of Alexandria. It is also given by Proclus. Required to draw from P a perpen- dicular to AP. Take X anywhere on '— the line and erect A'FXto AP in the usual manner. Bisect Z.PXY by the line XM. On XY take XN = XP, and draw NM ± to AF. Then draw PM. The proof is evident. These may at the proper time be given as interesting variants of the usual solution. Problem. To bisect a given line. Euclid said " finite straight line," but this wording is not commonly followed, because it will be inferred that the line is finite if it is to be bisected, and we use " line " alone to mean a straight line. Euclid's plan was to con- struct an equilateral triangle (by his Proposition 1 of Book I) on the line as a base, and then to bisect the vertical angle. Proclus tells us that ApoUonius of Perga, who wrote the first great work on conic sections, used a plan which is substantially that which is commonly found in textbooks to-day, — constructing two isosceles tri- angles upon the line as a common base, and connecting their vertices. Pkoblem. To bisect a given angle. It should be noticed that in the usual solution two arcs intersect, and the point thus determined is connected with the vertex. Now these two arcs intersect twice, and since one of the points of intersection may be the vertex LEADING PROPOSITIONS OF BOOK II 215 itself, the other point of intersection must be taken. It is not, however, worth while to make much of this matter with pupils. Proclus calls attention to the possible sug- gestion that the point of intersection may be imagined to lie outside the angle, and he proceeds to show the absurdity ; but here, again, the subject is not one of value to beginners. He also contributes to the history of the trisection of an angle. Any angle is easily trisected by means of certain higher curves, such as the conchoid of Nicomedes (ca. 180 B.C.), the quadratrix of Hippias of Elis {ca. 420 B.C.), or the spiral of Archimedes (ca. 250 B.C.). But since this problem, stated algebraically, re- quires the solution of a cubic equation, and this involves, geometrically, findiag three points, we cannot solve the problem by means of straight lines and circles alone. In other words, the trisection of any angle, by the use of the straightedge and compasses alone, is impossible. Special angles may however be trisected. Thus, to trisect an angle of 90° we need only to construct an angle of 60°, and this can be done by constructing an equilateral tri- angle. But while we cannot trisect the angle, we may easily approximate trisection. For since, in the infinite geometric series ^ + -j + ^V + li^ "* ' * = a -^ (1 — r), we have ^ = \-^\ = \- In other words, if we add ^ of the angle, \ of the angle, -^-^ of the angle, and so on, we approach as a limit -I of the angle ; but all of these fractions can be obtained by repeated bisec- tions, and hence by bisections we may approximate the trisection. The approximate bisection (or any other division) of an angle may of course be effected by the help of the protractor and a straightedge. The geometric method is, however, usually more accurate, and it is advantageous 216 THE TEACHING OF GEOMETRY to have the pupils try both plans, say for bisecting an angle of about 491-°. Applications of this problem are numerous. It may be desired, for example, to set a lamp-post on a line bisect- ing the angle formed by two streets that come together a little unsymmetrically, as here shown, in which case the bisect- ing line can easily be run by the use of a measuring tape, or even of a stout cord. A more interesting illustration is, however, the following: Let the pupils set a stake, say about 5 feet high, at a point N on the school grounds about 9 a.m., and carefully measure the length of the shadow, NW, placing a small wooden pin at W. Then about 3 p.m. let them watch until the shadow NE is exactly the same length that it was when W was fixed, and then place a small wooden pin at E. If the work has been very carefully done, and they take the tape and bisect the line WE, thus fixing the line NS, they will have a north and south line. If this is marked out for a short distance from iV, then when the shadow falls on NS, it will be noon by sun time (not standard time) at the school. Peoblem. From a given point in a given line, to draw a line making an angle equal to a given angle. Proclus says that Eudemus attributed to CEnopides the discovery of the solution which Euclid gave, and which is substantially the one now commonly seen in textbooks. The problem was probably solved in some fashion before the time of CEnopides, however. The object of the prob- lem is primarily to enable us to draw a line parallel to a given line. LEADING PROPOSITIONS OF BOOK II 217 Practically, the drawing of one line parallel to another is usually effected by means of a parallel ruler (see page 191), or by the use of draftsmen's triangles, as here shown, or even more commonly by the use of a T-square, such as is here seen. This illustration shows two T-squares used for draw- ing lines parallel to the sides of a board upon which the drawing paper is fastened.^ ^ An ingenious instrument de- scribed by Baron Dupin is illus- trated below. To the bar A is fastened the slid- ing check B. A movable check D may be fastened by a screw C. A sharp point is fixed in B, so that as D slides along the edge of a board, the point marks a line parallel to the edge. Moreover, Fand G are two brass arms of equal length joined by a pointed screw H that marks a line midway be- tween B and D. Furthermore, it is evident that ffwill draw a line bisecting any irregular board if the cheeks B and D are kept in contact with the irregular edges. Book II offers two general lines of application that may be introduced to advantage, preferably as additions to the textbook work. One of these has reference to topo- graphical drawing and related subjects, and the other to geometric design. As long as these can be introduced ^This illustration and the following two are from C. Dupin, " Mathematics Practically Applied," translated from the French by G. Birkbeck, Halifax, 1854. This is probably the most scholarly attempt ever made at constructing a " practical geometry." 218 THE TEACHING OF GEOMETRY to the pupil with an air of reality, they serve a good pur- pose, but if made a part of textbook work, they soon come to have less interest than the exercises of a more abstract character. If a teacher can relate the problems in topographical drawing to the pupil's home town, and can occasionally set some outdoor work of the nature here suggested, the results are usually salutary ; but if he reiterates only a half-dozen simple propositions time after time, with only slight changes in the nature of the application, then the results will not lead to a cultiva- tion of power in geometry, — a point which the writers on applied geometry usually fail to recognize. One of the simple applica- tions of this book relates to the rounding of corners in laying out streets in some of our modern towns where there is a desire to depart from the conventional square corner. It , is also used in lay- qj ing out park walks and drives. The figure in the middle of the page represents two streets, AP and BQ, that would, if pro- longed, intersect /B at C. It is re- quired to con- struct an arc so that they shall begin to curve at P and Q, where CP= CQ, and hence the "center of curvature " must be found. LEADING PROPOSITIONS OF BOOK 11 219 The problem is a common one in railroad work, only here AP is usually oblique to BQ ii they are produced to meet at C, as in the second figure on page 218. It is required to construct an arc so that the tracks shall begin to curve at P and Q, where CP = CQ. The problem becomes a little more complicated, and correspond- ingly more inter- esting, when we have to find the center of curva- ture for a street railway track that must turn a corner in such a way as to allow, say, exactly 5 feet from the point P, on account of a side- walk. The problem be- comes still more dif- ficult if we have two roads of different widths that we wish to join on a curve. Here the two centers of curvature are not the same, and the one road narrows to the other on the curve. The solutions will be understood from a study of the figures. The number of problems of this kind that can easily be made is limitless, and it is well to avoid the danger 220 THE TEACHING OF GEOMETRY of hobby riding on this or any similar topic. Therefore a single one will suffice to close this group. If a road ^ B, on an arc described X T about 0, is to be joined to road CD, described about 0', the arc BC should evidently be internally tan- gent to AB and externally tangent to CD. Hence the center is on BOX and O'CY, and is therefore at p. The problem becomes more real if we give some width to the roads in making the drawing, and imagine them in a park that is being laid out with drives. , j)/ It will be noticed that the above prob- lems require the erect- ing of perpendiculars, the bisecting of angles, and the application of the propositions on tan- gents. A somewhat differ- ent line of problems is that relating to the pass- ing of a circle through three given points. It is very easy to manu- facture problems of this kind that have a sem- blance of reality. For example, let it be required to plan a driveway from the G to the porch P so as to avoid a mass of rocks R, an arc LEADING PROPOSITIONS OF BOOK II 221 of a circle to be taken. Of course, if we allow pupils to use the Pythagorean Theorem at this time (and for metrical purposes this is entirely proper, because they have long been familiar with it), then we may ask not only for the drawing, but we may, for example, give the length from G to the point on K (which we may also call R), and the angle RGO as 60°, to find the radius. A second general line of exercises adapted to Book II is a continuation of the geometric drawing recommended as a preliminary to the work in demonstrative geometry. The copying or the making of designs requiring the de- scribing of circles, their inscription in or circumscription about triangles, and their construction in various posi- tions of tangency, has some value as applying the vari- ous problems studied in this book. For a number of years past, several enthusiastic teachers have made much of the designs found in Gothic windows, having their pupils make the outline drawings by the help of com- passes and straightedge. While such work has its value, it is liable soon to degenerate into purposeless formal- ism, and hence to lose interest by taking the vigorous mind of youth from the strong study of geometry to the weak manipulation of instruments. Nevertheless its value should be appreciated and conserved, and a few illustra- tions of these forms are given in order that the teacher may have examples from which to select. The best way of using this material is to offer it as supplementary work, using much or little, as may seem best, thus giving to it a freshness and interest that some have trouble in imparting to the regular book work. The best plan is to sketch rapidly the outline of a window on the blackboard, asking the pupils to make a rough drawing, and to bring in a mathematical drawing on the following day. 222 THE TEACHING OF GEOMETRY It might be said, for example, that in planning a Gothic win- dow this drawing is needed. The arc BC is drawn with ^ as a center and AB as & radius. The small arches are described with A, D, and B as centers and AD as. a radius. The center P is found by taking A and B as centers SbndAE as a radius. How may the points D, E, and F be found ? Draw the figure. From the study of the recti- linear figures suggested by such a simple pattern the properties of the equilateral triangle may be inferred. The Gothic window also offers some interesting pos- sibilities in connection with the study of the square. For example, the illustration given on page 223 shows a number of traceries involving the construction of a square, the bisecting of angles, and the describing of circles.^ The properties of the square, a figure now easily constructed by the pupils, are not numerous. What few there are may be brought out through the study of art forms, if desired. In case these forms are shown to a class, it is important that they should be selected from good de- signs. We have enough poor art in the world, so that geom- etry should not contribute any more. This illustration is a type of the best medieval Gothic parquetry.^ 1 This illustration and others of the same type used in this work are from the excellent drawings by R. W. Billings, in "The Infinity of Geometric Design Exemplified," London, 1849. 2 From H. Kolb, " Der Ornamentenschatz . . . aus alien Kunst^ Epochen," Stuttgart, 1883. The original is in the Church of Saint Anastasia in Verona. (ft" --^ ----- '"^M \ -''.-^i-....li.i I ..'.