,,the ,,teaching ,,of ,,geometry ,,by ,,david ,,eugene ,,smith ,,ginn ,,and ,,company ,,boston 4 ,,new ,,york 4 ,,chicago 4 ,,london ,,copyright1 #aiaa1 ,,by ,,david ,,eugene ,,smith ,,all ,,rights ,,reserved #iaa4f ,the ,athenaeum ,press ,,ginn ,,and ,,company 4 ,,proprietors ,,boston 4 ,u4,s4,a4 ,,preface ,a book upon the teaching of geometry may be planned in divers ways4 ,it may be written to exploit a new theory of geometry1 or a new method of presenting the science as we already have it4 ,on the other hand1 it may be ultraconservative1 making a plea for the ancient teaching and the ancient geometry4 ,it may be prepared for the purpose of setting forth the work as it now is1 or with the tempting but dangerous idea of prophecy4 ,it may appeal to the iconoclast by its spirit of destruction1 or to the disciples of laissez faire by its spirit of conserving what the past has bequeathed4 ,it may be written for the few who always lead1 or think they lead1 or for the many who are ranked by the few as followers4 ,and in view of these varied pathways into the joint domain of geometry and education1 a writer may well afford to pause before he sets his pen to paper1 and to decide with care the route that he will take4 ,at present in ,america we have a fairly well-defined body of matter in geometry1 and this occupies a fairly well-defined place in the curriculum4 ,there are not wanting many earnest teachers who would change both the matter and the place in a very radical fashion4 ,there are not wanting others1 also many in number1 who are content with things as they find them4 ,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 things1 contributing to this evolution as much as it can1 glad to know the best that others have to offer1 receptive of ideas that make for better teaching1 but out of sympathy with either the extreme of revolution or the extreme of stagnation4 ,it is for this larger class1 the great body of progressive teachers1 that this book is written4 ,it stands for vitalizing geometry in every legitimate way2 for improving the subject matter in such manner as not to destroy the pupil's interest2 for so teaching geometry as to make it appeal to pupils as strongly as any other subject in the curriculum2 but for the recognition of geometry for geometry's sake and not for the sake of a fancied utility that hardly exists4 ,expressing full appreciation of the desirability of establishing a motive for all studies1 so as to have the work proceed with interest and vigor1 it does not hesitate to express doubt as to certain motives that have been exploited1 nor to stand for such a genuine1 thought-compelling development of the science as is in harmony with the mental powers of the pupils in the ,american high school4 ,for this class of teachers the author hopes that the book will prove of service1 and that through its perusal they will come to admire the subject more and more1 and to teach it with greater interest4 ,it offers no panacea1 it champions no single method1 but it seeks to set forth plainly the reasons for teaching a geometry of the kind that we have inherited1 and for hoping for a gradual but definite improvement in the science and in the methods of its presentation4 ,,david ,,eugene ,,smith ,,contents ,,chapter ,,page ,i4 ,,certain ,,questions ,,now ,,at ,,issue #a ,,ii4 ,,why ,,geometry ,,is ,,studied #g ,,iii4 ,a ,,brief ,,history ,,of ,,geometry #bf ,,iv4 ,,development ,,of ,,the ,,teaching ,,of ,,geometry #dj ,v4 ,,euclid #dg ,,vi4 ,,efforts ,,at ,,improving ,,euclid #eg ,,vii4 ,,the ,,textbook ,,in ,,geometry #gj ,,viii4 ,,the ,,relation ,,of ,,algebra ,,to ,,geometry #hd ,,ix4 ,,the ,,introduction ,,to ,,geometry #ic ,x4 ,,the ,,conduct ,,of ,a ,,class ,,in ,,geometry #ajh ,,xi4 ,,the ,,axioms ,,and ,,postulates #aaf ,,xii4 ,,the ,,definitions ,,of ,,geometry #acb ,,xiii4 ,,how ,,to ,,attack ,,the ,,exercises #afj ,,xiv4 ,,book ,i ,,and ,,its ,,propositions #afe ,,xv4 ,,the ,,leading ,,propositions ,,of ,,book ,,ii #bja ,,xvi4 ,,the ,,leading ,,propositions ,,of ,,book ,,iii #bbg ,,xvii4 ,,the ,,leading ,,propositions ,,of ,,book ,,iv #beb ,,xviii4 ,,the ,,leading ,,propositions ,,of ,,book ,v #bfi ,,xix4 ,,the ,,leading ,,propositions ,,of ,,book ,,vi #bhi ,,xx4 ,,the ,,leading ,,propositions ,,of ,,book ,,vii #cjc ,,xxi4 ,,the ,,leading ,,propositions ,,of ,,book ,,viii #cba ,,index #cce ,,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 school4 ,it would be pleasant to feel that we are living in such an age1 but it is doubtful if the future historian of education will find this to be the case1 or that biographers will rank the leaders of our generation relatively as high as many who have passed away1 or that any great movements of the present will be found that measure up to certain ones that the world now recognizes as epoch-making4 ,every generation since the invention of printing has been a period of agitation in educational matters1 but out of all the noise and self-assertion1 out of all the pretense of the chronic revolutionist1 out of all the sham that leads to dogmatism1 so little is remembered that we are apt to feel that the past had no problems and was content simply to accept its inheritance4 ,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 agitator1 since it stirs us to action at finding ourselves in the midst of the skirmish2 but in another sense it is detrimental to our progress1 since we thereby tend to lose the idea of perspective1 and the coin comes to appear to our vision as large as the moon4 ,in considering a question like the teaching of geometry1 we at once find ourselves in the midst of a skirmish of this nature4 ,if we join thoughtlessly in the noise1 we may easily persuade ourselves that we are waging a mighty battle1 fighting for some stupendous principle1 doing deeds of great valor and of personal sacrifice4 ,if1 on the other hand1 we stand aloof and think of the present movement as merely a chronic effervescence1 fostered by the professional educator at the expense of the practical teacher1 we are equally shortsighted4 ,sir ,conan ,doyle expressed this sentiment most delightfully in these words3 ,the dead are such good company that one may come to think too little of the living4 ,it is a real and pressing danger with many of us that we should never find our own thoughts and our own souls1 but be ever obsessed by the dead4 ,in every generation it behooves the open-minded1 earnest1 progressive teacher to seek for the best in the way of improvement1 to endeavor to sift the few grains of gold out of the common dust1 to weigh the values of proposed reforms1 and to put forth his efforts to know and to use the best that the science of education has to offer4 ,this has been the attitude of mind of the real leaders in the school life of the past1 and it will be that of the leaders of the future4 ,with these remarks to guide us1 it is now proposed to take up the issues of the present day in the teaching of geometry1 in order that we may consider them calmly and dispassionately1 and may see where the opportunities for improvement lie4 ,at the present time1 in the educational circles of the ,united ,states1 questions of the following type are causing the chief discussion among teachers of geometry3 #a4 ,shall geometry continue to be taught as an application of logic1 or shall it be treated solely with reference to its applications8 #b4 ,if the latter is the purpose in view1 shall the propositions of geometry be limited to those that offer an opportunity for real application1 thus contracting the whole subject to very narrow dimensions8 #c4 ,shall a subject called geometry be extended over several years1 as is the case in ,europe1#a or shall the name be applied only to serious demonstrative geometry#b as given in the second year of the four-year high school course in the ,united ,states at present8 #d4 ,shall geometry be taught by itself1 or shall it be either mixed with algebra 7say a day of one subject followed by a day of the other7 or fused with it in the form of a combined mathematics8 #e4 ,shall a textbook be used in which the basal propositions are proved in full1 the exercises furnishing the opportunity for original work and being looked upon as the most important feature1 or shall one be employed in which the pupil is expected to invent the proofs for the basal propositions as well as for the exercises8 #f4 ,shall the terminology and the spirit of a modified ,euclid and ,legendre prevail in the future as they have in the past1 or shall there be a revolution in the use of terms and in the general statements of the propositions8 #g4 ,shall geometry be made a strong elective subject1 to be taken only by those whose minds are capable of serious work8 ,shall it be a required subject1 diluted to the comprehension of the weakest minds8 ,or is it now1 by proper teaching1 as suitable for all pupils as is any other required subject in the school curriculum8 ,and in any case1 will the various distinct types of high schools now arising call for distinct types of geometry8 ,this brief list might easily be amplified1 but it is sufficiently extended to set forth the trend of thought at the present time1 and to show that the questions before the teachers of geometry are neither particularly novel nor particularly serious4 ,these questions and others of similar nature are really side issues of two larger questions of far greater significance3 7#a7 ,are the reasons for teaching demonstrative geometry such that it should be a required subject1 or at least a subject that is strongly recommended to all1 whatever the type of high school8 7#b7 ,if so1 how can it be made interesting8 ,the present work is written with these two larger questions in mind1 although it considers from time to time the minor ones already mentioned1 together with others of a similar nature4 ,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 case1 and that for this type a different kind of mathematical training will naturally be developed4 ,it proceeds upon the theory1 however1 that for the normal mind1--for the boy or girl who is preparing to win out in the long run1--geometry will continue to be taught as demonstrative geometry1 as a vigorous thought-compelling subject1 and along the general lines that the experience of the world has shown to be the best4 ,soft mathematics is not interesting to this normal mind1 and a sham treatment will never appeal to the pupil2 and this book is written for teachers who believe in this principle1 who believe in geometry for the sake of geometry1 and who earnestly seek to make the subject so interesting that pupils will wish to study it whether it is required or elective4 ,the work stands for the great basal propositions that have come down to us1 as logically arranged and as scientifically proved as the powers of the pupils in the ,american high school will permit2 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 classes4 ,after all1 the problem of teaching any subject comes down to this3 ,get a subject worth teaching and then make every minute of it interesting4 ,pupils do not object to work if they like a subject1 but they do object to aimless and uninteresting tasks4 ,geometry is particularly fortunate in that the feeling of accomplishment comes with every proposition proved2 and1 given a class of fair intelligence1 a teacher must be lacking in knowledge and enthusiasm who cannot foster an interest that will make geometry stand forth as the subject that brings the most pleasure1 and that seems the most profitable of all that are studied in the first years of the high school4 ,continually to advance1 continually to attempt to make mathematics fascinating1 always to conserve the best of the old and to sift out and use the best of the new1 to believe that mankind is better served by nature's quiet and progressive changes than by earthquakes1#c to believe that geometry as geometry is so valuable and so interesting that the normal mind may rightly demand it1--this is to ally ourselves with progress4 ,continually to destroy1 continually to follow strange gods1 always to decry the best of the old1 and to have no well-considered aim in the teaching of a subject1--this is to join the forces of reaction1 to waste our time1 to be recreant to our trust1 to blind ourselves to the failures of the past1 and to confess our weakness as teachers4 ,it is with the desire to aid in the progressive movement1 to assist those who believe that real geometry should be recommended to all1 and to show that geometry is both attractive and valuable that this book is written4 ,,footnotes3 #a ,and really1 though not nominally1 in the ,united ,states1 where the first concepts are found in the kindergarten1 and where an excellent course in mensuration is given in any of our better class of arithmetics4 ,that we are wise in not attempting serious demonstrative geometry much earlier seems to be generally conceded4 #b ,the third stage of geometry as defined in the recent circular 7,no4 #gaa7 of the ,british ,board of ,education1 ,london1 #aiji4 #c ,the closing words of a sensible review of the ,british ,board of ,education circular 7,no4 #gaa71 on ,the ,teaching of ,geometry 7,london1 #aiji71 by ,h4 ,s4 ,hall in the ,school ,world1 #aiji1 p4 #bbb4 ,,chapter ,,ii ,,why ,,geometry ,,is ,,studied ,with geometry1 as with other subjects1 it is easier to set forth what are not the reasons for studying it than to proceed positively and enumerate the advantages4 ,although such a negative course is not satisfying to the mind as a finality1 it possesses definite advantages in the beginning of such a discussion as this4 ,whenever false prophets arise1 and with an attitude of pained superiority proclaim unworthy aims in human life1 it is well to show the fallacy of their position before proceeding to a constructive philosophy4 ,taking for a moment this negative course1 let us inquire as to what are not the reasons for studying geometry1 or1 to be more emphatic1 as to what are not the worthy reasons4 ,in view of a periodic activity in favor of the utilities of geometry1 it is well to understand1 in the first place1 that geometry is not studied1 and never has been studied1 because of its positive utility in commercial life or even in the workshop4 ,in ,america we commonly allow at least a year to plane geometry and a half year to solid geometry2 but all of the facts that a skilled mechanic or an engineer would ever need could be taught in a few lessons4 ,all the rest is either obvious or is commercially and technically useless4 ,we prove1 for example1 that the angles opposite the equal sides of a triangle are equal1 a fact that is probably quite as obvious as the postulate that but one line can be drawn through a given point parallel to a given line4 ,we then prove1 sometimes by the unsatisfactory process of reductio ad absurdum1 the converse of this proposition1--a fact that is as obvious as most other facts that come to our consciousness1 at least after the preceding proposition has been proved4 ,and these two theorems are perfectly fair types of upwards of one hundred sixty or seventy propositions comprising ,euclid's books on plane geometry4 ,they are generally not useful in daily life1 and they were never intended to be so4 ,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 teachers4 ,a ,greek writer1 ,stobaeus1 relates the story in these words3 ,some one who had begun to read geometry with ,euclid1 when he had learned the first theorem1 asked1 ,but what shall ,i get by learning these things8 ,euclid called his slave and said1 ,give him three obols1 since he must make gain out of what he learns4 ,whether true or not1 the story expresses the sentiment that runs through ,euclid's work1 and not improbably we have here a bit of real biography1--practically all of the personal ,euclid that has come down to us from the world's first great textbook maker4 ,it is well that we read the story occasionally1 and also such words as the following1 recently uttered#d by ,sir ,conan ,doyle1--words bearing the same lesson1 although upon a different theme3 ,in the present utilitarian age one frequently hears the question asked1 ,what is the use of it all8 as if every noble deed was not its own justification4 ,as if every action which makes for self-denial1 for hardihood1 and for endurance was not in itself a most precious lesson to mankind4 ,that people can be found to ask such a question shows how far materialism has gone1 and how needful it is that we insist upon the value of all that is nobler and higher in life4 ,an ,american statesman and jurist1 speaking upon a similar occasion#e1 gave utterance to the same sentiments in these words3 ,when the time comes that knowledge will not be sought for its own sake1 and men will not press forward simply in a desire of achievement1 without hope of gain1 to extend the limits of human knowledge and information1 then1 indeed1 will the race enter upon its decadence4 ,there have not been wanting1 however1 in every age1 those whose zeal is in inverse proportion to their experience1 who were possessed with the idea that it is the duty of the schools to make geometry practical4 ,we have them to-day1 and the world had them yesterday1 and the future shall see them as active as ever4 ,these people do good to the world1 and their labors should always be welcome1 for out of the myriad of suggestions that they make a few have value1 and these are helpful both to the mathematician and the artisan4 ,not infrequently they have contributed material that serves to make geometry somewhat more interesting1 but it must be confessed that most of their work is merely the threshing of old straw1 like the work of those who follow the will-o'-the-wisp of the circle squarers4 ,the medieval astrologers wished to make geometry more practical1 and so they carried to a considerable length the study of the star polygon1 a figure that they could use in their profession4 ,the cathedral builders1 as their art progressed1 found that architectural drawings were more exact if made with a single opening of the compasses1 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 science4 ,later1 and about the beginning of the sixteenth century1 the revival of art1 and particularly the great development of painting1 led to the practical application of geometry to the study of perspective and of those curves#f that occur most frequently in the graphic arts4 ,the sixteenth and seventeenth centuries witnessed the publication of a large number of treatises on practical geometry1 usually relating to the measuring of distances and partly answering the purposes of our present trigonometry4 ,such were the well-known treatises of ,belli 7#aefi71 ,cataneo 7#aefg71 and ,bartoli 7#aehi74g ,the period of two centuries from about #afjj to about #ahjj was quite as much given to experiments in the creation of a practical geometry as is the present time1 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 mathematician1 ,legendre1 to publish his geometry in #agid1--;a work that soon replaced ,euclid in the schools of ,america4 ,it thus appears that the effort to make geometry practical is by no means new4 ,euclid knew of it1 the ,middle ,ages contributed to it1 that period vaguely styled the ,renaissance joined in the movement1 and the first three centuries of printing contributed a large literature to the subject4 ,out of all this effort some genuine good remains1 but relatively not very much4#h ,and so it will be with the present movement2 it will serve its greatest purpose in making teachers think and read1 and in adding to their interest and enthusiasm and to the interest of their pupils2 but it will not greatly change geometry1 because no serious person ever believed that geometry was taught chiefly for practical purposes1 or was made more interesting or valuable through such a pretense4 ,changes in sequence1 in definitions1 and in proofs will come little by little2 but that there will be any such radical change in these matters in the immediate future1 as some writers have anticipated1 is not probable4#i ,a recent writer of much acumen#aj has summed up this thought in these words3 ,not one tenth of the graduates of our high schools ever enter professions in which their algebra and geometry are applied to concrete realities2 not one day in three hundred sixty-five is a high school graduate called upon to apply1 as it is called1 an algebraic or a geometrical proposition4444 ,why1 then1 do we teach these subjects1 if this alone is the sense of the word practical6444 ,to me the solution