^..iM Gothic Designs EJiPLOvixii Cikules and Bisected Angles 223 224 THE TEACHING OP GEOMETRY Even simple designs of a semipuzzling nature have their advantage in this connection. In the following example the inner square contains all of the triangles, the letters showing where they may be fitted. ^ Still more elaborate designs, based chiefly upon the square and circle, are shown in the window traceries on page 225, and others will be given in connection with the study of the regular polygons. Designs like the figure below are typical of the simple forms, based on the square and circle, that pupils may profitably incorporate in any work in art design that they may be doing at the time they are studying the circle and the problems relating to perpen- diculars and squares. Among the applications of the problem to draw a tan- gent to a given circle is the case of the common tangents to two given circles. Some authors give this as a basal problem, although it is more commonly given as an exercise or a corollary. One of the most obvious applications of the idea is that relating to the transmission of circular motion by means of a band over two wheels,^ ^and B, as shown on page 226. 1 From J. Bennett, " The Arcanum ... A Concise Theory of Prac- ticable Geometry," London, 1838, one of the many books that have assumed to revolutionize geometry by making it practical. 2 The figures are from Dupin, loc. cit. ^ Gothic Designs employing Circles and Bisected Angles 225 226 THE TEACHING OF GEOMETRY The band may either not be crossed (the case of the two exterior tangents), or be crossed (the interior tangents), the latter allowing the wheels to turn in opposite directions. In ease the band is liable to change its length, on account of stretching or variation in heat or moisture, >--;«wilf a third wheel, Z), is used. "We then have the case of tangents to three pairs of circles. Illustrations of this nature make the exercise on the drawing of common tangents to two circles assume an appearance of genuine reality that is of advantage to the work. CHAPTER XVI THE LEADING PROPOSITIONS OF BOOK III In the American textbooks Book III is usually as- signed to proportion. It is therefore necessary at the beginning of this discussion to consider what is meant by ratio and proportion, and to compare the ancient and the modern theories. The subject is treated by Euclid in his Book V, and an anonytaious commentator has told us that it " is the discovery of Eudoxus, the teacher of Plato." Now proportion had been known long before the time of Eudoxus (408-355 B.C.), but it was numer- ical proportion, and as such it had been studied by the Pythagoreans. They were also the first to study seriously the incommensurable number, and with this study the treatment of proportion from the standpoint of rational numbers lost its scientific position with respect to geom- etry. It was because of this that Eudoxus worked out a theory of geometric proportion that was uidependent of number as an expression of ratio. The following four definitions from Euclid are the basal ones of the ancient theory : A ratio is a sort of relation in respect of size between two magnitudes of the same kind. Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples 227 228 THE TEACHING OF GEOMETRY whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equi- multiples respectively taken in corresponding order. Let magnitudes which have the same ratio be called propor- tional.*^ Of these, the first is so loose in statement as often to have been thought to be an interpolation of some later writer. It was probably, however, put into the original for the sake of completeness, to have some kind of state- ment concerning ratio as a preliminary to the important definition of quantities in the same ratio. Like the defi- nition of " straight line," it was not intended to be taken seriously as a mathematical statement. The second definition is intended to exclude zero and infinite magnitudes, and to show that incommensurable magnitudes are included. The third definition is the essential one of the ancient theory. It defines what is meant by saying that magni- tudes are in the same ratio ; in other words, it defines a proportion. Into the merits of the definition it is not proposed to enter, for the reason that it is no longer met in teaching in America, and is practically abandoned even where the rest of Euclid's work is in use. It should be said, however, that it is scientifically correct, that it covers the case of incommensurable magnitudes as well as that of commensurable ones, and that it is the Greek forerunner of the modern theories of irrational numbers. As compared with the above treatment, the one now given in textbooks is unscientific. We define ratio as "the quotient of the numerical measures of two quantities of the same kind," and proportion as "an equality of ratios." ^ For a very full discussion of these four definitions see Heath's "Euclid," Vol. II, p. 116, and authorities there cited. LEADING PROPOSITIONS OF BOOK III 229 But what do we mean by the quotient, say of V2 by VS ? And when we multiply a ratio by V5, what is the mean- uig of this operation ? If we say that V2 : Vs means a quotient, what meaning shall we assign to " quotient " ? If it is the number that shows how many times one num- ber is contained in another, how many times is Vs con- tained in V 2 ? If to multiply is to take a number a certain number of times, how many times do we take it when we multiply by V5 ? We certainly take it more than 2 times and less than 3 times, but what meaning can we assign to VS times ? It will thus be seen that our treatment of proportion assumes that we already know the theory of irrationals and can apply it to geo- metric magnitudes, while the ancient treatment is inde- pendent of this theory. Educationally, however, we are forced to proceed as we do. Just as Dedekind's theory of numbers is a simple one for college students, so is the ancient theory of pro- portion ; but as the former is not suited to pupils in the high school, so the latter must be relegated to the college classes. And in this we merely harmonize educational progress with world progress, for the numerical theory of proportion long preceded the theory of Eudoxus. The ancients made much of such terms as duplicate, triplicate, alternate, and inverse ratio, and also such as composition, separation, and conversion of ratio. These entered into such propositions as, "If four magnitudes are proportional, they will also be proportional alter- nately." In later works they appear in the form of "proportion by composition," "by division," and "by composition and division." None of these is to-day of much importance, since modern symbolism has greatly simplified the ancient expressions, and in particular the 230 THE TEACHING OF GEOMETRY proposition concerning " composition and division " is no longer a basal theorem in geometry. Indeed, if our course of study were properly arranged, we might well relegate the whole theory of proportion to algebra, allowing this to precede the work in geometry. We shall now consider a few of the principal propo- sitions of Book III. Theorem. If a line is drawn through two sides of a triangle parallel to the third side, it divides those sides proportionally. In addition to the nsual proof it is instructive to con- sider in class the cases in which the parallel is drawn tlirough the two sides produced, either below the base or above the vertex, and also in which the parallel is drawn through the vertex. Theorem. The bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides. The proposition relating to the bisector of an exterior angle may be considered as a part of this one, but it is usually treated separately in order that the proof shall appear less involved, although the two are discussed to- gether at this time. The proposition relating to the ex- terior angle was recognized by Pappus of Alexandria. If ^SC is the given triangle, and CP^, CP^ are respectively the internal and external bisectors, then AB is divided har- monically by Pj and P^. .■.AP,:P^B = AP,:P,B. .-. AP^ : P,B = AP, - P^P, : P^P^ - P,B, and this is the criterion for the harmonic progression still seen in many algebras. For, letting AP^ = a, P^P^ = b, P^B = c, we have a a — h LEADING PROPOSITIONS OF BOOK III 231 which is also derived from taking the reciprocals of a, h, c, and placing them in an arithmetical progression, thus : whence b a c b a — b b — c ab he % — b ah a b — c be c This is the reason why the line AB is, said to be divided har- monically. The line P^P^ is also called the harmonie mean between AP^ and P^B, and the points A, P^, B, P^ are said to form an harmonie range. It may be noted that ZP^CP^, being made up of halves of two supplementary angles, is a right angle. Furthermore, if the ratio CA : CB is given, and ^iJ is given, then Pj and P^ are both fixed. Hence C must lie on a semicircle with P^P^ as a diameter, and therefore the locus of a point such that its distances from two given points are in a given ratio is a circle. This fact, Pappus tells us, was known to ApoUonius. At this point it is customary to define similar poly- gons as such as have their corresponding angles equal and their corresponding sides proportional. Aristotle gave substantially this definition, • saying that such fig- ures have " their sides proportional and their angles 232 THE TEACHING OF GEOMETRY equal." Euclid improved upon this by saying that they must "have their angles severally equal and the sides about the equal angles proportional." Our present phraseology seems clearer. Instead of " corresponding angles" we may say "homologous angles," but there seems to be no reason for using the less familiar word. It is more general to proceed by first considering similar figures instead of similar polygons, thus including the most obviously sim- ilar of all figures, — two circles ; but such a procedure is felt to be too difficult by many teachers. By this plan we first define similar sets of points, A^, A^, A^ as such that A^A^, and A^O = £,0 GA- ., and B^, B^, B^, ■■■, are concurrent in 0, CO=-.. Here the constant ratio A^O: A^O is called the ratio of similitude, and is called the center of similitude. Having defined similar sets of points, we then define similar figures as those figures whose points form similar sets. Then the two circles, the four triangles, and the three quadrilaterals LEADING PROPOSITIOXS OF BOOK III 233 respectively are similar figures. If the ratio of similitude is 1, the similar figures become symmetric figures, and they are therefore congruent. All of the propositions relating to similar figures can be proved from this definition, but it is customary to use the Greek one instead. Among the interesting applications of similarity is the case of a shadow, as here shown, where the light is the center of simili- tude. It is also well known to most high- ^i school pupils that in Qi a camera the lens reverses the image. The mathematical arrangement is here F' shown, the lens inclos- ing the center of simil- itude. The proposition may also be appUed to the enlargement of maps and working drawings. The propositions con- cerning similar figures have no particularly interesting history, nor do they present any difficulties that call for discussion. In schools where there is a little time for trigonometry, teachers sometimes find it helpful to begin such work at this time, since all of the trigonometric functions depend upon the properties of similar trian- gles, and a brief explanation of the simplest trigono- metric functions may add a little interest to the work. In the present state of our curriculum we cannot do more than mention the matter as a topic of general interest in this connection. 234 THE TEACHING OF GEOMETRY It is a mistaken idea that geometry is a prerequisite to trigonometry. We can get along very well in teach- ing trigonometry if we have three propositions : (1) the one about the sum of the angles of a triangle ; (2) the Pythagorean Theorem ; (3) the one that asserts that two right triangles are similar if an acute angle of the one equals an acute angle of the other. For teachers who may care to make a little digression at this time, the following brief statement of a few of the facts of trigo- nometry may be of value : In the right triangle OAB we shall let AB = y, OA = x, OB = r, thus adopting the letters of higher mathematics. Then, so long as ZO remains the same, such ^ ratios as - , - , etc., will remain the same, X r whatever is the size of the triangle. Some of these ratios have special names. For example, we call - the sine of 0, and we write sin = - ;' r r - the cosine of O, and we write cos = - ; r T - the tangent of 0, and we write tan = -• Now because sin = -, therefore )• sin = y; r X and because cos = -, therefore r cos = x; r y and because tan = -> therefore x tan = y. X Hence, if we knew the values of sin 0, cos 0, and tan for the various angles, we could find x, y, or r if we knew any one of them. Now the values of the sine, cosine, and tangent {functions of the angles, as they are called) have been computed for the various angles, and some interest may be developed by obtaining them LEADING PROPOSITIONS OF BOOK III 235 by actual measurement, using the protractor and squared paper. Some of those needed for such angles as a pupil in geometry is likely to use are as follows : Angle Sine Cosine Tangent Angle Sine Cosine Tangent 5° .087 .996 .087 50° .766 .643 1.192 10° .174 .985 .176 55° .819 .574 1.428 15° .259 .966 .268 60° .866 .500 1.732 20° .342 .940 .364 65° .906 .423 2.145 25° .423 .906 .466 70° .940 .342 , 2.748 30° .500 .866 .577 75° .966 .259 3.732 35° .574 .819 .700 80° .985 .174 5.671 40° .643 .766 .839 85° .996 .087 11.430 45° .707 .707 1.000 90° 1.00 .000 00 It will of course be understood that the values are correct only to the nearest thousandth. Thus the cosine of 5° is 0.99619, and the sine of 85° is 0.99619. The entire table can be copied by a class in five minutes if a teacher wishes to introduce this phase of the work, and the author has frequently assigned the comput- ing of a simpler table as a class exercise. Referring to the figure, if we know that ?