of this paradox consists in boldly confronting the dilemma1 and in saying that our conception of the practical utility of those studies must be readjusted1 and that we have frankly to face the truth that the practical ends we seek are in a sense ideal practical ends1 yet such as have1 after all1 an eminently utilitarian value in the intellectual sphere4 ,he quotes from ,c4 ,s4 ,jackson1 a progressive contemporary teacher of mechanics in ,england1 who speaks of pupils confusing millimeters and centimeters in some simple computation1 and who adds3 ,there is the enemy6 ,the real enemy we have to fight against1 whatever we teach1 is carelessness1 inaccuracy1 forgetfulness1 and slovenliness4 ,that battle has been fought and won with diverse weapons4 ,it has1 for instance1 been fought with ,latin grammar before now1 and won4 ,i say that because we must be very careful to guard against the notion that there is any one panacea for this sort of thing4 ,it borders on quackery to say that elementary physics will cure everything4 ,and of course the same thing may be said for mathematics4 ,nevertheless it is doubtful if we have any other subject that does so much to bring to the front this danger of carelessness1 of slovenly reasoning1 of inaccuracy1 and of forgetfulness as this science of geometry1 which has been so polished and perfected as the centuries have gone on4 ,there have been those who did not proclaim the utilitarian value of geometry1 but who fell into as serious an error1 namely1 the advocating of geometry as a means of training the memory4 ,in times not so very far past1 and to some extent to-day1 the memorizing of proofs has been justified on this ground4 ,this error has1 however1 been fully exposed by our modern psychologists4 ,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 before1 and that the learning of proofs verbatim is of no assistance whatever in retaining matter that is helpful in other lines of work4 ,geometry1 therefore1 as a training of the memory is of no more value than any other subject in the curriculum4 ,if geometry is not studied chiefly because it is practical1 or because it trains the memory1 what reasons can be adduced for its presence in the courses of study of every civilized country8 ,is it not1 after all1 a mere fetish1 and are not those virulent writers correct who see nothing good in the subject save only its utilities8#aa ,of this type one of the most entertaining is ,william ,j4 ,locke1#ab whose words upon the subject are well worth reading3 444 ,i earned my living at school slavery1 teaching to children the most useless1 the most disastrous1 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 mathematics4 ,there is no more reason for any human being on ,god's earth to be acquainted with the binomial theorem or the solution of triangles1 unless he is a professional scientist1--when he can begin to specialize in mathematics at the same age as the lawyer begins to specialize in law or the surgeon in anatomy1--than for him to be expert in ,choctaw1 the ,cabala1 or the ,book of ,mormon4 ,i look back with feelings of shame and degradation to the days when1 for a crust of bread1 ,i prostituted my intelligence to wasting the precious hours of impressionable childhood1 which could have been filled with so many beautiful and meaningful things1 over this utterly futile and inhuman subject4 ,it trains the mind1--it teaches boys to think1 they say4 ,it doesn't4 ,in reality it is a cut-and-dried subject1 easy to fit into a school curriculum4 ,its sacrosanctity saves educationalists an enormous amount of trouble1 and its chief use is to enable mindless young men from the universities to make a dishonest living by teaching it to others1 who in their turn may teach it to a future generation4 ,to be fair we must face just such attacks1 and we must recognize that they set forth the feelings of many honest people4 ,one is tempted to inquire if ,mr4 ,locke could have written in such an incisive style if he had not1 as was the case1 graduated with honors in mathematics at one of the great universities4 ,but he might reply that if his mind had not been warped by mathematics1 he would have written more temperately1 so the honors in the argument would be even4 ,much more to the point is the fact that ,mr4 ,locke taught mathematics in the schools of ,england1 and that these schools do not seem to the rest of the world to furnish a good type of the teaching of elementary mathematics4 ,no country goes to ,england for its model in this particular branch of education1 although the work is rapidly changing there1 and ,mr4 ,locke pictures a local condition in teaching rather than a general condition in mathematics4 ,few visitors to the schools of ,england would care to teach mathematics as they see it taught there1 in spite of their recognition of the thoroughness of the work and the earnestness of many of the teachers4 ,it is also of interest to note that the greatest protests against formal mathematics have come from ,england1 as witness the utterances of such men as ,sir ,william ,hamilton and ,professors ,perry1 ,minchin1 ,henrici1 and ,alfred ,lodge4 ,it may therefore be questioned whether these scholars are not unconsciously protesting against the ,english methods and curriculum rather than against the subject itself4 ,when ,professor ,minchin says that he had been through the six books of ,euclid without really understanding an angle1 it is ,euclid's text and his own teacher that are at fault1 and not geometry4 ,before considering directly the question as to why geometry should be taught1 let us turn for a moment to the other subjects in the secondary curriculum4 ,why1 for example1 do we study literature8 ,it does not lower the price of bread1 as ,malherbe remarked in speaking of the commentary of ,bachet on the great work of ,diophantus4 ,is it for the purpose of making authors8 ,not one person out of ten thousand who study literature ever writes for publication4 ,and why do we allow pupils to waste their time in physical education8 ,it uses valuable hours1 it wastes money1 and it is dangerous to life and limb4 ,would it not be better to set pupils at sawing wood8 ,and why do we study music8 ,to give pleasure by our performances8 ,how many who attempt to play the piano or to sing give much pleasure to any but themselves1 and possibly their parents8 ,the study of grammar does not make an accurate writer1 nor the study of rhetoric an orator1 nor the study of meter a poet1 nor the study of pedagogy a teacher4 ,the study of geography in the school does not make travel particularly easier1 nor does the study of biology tend to populate the earth4 ,so we might pass in review the various subjects that we study and ought to study1 and in no case would we find utility the moving cause1 and in every case would we find it difficult to state the one great reason for the pursuit of the subject in question1--and so it is with geometry4 ,what positive reasons can now be adduced for the study of a subject that occupies upwards of a year in the school course1 and that is1 perhaps unwisely1 required of all pupils8 ,probably the primary reason1 if we do not attempt to deceive ourselves1 is pleasure4 ,we study music because music gives us pleasure1 not necessarily our own music1 but good music1 whether ours1 or1 as is more probable1 that of others4 ,we study literature because we derive pleasure from books2 the better the book the more subtle and lasting the pleasure4 ,we study art because we receive pleasure from the great works of the masters1 and probably we appreciate them the more because we have dabbled a little in pigments or in clay4 ,we do not expect to be composers1 or poets1 or sculptors1 but we wish to appreciate music and letters and the fine arts1 and to derive pleasure from them and to be uplifted by them4 ,at any rate1 these are the nobler reasons for their study4 ,so it is with geometry4 ,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 perfected1 and we are uplifted by it4 ,to deny that our pupils derive this pleasure from the study is to confess ourselves poor teachers1 for most pupils do have positive enjoyment in the pursuit of geometry1 in spite of the tradition that leads them to proclaim a general dislike for all study4 ,this enjoyment is partly that of the game1--the playing of a game that can always be won1 but that cannot be won too easily4 ,it is partly that of the aesthetic1 the pleasure of symmetry of form1 the delight of fitting things together4 ,but probably it lies chiefly in the mental uplift that geometry brings1 the contact with absolute truth1 and the approach that one makes to the ,infinite4 ,we are not quite sure of any one thing in biology2 our knowledge of geology is relatively very slight1 and the economic laws of society are uncertain to every one except some individual who attempts to set them forth2 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 triangle1 and it will be so after this world is dead2 and the inhabitant of ,mars1 if he exists1 probably knows its truth as we know it4 ,the uplift of this contact with absolute truth1 with truth eternal1 gives pleasure to humanity to a greater or less degree1 depending upon the mental equipment of the particular individual2 but it probably gives an appreciable amount of pleasure to every student of geometry who has a teacher worthy of the name4 ,first1 then1 and foremost as a reason for studying geometry has always stood1 and will always stand1 the pleasure and the mental uplift that comes from contact with such a great body of human learning1 and particularly with the exact truth that it contains4 ,the teacher who is imbued with this feeling is on the road to success1 whatever method of presentation he may use2 the one who is not imbued with it is on the road to failure1 however logical his presentation or however large his supply of practical applications4 ,subordinate to these reasons for studying geometry are many others1 exactly as with all other subjects of the curriculum4 ,geometry1 for example1 offers the best developed application of logic that we have1 or are likely to have1 in the school course4 ,this does not mean that it always exemplifies perfect logic1 for it does not2 but to the pupil who is not ready for logic1 per se1 it offers an example of close reasoning such as his other subjects do not offer4 ,we may say1 and possibly with truth1 that one who studies geometry will not reason more clearly on a financial proposition than one who does not2 but in spite of the results of the very meager experiments of the psychologists1 it is probable that the man who has had some drill in syllogisms1 and who has learned to select the essentials and to neglect the nonessentials in reaching his conclusions1 has acquired habits in reasoning that will help him in every line of work4 ,as part of this 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 writers4 ,upon this same topic an ,english writer1 in one of the sanest of recent monographs upon the subject1#ac has expressed his views in the following words3 ,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 impressions1 that he has then eliminated from their consideration all irrelevant impressions 7in other words1 acquired a subjective command of these impressions71 that he has developed on the basis of these impressions an ordered and continuous system of logical deduction1 and finally that he is capable of expressing the nature of these impressions and his deductions therefrom in terms simple and free from ambiguity4 ,now the slightest consideration will convince any one not already conversant with the idea1 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 labor2 consequently a full training in the performance of such sequences must be regarded as forming an essential part of any education worthy of the name4 ,moreover1 the full appreciation of such processes has a higher value than is contained in the mental training involved1 great though this be1 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 faculties2 or1 again1 that part which the appreciation of a body of religious doctrine plays for the ethical aspirations4 ,now geometry is not the sole possible basis for inculcating this appreciation4 ,logic is an alternative for adults1 provided that the individual is possessed of sufficient wide1 though rough1 experience on which to base his reasoning4 ,geometry is1 however1 highly desirable in that the objective bases are so simple and precise that they can be grasped at an early age1 that the amount of training for the imagination is very large1 that the deductive processes are not beyond the scope of ordinary boys1 and finally that it affords a better basis for exercise in the art of simple and exact expression than any other possible subject of a school course4 ,are these results really secured by teachers1 however1 or are they merely imagined by the pedagogue as a justification for his existence8 ,do teachers have any such appreciation of geometry as has been suggested1 and even if they have it1 do they impart it to their pupils8 ,in reply it may be said1 probably with perfect safety1 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 education4 ,what teacher appreciates fully the beauties of ,in ,memoriam1 or of ,hamlet1 or of ,paradise ,lost1 and what one inspires his pupils with all the nobility of these world classics8 ,what teacher sees in biology all the grandeur of the evolution of the race1 or imparts to his pupils the noble lessons of life that the study of this subject should suggest8 ,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 contain8 ,and what teacher of ,french succeeds in bringing a pupil to carry on a conversation1 to read a ,french magazine1 to see the history imbedded in the words that are used1 to realize the charm and power of the language1 or to appreciate to the full a single classic8 ,in other words1 none of us fully appreciates his subject1 and none of us can hope to bring his pupils to the ideal attitude toward any part of it4 ,but it is probable that the teacher of geometry succeeds relatively better than the teacher of other subjects1 because the science has reached a relatively higher state of perfection4 ,the body of truth in geometry has been more clearly marked out1 it has been more successfully fitted together1 its lesson is more patent1 and the experience of centuries has brought it into a shape that is more usable in the school4 ,while1 therefore1 we have all kinds of teaching in all kinds of subjects1 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 curriculum4 ,but is this not mere conjecture8 ,what are the results of scientific investigation of the teaching of geometry8 ,unfortunately there is little hope from the results of such an inquiry1 either here or in other fields4 ,we cannot first weigh a pupil in an intellectual or moral balance1 then feed him geometry1 and then weigh him again1 and then set back his clock of time and begin all over again with the same individual4 ,there is no before taking and after taking of a subject that extends over a year or two of a pupil's life4 ,we can weigh utilities roughly1 we can estimate the pleasure of a subject relatively1 but we cannot say that geometry is worth so many dollars1 and history so many1 and so on through the curriculum4 ,the best we can do is to ask ourselves what the various subjects1 with teachers of fairly equal merit1 have done for us1 and to inquire what has been the experience of other persons4 ,such an investigation results in showing that1 with few exceptions1 people who have studied geometry received as much of pleasure1 of inspiration1 of satisfaction1 of what they call training from geometry as from any other subject of study1--given teachers of equal merit1--and that they would not willingly give up the something which geometry brought to them4 ,if this were not the feeling1 and if humanity believed that geometry is what ,mr4 ,locke's words would seem to indicate1 it would long ago have banished it from the schools1 since upon this ground rather than upon the ground of utility the subject has always stood4 ,these seem to be the great reasons for the study of geometry1 and to search for others would tend to weaken the argument4 ,at first sight they may not seem to justify the expenditure of time that geometry demands1 and they may seem unduly to neglect the argument that geometry is a stepping-stone to higher mathematics4 ,each of these points1 however1 has been neglected purposely4 ,a pupil has a number of school years at his disposal2 to what shall they be devoted8 ,to literature8 ,what claim has letters that is such as to justify the exclusion of geometry8 ,to music1 or natural science1 or language8 ,these are all valuable1 and all should be studied by one seeking a liberal education2 but for the same reason geometry should have its place4 ,what subject1 in fine1 can supply exactly what geometry does8 ,and if none1 then how can the pupil's time be better expended than in the study of this science8#ad ,as to the second point1 that a claim should be set forth that geometry is a sine qua non to higher mathematics1 this belief is considerably exaggerated because there are relatively few who proceed from geometry to a higher branch of mathematics4 ,this argument would justify its status as an elective rather than as a required subject4 ,let us then stand upon the ground already marked out1 holding that the pleasure1 the culture1 the mental poise1 the habits of exact reasoning that geometry brings1 and the general experience of mankind upon the subject are sufficient to justify us in demanding for it a reasonable amount of time in the framing of a curriculum4 ,let us be fair in our appreciation of all other branches1 but let us urge that every student may have an opportunity to know of real geometry1 say for a single year1 thereafter pursuing it or not1 according as we succeed in making its value apparent1 or fail in our attempt to present worthily an ancient and noble science to the mind confided to our instruction4 ,the shortsightedness of a narrow education1 of an education that teaches only machines to a prospective mechanic1 and agriculture to a prospective farmer1 and cooking and dressmaking to the girl1 and that would exclude all mathematics that is not utilitarian in the narrow sense1 cannot endure4 ,the community has found out that such schemes may be well fitted to give the children a good time in school1 but lead them to a bad time afterward4 ,life is hard work1 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 immediately1 they have missed the most valuable lesson of their school years4 ,the little practical information they could have learned at any time2 the energy of attention and concentration can no longer be learned if the early years are wasted4 ,however narrow and commercial the standpoint which is chosen may be1 it can always be found that it is the general education which pays best1 and the more the period of cultural work can be expanded the more efficient will be the services of the school for the practical services of the nation4#ae ,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 interesting and to vitalize it by establishing as strong motives as possible for its study4 ,let the home1 the workshop1 physics1 art1 play1--all contribute their quota of motive to geometry as to all mathematics and all other branches4 ,but let us never forget that geometry has a raison d'etre beyond all this1 and that these applications are sought primarily for the sake of geometry1 and that geometry is not taught primarily for the sake of these applications4 ,when we consider how often geometry is attacked by those who profess to be its friends1 and how teachers who have been trained in mathematics occasionally seem to make of the subject little besides a mongrel course in drawing and measuring1 all the time insisting that they are progressive while the champions of real geometry are reactionary1 it is well to read some of the opinions of the masters4 ,the following quotations may be given occasionally in geometry classes as showing the esteem in which the subject has been held in various ages1 and at any rate they should serve to inspire the teacher to greater love for his subject4 ,the enemies of geometry1 those who know it only imperfectly1 look upon the theoretical problems1 which constitute the most difficult part of the subject1 as mental games which consume time and energy that might better be employed in other ways4 ,such a belief is false1 and it would block the progress of science if it were credible4 ,but aside from the fact that the speculative problems1 which at first sight seem barren1 can often be applied to useful purposes1 they always stand as among the best means to develop and to express all the forces of the human intelligence4--,,abbe ,,bossut4 ,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 geometry4--,,condorcet4 ,if mathematical heights are hard to climb1 the fundamental principles lie at every threshold1 and this fact allows them to be comprehended by that common sense which ,descartes declared was apportioned equally among all men4--,,collet4 ,it may seem strange that geometry is unable to define the terms which it uses most frequently1 since it defines neither movement1 nor number1 nor space1---the three things with which it is chiefly concerned4 ,but we shall not be surprised if we stop to consider that this admirable science concerns only the most simple things1 and the very quality that renders these things worthy of study renders them incapable of being defined4 ,thus the very lack of definition is rather an evidence of perfection than a defect1 since it comes not from the obscurity of the terms1 but from the fact that they are so very well known4--,,pascal4 ,god eternally geometrizes4--,,plato4 ,god is a circle of which the center is everywhere and the circumference nowhere4--,,rabelais4 ,without mathematics no one can fathom the depths of philosophy4 ,without philosophy no one can fathom the depths of mathematics4 ,without the two no one can fathom the depths of anything4--,,bordas-,,demoulin4 ,we may look upon geometry as a practical logic1 for the truths which it studies1 being the most simple and most clearly understood of all truths1 are on this account the most susceptible of ready application in reasoning4--,d',,alembert4 ,the advance and the perfecting of mathematics are closely joined to the prosperity of the nation4--,,napoleon4 ,hold nothing as certain save what can be demonstrated4--,,newton4 ,to measure is to know4--,,kepler4 ,the method of making no mistake is sought by every one4 ,the logicians profess to show the way1 but the geometers alone ever reach it1 and aside from their science there is no genuine demonstration4--,,pascal4 ,the taste for exactness1 the impossibility of contenting one's self with vague notions or of leaning upon mere hypotheses1 the necessity for perceiving clearly the connection between certain propositions and the object in view1--these are the most precious fruits of the study of mathematics4--,,lacroix4 ,bibliography4 ,smith1 ,the ,teaching of ,elementary ,mathematics1 p4 #bcd1 ,new ,york1 #aijj2 ,henrici1 ,presidential ,address before the ,british ,association1 ,nature1 ,vol4 ,,xxviii1 p4 #dig2 ,hill1 ,educational ,value of ,mathematics1 ,educational ,review1 ,vol4 ,,ix1 p4 #cdi2 ,young1 ,the ,teaching of ,mathematics1 p4 #i1 ,new ,york1 #aijg4 ,the closing quotations are from ,rebiere1 ,mathematiques et ,mathematiciens1 ,paris1 #ahic4 ,,footnotes3 #d ,in an address in ,london1 ,june #ae1 #aiji1 at a dinner to ,sir ,ernest ,shackelton4 #e ,governor ,hughes1 now ,justice ,hughes1 of ,new ,york1 at the ,peary testimonial on ,february #h1 #aiaj1 at ,new ,york ,city4 #f ,the first work upon this subject1 and indeed the first printed treatise on curves in general1 was written by the famous artist of ,nuernberg1 ,albrecht ,duerer4 #g ,several of these writers are mentioned in ,chapter ,,iv4 #h ,if any reader chances upon ,george ,birkbeck's ,english translation of ,charles ,dupin's ,mathematics ,practically ,applied1 ,halifax1 #ahed1 he will find that ,dupin gave more good applications of geometry than all of our ,american advocates of practical geometry combined4 #i ,see1 for example1 ,henrici's ,congruent ,figures1 ,london1 #ahgi1 and the review of ,borel's ,elements of ,mathematics1 by ,professor ,sisam in the ,bulletin of the ,american ,mathematical ,society1 ,july1 #aiaj1 a matter discussed later in this work4 #aj ,t4 ,j4 ,mc,cormack1 ,why do we study ,mathematics3 a ,philosophical and ,historical ,retrospect1 p4 #i1 ,cedar ,rapids1 ,iowa1 #aiaj4 #aa ,of the fair and candid arguments against the culture value of mathematics1 one of the best of the recent ones is that by ,g4 ,f4 ,swain1 in the ,atti del ,,iv ,congresso ,internazionale dei ,matematici1 ,rome1 #aiji1 ,vol4 ,,iii1 p4 #cfa4 ,the literature of this school is quite extensive1 but ,perry's ,england's ,neglect of ,science1 ,london1 #aijj1 and ,discussion on the ,teaching of ,mathematics1 ,london1 #aija1 are typical4 #ab ,in his novel1 ,the ,morals of ,marcus ,ordeyne4 #ac ,g4 ,w4 ,l4 ,carson1 ,the ,functions of ,geometry as a ,subject of ,education1 p4 #c1 ,tonbridge1 #aiaj4 #ad ,it may well be1 however1 that the growing curriculum may justify some reduction in the time formerly assigned to geometry1 and any reasonable proposition of this nature should be fairly met by teachers of mathematics4 #ae ,professor ,muensterberg1 in the ,metropolitan ,magazine for ,july1 #aiaj4 ,,chapter ,,iii ,a ,,brief ,,history ,,of ,,geometry ,the geometry of very ancient peoples was largely the mensuration of simple areas and solids1 such as is taught to children in elementary arithmetic to-day4 ,they early learned how to find the area of a rectangle1 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 solids4 ,the earliest documents that we have relating to geometry come to us from ,babylon and ,egypt4 ,those from ,babylon are written on small clay tablets1 some of them about the size of the hand1 these tablets afterwards having been baked in the sun4 ,they show that the ,babylonians of that period knew something of land measures1 and perhaps had advanced far enough to compute the area of a trapezoid4 ,for the mensuration of the circle they later used1 as did the early ,hebrews1 the value pi #c4 ,a tablet in the ,british ,museum shows that they also used such geometric forms as triangles and circular segments in astrology or as talismans4 ,the ,egyptians must have had a fair knowledge of practical geometry long before the date of any mathematical treatise that has come down to us1 for the building of the pyramids1 between #cjjj and #bdjj ,b4,c41 required the application of several geometric principles4 ,some knowledge of surveying must also have been necessary to carry out the extensive plans for irrigation that were executed under ,amenemhat ,,iii1 about #bbjj ,b4,c4 ,the first definite knowledge that we have of ,egyptian mathematics comes to us from a manuscript copied on papyrus1 a kind of paper used about the ,mediterranean in early times4 ,this copy was made by one ,aah-mesu 7,the ,moon-born71 commonly called ,ahmes1 who probably flourished about #agjj ,b4,c4 ,the original from which he copied1 written about #bcjj ,b4,c41 has been lost1 but the papyrus of ,ahmes1 written nearly four thousand years ago1 is still preserved1 and is now in the ,british ,museum4 ,in this manuscript1 which is devoted chiefly to fractions and to a crude algebra1 is found some work on mensuration4 ,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 sides2 and that the area of a trapezoid having bases b1 b'1 and the nonparallel sides each equal to a1 is #a_/b;a7b b'74 ,one noteworthy advance appears1 however4 ,ahmes gives a rule for finding the area of a circle1 substantially as follows3 ,multiply the square on the radius by 7#af_/i7b1 which is equivalent to taking for pi the value #c4afje4 ,this papyrus also contains some treatment of the mensuration of solids1 particularly with reference to the capacity of granaries4 ,there is also some slight mention of similar figures1 and an extensive treatment of unit fractions1--fractions that were quite universal among the ancients4 ,in the line of algebra it contains a brief treatment of the equation of the first degree with one unknown1 and of progressions4#af ,herodotus tells us that ,sesostris1 king of ,egypt1#ag divided the land among his people and marked out the boundaries after the overflow of the ,nile1 so that surveying must have been well known in his day4 ,indeed1 the harpedonaptae1 or rope stretchers1 acquired their name because they stretched cords1 in which were knots1 so as to make the right triangle #c1 #d1 #e1 when they wished to erect a perpendicular4 ,this is a plan occasionally used by surveyors to-day1 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 proposition4 ,from ,egypt1 and possibly from ,babylon1 geometry passed to the shores of ,asia ,minor and ,greece4 ,the scientific study of the subject begins with ,thales1 one of the ,seven ,wise ,men of the ,grecian civilization4 ,born at ,miletus1 not far from ,smyrna and ,ephesus1 about #fdj ,b4,c41 he died at ,athens in #edh ,b4,c4 ,he spent his early manhood as a merchant1 accumulating the wealth that enabled him to spend his later years in study4 ,he visited ,egypt1 and is said to have learned such elements of geometry as were known there4 ,he founded a school of mathematics and philosophy at ,miletus1 known from the country as the ,ionic ,school4 ,how elementary the knowledge of geometry then was may be understood from the fact that tradition attributes only about four propositions to ,thales1--7#a7 that vertical angles are equal1 7#b7 that equal angles lie opposite the equal sides of an isosceles triangle1 7#c7 that a triangle is determined by two angles and the included side1 7#d7 that a diameter bisects the circle1 and possibly the propositions about the angle-sum of a triangle for special cases1 and the angle inscribed in a semicircle4#ah ,the greatest pupil of ,thales1 and one of the most remarkable men of antiquity1 was ,pythagoras4 ,born probably on the island of ,samos1 just off the coast of ,asia ,minor1 about the year #ehj ,b4,c41 ,pythagoras set forth as a young man to travel4 ,he went to ,miletus and studied under ,thales1 probably spent several years in ,egypt1 very likely went to ,babylon1 and possibly went even to ,india1 since tradition asserts this and the nature of his work in mathematics suggests it4 ,in later life he went to a ,greek colony in southern ,italy1 and at ,crotona1 in the southeastern part of the peninsula1 he founded a school and established a secret society to propagate his doctrines4 ,in geometry he is said to have been the first to demonstrate the proposition that the square on the hypotenuse is equal to the sum of the squares upon the other two sides of a right triangle4 ,the proposition was known in ,india and ,egypt before his time1 at any rate for special cases1 but he seems to have been the first to prove it4 ,to him or to his school seems also to have been due the construction of the regular pentagon and of the five regular polyhedrons4 ,the construction of the regular pentagon requires the dividing of a line into extreme and mean ratio1 and this problem is commonly assigned to the ,pythagoreans1 although it played an important part in ,plato's school4 ,pythagoras is also said to have known that six equilateral triangles1 three regular hexagons1 or four squares1 can be placed about a point so as just to fill the #cfj deg41 but that no other regular polygons can be so placed4 ,to his school is also due the proof for the general case that the sum of the angles of a triangle equals two right angles1 the first knowledge of the size of each angle of a regular polygon1 and the construction of at least one star-polygon1 the star-pentagon1 which became the badge of his fraternity4 ,the brotherhood founded by ,pythagoras proved so offensive to the government that it was dispersed before the death of the master4 ,pythagoras fled to ,megapontum1 a seaport lying to the north of ,crotona1 and there he died about #eja ,b4,c4#ai ,illustration3 ,,fanciful ,,portrait ,,of ,,pythagoras ,calandri's ,arithmetic1 #adia ,for two centuries after ,pythagoras geometry passed through a period of discovery of propositions4 ,the state of the science may be seen from the fact that ,oenopides of ,chios1 who flourished about #dfe ,b4,c41 and who had studied in ,egypt1 was celebrated because he showed how to let fall a perpendicular to a line1 and how to make an angle equal to a given angle4 ,a few years later1 about #ddj ,b4,c41 ,hippocrates of ,chios wrote the first ,greek textbook on mathematics4 ,he knew that the areas of circles are proportional to the squares on their radii1 but was ignorant of the fact that equal central angles or equal inscribed angles intercept equal arcs4 ,antiphon and ,bryson1 two ,greek scholars1 flourished about #dcj ,b4,c4 ,the former attempted to find the area of a circle by doubling the number of sides of a regular inscribed polygon1 and the latter by doing the same for both inscribed and circumscribed polygons4 ,they thus approximately exhausted the area between the polygon and the circle1 and hence this method is known as the method of exhaustions4 ,about #dbj ,b4,c4 ,hippias of ,elis invented a certain curve called the quadratrix1 by means of which he could square the circle and trisect any angle4 ,this curve cannot be constructed by the unmarked straightedge and the compasses1 and when we say that it is impossible to square the circle or to trisect any angle1 we mean that it is impossible by the help of these two instruments alone4 ,during this period the great philosophic school of ,plato 7#dbi-cdh ,b4,c47 flourished at ,athens1 and to this school is due the first systematic attempt to create exact definitions1 axioms1 and postulates1 and to distinguish between elementary and higher geometry4 ,it was at this time that elementary geometry became limited to the use of the compasses and the unmarked straightedge1 which took from this domain the possibility of constructing a square equivalent to a given circle 7squaring the circle71 of trisecting any given angle1 and of constructing a cube that should have twice the volume of a given cube 7duplicating the cube71 these being the three famous problems of antiquity4 ,plato and his school interested themselves with the so-called ,pythagorean numbers1 that is1 with numbers that would represent the three sides of a right triangle and hence fulfill the condition that a#b b#b c#b4 ,pythagoras had already given a rule that would be expressed in modern form1 as #a_/d7m#b #a7b m#b #a_/d7m#b - #a7b4 ,the school of ,plato found that 7#a_/bm7#b #ab m#b 7#a_/bm7#b - #ab4 ,by giving various values to m1 different ,pythagorean numbers may be found4 ,plato's nephew1 ,speusippus 7about #cej ,b4,c471 wrote upon this subject4 ,such numbers were known1 however1 both in ,india and in ,egypt1 long before this time4 ,one of ,plato's pupils was ,philippus of ,mende1 in ,egypt1 who flourished about #chj ,b4,c4 ,it is said that he discovered the proposition relating to the exterior angle of a triangle4 ,his interest1 however1 was chiefly in astronomy4 ,another of ,plato's pupils was ,eudoxus of ,cnidus 7#djh-cee ,b4,c474 ,he elaborated the theory of proportion1 placing it upon a thoroughly scientific foundation4 ,it is probable that ,book ,v of ,euclid1 which is devoted to proportion1 is essentially the work of ,eudoxus4 ,by means of the method of exhaustions of ,antiphon and ,bryson he proved that the pyramid is one third of a prism1 and the cone is one third of a cylinder1 each of the same base and the same altitude4 ,he wrote the first textbook known on solid geometry4 ,the subject of conic sections starts with another pupil of ,plato's1 ,menaechmus1 who lived about #cej ,b4,c4 ,he cut the three forms of conics 7the ellipse1 parabola1 and hyperbola7 out of three different forms of cone1--the acute-angled1 right-angled1 and obtuse-angled1--not noticing that he could have obtained all three from any form of right circular cone4 ,it is interesting to see the far-reaching influence of ,plato4 ,while primarily interested in philosophy1 he laid the first scientific foundations for a system of mathematics1 and his pupils were the leaders in this science in the generation following his greatest activity4 ,the great successor of ,plato at ,athens was ,aristotle1 the teacher of ,alexander the ,great4 ,he also was more interested in philosophy than in mathematics1 but in natural rather than mental philosophy4 ,with him comes the first application of mathematics to physics in the hands of a great man1 and with noteworthy results4 ,he seems to have been the first to represent an unknown quantity by letters4 ,he set forth the theory of the parallelogram of forces1 using only rectangular components1 however4 ,to one of his pupils1 ,eudemus of ,rhodes1 we are indebted for a history of ancient geometry1 some fragments of which have come down to us4 ,the first great textbook on geometry1 and the greatest one that has ever appeared1 was written by ,euclid1 who taught mathematics in the great university at ,alexandria1 ,egypt1 about #cjj ,b4,c4 ,alexandria was then practically a ,greek city1 having been named in honor of ,alexander the ,great1 and being ruled by the ,greeks4 ,in his work ,euclid placed all of the leading propositions of plane geometry then known1 and arranged them in a logical order4 ,most geometries of any importance written since his time have been based upon ,euclid1 improving the sequence1 symbols1 and wording as occasion demanded4 ,he also wrote upon other branches of mathematics besides elementary geometry1 including a work on optics4 ,he was not a great creator of mathematics1 but was rather a compiler of the work of others1 an office quite as difficult to fill and quite as honorable4 ,euclid did not give much solid geometry because not much was known then4 ,it was to ,archimedes 7#bhg-bab ,b4,c471 a famous mathematician of ,syracuse1 on the island of ,sicily1 that some of the most important propositions of solid geometry are due1 particularly those relating to the sphere and cylinder4 ,he also showed how to find the approximate value of pi by a method similar to the one we teach to-day1 proving that the real value lay between #c #a_/g and #c #aj_/ga4 ,the story goes that the sphere and cylinder were engraved upon his tomb1 and ,cicero1 visiting ,syracuse many years after his death1 found the tomb by looking for these symbols4 ,archimedes was the greatest mathematical physicist of ancient times4 ,the ,greeks contributed little more to elementary geometry1 although ,apollonius of ,perga1 who taught at ,alexandria between #bej and #bjj ,b4,c41 wrote extensively on conic sections1 and ,hypsicles of ,alexandria1 about #aij ,b4,c41 wrote on regular polyhedrons4 ,hypsicles was the first ,greek writer who is known to have used sexagesimal fractions1--the degrees1 minutes1 and seconds of our angle measure4 ,zenodorus 7#ahj ,b4,c47 wrote on isoperimetric figures1 and his contemporary1 ,nicomedes of ,gerasa1 invented a curve known as the conchoid1 by means of which he could trisect any angle4 ,another contemporary1 ,diocles1 invented the cissoid1 or ivy-shaped curve1 by means of which he solved the famous problem of duplicating the cube1 that is1 constructing a cube that should have twice the volume of a given cube4 ,the greatest of the ,greek astronomers1 ,hipparchus 7#ahj-abe ,b4,c471 lived about this period1 and with him begins spherical trigonometry as a definite science4 ,a kind of plane trigonometry had been known to the ancient ,egyptians4 ,the ,greeks usually employed the chord of an angle instead of the half chord 7sine71 the latter having been preferred by the later ,arab writers4 ,the most celebrated of the later ,greek physicists was ,heron of ,alexandria1 formerly supposed to have lived about #ajj ,b4,c41 but now assigned to the first century ,a4,d4 ,his contribution to geometry was the formula for the area of a triangle in terms of its sides a1 b1 and c1 with s standing for the semiperimeter #a_/b7;a b c74 ,the formula is sqrt7s7s-a77s-b77s-c774 ,probably nearly contemporary with ,heron was ,menelaus of ,alexandria1 who wrote a spherical trigonometry4 ,he gave an interesting proposition relating to plane and spherical triangles1 their sides being cut by a transversal4 ,for the plane triangle ,,abc1 the sides a1 b1 and c being cut respectively in ,x1 ,y1 and ,z1 the theorem asserts substantially that 7,,az_/,,bz7 4 7,,bx_/,,cx7 4 7,,cy_/,,ay7 #a4 ,the most popular writer on astronomy among the ,greeks was ,ptolemy 7,claudius ,ptolemaeus1 #hg-afe ,a4,d471 who lived at ,alexandria4 ,he wrote a work entitled ,megale ,syntaxis 7,the ,great ,collection71 which his followers designated as ,megistos 7greatest71 on which account the ,arab translators gave it the name ,almagest 7al meaning the74 ,he advanced the science of trigonometry1 