• = 30 and ZO = 40°, then since y = r sin 0, we have ?/ = 30 x 0.643 = 19.29. If we know that a: = 60 and ZO = 35°, then since y = x tan 0, we have X y = QO X 0.7 — 42. We may also find r, for cos = -, whence X GO '' cos 0.819 = 73.26. Therefore, if we could easily measure zlO and could measure the distance x, we could find the height of a istj THE TEACIIIXG OF GEOJIETKY Ijuilding y. In trigonometry we use a transit for meas- uring angles, but it is easy to measure tlieni with sufficient accuracy for illustrative purposes by placing an ordinary paper protractor upon something level, so that the center comes at the edge, and then sighting along a ruler held w M ft i iii iMii i '' ' • i iii r" ^ i rr""'" ! ' A Qi:aiikant of the Sixteenth Centurv Finaeiis's '* De re et praxi geometrica," Paris, 15r>ti against it, so as to find the angle of elevation of a build- ing. We may then measure the distance to the building and apply the formula y = x tan 0. It should always be understood that expensive appa- ratus is not necessary for such illustrative work. The telescope used on the transit is only three hundred years LEADIXG TKOPOSITIOXS (_)F BOOK III 237 old, and the world got along very well with its trigo- nometry before that was mvented. So a little ingenuity will enable any one to make from cheap protractors about as satisfactory instruments as the world used before 1600. A QlfADRANT 01" THE SEVENTEENTH CeNTCKY In order that this may be the more fully appreciated, a few illustrations are here given, showing the old instru- ments and methods used in practical surveyuig before the eighteenth century. The illustration on page 236 shows a simple form of the quadrant, an instrument easily made bjr a pupil who 238 THE TEACHING OF GEOMETRY may be interested iii outdoor work. It was the common surveying instrument of the early days. A more elab- orate example is seen in the illustration, on page 237, of a seventeenth-century brass specimen in the author's collection.! A Quadrant of the Seventeenth Century Bartoli's " Del modo di misurare," Venice, 1689 Another type, easily m,ade by pupils, is shown in the above illustration from Bartoli, 1689. Such instruments were usually made of wood, brass, or ivory.^ Instruments for the running of lines perpendicular to other lines were formerly common, and are easily made. They suffice, as the following illustration shows, for surveying an ordinary field. 1 These two and several which follow are from Stark, loc. cit. 2 The author has a beautiful ivory specimen of the sixteenth century. LEADIXi; rUOrOSITIONS OF BOOK 111 2iQ The quadrant was practicall}' used for all sorts cif outdoor measurmg. For example, the illustration from Surveying Instrumknt of the Eighteenth Century N. Biou's " Traite de la construction , ties instrumens de mathema- tique, Finaeus, on this page, sliows how it was used for altitudes, and the one reproduced on page 240 sliows how it was used for measuring depths. A similar instru- ment from the work ' of r>ettinus is given on page 241, the dis- tance of a ship being found hy construct- ing an isosceles trian- gle. A more elaborate form, with a pendu- lum attachment, is seen in the illustra- The Hasue, 1723 The Quadrant used eor Altitudes Finaeus's " De re et praxi geometrica, " Paris, 1556 tion from De Judaeis, which also appears on page 241. 240 THE TEACHING OF UEOMETUY Alia enirdrm obfeniaiioFiis Nocaniium. Srcufidus mot dusmeiicndl p'ofunda.ptr quvlrantcm. ORONTII FINEI DELPH. j^primi clemciitorumEucbdis facile manifcfl-atur . &: angulus A B H , an* euloAC F cflaequalisCnamutcrcpredh]s)igiturper 4fextidufdcmEuclidis, ft ficut H B ad B A, itaT G putei ladtudoad G A compoGtam ex G B 8c B A longi^ rudinem.fiue profundicatcm. Sic exempli oratia B H 20 partium^qualium latus quadratf eft 6o:b e aute me* tiatur,& fie in exemplum 6 cubitorum,tot etiam cubitonim crit C F:funt cnim la tera peralklogrammi B E F G oppo(Tta,quajper ^4 dufdem primi funt inuicem asq ualia.Duc igitur 6 in 6o,fient ^^orquze iuide per lo, &.habebis pro quotie<: re iS.Tot igitur cubicore crit A G: a quafi dempferis A B trium ucr^ bi oratia cubitorum, relinquptur B defyderata & in profundum depnTa pucei logiiudo ly cubiroR!, IDEM Q.V O Q.V E SIC O B.= tinebis. Mecirc H E: Htc^ exempli caufa J cubicoru. DeindemultiJ? plica 5 per 6o,fient jooihxc diui per lOjproducentur ly.uelut an* tea.Bina nancp triangula A B H ec H E F funi rurfum aequiangula- quoniam angulus A H B angulo E H F ad uerricem pofiro , per ly primi Euclidis eft ^qualis.ice re-' RELIQ.VVMETS, VT 4 eandem rerum in profundudcs prefTarum, per uulgaiu quadra^ tem mcn'ri doceamus altitudine. Sic itacp puteus circularis E F c HjCuius diamerer fit E F,aut illi apqualiSG H.Adplica igitur qua^ dratem ipfi purei orificio: in hue modu,ut finis lateris A D ad datu pUHfflum E conftiruatur. Leua poftmodu,autdcprime quadra^i temCh'bero Temper dcmi/To per:= pcndiculo)donec radius uifualis per ambo foramina pinnacidioRi ad infcriorem&e diametrofi- gnaturcrminSHperducat.Quo fafto & immoto quadrate, uide in qua The Quadkast used for Depths Finaeus's " Protomathesis," Paris, 1532 LJ'ADIXC; rUOPOSITIOXS OF BOOK III 241 A QlAIIEAXT OF THE SIXTEENTH CEXTfEV De Judaeis's " De qiiarlrante geometiico," Niiriiberg, IHM The Qlaijuast usei> iou Distances Bettinus's "Apiaria universae philosophiae luathematk-ae," Bologna, 1(J45 24i THE TEACHING OF GEOMETRY Tlie quadrant finall}' developed into the octant, as sliown in the following illustration from Jloffniann, and this in turn developed into the sextant, which is now used hy all navigators. Huit'maim's " De Octantis," Jena, 1512 In connection with this general subject the use of the speculum (mirror) in measuring heights should be mentioned. The illustration given on page 243 shows how in early days a simple device was used for this pur- jxjse. Two similar triangles are formed in this way, and we have only to measure the height of the eye above the ground, and the distances of the mirror from the tower and the observer, to have three terms of a proportion. All of tliese instruments are easily made. The mir- I'or is always at hand, and a paper protractor on a LEADING PROPOSITIONS OF BOOK III 243 piece of board, with a plumb line attached, serves as a quadrant. For a few cents, and by the expendi- ture of an hour or so, a school can have almost as good instruments - - as the ordinary surveyor had be- fore the nineteenth century. A well-known method of meas- uring the distance across a stream is illustrated in the figure below, where the distance from A to gome pointPis required. The Speculum Finaeus's " De re et praxi geometrica, " Paris, 1556 Run a line from A to Chj standing at C in line with A and P. Then run two perpendiculars from A and C by any of the meth- ods already given, — sighting on a protractor or along the edge of a book if no better means are at hand. Then sight from some point D, on CD, to P, putting a stake at B. Then run the perpendicular BE. Since DE : EB = BA : AP, and since we can measure DE, EB, and BA with the tape, we can compute the distance AP. There are many variations of this scheme of measuring distances by means of similar triangles, and pupils may be encouraged to try some of them. Other figures are suggested on page 244, and the triangles need not be confined to those having a right angle. 244 THE TEACHING OF GEOMETRY A very simple illustration of the use of similar triangles is found in one of the stories told of Thales. It is re- lated that he found the height of the pyramids by meas- uring their shadow at the instant when his own shadow just equaled his height. He thus had the case of two similar isosceles triangles. This is an interesting exercise which may be tried about the time that pupils are leav- ing school in the afternoon. Another application of the same principle is seen in a method often taken for measuring the height of a tree. The observer has a large right triangle made of wood. Such a triangle is shown in the picture, in which AB=BC. He holds AB level and walks toward the tree until he just sees the top along A C. Then because AB = BC, and AB:BC=AD:DE, the height above D will equal the distance AD. Questions like the fol- "■■ lowing may be given ^4 to the class : i^Jr -B 1. What is the height of the tree in the picture if the ««^U»MA'.-■tt^v.>av.JkJ,.^'?^■''t^i3ig^.^lMW••i«:S' triangle is 5 ft. 4 in. from the ground, and AD is 23 ft. 8 in. ? 2. Suppose a triangle is used which has AB = twice BC. What is the height if AD = 75 ft. ? LEADING PROPOSITIONS OF BOOK III 245 There are many variations of this principle. One con- sists in measuring the shadows of a tree and a staff at the same time. The height of the staff being known, the height of the tree is found by proportion. Another con- sists in sighting from the ground, across a mark on an upright staff, to the top of the tree. The height of the mark being known, and the distances from the eye to the staff and to the tree being measured, the height of the tree is found. An instrument sold by dealers for the measuring of heights is known as the hypsometer. It is made of brass, and is of the form here shown. •The base is graduated in equal divisions, say 50, and the upright bar is similarly divided. At the ends of the hinged radius are two sights. If the observer stands 50 feet from a tree and sights at the top, so that the hinged radius cuts the upright bar at 27, then he knows at once that the tree is 27 feet high. It is easy for a class to make a fairly good instrument of this kind out of stiff pasteboard. An interesting application of the theorem relating to similar triangles is this : Extend your arm and point to a distant object, closing your left eye and sighting across your finger tip with your right eye. Now keep the finger in the same position and sight with your left eye. The finger will then seem to be pointing to an object some distance to the right of the one at which you were point- ing. If you can estimate the distance between these two objects, which can often be done with a fair degree of accuracy when there are houses intervening, then you will be able to tell approximately your distance from the 246 THE TEACHING OE GEOMETRY objects, for it will be ten times the estimated distance between them. The finding of the reason for this by measuring the distance between the pupils of the two eyes, and the distance from the eye to the finger tip, and then drawing the figure, is an interesting exercise. Perhaps some pupil who has read Thoreau's descrip- tions of outdoor life may be iaterested in what he says of his crude mathematics. He writes, " I borrowed the plane and square, level and dividers, of a carpenter, and with a shingle contrived a rude sort of a quadrant, with pins for sights and pivots." With this he measured the heights of a cliff on the Massachusetts coast, and with similar home-made or school-made instruments a pupil in geometry can measure most of the heights and dis- tances in which he is interested. Theorem. If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse : 1. The triangles thus formed are similar to the given triangle, and are similar to each other. 2. The perpendicular is the mean proportional between the segments of the hypotenuse. 3. Each of the other sides is the mean proportional between the hypotenuse and the segment of the hypotenuse adjacent to that side. To this important proposition there is one corollary of particular interest, namely, The perpendicular from any point on a circle to a diameter is the mean proportional between the segments of the diameter. By means of this corollary we can easily construct a line whose numerical value is the square root of any number we please. Thus we may make AD = 2 in., DB = 3 in., and erect DC ± to AB. Then the length of DC will be V6 in., and we may- find V6 approximately by measuring DC. LEADING PROPOSITIONS OF BOOK III 247 Furthermore, if we introduce negative magnitudes into geom- etry, and let DB = + 3 and DA = — 2, then DC will equal V— 6. In other words, we have a justification for representing imaginary quantities by lines perpendicular to the line on which we repre- sent real quantities, as is done in the graphic treatment of imaginaries in algebra. It is an interesting exercise to have a class find, to one decimal place, by measuring as above, the value of V2, VS, V5, and V9, the last being integral. If, as is not usually the case, the class has studied the complex number, the absolute value of V— 6, V — 7, . . . , may be found in the same way. A practical illustration of the value of the above theorem is seen in a method for finding distances that is frequently described in early printed books. It seems to have come from the Roman surveyors. If a carpenter's square is put on top of an upright stick, as here shown, and an observer sights along the arms to a distant point B and a point A near the stick, then the two triangles are similar. Hence AD:DC=DC:DB. Hence, ii AD and DC are measured, DB can be found. The experiment is an interesting and instructive one for a class, especially as the square can easily be made out of heavy pasteboard. Theorem. If two chords intersect within a circle, the product of the segments of the one is equal to the product of the segments of the other. Theoeem. If from a point without a circle a secant and a tangent are drawn, the tangent is the mean propor- tional between the secant and its external segment. 248 THE TEACHING OF GEOMETRY Corollary. If from a point without a circle a secant is drawn, the product of the secant and its external segment is constant in whatever direction the secant is drawn. These two propositions and the corollary are all parts of one general proposition : If through a point a line is drawn cutting a circle, the product of the segments of the line is constant. If P is within the circle, then xx'= yf; HP is on the circle, then X and y become 0, and ■ x'= ■ y'= ; if P is at P^, then X and y, having passed through 0, may be considered negative if we wish, although the two p negative signs would cancel _ out in the equation ; if P is at P^, then y = y', and we have xx'= y% ov x : y = y : x', as stated in the proposition. We thus have an ex- cellent example of the Principle of Continuity, and classes are always interested to consider the result of letting F assume various positions. Among the pos- sible cases is the one of two tangents from an exter- nal point, and the one where P is at the center of the circle. Students should frequently be questioned as to the meaning of " product of lines." The Greeks always used " rectangle of lines," but it is entirely legitimate to speak of "product of lines," provided we define the. expression consistently. Most writers do this, saying that by the product of lines is meant the product of their numerical values, a subject already discussed at the beginning of this chapter. Theorem. The sqaare on the bisector of an angle of a triangle is equal to the product of the sides of this angle LEADING PROPOSITIONS OF BOOK III 249 diminished hy the product of the segments made hy the bisector upon the third side of the triangle. This proposition enables us to compute the length of a bisector of a triangle if the lengths of the sides are known. For, in this figure, let a = 3, b = 5, and c Then ' .' x : v = b : a, and y - we have x:y-. X 6- ■ X .•.3a; = 30-5.r. .■..'-■ = 3|,2/ = 2i. By the theorem, z' = ab — xy = 15 - 8A = 6 .•.z=V6: Theorem. In any triangle the product of two sides is equal to the product of the diameter of the circumscribed circle by the altitude upon the third side. This enables us, after the Pythagorean Theorem has been studied, to compute the length of the diameter of the circumscribed circle in terms of the three sides. For if we designate the sides by a, b, and e, as usual, and let CD=d and PB = x, then But CP^ = «2 - X-' = b^- (c- xf. A C^ h / ^"" •.a'^ — x^-b^-c'^ + 2 ex - x\ „2 _ J2 + c2 v-y ■■■^^'--^^^^1- CP-d^ ab. _ 2abc Vi a'c" - (cP- y^ + c r 250 THE TEACHING OF GEOMETRY This is not available at this time, however, because the Pythagorean Theorem has not been proved. These two propositions are merely special cases of the following general theorem, which may be given as an interesting exercise : If ABC is an inscribed triangle, and through C- there are drawn two straight lines CD, tneeting AB in D, and CP, meeting the circle in P, with angles A CD and PCB equal, then AC X BC will equal CD x CP, Fig. 4 Fig. 1 is the general case where D falls between A and B. If CP is a diameter, it reduces to the second figure given on page 249. If CP bisects Z A CB, we have Fig. 3, from which may be proved the proposition given at the foot of page 248. If D lies on BA pro- duced, we have Fig. 2. If D lies on AB produced, we have Fig. 4. This general proposition is proved by showing that A ADC and PBC are similar, exactly as in the second proposition given on page 249. These theorems are usually followed by problems of construction, of which only one has great interest, namely. To divide a given line in extreme and mean ratio. The purpose of this problem is to prepare for the con- struction of the regular decagon and pentagon. The division of a line in extreme and mean ratio is called " the golden section," and is probably " the section " mentioned by Proclus when he says that Eudoxus "greatly added to the number of the theorems which LEADING PROPOSITIONS OF BOOK III 251 Plato originated regarding the section." The expression " golden section " is not old, however, and its origin is uncertain. If a line AB is divided in golden section at P, we have Therefore, ii AB = a, and AP = x,-we have a (a — x) = x% or x^ + aa; — a" = ; whence x = + - Vo 2 2 = a(1.118-0.5) = 0.618 a, the other root representing the external point. That is, X = about 0.6 a, and a — x = about 0.4 a, and a is therefore divided in about the i-atio of 2:3. There has been a great deal written upon the aesthetic features of the golden section. It is claimed that a line is most harmoniously divided when it is either bisected or divided in extreme and mean ratio. A painting has the strong feature in the center, or more often at a point about 0.4 of the distance from one side, that is, at the golden section of the width of the picture. It is said that in nature this same harmony is found, as in the division of the veins of such leaves as the ivy and fern. CHAPTER XVII THE LEADING PROPOSITIONS OF BOOK IV Book IV treats of the area of polygons, and offers a large number of practical applications. Since the number of applications to the measuring of areas of various kinds of polygons is unlimited, while in the first three books these applications are not so obvious, less effort is made in this chapter to suggest practical problems to the teach- ers. The survey of the school grounds or of vacant lots in the vicinity offers all the outdoor work that is needed to make Book IV seem very important. Theoeem. Two rectangles having equal altitudes are to each other as their bases. Euclid's statement (Book VI, Proposition 1) was as follows : Triangles and parallelograms which are under the same height are to one arwther as their bases. Our plan of treating the two figures separately is manifestly better from the educational standpoint. In the modern treatment by limits the proof is divided into two parts : first, for commensurable bases ; and sec- ond, for incommensurable ones. Of these the second may well be omitted, or merely be read over by the teacher and class and the reasons explained. In general, it is doubtful if the majority of an American class in geom- etry get much out of the incommensurable case. Of course, with a bright class a teacher may well afford to take it as it is given in the textbook, but the important 252 LEADING PROPOSITIONS OP BOOK IV 253 thing is that the commensurable case should be proved and the incommensiirable one recognized. Euclid's treatment of proportion was so rigorous that no special treatment of the incommensurable was neces- sary. The French geometer, Legendre, gave a rigorous proof by reductio ad absurdum. In America the pupils are hardly ready for these proofs, and so our treatment by limits is less rigorous than these earlier ones. Theorem. The area of a rectangle is equal to the product of its base by its altitude. The easiest way to introduce this is to mark a rec- tangle, with commensurable sides, on squared paper, and count up the squares ; or, what is more convenient, to draw the rectangle and mark the area off in squares. It is interesting and valuable to a class to have its attention called to the fact that the perimeter of a rec- tangle is no criterion as to the area. Thus, if a rectan- gle has an area of 1 square foot and is only -^^-^ of an inch high, the perimeter is over 2 miles. The story of how Indians were induced to sell their land by measur- ing the perimeter is a very old one. Proclus speaks of travelers who described the size of cities by the perim- eters, and of men who cheated others by pretending to give them as much land as they themselves had, when really they made only the perimeters equal. Thucydides estimated the size of Sicily by the time it took to sail round it. Pupils will be interested to know in this con- nection that of polygons having the same perimeter and the same number of sides, the one having equal sides and equal angles is the greatest, and that of plane fig- ures having the same perimeter, the circle is the greatest. These facts were known to the Greek writers, Zenodorus (ca. 150 B.C.) and Proclus (410-485 a.d.). 254 THE TEACHING OF GEOMETRY The surfaces of rectangular solids may now be found, there being an advantage in thus incidentally connecting plane and solid geometry wherever it is natural to do so. Theorem. The area of a parallelogram is equal to the product of its base hy its altitude. The best way to introduce this theorem is to cut a parallelogram from paper, and then, with the class, sep- arate it into two parts by a cut perpendicular to the base. The two parts may then be fitted together to make a rectangle. In particular, if we cut off a triangle from one end and fit it on the other, we have the basis for the proof of the textbooks. The use of squared paper for such a proposition is not wise, since it makes the meas- urement appear to be merely an approximation. The cut- ting of the paper is in every way more satisfactory. Theorem. The area of a triangle is equal to half the product of its base by its altitude. Of course, the Greeks would never have used the wording of either of these two propositions. Euclid, for example, gives this one as follows : If a parallelogram have the same base with a triangle and be in the same par- allels, the parallelogram is double of the triangle. As to the parallelogram, he simply says it is equal to a parallel- ogram of equal base and " in the same parallels," which makes it equal to a rectangle of the same base and the same altitude. The number of applications of these two theorems is so great that the teacher will not be at a loss to find genuine ones that appeal to the class. Teachers may now introduce pyramids, requiring the areas of the tri- angular faces to be found. The Ahmes papyrus (ca. 1700 b.c) gives the area of an isosceles triangle as ^ bs, where s is one of the equal LEADING PROPOSITIONS OF BOOK IV 255 sides, thus taking s for the altitude. This shows the primitive state of geometry at that time. Theorem:. The area of a trapezoid is equal to half the sum of its bases multiplied hy the altitude. An interesting variation of the ordinary proof is made by placihg a trapezoid T, congruent to T, in the position here shown. The parallelogram , ' t formed equals a(b + V), I and therefore / 1 = a — ■ T' b h' The proposition should be discussed for the case h = h', when it reduces to the one about the area of a parallelogram. If h'— 0, the trapezoid reduces to a triangle, and T = a-- This proposition is the basis of the theory of land sur- veying, a piece of land being, for purposes of measure- ment, divided into trapezoids and triangles, the latter being, as we have seen, a kind of special trapezoid. The proposition is not in Euclid, but is given by Proclus in the fifth century. The term "isosceles trapezoid" is used to mean a trapezoid with two opposite sides equal, but not parallel. The area of such a figure was incorrectly given by the Ahmes papyrus as |-(6 + 6')s, where s is one of the equal sides. This amounts to taking s = a. The proposition is particularly important in the sur- veying of an irregular field such as is found in hilly districts. It is customary to consider the field as a poly- gon, and to draw a meridian line, letting fall perpendic- ulars upon it from the vertices, thus forming triangles and trapezoids that can easily be measured. An older plan, but one better suited to the use of pupils who may be working only with the tape, is given on page 99. 