but did not contribute to geometry4 ,at the close of the third century ,pappus of ,alexandria 7#bie ,a4,d47 wrote on geometry1 and one of his theorems1 a generalized form of the ,pythagorean proposition1 is mentioned in ,chapter ,,xvi of this work4 ,only two other ,greek writers on geometry need be mentioned4 ,theon of ,alexandria 7#cgj ,a4,d471 the father of the ,hypatia who is the heroine of ,charles ,kingsley's well-known novel1 wrote a commentary on ,euclid to which we are indebted for some historical information4 ,proclus 7#daj-dhe ,a4,d47 also wrote a commentary on ,euclid1 and much of our information concerning the first ,book of ,euclid is due to him4 ,the ,east did little for geometry1 although contributing considerably to algebra4 ,the first great ,hindu writer was ,aryabhatta1 who was born in #dgf ,a4,d4 ,he gave the very close approximation for pi1 expressed in modern notation as #c4adaf4 ,he also gave rules for finding the volume of the pyramid and sphere1 but they were incorrect1 showing that the ,greek mathematics had not yet reached the ,ganges4 ,another ,hindu writer1 ,brahmagupta 7born in #eih ,a4,d471 wrote an encyclopedia of mathematics4 ,he gave a rule for finding ,pythagorean numbers1 expressed in modern symbols as follows3 #a_/d77p#b_/q7 q7#b #a_/d77p#b_/q7 - q7#b p#b4 ,he also generalized ,heron's formula by asserting that the area of an inscribed quadrilateral of sides a1 b1 c1 d1 and semiperimeter s1 is sqrt77s - a77s - b77s - c77s - d774 ,the ,arabs1 about the time of the ,arabian ,nights ,tales 7#hjj ,a4,d471 did much for mathematics1 translating the ,greek authors into their language and also bringing learning from ,india4 ,indeed1 it is to them that modern ,europe owed its first knowledge of ,euclid4 ,they contributed nothing of importance to elementary geometry1 however4 ,the greatest of the ,arab writers was ,mohammed ibn ,musa al-,khowarazmi 7#hbj ,a4,d474 ,he lived at ,bagdad and ,damascus4 ,although chiefly interested in astronomy1 he wrote the first book bearing the name algebra 7,al-jabr wa'l-muqabalah1 ,restoration and ,equation71 composed an arithmetic using the ,hindu numerals1#bj and paid much attention to geometry and trigonometry4 ,euclid was translated from the ,arabic into ,latin in the twelfth century1 ,greek manuscripts not being then at hand1 or being neglected because of ignorance of the language4 ,the leading translators were ,athelhard of ,bath 7#aabj71 an ,english monk2 ,gherard of ,cremona 7#aafj71 an ,italian monk2 and ,johannes ,campanus 7#abej71 chaplain to ,pope ,urban ,,iv4 ,the greatest ,european mathematician of the ,middle ,ages was ,leonardo of ,pisa#ba 7ca4 #aagj-abej74 ,he was very influential in making the ,hindu-,arabic numerals known in ,europe1 wrote extensively on algebra1 and was the author of one book on geometry4 ,he contributed nothing to the elementary theory1 however4 ,the first edition of ,euclid was printed in ,latin in #adhb1 the first one in ,english appearing in #aegj4 ,our symbols are modern1 and - first appearing in a ,german work in #adhi2 in ,recorde's ,whetstone of ,witte in #aeeg2 and in the works of ,harriot 7#aefj-afba72 and x in a publication by ,oughtred 7#aegd-affj74 ,the most noteworthy advance in geometry in modern times was made by the great ,french philosopher ,descartes1 who published a small work entitled ,la ,geometrie in #afcg4 ,from this springs the modern analytic geometry1 a subject that has revolutionized the methods of all mathematics4 ,most of the subsequent discoveries in mathematics have been in higher branches4 ,to the great ,swiss mathematician ,euler 7#agjg-aghc7 is due1 however1 one proposition that has found its way into elementary geometry1 the one showing the relation between the number of edges1 vertices1 and faces of a polyhedron4 ,there has of late arisen a modern elementary geometry devoted chiefly to special points and lines relating to the triangle and the circle1 and many interesting propositions have been discovered4 ,the subject is so extensive that it cannot find any place in our crowded curriculum1 and must necessarily be left to the specialist4#bb ,some idea of the nature of the work may be obtained from a mention of a few propositions3 ,the medians of a triangle are concurrent in the centroid1 or center of gravity of the triangle4 ,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 triangle4 ,the common chord of two intersecting circles is a special case of their radical axis1 and tangents to the circles from any point on the radical axis are equal4 ,if ,o is the orthocenter of the triangle ,,abc1 and ,x1 ,y1 ,z are the feet of the perpendiculars from ,a1 ,b1 ,c respectively1 and ,p1 ,q1 ,r are the mid-points of a1 b1 c respectively1 and ,l1 ,m1 ,n are the mid-points of ,,oa1 ,,ob1 ,,oc respectively2 then the points ,l1 ,m1 ,n2 ,p1 ,q1 ,r2 ,x1 ,y1 ,z all lie on a circle1 the nine points circle4 ,in the teaching of geometry it adds a human interest to the subject to mention occasionally some of the historical facts connected with it4 ,for this reason this brief sketch will be supplemented by many notes upon the various important propositions as they occur in the several books described in the later chapters of this work4 ,,footnotes3 #af ,it was published in ,german translation by ,a4 ,eisenlohr1 ,ein mathematisches ,handbuch der alten ,aegypter1 ,leipzig1 #ahgg1 and in facsimile by the ,british ,museum1 under the title1 ,the ,rhind ,papyrus1 in #ahih4 #ag ,generally known as ,rameses ,,ii4 ,he reigned in ,egypt about #acej ,b4,c4 #ah ,two excellent works on ,thales and his successors1 and indeed the best in ,english1 are the following3 ,g4 ,j4 ,allman1 ,greek ,geometry from ,thales to ,euclid1 ,dublin1 #ahhi2 ,j4 ,gow1 ,a ,history of ,greek ,mathematics1 ,cambridge1 #ahhd4 ,on all mathematical subjects the best general history is that of ,m4 ,cantor1 ,geschichte der ,mathematik1 #d vols1 ,leipzig1 #ahhj-aijh4 #ai ,another good work on ,greek geometry1 with considerable material on ,pythagoras1 is by ,c4 ,a4 ,bretschneider1 ,die ,geometrie und die ,geometer vor ,eukleides1 ,leipzig1 #ahgj4 #bj ,smith and ,karpinski1 ,the ,hindu-,arabic ,numerals1 ,boston1 #aiaa4 #ba ,for a sketch of his life see ,smith and ,karpinski1 loc4 cit4 #bb ,those who care for a brief description of this phase of the subject may consult ,j4 ,casey1 ,a ,sequel to ,euclid1 ,dublin1 fifth edition1 #ahhh2 ,w4 ,j4 ,m',clelland1 ,a ,treatise on the ,geometry of the ,circle1 ,new ,york1 #ahia2 ,m4 ,simon1 ,ueber die ,entwicklung der ,elementar-,geometrie im ,,xix4 ,jahrhundert1 ,leipzig1 #aijf4 ,,chapter ,,iv ,,development ,,of ,,the ,,teaching ,,of ,,geometry ,we know little of the teaching of geometry in very ancient times1 but we can infer its nature from the teaching that is still seen in the native schools of the ,east4 ,here a man1 learned in any science1 will have a group of voluntary students sitting about him1 and to them he will expound the truth4 ,such schools may still be seen in ,india1 ,persia1 and ,china1 the master sitting on a mat placed on the ground or on the floor of a veranda1 and the pupils reading aloud or listening to his words of exposition4 ,in ,egypt geometry seems to have been in early times mere mensuration1 confined largely to the priestly caste4 ,it was taught to novices who gave promise of success in this subject1 and not to others1 the idea of general culture1 of training in logic1 of the cultivation of exact expression1 and of coming in contact with truth being wholly wanting4 ,in ,greece it was taught in the schools of philosophy1 often as a general preparation for philosophic study4 ,thus ,thales introduced it into his ,ionic school1 ,pythagoras made it very prominent in his great school at ,crotona in southern ,italy 7,magna ,graecia71 and ,plato placed above the door of his ,academia the words1 ,let no one ignorant of geometry enter here1--a kind of entrance examination for his school of philosophy4 ,in these gatherings of students it is probable that geometry was taught in much the way already mentioned for the schools of the ,east1 a small group of students being instructed by a master4 ,printing was unknown1 papyrus was dear1 parchment was only in process of invention4 ,paper such as we know had not yet appeared1 so that instruction was largely oral1 and geometric figures were drawn by a pointed stick on a board covered with fine sand1 or on a tablet of wax4 ,but with these crude materials there went an abundance of time1 so that a number of great results were accomplished in spite of the difficulties attending the study of the subject4 ,it is said that ,hippocrates of ,chios 7ca4 #ddj ,b4,c47 wrote the first elementary textbook on mathematics and invented the method of geometric reduction1 the replacing of a proposition to be proved by another which1 when proved1 allows the first one to be demonstrated4 ,a little later ,eudoxus of ,cnidus 7ca4 #cge ,b4,c471 a pupil of ,plato's1 used the reductio ad absurdum1 and ,plato is said to have invented the method of proof by analysis1 an elaboration of the plan used by ,hippocrates4 ,thus these early philosophers taught their pupils not facts alone1 but methods of proof1 giving them power as well as knowledge4 ,furthermore1 they taught them how to discuss their problems1 investigating the conditions under which they are capable of solution4 ,this feature of the work they called the diorismus1 and it seems to have started with ,leon1 a follower of ,plato4 ,between the time of ,plato 7ca4 #djj ,b4,c47 and ,euclid 7ca4 #cjj ,b4,c47 several attempts were made to arrange the accumulated material of elementary geometry in a textbook4 ,plato had laid the foundations for the science1 in the form of axioms1 postulates1 and definitions1 and he had limited the instruments to the straightedge and the compasses4 ,aristotle 7ca4 #cej ,b4,c47 had paid special attention to the history of the subject1 thus finding out what had already been accomplished1 and had also made much of the applications of geometry4 ,the world was therefore ready for a good teacher who should gather the material and arrange it scientifically4 ,after several attempts to find the man for such a task1 he was discovered in ,euclid1 and to his work the next chapter is devoted4 ,after ,euclid1 ,archimedes 7ca4 #bej ,b4,c47 made his great contributions4 ,he was not a teacher like his illustrious predecessor1 but he was a great discoverer4 ,he has left us1 however1 a statement of his methods of investigation which is helpful to those who teach4 ,these methods were largely experimental1 even extending to the weighing of geometric forms to discover certain relations1 the proof being given later4 ,here was born1 perhaps1 what has been called the laboratory method of the present4 ,of the other ,greek teachers we have but little information as to methods of imparting instruction4 ,it is not until the ,middle ,ages that there is much known in this line4 ,whatever of geometry was taught seems to have been imparted by word of mouth in the way of expounding ,euclid1 and this was done in the ancient fashion4 ,the early ,church leaders usually paid no attention to geometry1 but as time progressed the quadrivium1 or four sciences of arithmetic1 music1 geometry1 and astronomy1 came to rank with the trivium 7grammar1 rhetoric1 dialectics71 the two making up the seven liberal arts4 ,all that there was of geometry in the first thousand years of ,christianity1 however1 at least in the great majority of ,church schools1 was summed up in a few definitions and rules of mensuration4 ,gerbert1 who became ,pope ,sylvester ,,ii in #iii ,a4,d41 gave a new impetus to geometry by discovering a manuscript of the old ,roman surveyors and a copy of the geometry of ,boethius1 who paraphrased ,euclid about #ejj ,a4,d4 ,he thereupon wrote a brief geometry1 and his elevation to the papal chair tended to bring the study of mathematics again into prominence4 ,geometry now began to have some place in the ,church schools1 naturally the only schools of high rank in the ,middle ,ages4 ,the study of the subject1 however1 seems to have been merely a matter of memorizing4 ,geometry received another impetus in the book written by ,leonardo of ,pisa in #abbj1 the ,practica ,geometriae4 ,euclid was also translated into ,latin about this time 7strangely enough1 as already stated1 from the ,arabic instead of the ,greek71 and thus the treasury of elementary geometry was opened to scholars in ,europe4 ,from now on1 until the invention of printing 7ca4 #adej71 numerous writers on geometry appear1 but1 so far as we know1 the method of instruction remained much as it had always been4 ,the universities began to appear about the thirteenth century1 and ,sacrobosco1 a well-known medieval mathematician1 taught mathematics about #abej in the ,university of ,paris4 ,in #accf this university decreed that mathematics should be required for a degree4 ,in the thirteenth century ,oxford required six books of ,euclid for one who was to teach1 but this amount of work seems to have been merely nominal1 for in #adej only two books were actually read4 ,the universities of ,prague 7founded in #acej7 and ,vienna 7statutes of #achi7 required most of plane geometry for the teacher's license1 although ,vienna demanded but one book for the bachelor's degree4 ,so1 in general1 the universities of the thirteenth1 fourteenth1 and fifteenth centuries required less for the degree of master of arts than we now require from a pupil in our ,american high schools4 ,on the other hand1 the university students were younger than now1 and were really doing only high school work4 ,the invention of printing made possible the study of geometry in a new fashion4 ,it now became possible for any one to study from a book1 whereas before this time instruction was chiefly by word of mouth1 consisting of an explanation of ,euclid4 ,the first ,euclid was printed in #adhb1 at ,venice1 and new editions and variations of this text came out frequently in the next century4 ,practical geometries became very popular1 and the reaction against the idea of mental discipline threatened to abolish the old style of text4 ,it was argued that geometry was uninteresting1 that it was not sufficient in itself1 that boys needed to see the practical uses of the subject1 that only those propositions that were capable of application should be retained1 that there must be a fusion between the demands of culture and the demands of business1 and that every man who stood for mathematical ideals represented an obsolete type4 ,such writers as ,finaeus 7#aeef71 ,bartoli 7#aehi71 ,belli 7#aefi71 and ,cataneo 7#aefg71 in the sixteenth century1 and ,capra 7#afgh71 ,gargiolli 7#afee71 and many others in the seventeenth century1 either directly or inferentially1 took this attitude towards the subject1--exactly the attitude that is being taken at the present time by a number of teachers in the ,united ,states4 ,as is always the case1 to such an extreme did this movement lead that there was a reaction that brought the ,euclid type of book again to the front1 and it has maintained its prominence even to the present4 ,the study of geometry in the high schools is relatively recent4 ,the ,gymnasium 7classical school preparatory to the university7 at ,nuernberg1 founded in #aebf1 and the ,cathedral school at ,wuerttemberg 7as shown by the curriculum of #aeef7 seem to have had no geometry before #afjj1 although the ,gymnasium at ,strassburg included some of this branch of mathematics in #aegh1 and an elective course in geometry was offered at ,zwickau1 in ,saxony1 in #aeba4 ,in the seventeenth century geometry is found in a considerable number of secondary schools1 as at ,coburg 7#afje71 ,kurfalz 7#afae1 elective71 ,erfurt 7#afdc71 ,gotha 7#afje71 ,giessen 7#afje71 and numerous other places in ,germany1 although it appeared but rarely in the secondary schools of ,france before the eighteenth century4 ,in ,germany the ,realschulen--schools with more science and less classics than are found in the ,gymnasium--came into being in the eighteenth century1 and considerable effort was made to construct a course in geometry that should be more practical than that of the modified ,euclid4 ,at the opening of the nineteenth century the ,prussian schools were reorganized1 and from that time on geometry has had a firm position in the secondary schools of all ,germany4 ,in the eighteenth century some excellent textbooks on geometry appeared in ,france1 among the best being that of ,legendre 7#agid71 which influenced in such a marked degree the geometries of ,america4 ,soon after the opening of the nineteenth century the lycees of ,france became strong institutions1 and geometry1 chiefly based on ,legendre1 was well taught in the mathematical divisions4 ,a worthy rival of ,legendre's geometry was the work of ,lacroix1 who called attention continually to the analogy between the theorems of plane and solid geometry1 and even went so far as to suggest treating the related propositions together in certain cases4 ,in ,england the preparatory schools1 such as ,rugby1 ,harrow1 and ,eton1 did not commonly teach geometry until quite recently1 leaving this work for the universities4 ,in ,christ's ,hospital1 ,london1 however1 geometry was taught as early as #afha1 from a work written by several teachers of prominence4 ,the highest class at ,harrow studied ,euclid and vulgar fractions one period a week in #ahbi1 but geometry was not seriously studied before #ahcg4 ,in the ,edinburgh ,academy as early as #ahhe1 and in ,rugby by #ahci1 plane geometry was completed4 ,not until #ahdd did ,harvard require any plane geometry for entrance4 ,in #ahee ,yale required only two books of ,euclid4 ,it was therefore from #ahej to #ahge that plane geometry took a definite place in the ,american high school4 ,solid geometry has not been generally required for entrance to any eastern college1 although in the ,west this is not the case4 ,the ,east teaches plane geometry more thoroughly1 but allows a pupil to enter college or to go into business with no solid geometry4 ,given a year to the subject1 it is possible to do little more than cover plane geometry2 with a year and a half the solid geometry ought easily to be covered also4 ,bibliography4 ,stamper1 ,a ,history of the ,teaching of ,elementary ,geometry1 ,new ,york1 #aiji1 with a very full bibliography of the subject2 ,cajori1 ,the ,teaching of ,mathematics in the ,united ,states1 ,washington1 #ahij2 ,cantor1 ,geschichte der ,mathematik1 ,vol4 ,,iv1 p4 #cba1 ,leipzig1 #aijh2 ,schotten1 ,inhalt und ,methode des planimetrischen ,unterrichts1 ,leipzig1 #ahij4 ,,chapter ,v ,,euclid ,it is fitting that a chapter in a book upon the teaching of this subject should be devoted to the life and labors of the greatest of all textbook writers1 ,euclid1--a man whose name has been1 for more than two thousand years1 a synonym for elementary plane geometry wherever the subject has been studied4 ,and yet when an effort is made to pick up the scattered fragments of his biography1 we are surprised to find how little is known of one whose fame is so universal4 ,although more editions of his work have been printed than of any other book save the ,bible1#bc we do not know when he was born1 or in what city1 or even in what country1 nor do we know his race1 his parentage1 or the time of his death4 ,we should not feel that we knew much of the life of a man who lived when the ,magna ,charta was wrested from ,king ,john1 if our first and only source of information was a paragraph in the works of some historian of to-day2 and yet this is about the situation in respect to ,euclid4 ,proclus of ,alexandria1 philosopher1 teacher1 and mathematician1 lived from #daj to #dhe ,a4,d41 and wrote a commentary on the works of ,euclid4 ,in his writings1 which seem to set forth in amplified form his lectures to the students in the ,neoplatonist ,school of ,alexandria1 ,proclus makes this statement1 and of ,euclid's life we have little else3 ,not much younger than these#bd is ,euclid1 who put together the ,elements1 collecting many of the theorems of ,eudoxus1 perfecting many of those of ,theaetetus1 and also demonstrating with perfect certainty what his predecessors had but insufficiently proved4 ,he flourished in the time of the first ,ptolemy1 for ,archimedes1 who closely followed this ruler1#be speaks of ,euclid4 ,furthermore it is related that ,ptolemy one time demanded of him if there was in geometry no shorter way than that of the ,elements1 to whom he replied that there was no royal road to geometry4#bf ,he was therefore younger than the pupils of ,plato1 but older than ,eratosthenes and ,archimedes2 for the latter were contemporary with one another1 as ,eratosthenes somewhere says4#bg ,thus we have in a few lines1 from one who lived perhaps seven or eight hundred years after ,euclid1 nearly all that is known of the most famous teacher of geometry that ever lived4 ,nevertheless1 even this little tells us about when he flourished1 for ,hermotimus and ,philippus were pupils of ,plato1 who died in #cdg ,b4,c41 whereas ,archimedes was born about #bhg ,b4,c4 and was writing about #bej ,b4,c4 ,furthermore1 since ,ptolemy ,i reigned from #cjf to #bhc ,b4,c41 ,euclid must have been teaching about #cjj ,b4,c41 and this is the date that is generally assigned to him4 ,euclid probably studied at ,athens1 for until he himself assisted in transferring the center of mathematical culture to ,alexandria1 it had long been in the ,grecian capital1 indeed since the time of ,pythagoras4 ,moreover1 numerous attempts had been made at ,athens to do exactly what ,euclid succeeded in doing1--to construct a logical sequence of propositions2 in other words1 to write a textbook on plane geometry4 ,it was at ,athens1 therefore1 that he could best have received the inspiration to compose his ,elements4#bh ,after finishing his education at ,athens it is quite probable that he1 like other savants of the period1 was called to ,alexandria by ,ptolemy ,soter1 the king1 to assist in establishing the great school which made that city the center of the world's learning for several centuries4 ,in this school he taught1 and here he wrote the ,elements and numerous other works1 perhaps ten in all4 ,although the ,greek writers who may have known something of the life of ,euclid have little to say of him1 the ,arab writers1 who could have known nothing save from ,greek sources1 have allowed their imaginations the usual latitude in speaking of him and of his labors4 ,thus ,al-,qift4i1 who wrote in the thirteenth century1 has this to say in his biographical treatise ,ta'rikh al-,h4ukama3 ,euclid1 son of ,naucrates1 grandson of ,zenarchus1 called the author of geometry1 a ,greek by nationality1 domiciled at ,damascus1 born at ,tyre1 most learned in the science of geometry1 published a most excellent and most useful work entitled ,the ,foundation or ,elements of ,geometry1 a subject in which no more general treatise existed before among the ,greeks2 nay1 there was no one even of later date who did not walk in his footsteps and frankly profess his doctrine4 ,this is rather a specimen of the ,arab tendency to manufacture history than a serious contribution to the biography of ,euclid1 of whose personal history we have only the information given by ,proclus4 ,illustration3 ,,euclid ,from an old print ,euclid's works at once took high rank1 and they are mentioned by various classical authors4 ,cicero knew of them1 and ,capella 7ca4 #dgj ,a4,d471 ,cassiodorius 7ca4 #eae ,a4,d471 and ,boethius 7ca4 #dhj-ebd ,a4,d47 were all more or less familiar with the ,elements4 ,with the advance of the ,dark ,ages1 however1 learning was held in less and less esteem1 so that ,euclid was finally forgotten1 and manuscripts of his works were either destroyed or buried in some remote cloister4 ,the ,arabs1 however1 whose civilization assumed prominence from about #gej ,a4,d4 to about #aejj1 translated the most important treatises of the ,greeks1 and ,euclid's ,elements among the rest4 ,one of these ,arabic editions an ,english monk of the twelfth century1 one ,athelhard 7,a,ethelhard7 of ,bath1 found and translated into ,latin 7ca4 #aabj ,a4,d474 ,a little later ,gherard of ,cremona 7#aaad-aahg7 made a new translation from the ,arabic1 differing in essential features from that of ,athelhard1 and about #abfj ,johannes ,campanus made still a third translation1 also from ,arabic into ,latin4#bi ,there is reason to believe that ,athelhard1 ,campanus1 and ,gherard may all have had access to an earlier ,latin translation1 since all are quite alike in some particulars while diverging noticeably in others4 ,indeed1 there is an old ,english verse that relates3 ,the clerk ,euclide on this wyse hit fonde ,thys craft of gemetry yn ,egypte londe 444 ,thys craft com into ,england1 as y yow say1 ,yn tyme of good ,kyng ,adelstone's day4 ,if this be true1 ,euclid was known in ,england as early as #ibd-idj ,a4,d4 ,without going into particulars further1 it suffices to say that the modern knowledge of ,euclid came first through the ,arabic into the ,latin1 and the first printed edition of the ,elements 7,venice1 #adhb7 was the ,campanus translation4 ,greek manuscripts now began to appear1 and at the present time several are known4 ,there is a manuscript of the ninth century in the ,bodleian library at ,oxford1 one of the tenth century in the ,vatican1 another of the tenth century in ,florence1 one of the eleventh century at ,bologna1 and two of the twelfth century at ,paris4 ,there are also fragments containing bits of ,euclid in ,greek1 and going back as far as the second and third century ,a4,d4 ,the first modern translation from the ,greek into the ,latin was made by ,zamberti 7or ,zamberto71#cj and was printed at ,venice in #aeac4 ,the first translation into ,english was made by ,sir ,henry ,billingsley and was printed in #aegj1 sixteen years before he became ,lord ,mayor of ,london4 ,proclus1 in his commentary upon ,euclid's work1 remarks3 ,in the whole of geometry there are certain leading theorems1 bearing to those which follow the relation of a principle1 all-pervading1 and furnishing proofs of many properties4 ,such theorems are called by the name of elements1 and their function may be compared to that of the letters of the alphabet in relation to language1 letters being indeed called by the same name in ,greek ,greek3 stoicheia1 stoicheia4#ca ,this characterizes the work of ,euclid1 a collection of the basic propositions of geometry1 and chiefly of plane geometry1 arranged in logical sequence1 the proof of each depending upon some preceding proposition1 definition1 or assumption 7axiom or postulate74 ,the number of the propositions of plane geometry included in the ,elements is not entirely certain1 owing to some disagreement in the manuscripts1 but it was between one hundred sixty and one hundred seventy-five4 ,it is possible to reduce this number by about thirty or forty1 because ,euclid included a certain amount of geometric algebra2 but beyond this we cannot safely go in the way of elimination1 since from the very nature of the ,elements these propositions are basic4 ,the efforts at revising ,euclid have been generally confined1 therefore1 to rearranging his material1 to rendering more modern his phraseology1 and to making a book that is more usable with beginners if not more logical in its presentation of the subject4 ,while there has been an improvement upon ,euclid in the art of bookmaking1 and in minor matters of phraseology and sequence1 the educational gain has not been commensurate with the effort put forth4 ,with a little modification of ,euclid's semi-algebraic ,book ,,ii and of his treatment of proportion1 with some scattering of the definitions and the inclusion of well-graded exercises at proper places1 and with attention to the modern science of bookmaking1 the ,elements would answer quite as well for a textbook to-day as most of our modern substitutes1 and much better than some of them4 ,it would1 moreover1 have the advantage of being a classic1--somewhat the same advantage that comes from reading ,homer in the original instead of from ,pope's metrical translation4 ,this is not a plea for a return to the ,euclid text1 but for a recognition of the excellence of ,euclid's work4 ,the distinctive feature of ,euclid's ,elements1 compared with the modern ,american textbook1 is perhaps this3 ,euclid begins a book with what seems to him the easiest proposition1 be it theorem or problem2 upon this he builds another2 upon these a third1 and so on1 concerning himself but little with the classification of propositions4 ,furthermore1 he arranges his propositions so as to construct his figures before using them4 ,we1 on the other hand1 make some little attempt to classify our propositions within each book1 and we make no attempt to construct our figures before using them1 or at least to prove that the constructions are correct4 ,indeed1 we go so far as to study the properties of figures that we cannot construct1 as when we ask for the size of the angle of a regular heptagon4 ,thus ,euclid begins ,book ,i by a problem1 to construct an equilateral triangle on a given line4 ,his object is to follow this by problems on drawing a straight line equal to a given straight line1 and cutting off from the greater of two straight lines a line equal to the less4 ,he now introduces a theorem1 which might equally well have been his first proposition1 namely1 the case of the congruence of two triangles1 having given two sides and the included angle4 ,by means of his third and fourth propositions he is now able to prove the pons asinorum1 that the angles at the base of an isosceles triangle are equal4 ,we1 on the other hand1 seek to group our propositions where this can conveniently be done1 putting the congruent propositions together1 those about inequalities by themselves1 and the propositions about parallels in one set4 ,the results of the two arrangements are not radically different1 and the effect of either upon the pupil's mind does not seem particularly better than that of the other4 ,teachers who have used both plans quite commonly feel that1 apart from ,books ,,ii and ,v1 ,euclid is nearly as easily understood as our modern texts1 if presented in as satisfactory dress4 ,the topics treated and the number of propositions in the plane geometry of the ,elements are as follows3 ,book ,i4 ,rectilinear figures #dh ,book ,,ii4 ,geometric algebra #ad ,book ,,iii4 ,circles #cg ,book ,,iv4 ,problems about circles #af ,book ,v4 ,proportion #be ,book ,,vi4 ,applications of proportion #cc ---- #agc ,of these we now omit ,euclid's ,book ,,ii1 because we have an algebraic symbolism that was unknown in his time1 although he would not have used it in geometry even had it been known4 ,thus his first proposition in ,book ,,ii is as follows3 ,if there be two straight lines1 and one of them be cut into any number of segments whatever1 the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments4 ,this amounts to saying that if x p q r 4441 then ax ap aq ar 4444 ,we also materially simplify ,euclid's ,book ,v4 ,he1 for example1 proves that ,if four magnitudes be proportional1 they will also be proportional alternately4 ,this he proves generally for any kind of magnitude1 while we merely prove it for numbers having a common measure4 ,we say that we may substitute for the older form of proportion1 namely1 a 3 b c 3 d1 the fractional form a_/b c_/d4 ,from this we have ad bc4 ,whence a_/c b_/d4 ,in this work we assume that we may multiply equals by b and d4 ,but suppose b and d are cubes1 of which1 indeed1 we do not even know the approximate numerical measure2 what shall we do8 ,to ,euclid the multiplication by a cube or a polygon or a sphere would have been entirely meaningless1 as it always is from the standpoint of pure geometry4 ,hence it is that our treatment of proportion has no serious standing in geometry as compared with ,euclid's1 and our only justification for it lies in the fact that it is easier4 ,euclid's treatment is much more rigorous than ours1 but it is adapted to the comprehension of only advanced students1 while ours is merely a confession1 and it should be a frank confession1 of the weakness of our pupils1 and possibly1 at times1 of ourselves4 ,if we should take ,euclid's ,books ,,ii and ,v for granted1 or as sufficiently evident from our study of algebra1 we should have remaining only one hundred thirty-four propositions1 most of which may be designated as basal propositions of plane geometry4 ,revise ,euclid as we will1 we shall not be able to eliminate any large number of his fundamental truths1 while we might do much worse than to adopt these one hundred thirty-four propositions in toto as the bases1 and indeed as the definition1 of elementary plane geometry4 ,bibliography4 ,heath1 ,the ,thirteen ,books of ,euclid's ,elements1 #c vols41 ,cambridge1 #aijh2 ,frankland1 ,the ,first ,book of ,euclid1 ,cambridge1 #aijf2 ,smith1 ,dictionary of ,greek and ,roman ,biography1 article ,eukleides2 ,simon1 ,euclid und die sechs planimetrischen ,buecher1 ,leipzig1 #aija2 ,gow1 ,history of ,greek ,mathematics1 ,cambridge1 #ahhd1 and any of the standard histories of mathematics4 ,both ,heath and ,simon give extensive bibliographies4 ,the latest standard ,greek and ,latin texts are ,heiberg's1 published by ,teubner of ,leipzig4 ,,footnotes3 #bc ,riccardi1 ,saggio di una bibliografia ,euclidea1 ,part ,i1 p4 #c1 ,bologna1 #ahhg4 ,riccardi lists well towards two thousand editions4 #bd ,hermotimus of ,colophon and ,philippus of ,mende4 #be ,literally1 ,who closely followed the first1 i4e4 the first ,ptolemy4 #bf ,menaechmus is said to have replied to a similar question of ,alexander the ,great3 ,o ,king1 through the country there are royal roads and roads for common citizens1 but in geometry there is one road for all4 #bg ,this is also shown in a letter from ,archimedes to ,eratosthenes1 recently discovered by ,heiberg4 #bh ,on this phase of the subject1 and indeed upon ,euclid and his propositions and works in general1 consult ,t4 ,l4 ,heath1 ,the ,thirteen ,books of ,euclid's ,elements1 #c vols41 ,cambridge1 #aijh1 a masterly treatise of which frequent use has been made in preparing this work4 #bi ,a contemporary copy of this translation is now in the library of ,george ,a4 ,plimpton1 ,esq41 of ,new ,york4 ,see the author's ,rara ,arithmetica1 p4 #dcc1 ,boston1 #aiji4 #cj ,a beautiful vellum manuscript of this translation is in the library of ,george ,a4 ,plimpton1 ,esq41 of ,new ,york4 ,see the author's ,rara ,arithmetica1 p4 #dha1 ,boston1 #aiji4 #ca ,heath1 loc4 cit41 ,vol4 ,i1 p4 #aad4 ,,chapter ,,vi ,,efforts ,,at ,,improving ,,euclid ,from time to time an effort is made by some teacher1 or association of teachers1 animated by a serious desire to improve the instruction in geometry1 to prepare a new syllabus that shall mark out some royal road1 and it therefore becomes those who are interested in teaching to consider with care the results of similar efforts in recent years4 ,there are many questions which such an attempt suggests3 ,what is the real purpose of the movement8 ,what will the teaching world say of the result8 ,shall a reckless1 ill-considered radicalism dominate the effort1 bringing in a distasteful terminology and symbolism merely for its novelty1 insisting upon an ultralogical treatment that is beyond the powers of the learner1 rearranging the subject matter to fit some narrow notion of the projectors1 seeking to emasculate mathematics by looking only to the applications1 riding some little hobby in the way of some particular class of exercises1 and cutting the number of propositions to a minimum that will satisfy the mere demands of the artisan8 ,such are some of the questions that naturally arise in the mind of every one who wishes well for the ancient science of geometry4 ,it is not proposed in this chapter to attempt to answer these questions1 but rather to assist in understanding the problem by considering the results of similar attempts4 ,if it shall be found that syllabi have been prepared under circumstances quite as favorable as those that obtain at present1 and if these syllabi have had little or no real influence1 then it becomes our duty to see if new plans may be worked out so as to be more successful than their predecessors4 ,if the older attempts have led to some good1 it is well to know what is the nature of this good1 to the end that new efforts may also result in something of benefit to the schools4 ,it is proposed in this chapter to call attention to four important syllabi1 setting forth briefly their distinguishing features and drawing some conclusions that may be helpful in other efforts of this nature4 ,in ,england two noteworthy attempts have been made within a century1 looking to a more satisfactory sequence and selection of propositions than is found in ,euclid4 ,each began with a list of propositions arranged in proper sequence1 and each was thereafter elaborated into a textbook4 ,neither accomplished fully the purpose intended1 but each was instrumental in provoking healthy discussion and in improving the texts from which geometry is studied4 ,the first of these attempts was made by ,professor ,augustus de ,morgan1 under the auspices of the ,society for the ,diffusion of ,useful ,knowledge1 and it resulted in a textbook1 including plane1 solid1 and spherical geometry1 in six books4 ,according to ,de ,morgan's plan1 plane geometry consisted of three books1 the number of propositions being as follows3 ,book ,i4 ,rectilinear figures #fj ,book ,,ii4 ,ratio1 proportion1 applications #fi ,book ,,iii4 ,the circle #fe ---- ,total for plane geometry #aid ,of the #aid propositions ,de ,morgan selected #aad with their corollaries as necessary for a beginner who is teaching himself4 ,in solid geometry the plan was as follows3 ,book ,,iv4 ,lines in different planes1 solids contained by planes #eb ,book ,v4 ,cylinder1 cone1 sphere #be ,book ,,vi4 ,figures on a sphere #db ---- ,total for solid geometry #aai ,of these #aai propositions ,de ,morgan selected #gf with their corollaries as necessary for a beginner1 thus making #aij necessary propositions out of #cje desirable ones1 besides the corollaries in plane and solid geometry4 ,in other words1 of the desirable propositions he considered that about two thirds are absolutely necessary4 ,it is interesting to note1 however1 that he summed up the results of his labors by saying3 ,it will be found that the course just laid down1 excepting the sixth book of it only1 is not of much greater extent1 nor very different in point of matter from that of ,euclid1 whose ,elements have at all times been justly esteemed a model not only of easy and progressive instruction in geometry1 but of accuracy and perspicuity in reasoning4 ,de ,morgan's effort1 essentially that of a syllabus-maker rather than a textbook writer1 although it was published under the patronage of a prominent society with which were associated the names of men like ,henry ,hallam1 ,rowland ,hill1 ,lord ,john ,russell1 and ,george ,peacock1 had no apparent influence on geometry either in ,england or abroad4 ,nevertheless the syllabus was in many respects excellent2 it rearranged the matter1 it classified the propositions1 it improved some of the terminology1 and it reduced the number of essential propositions2 it had the assistance of ,de ,morgan's enthusiasm and of the society with which he was so prominently connected1 and it was circulated with considerable generosity throughout the ,english-speaking world2 but in spite of all this it is to-day practically unknown4 ,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 purpose1 the ,association for the ,improvement of ,geometrical ,teaching4 ,this society was composed of many of the most progressive teachers in ,england1 and it included in its membership men of high standing in mathematics in the universities4 ,as a result of their labors a syllabus was prepared1 which was elaborated into a textbook1 and in #ahhi a revised syllabus was issued4 ,as to the arrangement of matter1 the syllabus departs from ,euclid chiefly by separating the problems from the theorems1 as is the case in our ,american textbooks1 and in improving the phraseology4 ,the course is preceded by some simple exercises in the use of the compasses and ruler1 a valuable plan that is followed by many of the best teachers everywhere4 ,considerable attention is paid to logical processes before beginning the work1 such terms as contrapositive and obverse1 and such rules as the rule of conversion and the rule of identity being introduced before any propositions are considered4 ,the arrangement of the work and the number of propositions in plane geometry are as follows3 ,book ,i4 ,the straight line #ea ,book ,,ii4 ,equality of areas #ai ,book ,,iii4 ,the circle #db ,book ,,iv4 ,ratio and proportion #cb ,book ,v4 ,proportion #bd ---- ,total for plane geometry #afh ,here1 then1 is the result of several years of labor by a somewhat radical organization1 fostered by excellent mathematicians1 and carried on in a country where elementary geometry is held in highest esteem1 and where ,euclid was thought unsuited to the needs of the beginner4 ,the number of propositions remains substantially the same as in ,euclid1 and the introduction of some unusable logic tends to counterbalance the improvement in sequence of the propositions4 ,the report provoked thought2 it shook the ,euclid stronghold2 it was probably instrumental in bringing about the present upheaval in geometry in ,england1 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 advocates4 ,the same association published later1 and republished in #aije1 a ,report on the ,teaching of ,geometry1 in which it returned to ,euclid1 modifying the ,elements by omitting certain propositions1 changing the order and proof of others1 and introducing a few new theorems4 ,it seems to reduce the propositions to be proved in plane geometry to about one hundred fifteen1 and it recommends the omission of the incommensurable case4 ,this number is1 however1 somewhat misleading1 for ,euclid frequently puts in one proposition what we in ,america1 for educational reasons1 find it better to treat in two1 or even three1 propositions4 ,this report1 therefore1 reaches about the same conclusion as to the geometric facts to be mastered as is reached by our later textbook writers in ,america4 ,it is not extreme1 and it stands for good mathematics4 ,in the ,united ,states the influence of our early wars with ,england1 and the sympathy of ,france at that time1 turned the attention of our scholars of a century ago from ,cambridge to ,paris as a mathematical center4 ,the influx of ,french mathematics brought with it such works as ,legendre's geometry 7#agid7 and ,bourdon's algebra1 and made known the texts of ,lacroix1 ,bertrand1 and ,bezout4 ,legendre's geometry was the result of the efforts of a great mathematician at syllabus-making1 a natural thing in a country that had early broken away from ,euclid4 ,legendre changed the ,greek sequence1 sought to select only propositions that are necessary to a good understanding of the subject1 and added a good course in solid geometry4 ,his arrangement1 with the number of propositions as given in the ,davies translation1 is as follows3 ,book ,i4 ,rectilinear figures #ca ,book ,,ii4 ,ratio and proportion #ad ,book ,,iii4 ,the circle #dh ,book ,,iv4 ,proportions of figures and areas #ea ,book ,v4 ,polygons and circles #ag ---- ,total for plane geometry #afa ,legendre made1 therefore1 practically no reduction in the number of ,euclid's propositions1 and his improvement on ,euclid consisted chiefly in his separation of problems and theorems1 and in a less rigorous treatment of proportion which boys and girls could comprehend4 ,d',alembert had demanded that the sequence of propositions should be determined by the order in which they had been discovered1 but ,legendre wisely ignored such an extreme and gave the world a very usable book4 ,the principal effect of ,legendre's geometry in ,america was to make every textbook writer his own syllabus-maker1 and to put solid geometry on a more satisfactory footing4 ,the minute we depart from a standard text like ,euclid's1 and have no recognized examining body1 every one is free to set up his own standard1 always within the somewhat uncertain boundary prescribed by public opinion and by the colleges4 ,the efforts of the past few years at syllabus-making have been merely attempts to define this boundary more clearly4 ,of these attempts two are especially worthy of consideration as having been very carefully planned and having brought forth such definite results as to appeal to a large number of teachers4 ,other syllabi have been made and are familiar to many teachers1 but in point of clearness of purpose1 conciseness of expression1 and form of publication they have not been such as to compare with the two in question4 ,the first of these is the ,harvard syllabus1 which is placed in the hands of students for reference when trying the entrance examinations of that university1 a plan not followed elsewhere4 ,it sets forth the basal propositions that should form the essential part of the student's preparation1 and that are necessary and sufficient for proving any original proposition 7to take the common expression7 that may be set on the examination4 ,the propositions are arranged by books as follows3 ,book ,i4 ,angles1 triangles1 parallels #be ,book ,,ii4 ,the circle1 angle measure #ah ,book ,,iii4 ,similar polygons #aj ,book ,,iv4 ,area of polygons #h ,book ,v4 ,polygons and circle measure #aa ,constructions #ba ,ratio and proportion #f ---- ,total for plane geometry #ii ,the total for solid geometry is #gi propositions1 or #agh for both plane and solid geometry4 ,this is perhaps the most successful attempt that has been made at reaching a minimum number of propositions4 ,it might well be further reduced1 since it includes the proposition about two adjacent angles formed by one line meeting another1 and the one about the circle as the limit of the inscribed and circumscribed regular polygons4 ,the first of these leads a beginner to doubt the value of geometry1 and the second is beyond the powers of the majority of students4 ,as compared with the syllabus reported by a ,wisconsin committee in #aijd1 for example1 here are #ii propositions against #acb4 ,on the other hand1 a committee appointed by the ,central ,association of ,science and ,mathematics ,teachers reported in #aiji a syllabus with what seems at first sight to be a list of only #ei propositions in plane geometry4 ,this number is fictitious1 however1 for the reason that numerous converses are indicated with the propositions1 and are not included in the count1 and directions are given to include related theorems and problems dealing with the length and area of a circle1 so that in some cases one proposition is evidently intended to cover several others4 ,this syllabus is therefore lacking in definiteness1 so that the ,harvard list stands out as perhaps the best of its type4 ,the second noteworthy recent attempt in ,america is that made by a committee of the ,association of ,mathematical ,teachers in ,new ,england4 ,this committee was organized in #aijd4 ,it held sixteen meetings and carried on a great deal of correspondence4 ,as a result1 it prepared a syllabus arranged by topics1 the propositions of solid geometry being grouped immediately after the corresponding ones of plane geometry4 ,for example1 the nine propositions on congruence in a plane are followed by nine on congruence in space4 ,as a result1 the following summarizes the work in plane geometry3 ,congruence in a plane #i ,equivalence #c ,parallels and perpendiculars #i ,symmetry #bj ,angles #ae ,tangents #d ,similar figures #ah ,inequalities #h ,lengths and areas #ag ,loci #b ,concurrent lines #e ---- ,total for plane geometry #aaj ,not so conventional in arrangement as the ,harvard syllabus1 and with a few propositions that are evidently not basal to the same extent as the rest1 the list is nevertheless a very satisfactory one1 and the parallelism shown between plane and solid geometry is suggestive to both student and teacher4 ,on the whole1 however1 the ,harvard selection of basal propositions is perhaps as satisfactory as any that has been made1 even though it appears to lack a factor of safety1 and it is probable that any further reduction would be unwise4 ,what1 now1 has been the effect of all these efforts8 ,what teacher or school would be content to follow any one of these syllabi exactly8 ,what textbook writer would feel it safe to limit his regular propositions to those in any one syllabus8 ,these questions suggest their own answers1 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 view4 ,this depends1 however1 on what this end is conceived to be4 ,if the purpose has been to cut out a very large number of the propositions that are found in ,euclid's plane geometry1 the effort has not been successful4 ,we may reduce this number to about one hundred thirty1 but in general1 whatever a syllabus may give as a minimum1 teachers will favor a larger number than is suggested by the ,harvard list1 for the purpose of exercise in the reading of mathematics if for no other reason4 ,the ,french geometer1 ,lacroix1 who wrote more than a century ago1 proposed to limit the propositions to those needed to prove other important ones1 and those needed in practical mathematics4 ,if to this we should add those that are used in treating a considerable range of exercises1 we should have a list of about one hundred thirty4 ,but this is not the real purpose of these syllabi1 or at most it seems like a relatively unimportant one4 ,the purpose that has been attained is to stop the indefinite increase in the number of propositions that would follow from the recent developments in the geometry of the triangle and circle1 and of similar modern topics1 if some such counter-movement as this did not take place4 ,if the result is1 as it probably will be1 to let the basal propositions of ,euclid remain about as they always have been1 as the standards for beginners1 the syllabi will have accomplished a worthy achievement4 ,if1 in addition1 they furnish an irreducible minimum of propositions to which a student may have access if he desires it1 on an examination1 as was intended in the case of the ,harvard and the ,new ,england ,association syllabi1 the achievement may possibly be still more worthy4 ,in preparing a syllabus1 therefore1 no one should hope to bring the teaching world at once to agree to any great reduction in the number of basal propositions1 nor to agree to any radical change of terminology1 symbolism1 or sequence4 ,rather should it be the purpose to show that we have enough topics in geometry at present1 and that the number of propositions is really greater than is absolutely necessary1 so that teachers shall not be led to introduce any considerable number of propositions out of the large amount of new material that has recently been accumulating4 ,such a syllabus will always accomplish a good purpose1 for at least it will provoke thought and arouse interest1 but any other kind is bound to be ephemeral4#cb ,besides the evolutionary attempts at rearranging and reducing in number the propositions of ,euclid1 there have been very many revolutionary efforts to change his treatment of geometry entirely4 ,the great ,french mathematician1 ,d',alembert1 for example1 in the eighteenth century1 wished to divide geometry into three branches3 7#a7 that dealing with straight lines and circles1 apparently not limited to a plane2 7#b7 that dealing with surfaces2 and 7#c7 that dealing with solids4 ,so ,meray in ,france and ,de ,paolis#cc in ,italy have attempted to fuse plane and solid geometry1 but have not produced a system that has been particularly successful4 ,more recently ,bourlet1 ,grevy1 ,borel1 and others in ,france have produced several works on the elements of mathematics that may lead to something of value4 ,they place intuition to the front1 favor as much applied mathematics as is reasonable1 to all of which ,american teachers would generally agree1 but they claim that the basis of elementary geometry in the future must be the investigation of the group of motions4 ,it is1 of course1 possible that certain of the notions of the higher mathematical thought of the nineteenth century may be so simplified as to be within the comprehension of the tyro in geometry1 and we should be ready to receive all efforts of this kind with open mind4 ,these writers have not however produced the ideal work1 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 ,treutlein1 and ,h4 ,mueller1 and ,schlegel a few years ago in ,germany1 and of ,veronese in ,italy4 ,all such efforts1 however1 should be welcomed and tried out1 although so far as at present appears there is nothing in sight to replace a well-arranged1 vitalized1 simplified textbook based upon the labors of ,euclid and ,legendre4 ,the most broad-minded of the great mathematicians who have recently given attention to secondary problems is ,professor ,klein of ,goettingen4 ,he has had the good sense to look at something besides the mere question of good mathematics4#cd ,thus he insists upon the psychologic point of view1 to the end that the geometry shall be adapted to the mental development of the pupil1--a thing that is apparently ignored by ,meray 7at least for the average pupil71 and1 it is to be feared1 by the other recent ,french writers4 ,he then demands a careful selection of the subject matter1 which in our ,american schools would mean the elimination of propositions that are not basal1 that is1 that are not used for most of the exercises that one naturally meets in elementary geometry and in applied work4 ,he further insists upon a reasonable correlation with practical work to which every teacher will agree so long as the work is really or even potentially practical4 ,and finally he asks that we look with favor upon the union of plane and solid geometry1 and of algebra and geometry4 ,he does not make any plea for extreme fusion1 but presumably he asks that to which every one of open mind would agree1 namely1 that whenever the opportunity offers in teaching plane geometry1 to open the vision to a generalization in space1 or to the measurement of well-known solids1 or to the use of the algebra that the pupil has learned1 the opportunity should be seized4 ,,footnotes3 #cb ,the author is a member of a committee that has for more than a year been considering a syllabus in geometry4 ,this committee will probably report sometime during the year #aiaa4 ,at the present writing it seems disposed to recommend about the usual list of basal propositions4 #cc ,elementi di ,geometria1 ,milan1 #ahhd4 #cd ,see his ,elementarmathematik vom hoeheren ,standpunkt aus1 ,part ,,ii1 ,leipzig1 #aiji4 ,,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 generations4 ,in general1 in other countries1 geometry is not taught to mixed classes of boys and girls4 ,furthermore1 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 life4 ,in ,america it is not alone the boy who is interested in education in general1 or in mathematics in particular1 who studies geometry1 and who joins with others of like tastes in this pursuit1 but it is often the boy and the girl who are not compelled to go out and work1 and who fill the years of youth with a not over-strenuous school life4 ,it is therefore clear that we cannot hold the interest of such pupils by the study of ,euclid alone4 ,geometry must1 for them1 be less formal than it was half a century ago4 ,we cannot expect to make our classes enthusiastic merely over a logical sequence of proved propositions4 ,it becomes necessary to make the work more concrete1 and to give a much larger number of simple exercises in order to create the interest that comes from independent work1 from a feeling of conquest1 and from a desire to do something original4 ,if we would cast a glamor over the multiplication table1 as an admirer of ,macaulay has said that the latter could do1 we must have the facilities for so doing4 ,it therefore becomes necessary in weighing the merits of a textbook to consider3 7#a7 if the number of proved propositions is reduced to a safe minimum2 7#b7 if there is reasonable opportunity to apply the theory1 the actual applications coming best1 however1 from the teacher as an outside interest2 7#c7 if there is an abundance of material in the way of simple exercises1 since such material is not so readily given by the teacher as the seemingly local applications of the propositions to outdoor measurements2 7#d7 if the book gives a reasonable amount of introductory work in the use of simple and inexpensive instruments1 not at that time emphasizing the formal side of the subject2 7#e7 if there is afforded some opportunity to see the recreative side of the subject1 and to know a little of the story of geometry as it has developed from ancient to modern times4 ,but this does not mean that there is to be a geometric cataclysm4 ,it means that we must have the same safe1 conservative evolution in geometry that we have in other subjects4 ,geometry is not going to degenerate into mere measuring1 nor is the ancient sequence going to become a mere hodge-podge without system and with no incentive to strenuous effort4 ,it is now about fifteen hundred years since ,proclus laid down what he considered the essential features of a good textbook1 and in all of our efforts at reform we cannot improve very much upon his statement4 ,it is essential1 he says1 that such a treatise should be rid of everything superfluous1 for the superfluous is an obstacle to the acquisition of knowledge2 it should select everything that embraces the subject and brings it to a focus1 for this is of the highest service to science2 it must have great regard both to clearness and to conciseness1 for their opposites trouble our understanding2 it must aim to generalize its theorems1 for the division of knowledge into small elements renders it difficult of comprehension4 ,it being prefaced that we must make the book more concrete in its applications1 either directly or by suggesting seemingly practical outdoor work2 that we must increase the number of simple exercises calling for original work2 that we must reasonably reduce the number of proved propositions2 and that we must not allow the good of the ancient geometry to depart1 let us consider in detail some of the features of a good1 practical1 common-sense textbook4 ,the early textbooks in geometry contained only the propositions1 with the proofs in full1 preceded by lists of definitions and assumptions 7axioms and postulates74 ,there were no exercises1 and the proofs were given in essay form4 ,then came treatises with exercises1 these exercises being grouped at the end of the work or at the close of the respective books4 ,the next step was to the unit page1 arranged in steps to aid the eye1 one proposition to a page whenever this was possible4 ,some effort was made in this direction in ,france about two hundred years ago1 but with no success4 ,the arrangement has so much to commend it1 however1 the proof being so much more easily followed by the eye than was the case in the old-style works1 that it has of late been revived4 ,in this respect the ,wentworth geometry was a pioneer in ,america1 and so successful was the effort that this type of page has been adopted1 as far as the various writers were able to adopt it1 in all successful geometries that have appeared of late years in this country4 ,as a result1 the ,american textbooks on this subject are more helpful and pleasing to the eye than those found elsewhere4 ,the latest improvements in textbook-making have removed most of the blemishes of arrangement that remained1 scattering the exercises through the book1 grading them with greater care1 and making them more modern in character4 ,but the best of the latest works do more than this4 ,they reduce the number of proved theorems and increase the number of exercises1 and they simplify the proofs whenever possible and eliminate the most difficult of the exercises of twenty-five years ago4 ,it would be possible to carry this change too far by putting in only half as many1 or a quarter as many1 regular propositions1 but it should not be the object to see how the work can be cut down1 but to see how it can be improved4 ,what should be the basis of selection of propositions and exercises8 ,evidently the selection must include the great basal propositions that are needed in mensuration and in later mathematics1 together with others that are necessary to prove them4 ,euclid's one hundred seventy-three propositions of plane geometry were really upwards of one hundred eighty1 because he several times combined two or more in one4 ,these we may reduce to about one hundred thirty with perfect safety1 or less than one a day for a school year1 but to reduce still further is undesirable as well as unnecessary4 ,it would not be difficult to dispense with a few more2 indeed1 we might dispense with thirty more if we should set about it1 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 offered4 ,but let it be clear that if we teach #ajj instead of #acj1 our results are liable to be about #ajj_/acj as satisfactory4 ,we may theorize on pedagogy as we please1 but geometry will pay us about in proportion to what we give4 ,and as to the exercises1 what is the basis of selection8 ,in general1 let it be said that any exercise that pretends to be real should be so1 and that words taken from science or measurements do not necessarily