256 THE TEACHING OF GEOMETRY Theorem. The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. This proposition may be omitted as far as its use in plane geometry is concerned, for we can prove the next proposition here given without using it. In solid geom- etry it is used only in a proposition relating to the volumes of two triangular pyramids having a common trihedral angle, and this is usually omitted. But the theo- rem is so simple that it takes but little time, and it adds greatly to the student's appreciation of similar triangles. It not only simplifies the next one here given, but teachers can at once deduce the latter from it as a special case by asking to what it reduces if a second angle of one tri- angle is also equal to a second angle of the other triangle. It is helpful to give numerical values to the sides of a few triangles having such equal angles, and to find the numerical ratio of the areas. Theorem. The areas of two similar triangles are to each other as the squares on any two corresponding sides. This may be proved independently of the preceding proposition C by drawing the altitudes p and p'. Then AACC _ cp AA'B'C c'p' But by similar triangles. . l^ABC AA'B'C P ' — y' P and so for other sides. ■^' -B This proof is unnecessarily long, however, because of the introduction of the altitudes. LEADING PROPOSITIONS OF BOOK IV 257 In this and several other propositions in Book IV occurs the expression " the square on a line." We have, in our departure from Euclid, treated a line either as a geometric figure or as a number (the length of the line), as was the more convenient. Of course if we are speak- ing of a line, the preferable expression is " square on the line," whereas if we speak of a number, we say " square of the number." In the case of a rectangle of two lines we have come to speak of the "product of the lines," meaning the product of their numerical values. We are therefore not as accurate in our phraseology as Euclid, and we do not pretend to be, for reasons already given. But when it comes to " square on a line " or " square of a line," the former is the one demanding no explanation or apology, and it is even better understood than the latter. Theoeem. The areas of two similar polygons are to each other as the squares on any two corresponding sides. This is a proposition of great importance, and in. due time the pupil sees that it applies to circles, with the nec- essary change of the word " sides " to " lines." It is well to ask a few questions like the following : If one square is twice as high as another, how do the areas compare ? If the side of one equilateral triangle is three times as long as that of another, how do the perimeters compare ? how do the areas compare ? If the area of one square is twenty-five times the area of another square, the side of the fii'st is how many times as long as the side of the second ? If a photograph is enlarged so that a tree is four times as high as it was before, what is the ratio of corresponding dimensions ? The area of the enlarged photograph is how many times as great as the area of the original ? 258 THE TEACHING OF GEOMETRY Theorem. The square on the hypotenuse of a right tri- angle is equivalent to the sum of the squares on the other two sides. Of all the propositions of geometry this is the most famous and perhaps the most valuable. Trigonometry is based chiefly upon two facts of plane geometry : (1) in similar triangles the corresponding sides are proportional, and (2) this proposition. In mensuration, in general, this proposition enters more often than any others, except those on the measuring of the rectangle and triangle. It is proposed, therefore, to devote considerable space to speaking of the history of the theorem, and to certain proofs that may profitably be suggested from time to time to different classes for the purpose of adding in- terest to the work. Proclus, the old Greek commentator on Euclid, has this to say of the history : " If we listen to those who wish to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honor of his discovery. But for my part, while I admire those who first observed the truth of this theorem, I marvel more at the writer of the ' Elements ' (Euchd), not only because he made it fast by a most lucid demonstration, but because he compelled assent to the still more general theorem by the irrefra- gable arguments of science in Book VI. For in that book he proves, generally, that in right triangles the figure on the side subtending the right angle is equal to the similar and similarly placed figures described on the sides about the right angle." Now it appears from this that Proclus, in the fifth century A.D., thought that Pythagoras discovered the proposition in the sixth cen- tury B.C., that the usual proof, as given in most of our LEADING PROPOSITIONS OF BOOK IV 259 American textbooks, was due to Euclid, and that the generahzed form was also due to the latter. For it should be made known to students that the proposition is true not only for squares, but for any similar figures, such as equilateral triangles, parallelograms, semicircles, and irregular figures, provided they are similarly placed on the three sides of the right triangle. Besides Proclus, Plutarch testifies to the fact that Pythagoras was the discoverer, saying that "Pythag- oras sacrificed an ox on the strength of his proposi- tion as ApoUodotus says," but saying that there were two possible propositions to which this refers. This ApoUodotus was probably ApoUodorus, surnamed Lo- gisticus (the Calculator), whose date is quite uncertain, and who speaks in some verses of a "famous proposi- tion " discovered by Pythagoras, and all tradition makes this the one. Cicero, who comments upon these verses, does not question the discovery, but doubts the story of the sacrifice of the ox. Of other early writers, Diogenes Laertius, whose date is entirely uncertain (perhaps the second century a.d.), and Athenseus (third century a.d.) may be mentioned as attributing the theorem to Pythag- oras, while Heron (first century A.D.) says that he gave a rule for forming right triangles with rational in- tegers for the sides, like 3, 4, 5, where 3^ 4- 4^ = 5^. It should be said, however, that the Pythagorean origin has been doubted, notably in an article by H. Vogt, pub- lished in the Bibliotheca Maihematica in 1908 (Vol. IX (3), p. 15), entitled "Die Geometrie des Pythagoras," and by G. Junge, in his work entitled " Wann haben die Griechen das Irrationale entdeckt?" (Halle, 1907). These writers claim that all the authorities attributing the proposition to Pythagoras are centuries later than 260 THE TEACHING OF GEOMETRY his time, and are open to grave suspicion. Nevertheless it is hardly possible that such a general tradition, and one so universally accepted, should have arisen without good foundation. The evidence has been carefully studied by Heath in his " Euclid," who concludes with these words : " On the whole, therefore, I see no suffi- cient reason to question the tradition that, so far as Greek geometry is concerned . . . , Pythagoras was the first to introduce the theorem . . . and to give a general proof of it." That the fact was known earlier, probably without the general proof, is recognized by all modern writers. Pythagoras had studied in Egypt aad possibly in the East before he established his school at Crotona, in south- em Italy. In Egypt, at any rate, he could easily have found that a triangle with the sides 3, 4, 5, is a right triangle, and Vitruvius (first century B.C.) tells us that he taught this fact. The Egyptian harpedonaptae (rope stretchers) stretched ropes about pegs so as to make such a triangle for the purpose of laying out a right angle in their surveying, just as our surveyors do to-day. The great pyramids have an angle of slope such as is given by this triangle. Indeed, a papy- rus of the twelfth djoiasty, lately discov- ered at Kahun, in Egypt, refers to four of these triangles, such as 1^ + (|-)^ = (1^)'^. ^ This property seems to have been a matter of common knowledge long before Pythagoras, even as far east as China. He was, therefore, naturally led to attempt to prove the general property which had already been recognized for special cases, and in particular for the isosceles right triangle. LEADING PROPOSITIONS OF BOOK IV 261 How Pythagoras proved the proposition is not known. It has been thought that he used a proof by proportion, because Proclus says that Euclid gave a new style of proof, and Euclid does not use proportion for this pur- pose, while the -subject, in incomplete form, was highly esteemed by the Pythagoreans. Heath suggests that this is among the possibilities: i^ABC and APC are similar. .-.AB xAP = AC\ Similarly, AB x PB^BC". .-. AB(AP + PB) = AC''+ BC\ or AB^=AC^ + JBC\ ^ Others have thought that Pythagoras derived his proof from dissecting a square and showing that the square on the hypotenuse must equal the sum of the squares on the other two sides, in some such manner as this: ^--^^ b^. a'' / Fig. 1 Fig. 2 Here Fig. 1 is evidently A^ + 4 A. Fig. 2 is evidently a^ + J2+ 4 ^. i + 4 A = a^ + 6^ + 4 A, the A all being congruent. .- . li" = a^ + h\ 262 THE TEACHING OF GEOMETRY The great Hindu mathematician, Bhaskara (born 1114 A.D.), proceeds in a somewhat similar manner. He draws this figure, but gives no proof. It is evi- dent that he had in mind this relation : 7(2 = 4 . ^ + (J _ a)2 = a2 + I A somewhat similar proof can be based upon the following figure : If the four triangles, 1 + 2 + 3+4, are taken away, there remains the square on the hypote- nuse. But if we take away the two shaded rec- tangles, which equal the four triangles, there remain the squares on the two sides. Therefore the square on the hypotenuse must equal the sum of these two squares. It has long been thought that the truth of the prop- osition was first observed by seeing the tiles on the floors of ancient temples. If they were arranged as here shown, the proposition would be evident for the special case of an isosceles right triangle. The Hindus knew the proposition long before Bhaskara, however, and possibly before Pythagoras. It is referred to in the old religious poems of the Brahmans, the " Sulvasutras," but the date of these poems is so uncertain that it is impossible to state that they preceded the sixth century b.c.,^ in which Pythagoras lived. The "Sulvasutra" of Apastamba has 1 See, for example, G. B. Kaye, " The Source of Hindu Mathe- matics," in the Journal of the Royal Asiatic Society, July, 1910. LEADING PROPOSITIONS OF BOOK IV 263 a collection of rules, without proofs, for constructiag various figures. Among these is one for constructing right angles by stretching cords of the following lengths : 3, 4, 5 ; 12, 16, 20 ; 15, 20, 25 (the two latter being mul- tiples of the first); 5, 12, 13 ; 15, 36, 39 ; 8, 15, 17 ; 12, 35, 37. Whatever the date of these " Sulvasutras," there is no evidence that the Indians had a definite proof of the theorem, even though they, like the early Egyptians, recognized the general fact. It is always interesting to a class to see more than one proof of a famous theorem, and many teachers find it profitable to ask their pupils to work out proofs that are (to them) original, often suggesting the figure. Two of the best known historic proofs are here given. The first makes the Pythagorean Theorem a special case of a proposition due to Pappus (fourth century A.D.), relating to any kind of a triangle. Somewhat simplified, this proposition asserts that if ABC is any kind of triangle, and MC, NC are parallelograms on AC, BC, the opposite sides being produced to meet at P ; and if PC is produced making QR = PC; and if the parallel- ogram AT is con- structed, then AT= MC + NC. For MC = AP = AR, having equal bases and equal altitudes. Similarly, NC = QT. Adding, MC+NC=AT. If, now, ABC is a right triangle, and if MC and NC are squares, it is easy to show that AT is a, square, and the proposi- tion reduces to the Pythagorean Theorem. R T 264 THE TEACHING OF GEOMETRY \/\ The Arab writer, Al- Nairizi (died about 922 A.D.), attributes to Thabit ben Qurra (826-901 a.d.) a proof substantially as follows: The four triangles T can be proved congruent. Then if we take from the whole figure T and T", we have left the squares on the two sides of the right angle. If we take away the other two triangles instead, we have left the square on the hypotenuse. Therefore the former is' equivalent to the latter. A proof attributed to the great art- ist, Leonardo da Vinci (1452-1519), is as follows : The construction of the following figure is evident. It is easily shown that the four quadrilaterals ABMX, XNCA, SBCP, and SRQP are con- gruent. .-.ABMXNCA equals SBCPQRS but is not congru- ent to it, the congru- ent quadrilaterals being differently arranged. Subtract the con- gruent triangles MXN,ABC,RAQ, l < 43^'-^4^ mula for the weight (w) of a steel cylinder where d is the diameter and I is the length : w=0.07irdH. Taking TT = 3^, find the weight of the pin. The most elaborate study of the cylinder, cone, and sphere (the " three round bodies ") in the Greek litera- ture is that of Archimedes of Syracuse (on the island of Sicily), who lived in the third century B.C. Archimedes tells us, however, that Eudoxus (bom ca. 407 B.C.) discov- ered that any cone is one third of a cylinder of the same base and the same altitude. Tradition says that Archi- medes requested that a sphere and a cylinder be carved upon his tomb, and that this was done. Cicero relates that he discovered the tomb by means of these symbols. The tomb now shown to visitors in ancient Syracuse as LEADING PROPOSITIONS OP BOOK VII 315 that of Archimedes cannot be his, for it bears no such figures, and is not " outside the gate of Agrigentum," as Cicero describes. The cone is now introduced. A conic surface is easily illustrated to a class by taking a piece of paper and roll- ing it up into a cornucopia, the space inclosed being a conic space, a term that is sometimes convenient. The generation of a conic surface may be shown by taking a blackboard pointer and swinging it around by its tip so that the other end moves in a curve. If we consider a straight line as the limit of a curve, then the pointer may swing in a plane, and so a plane is the limit of a conic surface. If we swing the pointer about a point in the middle, we shall generate the two nappes of the cone, the conic space now being double. In practice the right circular cone, or cone of revolu- tion, is the important type, and special attention should be given to this form. Theokem. Every section of a cone made hy a plane passing through its vertex is a triangle. At this time, or in speaking of the preliminary defini- tions, reference should be made to the conic sections. Of these there are three great types : (1) the ellipse, where the cutting plane intersects all the elements on one side of the vertex ; a circle is a special form of the ellipse ; (2) the parabola, where the plane is parallel to an element ; (3) the hyperbola, where the plane cuts some of the elements on one side of the vertex, and the rest on the other side ; that is, where it cuts both nappes. It is to be observed that the ellipse may vary greatly in shape, from a circle to a very long ellipse, as the cutting plane changes from being perpendicular to the axis to being nearly parallel to an element. The instant it 316 THE TEACHING OF GEOMETRY becomes parallel to an element the ellipse changes sud- denly to a parabola. If the plane tips the slightest amount more, the section becomes an hyperbola. While these conic sections are not studied in elemen- tary geometry, the terms should be known for general information, particularly the ellipse and parabola. The study of the conic sections forms a large part of the work of analytic geometry, a subject in which the fig- ures resemble the graphic work in algebra, this having been taken from "analytics," as the higher subject is commonly called. The planets move about the sun in elliptic orbits, and Halley's comet that returned to view in 1909-1910 has for its path an enormous ellipse. Most comets seem to move in parabolas, and a body thrown into the air would take a parabolic path if it were not for the resistance of the atmosphere. Two of the sides of the triangle in this proposition constitute a special form of the hyperbola. The study of conic sections was brought to a high state by the Greeks. They were not known to the Py- thagoreans, but were discovered by Mensechmus in the fourth century B.C. This discovery is mentioned by Proclus, who says, " Further, as to these sections, the conies were conceived by Mensechmus." Since if the cutting plane is perpendicular to the axis the section is a circle, and if oblique it is an ellipse, a parabola, or an hyperbola, it follows that if light pro- ceeds from a point, the shadow of a circle is a circle, an ellipse, a parabola, or an hyperbola, depending on the position of the plane on which the shadow falls. It is interesting and instructive to a class to see these shadows, but of course not much time can be allowed for such work. At this point the chief thing is to have LEADING PROPOSITIONS OF BOOK VK 317 the names " ellipse " and " parabola," so often met in reading, understood. It is also of interest to pupils to see at this time the method of drawing an ellipse by means of a pencil stretching a string band that moves about two pins fastened in the paper. This is a practical method, and is familiar to all teachers who have studied analytic geometry. In designing elliptic arches, however, three circular arcs are often joined, as here shown, the result being approximately an elliptic arc. Here O is the center of arc BC,Moi arc AB, and N of arc CD. Since XF is per- pendicular to BM and BO, it is tangent to arcs AB and BC, so there is no abrupt turning at B, and similarly for C.^ V Theorem. The volume of a circular cone is equal to one third the product of its base hy its altitude. It is easy to prove this for noncircular cones as well, but since they are not met commonly in practice, they may be omitted in elementary geometry. The important formula at this time is v = |- irr^h. As already stated, this proposition was discovered by Eudoxus of Cnidus (bom ca. 407 B.C., died ca. 354 B.C.), a man who, as abeady stated, was born poor, but who became one of the most illustrious and most highly esteemed of all the Greeks of his time. Theoeem. The lateral area of a frustum of a cone of revolution is equal to half the sum of the circumferences of its bases multiplied by the slant height. 1 The illustration is from Dupin, loc. olt. 318 THE TEACHING- OF GEOMETRY An interesting case for a class to notice is that in which the upper base becomes zero and the frustum becomes a cone, the proposition being still true. If the upper base is equal to the lower base, the frustum becomes a cylin- der, and still the proposition remains true. The proposi- tion thus offers an excellent illustration of the elementary Principle of Continuity. Then follows, in most textbooks, a theorem relating to the volume of a frustum. In the case of a cone of revolution v = ^ irh (r^ + r'^ + rr'). Here if r'= 0, we have v = ■y'7rr%, the volume of a cone. If 7^= r, we have v = ^wh (r^ + r^ + r^) = irhr^, the volume of a cylinder. If one needs examples in mensuration beyond those given in a first-class textbook, they are easily found. The monument to Sir Christopher Wren, the professor of geometry in Cambridge University, who became the great architect of St. Paul's Cathedral in London, has a Latin inscription which means, " Reader, if you would see his monument, look about you." So it is with prac- tical examples in Book VII. Appended to this Book, or more often to the course in solid geometry, is frequently found a proposition known as Euler's Theorem. This is often considered too difficult for the average pupil and is therefore omitted. On account of its importance, however, in the theory of polyhedrons, some reference to it at this time may be helpful to the teacher. The theorem asserts that in any convex polyhedron the number of edges increased by two is equal to the number of vertices increased by the number of faces. In other words, that e + 2 = v +f. On account of its importance a proof will be given that differs from the one ordinarily found in textbooks. LEADING PROPOSITIONS OF BOOK VII 319 Let Sj, Sj, • • •, s„ be the number of sides of the various faces, and / the number of faces. Now since the sum of the angles of a polygon of >- sides is (s — 2) 180°, therefore the sum of the angles of all the faces is (Sj + s^ + Sg + ■ ■ • + s„ — 2/) 180°. But Sj + Sj + Sg + • • . + s„ is twice the number of edges, be- cause each edge belongs to two faces. .'. the sum of the angles of all the faces is (2 e - 2/) 180°, or (e -/) 360°. Since the polyhedron is convex, it is possible to find some outside point of view, P, from which some face, as ABODE, covers up the whole figure, as in this illustration. If we think of all the vertices projected on ABODE, by lines through P, the sum of the angles of all the faces will be the same as the sum of the angles of all their projections on ABODE. Calling ABODE Sj, and think- ^ ing of the projections as /^^"^N. traced by dotted lines y^ / \\s„^^ on the opposite side of y^ / \ ^^^ Sy, this sum is evidently ^/ ^,J. ^\ ^^^ equal to y^--'''' / \ " 'f\ -^C (1) the sum of the -^kT^ /'' \ / \ / angles in s^, or (sj — 2) \ ~^-^y'' \ / \ / 180°, plus \ / 'y'' \ / (2) the sum of the \ / _,.,— ' ""~~---,^ \ / angles on the other side a"" ~r ofsi,or(Si-2) 180°, plus (3) the sum of the angles about the various points shown as inside of Sj, of which there are v — Sj points, about each of which the sum of the angles is 360°, making (u — Sj) 360° in all. Adding, we have (Si-2) 180°+(Si-2) 180°-f(w-s,) 360°=[(Si-2)-)-(t!-Si)] 360° = (!)-2)360°. Equating the two sums already found, we have («-/)360°= (t)-2)360°, e-/=i>-2, e + 2 = w-h/. 320 THE TEACHING OF GEOMETRY This proof is too abstract for most pupils in the high school, but it is more scientific than those found in any ' of the elementary textbooks, and teachers will find it of service in relieving their own minds of any question as to the legitimacy of the theorem. Although this proposition is generally attributed to Euler, and was, indeed, rediscovered by him and pub- lished in 1752, it was known to the great French geometer Descartes, a fact that Leibnitz mentions.^ This theorem has a very practical application in the study of crystals, since it offers a convenient check on the count of faces, edges, and vertices. Some use of crys- tals, or even of polyhedrons cut from a piece of crayon, is desirable when studying Euler's proposition. The following illustrations of common forms of crystals may be used in this connection : The first represents two truncated pyramids placed base to base. Here e = 20, /=10, ?; = 12, so that e + 2 =f+ V. The second represents a crystal formed by replacing each edge of a cube by a plane, with the result that e = 40,/= 18, and v = 24. The third repre- sents a crystal formed by replacing each edge of an octahedron by a plane, it being easy to see that Euler's law still holds true. 1 For the historical bibliography consult G. Holzmiiller, Elemente der Stereometrie, Vol. I, p. 181, Leipzig, 1900. CHAPTER XXI THE LEADIJfG PROPOSITIONS OF BOOK VIII Book VIII treats of the sphere. Just as the cbcle may be defined either as a plane surface or as the bound- ing line which is the locus of a point in a plane at a given distance from a fixed point, so a sphere may be defined either as a solid or as the bounding surface which is the locus of a point in space at a given distance from a fixed point. In higher mathematics the circle is defined as the bounding line and the sphere as the bounding surface ; that is, each is defined as a locus. This view of the circle as a line is becoming quite, general in elementary geom- etry, it being the desire that students may not have to change definitions in passing from elementary to higher mathematics. The sphere is less frequently looked upon in geometry as a surface, and in popular usage it is always taken as a solid. Analogous to the postulate that a circle may be de- scribed with any given point as a center and any given line as a radius, is the postulate for constructing a sphere with any given center and any given radius. This pos- tulate is not so essential, however, as the one about the circle, because we are not so concerned with constructions here as we are in plane geometry. A good opportunity is offered for illustrating several of the definitions connected with the study of the sphere, such as great circle, axis, small circle, and pole, 321 322 THE TEACHING OF GEOMETRY by referring to geography. Indeed, the first three prop- ositions usually given in Book VIII have a direct bear- ing upon the study of the earth. Theoebm. a plane perpendicular to a radius at its extremity is tangent to the sphere. The student should always have his attention called to the analogue in plane geometry, where there is one. If here we pass a plane tlu-ough the radius in question, the figure formed on the plane will be that of a line tangent to a circle. If we revolve this about the line of the radius in question, as an axis, the circle will generate the sphere again, and the tangent line will generate the tangent plane. Theoeem. a sphere may he inscribed in any given tetrahedron. Here again we may form a corresponding proposition of plane geometry by passing a plane through any three points of contact of the sphere and the tetrahedron. We shall then form the figure of a circle inscribed in a tri- angle. And just as in the case of the triangle we may have escribed circles by producing the sides, so in the case of the tetrahedron we may have escribed spheres by producing the planes indefinitely and proceeding in the same way as for the inscribed sphere. The figure is difficult to draw, but it is not difficult to understand, particularly if we construct the tetrahedron out of pasteboard. Theoeem. A sphere may he circumscribed about any given tetrahedron. By producing one of the faces indefinitely it will cut the sphere in a circle, and the resulting figure, on the plane, will be that of the analogous proposition of plane geometry, the circle circumscribed, about a triangle. It LEADING PROPOSITIONS OF BOOK VIII 323 is easily proved from the proposition that the four per- pendiculars erected at the centers of the faces of a tetra- hedron meet in a point (are concurrent), the analogue of the proposition about the perpendicular bisectors of the sides of a triangle. Theorem. The intersection of two spherical surfaces is a circle whose plane is perpendicular to the line joining the centers of the surfaces and whose center is in that line. The figure suggests the case of two circles in plane geometry. In the case of two circles that do not inter- sect or touch, one not being within the other, there are four common tangents. If the circles touch, two close up into one. If one circle is wholly within the other, this last tangent disappears. The same thing exists in rela- tion to two spheres, and the analogous cases are formed by revolving the circles and tangents about the line through their centers. In plane geometry it is easily proved that if two circles intersect, the tangents from any point on their common chord produced are equal. For if the common chord is AB and the point P is taken on AB produced, then the square on any tangent from P is equal to PB x PA. The line PBA is sometimes called the radical axis. Similarly in this proposition concerning spheres, if from any point in the plane of the circle formed by the intersection of the two spherical surfaces lines are drawn tangent to either sphere, these tangents are equal. For it is easily proved that all tangents to the same sphere from an external point are equal, and it can be proved as in plane geometry that two tangents to the two spheres are equal. Among the interesting analogies between plane and solid geometry is the one relating to the four common 324 THE TEACHING OF GEOMETRY tangents to two circles. If the figure be revolved about the line of centers, the circles generate spheres and the tangents generate conical surfaces. To study this case for various sizes and positions of the two spheres is one of the most interesting generalizations of solid geometry. An application of the proposition is seen in the case of an eclipse, where the sphere (/ represents the moon, the earth, and S the sun. It is also seen in the case of the full moon, when S is on the other side of the earth. In this case the part MIN is fully illuminated by the moon, but the zone ABNM is only partly illuminated, as the figure shows.''