make the problem genuine4 ,to take a proposition and apply it in a manner that the world never sanctions is to indulge in deceit4 ,on the other hand1 wholly to neglect the common applications of geometry to handwork of various kinds is to miss one of our great opportunities to make the subject vital to the pupil1 to arouse new interest1 and to give a meaning to it that is otherwise wanting4 ,it should always be remembered that mental discipline1 whatever the phrase may mean1 can as readily be obtained from a genuine application of a theorem as from a mere geometric puzzle4 ,on the other hand1 it is evident that not more than #be per cent of propositions have any genuine applications outside of geometry1 and that if we are to attempt any applications at all1 these must be sought mainly in the field of pure geometry4 ,in the exercises1 therefore1 we seek to-day a sane and a balanced book1 giving equal weight to theory and to practice1 to the demands of the artisan and to those of the mathematician1 to the applications of concrete science and to those of pure geometry1 thus making a fusion of pure and applied mathematics1 with the latter as prominent as the supply of genuine problems permits4 ,the old is not all bad and the new is not all good1 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 new4 ,lest the nature of the exercises of geometry may be misunderstood1 it is well that we consider for a moment what constitutes a genuine application of the subject4 ,it is the ephemeral fashion just at present in ,america to call these genuine applications by the name of real problems4 ,the name is an unfortunate importation1 but that is not a matter of serious moment4 ,the important thing is that we should know what makes a problem real to the pupil of geometry1 especially as the whole thing is coming rapidly into disrepute through the mistaken zeal of some of its supporters4 ,a real problem is a problem that the average citizen may sometime be called upon to solve2 that1 if so called upon1 he will solve in the manner indicated2 and that is expressed in terms that are familiar to the pupil4 ,this definition1 which seems fairly to state the conditions under which a problem can be called real in the schoolroom1 involves three points3 7#a7 people must be liable to meet such a problem2 7#b7 in that case they will solve it in the way suggested by the book2 7#c7 it must be clothed in language familiar to the pupil4 ,for example1 let the problem be to find the dimensions of a rectangular field1 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 field4 ,as a real problem this is ridiculous1 since no one would ever meet such a case outside the puzzle department of a schoolroom4 ,again1 if by any stretch of a vigorous imagination any human being should care to find the area of a piece of glass1 bounded by the arcs of circles1 in a ,gothic window in ,york ,minster1 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 geometry4 ,and for the third point1 a problem is not real to a pupil simply because it relates to moments of inertia or the tensile strength of a steel bar4 ,indeed1 it is unreal precisely because it does talk of these things at a time when they are unfamiliar1 and properly so1 to the pupil4 ,it must not be thought that puzzle problems1 and unreal problems generally1 have no value4 ,all that is insisted upon is that such problems as the above are not real1 and that about #ij per cent of problems that go by this name are equally lacking in the elements that make for reality in this sense of the word4 ,for the other #aj per cent of such problems we should be thankful1 and we should endeavor to add to the number4 ,as for the great mass1 however1 they are no better than those that have stood the test of generations1 and by their pretense they are distinctly worse4 ,it is proper1 however1 to consider whether a teacher is not justified in relating his work to those geometric forms that are found in art1 let us say in floor patterns1 in domes of buildings1 in oilcloth designs1 and the like1 for the purpose of arousing interest1 if for no other reason4 ,the answer is apparent to any teacher3 ,it is certainly justifiable to arouse the pupil's interest in his subject1 and to call his attention to the fact that geometric design plays an important part in art2 but we must see to it that our efforts accomplish this purpose4 ,to make a course in geometry one on oilcloth design would be absurd1 and nothing more unprofitable or depressing could be imagined in connection with this subject4 ,of course no one would advocate such an extreme1 but it sometimes seems as if we are getting painfully near it in certain schools4 ,a pupil has a passing interest in geometric design4 ,he should learn to use the instruments of geometry1 and he learns this most easily by drawing a few such patterns4 ,but to keep him week after week on questions relating to such designs of however great variety1 and especially to keep him upon designs relating to only one or two types1 is neither sound educational policy nor even common sense4 ,that this enthusiastic teacher or that one succeeds by such a plan is of no significance2 it is the enthusiasm that succeeds1 not the plan4 ,the experience of the world is that pupils of geometry like to use the subject practically1 but that they are more interested in the pure theory than in any fictitious applications1 and this is why pure geometry has endured1 while the great mass of applied geometry that was brought forward some three hundred years ago has long since been forgotten4 ,the question of the real applications of the subject is considered in subsequent chapters4 ,in ,chapter ,,vi we considered the question of the number of regular propositions to be expected in the text1 and we have just considered the nature of the exercises which should follow those propositions4 ,it is well to turn our attention next to the nature of the proofs of the basal theorems4 ,shall they appear in full8 ,shall they be merely suggested demonstrations8 ,shall they be only a series of questions that lead to the proof8 ,shall the proofs be omitted entirely8 ,or shall there be some combination of these plans8 ,the natural temptation in the nervous atmosphere of ,america is to listen to the voice of the mob and to proceed at once to lynch ,euclid and every one who stands for that for which the ,elements has stood these two thousand years4 ,this is what some who wish to be considered as educators tend to do2 in the language of the mob1 to smash things2 to call reactionary that which does not conform to their ephemeral views4 ,it is so easy to be an iconoclast1 to think that cui bono is a conclusive argument1 to say so glibly that ,raphael was not a great painter1--to do anything but construct4 ,a few years ago every one must take up with the heuristic method developed in ,germany half a century back and containing much that was commendable4 ,a little later one who did not believe that the ,culture ,epoch ,theory was vital in education was looked upon with pity by a considerable number of serious educators4 ,a little later the man who did not think that the principle of ,concentration in education was a regula aurea was thought to be hopeless4 ,a little later it may have been that ,correlation was the saving factor1 to be looked upon in geometry teaching as a guiding beacon1 even as the fusion of all mathematics is the temporary view of a few enthusiasts to-day4#ce ,and just now it is vocational training that is the catch phrase1 and to many this phrase seems to sound the funeral knell of the standard textbook in geometry4 ,but does it do so8 ,does this present cry of the pedagogical circle really mean that we are no longer to have geometry for geometry's sake8 ,does it mean that a panacea has been found for the ills of memorizing without understanding a proof in the class of a teacher who is so inefficient as to allow this kind of work to go on8 ,does it mean that a teacher who does not see the human side of geometry1 who does not know the real uses of geometry1 and who has no faculty of making pupils enthusiastic over geometry1--that this teacher is to succeed with some scrappy1 weak1 pretending apology for a real work on the subject8 ,no one believes in stupid teaching1 in memorizing a textbook1 in having a book that does all the work for a pupil1 or in any of the other ills of inefficient instruction4 ,on the other hand1 no fair-minded person can condemn a type of book that has stood for generations until something besides the mere transient experiments of the moment has been suggested to replace it4 ,let us1 for example1 consider the question of having the basal propositions proved in full1 a feature that is so easy to condemn as leading to memorizing4 ,the argument in favor of a book with every basal proposition proved in full1 or with most of them so proved1 the rest having only suggestions for the proof1 is that the pupil has before him standard forms exhibiting the best1 most succinct1 most clearly stated demonstrations that geometry contains4 ,the demonstrations stand for the same thing that the type problems stand for in algebra1 and are generally given in full in the same way4 ,the argument against the plan is that it takes away the pupil's originality by doing all the work for him1 allowing him to merely memorize the work4 ,now if all there is to geometry were in the basal propositions1 this argument might hold1 just as it would hold in algebra in case there were only those exercises that are solved in full4 ,but just as this is not the case in algebra1 the solved exercises standing as types or as bases for the pupil's real work1 so the demonstrated proposition forms a relatively small part of geometry1 standing as a type1 a basis for the more important part of the work4 ,moreover1 a pupil who uses a syllabus is exposed to a danger that should be considered1 namely1 that of dishonesty4 ,any textbook in geometry will furnish the proofs of most of the propositions in a syllabus1 whatever changes there may be in the sequence1 and it is not a healthy condition of mind that is induced by getting the proofs surreptitiously4 ,unless a teacher has more time for the course than is usually allowed1 he cannot develop the new work as much as is necessary with only a syllabus1 and the result is that a pupil gets more of his work from other books and has less time for exercises4 ,the question therefore comes to this3 ,is it better to use a book containing standard forms of proof for the basal propositions1 and have time for solving a large number of original exercises and for seeking the applications of geometry8 ,or is it better to use a book that requires more time on the basal propositions1 with the danger of dishonesty1 and allows less time for solving originals8 ,to these questions the great majority of teachers answer in favor of the textbook with most of the basal propositions fully demonstrated4 ,in general1 therefore1 it is a good rule to use the proofs of the basal propositions as models1 and to get the original work from the exercises4 ,unless we preserve these model proofs1 or unless we supply them with a syllabus1 the habit of correct1 succinct self-expression1 which is one of the chief assets of geometry1 will tend to become atrophied4 ,so important is this habit that no system of education in which its performance is neglected can hope or profess to evolve men and women who are competent in the full sense of the word4 ,so long as teachers of geometry neglect the possibilities of the subject in this respect1 so long will the time devoted to it be in large part wasted1 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 done4#cf ,it is here that the chief danger of syllabus-teaching lies1 and it is because of this patent fact that a syllabus without a carefully selected set of model proofs1 or without the unnecessary expenditure of time by the class1 is a dangerous kind of textbook4 ,what shall then be said of those books that merely suggest the proofs1 or that give a series of questions that lead to the demonstrations8 ,there is a certain plausibility about such a plan at first sight4 ,but it is easily seen to have only a fictitious claim to educational value4 ,in the first place1 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 method4 ,the questions are so framed as to admit1 in most cases1 of only a single answer1 so that this answer might just as well be given instead of the question4 ,the pupil has therefore a proof requiring no more effort than is the case in the standard form of textbook1 but not given in the clear language of a careful writer4 ,furthermore1 the pupil is losing here1 as when he uses only a syllabus1 one of the very things that he should be acquiring1 namely1 the habit of reading mathematics4 ,if he met only syllabi without proofs1 or suggestive geometries1 or books that endeavored to question every proof out of him1 he would be in a sorry plight when he tried to read higher mathematics1 or even other elementary treatises4 ,it is for reasons such as these that the heuristic textbook has never succeeded for any great length of time or in any wide territory4 ,and finally1 upon this point1 shall the demonstrations be omitted entirely1 leaving only the list of propositions1--in other words1 a pure syllabus8 ,this has been sufficiently answered above4 ,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 found4 ,this is an arrangement that begins like the ordinary textbook and1 after the pupil has acquired the form of proof1 gradually merges into a syllabus1 so that there is no temptation to go surreptitiously to other books for help4 ,such a book1 if worked out with skill1 would appeal to an enthusiastic teacher1 and would accomplish the results claimed for the cruder forms of manual already described4 ,it would not be in general as safe a book as the standard form1 but with the right teacher it would bring good results4 ,in conclusion1 there are two types of textbook that have any hope of success4 ,the first is the one with all or a large part of the basal propositions demonstrated in full1 and with these propositions not unduly reduced in number4 ,such a book should give a large number of simple exercises scattered through the work1 with a relatively small number of difficult ones4 ,it should be modern in its spirit1 with figures systematically lettered1 with each page a unit as far as possible1 and with every proof a model of clearness of statement and neatness of form4 ,above all1 it should not yield to the demand of a few who are always looking merely for something to change1 nor should it in a reactionary spirit return to the old essay form of proof1 which hinders the pupil at this stage4 ,the second type is the semisyllabus1 otherwise with all the spirit of the first type4 ,in both there should be an honest fusion of pure and applied geometry1 with no exercises that pretend to be practical without being so1 with no forced applications that lead the pupil to measure things in a way that would appeal to no practical man1 with no merely narrow range of applications1 and with no array of difficult terms from physics and engineering that submerge all thought of mathematics in the slough of despond of an unknown technical vocabulary4 ,outdoor exercises1 even if somewhat primitive1 may be introduced1 but it should be perfectly understood that such exercises are given for the purpose of increasing the interest in geometry1 and they should be abandoned if they fail of this purpose4 ,bibliography4 ,for a list of standard textbooks issued prior to the present generation1 consult the bibliography in ,stamper1 ,history of the ,teaching of ,geometry1 ,new ,york1 #aijh4 ,,footnotes3 #ce ,for some classes of schools and under certain circumstances courses in combined mathematics are very desirable4 ,all that is here insisted upon is that any general fusion all along the line would result in weak1 insipid1 and uninteresting mathematics4 ,a beginning1 inspirational course in combined mathematics has a good reason for being in many high schools in spite of its manifest disadvantages1 and such a course may be developed to cover all of the required mathematics given in certain schools4 #cf ,carson1 loc4 cit41 p4 #ae4 ,,chapter ,,viii ,,the ,,relation ,,of ,,algebra ,,to ,,geometry ,from the standpoint of theory there is or need be no relation whatever between algebra and geometry4 ,algebra was originally the science of the equation1 as its name#cg indicates4 ,this means that it was the science of finding the value of an unknown quantity in a statement of equality4 ,later it came to mean much more than this1 and ,newton spoke of it as universal arithmetic1 and wrote an algebra with this title4 ,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 studied1 functions which it is not necessary to specify at this time4 ,the work relates chiefly to functions involving the idea of number4 ,in geometry1 on the other hand1 the work relates chiefly to form4 ,indeed1 in pure geometry number plays practically no part1 while in pure algebra form plays practically no part4 ,in #afhg the great ,french philosopher1 ,descartes1 wishing to picture certain algebraic functions1 wrote a work of about a hundred pages1 entitled ,la ,geometrie1 and in this he showed a correspondence between the numbers of algebra 7which may be expressed by letters7 and the concepts of geometry4 ,this was the first great step in the analytic geometry that finally gave us the graph in algebra4 ,since then there have been brought out from time to time other analogies between algebra and geometry1 always to the advantage of each science4 ,this has led to a desire on the part of some teachers to unite algebra and geometry into one science1 having simply a class in mathematics without these special names4 ,it is well to consider the advantages and the disadvantages of such a plan1 and to decide as to the rational attitude to be taken by teachers concerning the question at issue4 ,on the side of advantages it is claimed that there is economy of time and of energy4 ,if a pupil is studying formulas1 let the formulas of geometry be studied2 if he is taking up ratio and proportion2 let him do so for algebra and geometry at the same time2 if he is solving quadratics1 let him apply them at once to certain propositions concerning secants2 and if he is proving that 7a b7#b equals a#b #b;ab b#b1 let him do so by algebra and by geometry simultaneously4 ,it is claimed that not only is there economy in this arrangement1 but that the pupil sees mathematics as a whole1 and thus acquires more of a mastery than comes by our present tandem arrangement4 ,on the side of disadvantages it may be asked if the same arguments would not lead us to teach ,latin and ,greek together1 or ,latin and ,french1 or all three simultaneously8 ,if pupils should decline nouns in all three languages at the same time1 learn to count in all at the same time1 and begin to translate in all simultaneously1 would there not be an economy of time and effort1 and would there not be developed a much broader view of language8 ,now the fusionist of algebra and geometry does not like this argument1 and he says that the cases are not parallel1 and he tries to tell why they are not4 ,he demands that his opponent abandon argument by analogy and advance some positive reason why algebra and geometry should not be fused4 ,then his opponent says that it is not for him to advance any reason for what already exists1 the teaching of the two separately2 that he has only to refute the fusionist's arguments1 and that he has done so4 ,he asserts that algebra and geometry are as distinct as chemistry and biology2 that they have a few common points1 but not enough to require teaching them together4 ,he claims that to begin ,latin and ,greek at the same time has always proved to be confusing1 and that the same is true of algebra and geometry4 ,he grants that unified knowledge is desirable1 but he argues that when the fine arts of music and color work fuse1 and when the natural sciences of chemistry and physics are taught in the same class1 and when we follow the declension of a ,german noun by that of a ,french noun and a ,latin noun1 and when we teach drawing and penmanship together1 then it is well to talk of mixing algebra and geometry4 ,it is well1 before deciding such a question for ourselves 7for evidently we cannot decide it for the world71 to consider what has been the result of experience4 ,algebra and geometry were always taught together in early times1 as were