- Theorem. The sum of the sides of a spherical poly- gon is less than 360°. In all such cases the relation to the polyhedral angle should be made clear. This is done in the proofs usually given in the textbooks. It is easily seen that this is true only with the limitation set forth in most textbooks, that the spherical polygons considered are convex. Thus we might have a spherical triangle that is concave, with its base 359°, and its other two sides each 90°, the sum of the sides being 539°. Theorem. The sum of the angles of a spherical triangle is greater than 180° and less than 640°. 1 The illustration is from Dupin, loc. cit. LEADING PROPOSITIONS OF BOOK VIII 325 It is for the purpose of proving this important fact that polar triangles are introduced. This, proposition shows the relation of the spherical to the plane triangle. If our planes were in reality slightly curved, being small portions of enormous spherical surfaces, then the sum of the angles of a triangle would not be exactly 180°, but would exceed 180° by some amount depending on the curvature of the surface. Just as a being may be imag- ined as having only two dimensions, and living always on a plane surface (in a space of two dimensions), and having no conception of a space of three dimensions, so we may think of ourselves as living in a space of three dimensions but surrounded by a space of four dimen- sions. The flat being could not point to a third dimen- sion because he could not get out of his plane, and we cannot point to the fourth dimension because we cannot get out of our space. Now what the flat being thinks is his plane may be the surface of an enormous sphere in our three dimensions ; in other words, the space he lives in may curve through some higher space without his be- ing conscious of it. So our space may also curve through some higher space without our being conscious of it. If our planes have really some curvature, then the sum of the angles of our triangles has a slight excess over 180°. All this is mere speculation, but it may interest some student to know that the idea of fourth and higher dimensions enters largely into mathematical investigation to-day. Theorem. Two symmetric spherical triangles are equivalent. While it is not a subject that has any place iii a school, save perhaps for incidental conversation with some group of enthusiastic students, it may interest the teacher to consider this proposition in connection with the fourth 326 THE TEACHING OF GEOMETRY dimension just mentioned. Consider these triangles, where ZA = ZA',AB = A 'B', AC = A'C'.We prove them congruent by superposition, turning one over and plac- ing it upon the other. But su.p- „ pose we were beings in Flatland, beings with only two dimensions and without the power to point in any direction except in the plane we lived ia. We should then be unable to turn AA'B'C' over so that it could coincide with A ABC, and we should have to prove these triangles equivalent in some other way, probably by dividing them into isosceles triangles that could be superposed. Now it is the same thing with symmetric spherical tri- angles ; we cannot superpose them. But might it not be possible to do so if we could turn them through the fourth dimension exactly as we turn the Flatlander's triangle through our third dimension ? It is interesting to think about this possibility even though we carry it no further, and in these side lights on mathematics lies much of the fascination of the subject. Theorem. The shortest line that can he drawn on the surface of a sphere between tivo points is the minor arc of a great circle joining the two points. It is always interesting to a class to apply this prac- tically. By taking a terrestrial globe and drawing a great circle between the southern point of Ireland and New York City, we represent the shortest route for ships LEADING PROPOSITIONS OF BOOK VIII 327 crossing to England. Now if we notice where this great- circle arc cuts the various meridians and mark this on an ordinary Mercator's projection map, such as is found in any schoolroom, we shall find that the path of the ship does not make a straight line. Passengers at sea often do not understand why the ship's course on the map is not a straight line ; but the chief reason is that the ship is taking a great-circle arc, and this is not, in general, a straight line on a Mercator projection. The small circles of latitude are straight lines, and so are the meridians and the equator, but other great circles are represented by curved lines. Theorem. The area of the surface of a sphere is equal to the product of its diameter hy the circumference of a great circle. This leads to the remarkable formula, a = 4 irr^. That the area of the sphere, a curved surface, should exactly equal the sum of the areas of four great circles, plane surfaces, is the remarkable feature. This was one of the greatest discoveries of Archimedes (ca. 287-212 B.C.), who gives it as the thirty-fifth proposition of his treatise on the "Sphere and the Cylinder," and who mentions it specially in a letter to his friend Dositheus, a mathe- matician of some prominence. Archimedes also states that the surface of a sphere is two thirds that of the circumscribed cylinder, or the same as the curved sur- face of this cylinder. This is evident, since the cylin- dric surface of the cylinder is 2 Trr x 2 r, or 4 Trr^ and the two bases have an area 7rr^ + 7^r■^ making the total area 6 ttt^. Theorem. The area of a spherical triangle is equal to the area of a lune whose angle is half the triangle's spheri- cal excess. 328 THE TEACHING OF GEOMETRY This theorem, so important in finding areas on the earth's surface, should be followed by a considerable amount of computation of triangular areas, else it will be rather meaningless. Students tend to memorize a proof of this character, and in order to have the prop- osition mean what it should to them, they shoiild at once apply it. The same is true of the following prop- osition on the area of a spherical polygon. It is prob- able that neither of these propositions is very old; at any rate, they do not seem to have been known to the writers on elementary mathematics among the Greeks. Theoeem. The volume of a sphere is equal to the prod- uct of the area of its surface hy one third of its radius. This gives the formula v= ^ irr^. This is one of the greatest discoveries of Archimedes. He also found as a result that the volume of a sphere is two thirds the volume of the circumscribed cylinder. This is easily seen, since the volume of the cylinder is Trr^ X 2 r, or 2 tt/, and | ttt^ is J of 2 irr^. It was because of these discoveries on the sphere and cylinder that Archimedes wished these figures engraved upon his tomb, as has already been stated. The Roman general Marcellus con- quered Syracuse in 212 B.C., and at the sack of the city Archimedes was killed by an ignorant soldier. Marcel- lus carried out the wishes of Archimedes with respect to the figures on his tomb. The volume of a sphere can also be very elegantly found by means of a proposition known as Cavalieri's Theorem. This asserts that if two solids lie between parallel planes, and are such that the two sections made by any plane parallel to the given planes are equal in area, the solids are themselves equal in volume. Thus, if these solids have the same altitude, a, and if S and S' LEADING PROPOSITIONS OF BOOK VIII 329 are equal sections made by a plane parallel to MN, then the solids have the same volu;ne. The proof is sim- ple, since prisms of the same alti- tude, say -, and n on the bases Swxdi S' are equivalent, and the sums of n such prisms are the given solids; m and as n increases, the sums of the prisms approach the solids as their limits ; hence the volumes are equal. This proposition, which will now be applied to fmd- ing the volume of the sphere, was discovered by Bona- ventura .Cavalieri (1591 or 1598-1647). He was a Jesuit professor in the University of Bologna, and his best known work is his " Geometria Indivisilibus," which he wrote in 1626, at least in part, and published in 1635 (second edition, 1647). By means of the proposition it is also possible to prove several other theorems, as that the volumes of triangular pyramids of equivalent bases and equal altitudes are equal. To find the volume of a sphere, take the quadrant OPQ, in the square OPRQ- Then if this fig- ure is revolved about OP, OPQ will generate a hemisphere, OPR will generate a cone of volume 1 Trr' and OPRQ will generate a cylinder of volume irr«. Hence the figure generated hj ORQ will have a volume irr^-^Trr^ or 2 7rr^ which we will call X. Now OA = AB, and OC = ^X» ; also OC-OA =AC ,so that ^j,d ^-AD'-rUB' = .-AC\ 330 THE TEACHING OF GEOMETRY But wAB — itAB is the area of the ring generated by BD, a section of x, and ttA C is the corresponding section of the hemisphere. Hence, by Cavalieri's Theorem, l-Trt^ = the volume of the hemisphere. .'. ^ trr' = the volume of the sphere. In connectibn with the sphere some easy work in quadratics may be introduced even if the class has had only a year in algebra. For example, suppose a cube is inscribed in a hemisphere of radius r and we wish to find its edge, and thereby its surface and its volume. li X = the edge of the cube, the diagonal of the base must be X V^, and the prpjection of r (drawn from the center of the base to one of the vertices) on the base is half of this diagonal, a;V2 or—. Hence, by the Pythagorean Theorem, 2"' /!■■ and the total surface is 6 x^ = 4 /■", 2 /2 and the volume is nfl = -r^\i-- ^y 3 \3 L'ENVOI In the Valley of Youth, through which all wayfarers must pass on their journey from the Land of Mystery to the Land of the Infinite, there is a village where the pil- grim rests and indulges in various excursions for which the valley is celebrated. There also gather many guides in this spot, some of whom show the stranger all the various points of common interest, and others of whom take visitors to special points from which the views are of peculiar significance. As time has gone on new paths have opened, and new resting places have been made from which these views are best obtained. Some of the mountain peaks have been neglected in the past, but of late they too have been scaled, and paths have been hewn out that approach the summits, and many pilgrims ascend them and find that the result is abundantly worth the effort and the time. The effect of these several improvements has been a natural and usually friendly rivalry in the body of guides that show the way. The mountains have not changed, and the views are what they have always been. But there are not wanting those who say, "My mountain may not be as lofty as yours, but it is easier to ascend " ; or " There are quarries on my peak, and points of view from which a building may be seen in process of erec- tion, or a mill in operation, or a canal, while your moun- tain shows only a stretch of hills and valleys, and thus you will see that mine is the more profitable to visit." 331 332 THE TEACHING OF GEOMETRY Then there are guides who are themselves often weak of limb, and who are attached to numerous sand dunes, and these say to the weaker pilgrims, "Why tire your- selves climbing a rocky mountain when here are peaks whose summits you can reach with ease and from which the view is just as good as that from the most famous precipice ? " The result is not wholly disadvantageous, for many who pass through the valley are able to approach the summits of the sand dunes only, and would make progress with greatest difficulty should they attempt to scale a real mountain, although even for them it would be better to climb a little way where it is really worth the effort instead of spending all their efforts on the dunes. Then, too,' there have of late come guides who have shown much ingenuity by digging tunnels into some of the greatest mountains. These they have paved with smooth concrete, and have arranged for rubber-tired cars that run without jar to the heart of some mountain. Arrived there the pilgrim has a glance, as the car swiftly turns in a blaze of electric light, at a roughly painted panorama of the view from the summit, and he is assured by the guide that he has accomplished all that he would have done, had he laboriously climbed the peak itself. In the midst of all the advocacy of sand-dune climb- ing, and of rubber-tired cars to see a painted view, the great body of guides still climb their mountains with their little groups of followers, and the vigor of the ascent and the magnificence of the view still attract all who are strong and earnest, during their sojourn ia the Valley of Youth. Among the mountains that have for . ages attracted the pilgrims is Mons Latinus, usually called in the valley by the more pleasing name Latinat L'ENVOI 333 Mathematica, and Rhetorica, and Grammatica are also among the best known. A group known as Montes Naturales comprises Physica, Biologica, and Chemica, and one great peak with minor peaks about it is called by the people Philosophia. There are those who claim that these great masses of rock are too old to be climbed, as if that affected the view ; while others claim that the ascent is too difficult and that all who do not favor the sand dunes are reactionary. But this affects only a few who belong to the real mountains, and the others labor diligently to improve the paths and to lessen unnecessary toil, but they seek not to tear off the summits nor do they attend to the amusing attempts of those who sit by the hillocks and throw pebbles at the rocky sides of the mountains upon which they work. Geometry is a mountain. Vigor is needed for its ascent. The views all along the paths are magnificent. The effort of climbing is stimulating. A guide who points out the beauties, the grandeur, and the special places of interest commands the admiration of his group of pilgrims. One who fails to do this, who does not know the paths, who puts unnecessary burdens upon the pilgrim, or who blindfolds him in his progress, is unworthy of his position. The pretended guide who says that the painted panorama, seen from the rubber- tired car, is as good as the view from the summit is simply a fakir and is generally recognized as such. The mountain will stand ; it will not be used as a mere com- mercial quarry for building stone ; it will not be affected by pellets thrown from the little hillocks about ; but its paths will be freed from unnecessary flints, they will be 334 THE TEACHING OF GEOMETRY straightened where this can advantageously be done, and new paths on entirely novel plans will be made as time goes on, but these paths will be hewed out of rock, not made out of the dreams of a day. Every worthy guide will assist in all these efforts at betterment, and will urge the pilgrim at least to ascend a little way because of the fact that the same view cannot be obtained from other peaks ; but he will not take seriously the efforts of the fakir, nor will he listen with more than passing interest to him who proclaims the sand heap to be a Matterhom. INDEX Ahmes, 27, 254, 278, 306 Alexandrofi, 164 Algebra, 37, 84 Al-Khowarazmi, 37 Allman, G. J., 29 Almagest, 35 Al-Nairizi, 171, 193, 214, 264 Al-QiftI, 49 Analysis, 41, 161 Angle, 142, 155 ; trisection of, 31, 215 Anthonisz, Adriaen, 279 Antiphon, 31, 32, 276 ApoUodotus (ApoUodorus), 259 ApoUonius, 34, 214, 231 Applied problems, 75, 103, 178, 186, 192, 195, 203, 204, 209, 215, 217, 242, 267, 295, 317 Appreciation of geometry, 19 Arab geometry, 37, 51 Archimedes, 34, 42, 48, 139, 141, 215, 276, 278, 314, 327, 328 Aristaeus, 310 Aristotle, 33, 42, 134, 135, 137, 146, 154, 177, 209 Aryabhatta, 36, 279 Associations, syllabi of, 58, 60, 64 Assumptions, 116 Astrolabe, 172 Athelhard of Bath, 37, 51 Athenseus, 259 Axioms, 31, 41, 116 Babylon, 26, 272 Bartoli, 10, 44, 238 Belli, 10, 44, 172 Beltinus, 239, 241 Beltrami, 127 Bennett, J., 224 Bernoulli, 280 Bertrand, 62 Betz, 131 Bezout, 62 Bhaskara, 232, 268 Billings, R. "W., 222 Billingsiey, 52 Bion, 192, 239 Boethius, 43, 50 Bolyai, 128 Bonola, 128 Books of geometry, 165, 167, 201, , 227, 252, 269, 289, 303, 321 Bordas-Demoulin, 24 Borel, 11, 67, 196 Bosanquet, 272 Bossut, 23 Bourdon, 62 Bourlet, 67, 165, 196 Brahmagupta, 36, 268, 279 Bretschneider, C. A., 30 Brounoker, 280 Bruce, W. N., 199 Bryson, 31, 32, 276 Cajori, 46 Calandri, 30 Campanus, 37, 51, 135 Cantor, M., 29, 46 Capella, 50, 135 Capra, 44 Carson, G. "W. L., 18, 96, 114 Casey, J., 38 Cassiodorius, 50 Cataneo, 10, 44 Cavalieri, 136, 329 Chinese values of tt, 279 Church schools, 43 Cicero, 34, 50, 259, 314 Circle, 145, 201, 270, 287 ; squar- ing the, 31, 32, 277 Circumference, 145 335 336 THE TEACHING OF GEOMETRY Cissoid, 34 Class in geometry, 108 Clavius, 121 Colleges, geometry in the, 46 Collet, 24 Commensurable magnitudes, 206, 207 Conchoid, 34 Condorcet, 23 Cone, 315 Congruent, 151 Conic sections, 33, 315 Continuity, 212 Converse proposition, 175, 190, 191 Crelle, 142 Cube, duplicating the, 32, 307 Cylinder, 313 D'Alembert, 24, 67 Dase, 279 Decagon, 273 Definitions, 41, 132 De Judaeis, 239, 241 De Morgan, A., 58 De Paolis, 67 Descartes, 38, 84, 320 Diamefer, 146 Dihedral, 298 Diodes, 34 Diogenes Laertius, 259 Diorismus, 41 Direction, 150 Distance, 154 Doyle, Conan, 8 Drawing, 95, 221, 281 Duality, 173 Duhamel, 164 Dupin, 11, 217 Duplication problem, 32, 307 Durer, 10 Educational problems, 1 Egypt, 26, 40 Eisenlohr, 27 Engel arid Stackel, 128 England, 14, 46, 58, 60 Epicureans, 188 Equal, 151, 153 Equilateral, 147 Equivalent, 151 Eratosthenes, 48 Euclid, 33, 42, 43, 44, 119, 125, 135, 156, 165, 167 ft., 201 ff., et passim ; editions of, 47, 52 ; efforts at improving, 57 ; life of, 47; nature of his "Ele- ments," 52, 55 ; opinions of, 8 Eudemus, 33, 168, 171, 185, 216, 309 Eudoxus, 32, 41, 48, 227, 308, 314, 317 Euler, 38, 280, 318 Eutocius, 184 Exercises, nature of, 74, 103 ; , how to attack, 160 Exhaustions, method of, 31 Extreme and mean ratio, 250 Figures in geometry, 104, 107, 113 Finaeus, 44, 239, 240, 243 Fourier, 142 Fourth dimension, 326 Frankland, 56, 117, 127, 135, 159 Fusion, of algebra and geometry, 84 ; of geometry and tMgonom- etry, 91 Gargioli, 44 Gauss, 140, 274 Geminus, 126, 128, 149 Geometry, books of, 165, 167, .201, 227, 262, 269, 289, 303, 321 ; compared with other sub- jects, 14 ; introduction to, 93 ; modern, 38 ; of motion, 68, 196 ; reasons for teaching, 7; 15, 20 ; related to algebra, 84 ; text- books in, 70 Gerbert, 43 Gherard of Cremona, 37, 51 Gnomon, 212 Golden section, 250 Gothic windows, 75, 221 ff., 274, 282 Gow, J., 29, 56 Greece, 28, 40 Gregoire de St. Vincent, 267 Gregory, 280 Gr^vy, 67 Gymnasia, geometry in the, 45 INDEX 337 Hadamard, 164 Hamilton, W., 14 Harmonic division, 231 Harpedonaptae, 28 Harriot, 37 Harvard syllabus, 63 Heath, T. L., 49, 56, 119, 126, 127, 135, 149, 159, 170, 175, 228, 261 Hebrews, 26 Henrici, 0., 11, 14, 25, 164, 196 Henrici and Treutlein, 68, 164, 196 Hermits, 281 Herodotus, 28 Heron, 35, 137, 139, 141, 209, 259, 267 Hexagon, regular, 272 High schools, geometry in the, 45 Hilbert, 119, 131 Hipparchus, 35 Hippasus, 273, 309 Hippias, 31, 215 Hippocrates, 31, 41, 281 History of geometry, 26 Hobson, 166 Hoffmann, 242 Holzmuller, 320 Hughes, Justice, 9 Hypatia, 36 Hypsicles, 34 Hypsometer, 245 lamblichus, 273, 309 Illusions, optical, 100 Ingrami, 128 Instruments, 96, 178, 236 Introduction to geometry, 93 Ionic school, 28 Jackson, C. S., 12 Jones, W., 271 Junge, 259 Karagiannides, 128 Karpinski, 37 Kaye, G. B., 232 Kepler, 24, 149 Kingsley, C, 36 Klein, F., 68, 89 Kolb, 222 Lacroix, 24, 46, 62, 66 Langley, E. M., 291 Laplace, 101 Legendre, 10, 45, 62, 127, 128, 152 Leibnitz, 140, 150 Leon, 41 Leonardo da Vinci, 264 Leonardo of Pisa, 37, 43, 279 Lettering figures, 105 Limits, 207 Lindemann, 278, 281 Line defined, 137 Lobachevsky, 128 Loci, 163, 198 Locke, W. J., 13 Lodge, A., 14 Logic, 17, 104 Loonvs, 164 Ludolph van Ceulen, 279 Lyc^es, geometry in, 45 M'Clelland, 38 McCormack, T. J., 11 Measured by, 208 Memorizing, 12, 79 Mensechmus, 33, 316 Menelaus, 35 M6ray, 67, 68, 196, 289 Methods, 41, 42, 115 Metius, 279 Mikami, 264 Minchin, 14 Models, 93, 290 Modern geometry, 38 Mohammed ibn Musa, 37 Moore, E. H., 131 Mosaics, 274 MuUer, H., 68 Munsterberg, 22 Napoleon, 24, 287 Newton, 24 Nicomedes, 34, 215 Octant, 242 Oinopides, 31, 212, 216 Optical illusions, 100 Oughtred, 37 Paciuolo, 86 Pamphilius, 185 338 THE TEACHING OF GEOMETRY Pappus, 36, 230, 263 Parallelepiped, 303 Parallels, 149, 181 Parquetry, 222, 274 Pascal, 24, 159 Peletier, 169 Perigon, 151 Perry, J., 13, 14 Petersen, 164 Philippus of Mende, 32, 185 Philo, 178 Philolaus, 309 TT, 26, 27, 34, 36, 271, 278, 280 Plane, 140 Plato, 25, 31, 41, 48, 129, 136, 137, 309, 310 Playfair, 128 Pleasure of geometry, 16 . Plimpton, G. A., 51, 52 Plutarch, 259 Poinsot, 311 Point, 135 Polygons, 156, 252, 269, 274 Polyhedrons, 301, 303, 310 Pomodoro, 179 Pons asinorum, 174, 265 Posidonius, 128, 149 Postulates, 31, 41, 116, 125, 292 Practical geometry, 3, 7, 9, 44 Printing, effect of, 44 Prism, 303 Problems, applied, 75, 103, 178, 186, 192, 195, 203, 204, 209, 215, 217, 242, 267, 295, 317 Proclus, 86, 47, 48, 52, 71, 127, 128, 136, 137, 139, 140, 149, 155, 186, 188, 197, 212, 214, 253, 258, 310, 311 Projections, 300 Proofs in full, 79 Proportion, 32, 227 Psychology, 12, 20 Ptolemy, C, 85, 278 ; king, 48, 49 Pyramid, 307 Pythagoras, 29, 40, 258, 272, 273, 310 Pythagorean Theorem, 28, 36, 258 Pythagorean numbers, 82, 86, 268, 266 Pythagoreans, 136, 137, 185, 227, 269, 309 Quadrant, 236 Quadratrix, 31, 215 Quadrilaterals, 148, 157 Quadrivium, 42 Questions at issue, 3 Rabelais, 24 Radius, 153 Ratio, 205, 227 Real problem defined, 75, 103 Reasons for studying geometry, 7, 100 Rebifere, 25 Reciprocal propositions, 173 Recitation in geometry, 113 Recorde, 87 Rectilinear figures, 146 Reduotio ad absurdum, 41, 177 Regular polygons, 269 "Rhind Papyrus," 27 Rhombus and rhomboid, 148 Riccardi, 47 Riohter, 279 Roman surveyors, 247 Sacoheri, 127 Sacrobosco, 43 San Giovanni, 192 Sauvage, 164 Sayre, 272 Scalene, 147 Schlegel, 68 Schopenhauer, 121, 265 Schotten, 46, 135, 149 Sector, 154, 166 Segment, 154 Semicircle, 146 Shanks, 279 Similar figures, 232 Simon, 38, 56, 135 Simson, 142 Sisam, 11 Smith, D. E., 25, 37, 51, 52, 119, 131, 135, 159 Solid geometry, 289 Speusippus, 32 Sphere, 821 Square on a line, 257 Squaring the circle, 31, 82, 277 StSckel, 128 Stamper, 46, 83 INDEX 339 stark, W. E., 172, 238 Stereoscopic slides, 291 StpbsBus, 8 Straight angle, 151 Straight line, 138 Suggested proofs, 81 Sulvasutras, 232 Superposition, 169 Surface, 140 Swain, G. F., 13 Syllabi, 58, 60, 63, 64, 66, 67, 80, 82 Sylvester II, 43 Synthetic method, 161 Tangent, 154 Tartaglia, 158 Tatius, Achilles, 272 Teaching geometiy, reasons for, 7, 15, 20 ; development of, 40 Textbooks, 32, 33, 41, 70, 80, 82 Thales, 28, 168, 171, 185, 210 Thesetetus, 48, 310 Theon of Alexandria, 36 Thibaut, 185 Thoreau, 246 Trapezium, 148 Treutlein, 68, 164 Triangle, 147 Trigonometry, 234 Trisection problem, 31, 215 Trivium, 42 Universities, geometry in the, 43 Uselessness of mathematics, 13 Veblen, 131, 159 Vega, 279 Veronese, 68 Vieta, 279 Vogt, 259 Wallis, 127, 280 Young, J. W. A., 25, 131, 159, 277 Zamberti, 52 Zenodorus, 34, 253 ANNOUNCEMENTS BOOKS FOR TEACHERS List price Allen: Civics and Health I1.25 Brigham : Geographic Influences in American History . . Channing and Hart : Guide to the Study of American History Hall : Aspects of Child Life and Education .... . . Hodge : Nature Study and Life . . Johnson : Education by Plays and Games . . . Johnson : What to Do at Recess . . . Kern : Among Country Schools ... ... Mace : Method in History .... MacVicar : The Principles of Education . ... Moral Training in the Public Schools Prince : Courses of Studies and Methods of Teaching Scott : Social Education . . Tompkins : Philosophy of School Management . . Tompkins : Philosophy of Teaching . . Wiltse : Place of the Story in Early Education, and Other Essays. A Manual for Teachers 25 00 5° 50 90 25 25 00 60 25 75 25 75 75 50 FOR CLASS RECORDS .Comings; Complete Record^- Attendance and Scholarship Graded-School Edition 3° High- School Edition 3° Ginn and. Company ; Teacher's Class Books No. I 30 No. II 40 Twenty Weeks' Class Book 30 196 GINN AND COMPANY Publishers CIVICS AND HEALTH By WILLIAM H. ALLEN, Secretary of the Bureau of Municipal Research, New York City. With an Introduction by Professor William T. Sedgwick, Professor of Biology in the Massachusetts Institute of Technology List price, $1.25 Adopted by the Teachers' Reading Circles of Maryland, Kentucky, North Dakota, South Dakota, Okla- homa, Netv Mexico, South Carolina, Alabama, Arizona, Illi- nois, Michigan, Colorado, Texas, Virginia, Iowa, Arkansas, Wyoming, Missouri, Indiana, Nebraska, and Washington FOR Dr. Allen prevention is a text and the making of sound citizens a sermon. In " Civics and Health " he sounds a slogan which should awaken every community in this country to its opportunities in municipal reform. Every teacher who reads this book will gain a new sense of duty in matters of hygiene and sanitation. Civics and Health is entbiallingly interesting. It is humanized sociology. Cleaning up children by scientific illumination will appeal to every father and mother, every child lover who has' any patriotism or desire to learn how we as a people are to make moral-reform agita- tions fruitful through health of American children, and so establish health of national life. — Boston Transcript. This is one of the books we wish the law required every citizen to have in his bouse and to know by heart. Then, indeed, mankind would have made an immense stride forward. — Chicago Medical Recorder, The book is alive from cover to cover. It breathes reform but not of thfe platforirt variety. It abounds in ugly facts- but superabounds in the statement of best methods of getting rid of this ugliness. As claimed by the publishers, it is preeminently a book on " getting things done." — Hygiene and Physical Education^ Springfield, Mass. GINN AND COMPANY Publishers