trigonometry and astronomy4 ,the ,ahmes papyrus contains both primitive algebra and primitive geometry4 ,euclid's ,elements contains not only pure geometry1 but also a geometric algebra and the theory of numbers4 ,the early works of the ,hindus often fused geometry and arithmetic1 or geometry and algebra4 ,even the first great printed compendium of mathematics1 the ,suma of ,paciuolo 7#adid7 contained all of the branches of mathematics4 ,much of this later attempt was not1 however1 an example of perfect fusion1 but rather of assigning one set of chapters to algebra1 another to geometry1 and another to arithmetic4 ,so fusion1 more or less perfect1 has been tried over long periods1 and abandoned as each subject grew more complete in itself1 with its own language and its peculiar symbols4 ,but it is asserted that fusion is being carried on successfully to-day by more than one enthusiastic teacher1 and that this proves the contention that the plan is a good one4 ,books are cited to show that the arrangement is feasible1 and classes are indicated where the work is progressing along this line4 ,what1 then1 is the conclusion8 ,that is a question for the teacher to settle1 but it is one upon which a writer on the teaching of mathematics should not fear to express his candid opinion4 ,it is a fact that the ,greek and ,latin fusion is a fair analogy4 ,there are reasons for it1 but there are many more against it1 the chief one being the confusion of beginning two languages at once1 and the learning simultaneously of two vocabularies that must be kept separate4 ,it is also a fact that algebra and geometry are fully as distinct as physics and chemistry1 or chemistry and biology4 ,life may be electricity1 and a brief cessation of oxidization in the lungs brings death1 but these facts are no reasons for fusing the sciences of physics1 biology1 and chemistry4 ,algebra is primarily a theory of certain elementary functions1 a generalized arithmetic1 while geometry is primarily a theory of form with a highly refined logic to be used in its mastery4 ,they have a few things in common1 as many other subjects have1 but they have very many more features that are peculiar to the one or the other4 ,the experience of the world has led it away from a simultaneous treatment1 and the contrary experience of a few enthusiastic teachers of to-day proves only their own powers to succeed with any method4 ,it is easy to teach logarithms in the seventh school year1 but it is not good policy to do so under present conditions4 ,so the experience of the world is against the plan of strict fusion1 and no arguments have as yet been advanced that are likely to change the world's view4 ,no one has written a book combining algebra and geometry in this fashion that has helped the cause of fusion a particle2 on the contrary1 every such work that has appeared has damaged that cause by showing how unscientific a result has come from the labor of an enthusiastic supporter of the movement4 ,but there is one feature that has not been considered above1 and that is a serious handicap to any effort at combining the two sciences in the high school1 and this is the question of relative difficulty4 ,it is sometimes said1 in a doctrinaire fashion1 that geometry is easier than algebra1 since form is easier to grasp than function1 and that therefore geometry should precede algebra4 ,but every teacher of mathematics knows better than this4 ,he knows that the simplest form is easier to grasp than the simplest function1 but nevertheless that plane geometry1 as we understand the term to-day1 is much more difficult than elementary algebra for a pupil of fourteen4 ,the child studies form in the kindergarten before he studies number1 and this is sound educational policy4 ,he studies form1 in mensuration1 throughout his course in arithmetic1 and this1 too1 is good educational policy4 ,this kind of geometry very properly precedes algebra4 ,but the demonstrations of geometry1 the study by pupils of fourteen years of a geometry that was written for college students and always studied by them until about fifty years ago1--that is by no means as easy as the study of a simple algebraic symbolism and its application to easy equations4 ,if geometry is to be taught for the same reasons as at present1 it cannot advantageously be taught earlier than now without much simplification1 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 purpose4 ,when great mathematicians like ,professor ,klein speak of the fusion of all mathematics1 they speak from the standpoint of advanced students1 not for the teacher of elementary geometry4 ,it is therefore probable that simple mensuration will continue1 as a part of arithmetic1 to precede algebra1 as at present2 and that algebra into or through quadratics will precede geometry1#ch drawing upon the mensuration of arithmetic as may be needed2 and that geometry will follow this part of algebra1 using its principles as far as possible to assist in the demonstrations and to express and manipulate its formulas4 ,plane geometry1 or else a year of plane and solid geometry1 will probably1 in this country1 be followed by algebra1 completing quadratics and studying progressions2 and by solid geometry1 or a supplementary course in plane and solid geometry1 this work being elective in many1 if not all1 schools4#ci ,it is also probable that a general review of mathematics1 where the fusion idea may be carried out1 will prove to be a feature of the last year of the high school1 and one that will grow in popularity as time goes on4 ,such a plan will keep algebra and geometry separate1 but it will allow each to use all of the other that has preceded it1 and will encourage every effort in this direction4 ,it will accomplish all that a more complete fusion really hopes to accomplish1 and it will give encouragement to all who seek to modernize the spirit of each of these great branches of mathematics4 ,there is1 however1 a chance for fusion in two classes of school1 neither of which is as yet well developed in this country4 ,the first is the technical high school that is at present coming into some prominence4 ,it is not probable even here that the best results can be secured by eliminating all mathematics save only what is applicable in the shop1 but if this view should prevail for a time1 there would be so little left of either algebra or geometry that each could readily be joined to the other4 ,the actual amount of algebra needed by a foreman in a machine shop can be taught in about four lessons1 and the geometry or mensuration that he needs can be taught in eight lessons at the most4 ,the necessary trigonometry may take eight more1 so that it is entirely feasible to unite these three subjects4 ,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 literature1 but he would have enough for his immediate needs1 even though he had no appreciation of mathematics as a science4 ,if any one asks if this is not all that the school should give him1 it might be well to ask if the school should give only the ability to read1 without the knowledge of any good literature2 if it should give only the ability to sing1 without the knowledge of good music2 if it should give only the ability to speak1 without any training in the use of good language2 and if it should give a knowledge of home geography1 without any intimation that the world is round1--an atom in the unfathomable universe about us4 ,the second opportunity for fusion is possibly 7for it is by no means certain7 to be found in a type of school in which the only required courses are the initial ones4 ,these schools have some strong advocates1 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 interested4 ,whether or not this is sound educational policy need not be discussed at this time2 but if such a plan were developed1 it might be well to offer a somewhat superficial 7in the sense of abridged7 course that should embody a little of algebra1 a little of geometry1 and a little of trigonometry4 ,this would unconsciously become a bait for students1 and the result would probably be some good teaching in the class in question4 ,it is to be hoped that we may have some strong1 well-considered textbooks upon this phase of the work4 ,as to the fusion of trigonometry and plane geometry little need be said1 because the subject is in the doctrinaire stage4 ,trigonometry naturally follows the chapter on similar triangles1 but to put it there means1 in our crowded curriculum1 to eliminate something from geometry4 ,which1 then1 is better1--to give up the latter portion of geometry1 or part of it at least1 or to give up trigonometry8 ,some advocates have entered a plea for two or three lessons in trigonometry at this point1 and this is a feature that any teacher may introduce as a bit of interest1 as is suggested in ,chapter ,,xvi1 just as he may give a popular talk to his class upon the fourth dimension or the non-,euclidean geometry4 ,the lasting impression upon the pupil will be exactly the same as that of four lessons in ,sanskrit while he is studying ,latin4 ,he might remember each with pleasure1 ,latin being related1 as it is1 to ,sanskrit1 and trigonometry being an outcome of the theory of similar triangles4 ,but that either of these departures from the regular sequence is of any serious mathematical or linguistic significance no one would feel like asserting4 ,each is allowable on the score of interest1 but neither will add to the pupil's power in any essential feature4 ,each of these subjects is better taught by itself1 each using the other as far as possible and being followed by a review that shall make use of all4 ,it is not improbable that we may in due time have high schools that give less extended courses in algebra and geometry1 adding brief practical courses in trigonometry and the elements of the calculus2 but even in such schools it is likely to be found that geometry is best taught by itself1 making use of all the mathematics that has preceded it4 ,it will of course be understood that the fusion of algebra and geometry as here understood has nothing to do with the question of teaching the two subjects simultaneously1 say two days in the week for one and three days for the other4 ,this plan has many advocates1 although on the whole it has not been well received in this country4 ,but what is meant here is the actual fusing of algebra and geometry day after day1--a plan that has as yet met with only a sporadic success1 but which may be developed for beginning classes in due time4 ,,footnotes3 #cg ,al-jabr wa'l-muqabalah3 restoration and equation is a fairly good translation of the ,arabic4 #ch ,or be carried along at the same time as a distinct topic4 #ci ,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 geometry4 ,,chapter ,,ix ,,the ,,introduction ,,to ,,geometry ,there are two difficult crises in the geometry course1 both for the pupil and for the teacher4 ,these crises are met at the beginning of the subject and at the beginning of solid geometry4 ,once a class has fairly got into ,book ,i1 if the interest in the subject can be maintained1 there are only the incidental difficulties of logical advance throughout the plane geometry4 ,when the pupil who has been seeing figures in one plane for a year attempts to visualize solids from a flat drawing1 the second difficult place is reached4 ,teachers going over solid geometry from year to year often forget this difficulty1 but most of them can easily place themselves in the pupil's position by looking at the working drawings of any artisan1--usually simple cases in the so-called descriptive geometry4 ,they will then realize how difficult it is to visualize a solid from an unfamiliar kind of picture4 ,the trouble is usually avoided by the help of a couple of pieces of heavy cardboard or box board1 and a few knitting needles with which to represent lines in space4 ,if these are judiciously used in class for a few days1 until the figures are understood1 the second crisis is easily passed4 ,the continued use of such material1 however1 or the daily use of either models or photographs1 weakens the pupil1 even as a child is weakened by being kept too long in a perambulator4 ,such devices have their place2 they are useful when needed1 but they are pernicious when unnecessary4 ,just as the mechanic must be able to make and to visualize his working drawings1 so the student of solid geometry must be able to get on with pencil and paper1 representing his solid figures in the flat4 ,but the introduction to plane geometry is not so easily disposed of4 ,the pupil at that time is entering a field that is entirely unfamiliar4 ,he is only fourteen or fifteen years of age1 and his thoughts are distinctly not on geometry4 ,of logic he knows little and cares less4 ,he is not interested in a subject of which he knows nothing1 not even the meaning of its name4 ,he asks1 naturally and properly1 what it all signifies1 what possible use there is for studying geometry1 and why he should have to prove what seems to him evident without proof4 ,to pass him successfully through this stage has taxed the ingenuity of every real teacher from the time of ,euclid to the present2 and just as ,euclid remarked to ,king ,ptolemy1 his patron1 that there is no royal road to geometry1 so we may affirm that there is no royal road to the teaching of geometry4 ,nevertheless the experience of teachers counts for a great deal1 and this experience has shown that1 aside from the matter of technic in handling the class1 certain suggestions are of value1 and a few of these will now be set forth4 ,first1 as to why geometry is studied1 it is manifestly impossible successfully to explain to a boy of fourteen or fifteen the larger reasons for studying anything whatever4 ,when we confess ourselves honestly we find that these reasons1 whether in mathematics1 the natural sciences1 handwork1 letters1 the vocations1 or the fine arts1 are none too clear in our own minds1 in spite of any pretentious language that we may use4 ,it is therefore most satisfactory to anticipate the question at once1 and to set the pupils1 for a few days1 at using the compasses and ruler in the drawing of geometric designs and of the most common figures that they will use4 ,this serves several purposes3 it excites their interest1 it guards against the slovenly figures that so often lead them to erroneous conclusions1 it has a genuine value for the future artisan1 and it shows that geometry is something besides mere theory4 ,whether the textbook provides for it or not1 the teacher will find a few days of such work well spent1 it being a simple matter to supplement the book in this respect4 ,there was a time when some form of mechanical drawing was generally taught in the schools1 but this has given place to more genuine art work1 leaving it to the teacher of geometry to impart such knowledge of drawing as is a necessary preliminary to the regular study of the subject4 ,such work in drawing should go so far1 and only so far1 as to arouse an interest in geometric form without becoming wearisome1 and to familiarize the pupil with the use of the instruments4 ,he should be counseled about making fine lines1 about being careful in setting the point of his compasses on the exact center that he wishes to use1 and about representing a point by a very fine dot1 or1 preferably at first1 by two crossed lines4 ,unless these details are carefully considered1 the pupil will soon find that the lines of his drawings do not fit together1 and that the result is not pleasing to the eye4 ,the figures here given are good ones upon which to begin1 the dotted construction lines being erased after the work is completed4 ,they may be constructed with the compasses and ruler alone1 or the draftsman's ,t-square1 triangle1 and protractor may be used1 although these latter instruments are not necessary4 ,we should constantly remember that there is a danger in the slavish use of instruments and of such helps as squared paper4 ,just as ,euclid rode roughshod over the growing intellects of boys and girls1 so may instruments ride roughshod over their growing perceptions by interfering with natural and healthy intuitions1 and making them the subject of laborious measurement4#dj ,illustration ,the pupil who cannot see the equality of vertical angles intuitively better than by the use of the protractor is abnormal4 ,nevertheless it is the pupil's interest that is at stake1 together with his ability to use the instruments of daily life4 ,if1 therefore1 he can readily be supplied with draftsmen's materials1 and is not compelled to use them in a foolish manner1 so much the better4 ,they will not hurt his geometry if the teacher does not interfere1 and they will help his practical drawing2 but for obvious reasons we cannot demand that the pupil purchase what is not really essential to his study of the subject4 ,the most valuable single instrument of the three just mentioned is the protractor1 and since a paper one costs only a few cents and is often helpful in the drawing of figures1 it should be recommended to pupils4 ,there is also another line of work that often arouses a good deal of interest1 namely1 the simple field measures that can easily be made about the school grounds4 ,guarding against the ever-present danger of doing too much of such work1 of doing work that has no interest for the pupil1 of requiring it done in a way that seems unreal to a class1 and of neglecting the essence of geometry by a line of work that involves no new principles1--such outdoor exercises in measurement have a positive value1 and a plentiful supply of suggestions in this line is given in the subsequent chapters4 ,the object is chiefly to furnish a motive for geometry1 and for many pupils this is quite unnecessary4 ,for some1 however1 and particularly for the energetic1 restless boy1 such work has been successfully offered by various teachers as an alternative to some of the book work4 ,because of this value a considerable amount of such work will be suggested for teachers who may care to use it1 the textbook being manifestly not the place for occasional topics of this nature4 ,for the purposes of an introduction only a tape line need be purchased4 ,wooden pins and a plumb line can easily be made4 ,even before he comes to the propositions in mensuration in geometry the pupil knows1 from his arithmetic1 how to find ordinary areas and volumes1 and he may therefore be set at work to find the area of the school ground1 or of a field1 or of a city block4 ,the following are among the simple exercises for a beginner3 ,illustration #a4 ,drive stakes at two corners1 ,a and ,b1 of the school grounds1 putting a cross on top of each2 or make the crosses on the sidewalk1 so as to get two points between which to measure4 ,measure from ,a to ,b by holding the tape taut and level1 dropping perpendiculars when necessary by means of the plumb line1 as shown in the figure4 ,check the work by measuring from ,b back to ,a in the same way4 ,pupils will find that their work should always be checked1 and they will be surprised to see how the results will vary in such a simple measurement as this1 unless very great care is taken4 ,if they learn the lesson of accuracy thus early1 they will have gained much4 ,illustration #b4 ,take two stakes1 ,x1 ,y1 in a field1 preferably two or three hundred feet apart1 always marked on top with crosses so as to have exact points from which to work4 ,let it then be required to stake out or range the line from ,x to ,y by placing stakes at specified distances4 ,one boy stands at ,y and another at ,x1 each with a plumb line4 ,a third one takes a plumb line and stands at ,p1 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 ,y4 ,a stake is then driven at ,p1 and the pupil at ,x moves on to the stake ,p4 ,then ,q is located in the same way1 and then ,r1 and so on4 ,the work is checked by ranging back from ,y to ,x4 ,in some of the simple exercises suggested later it is necessary to range a line so that this work is useful in making measurements4 ,the geometric principle involved is that two points determine a straight line4 ,illustration ,illustration #c4 ,to test a perpendicular or to draw one line perpendicular to another in a field1 we may take a stout cord twelve feet long1 having a knot at the end of every foot4 ,if this is laid along four feet1 the ends of this part being fixed1 and it is stretched as here shown1 so that the next vertex is five feet from one of these ends and three feet from the other end1 a right angle will be formed4 ,a right angle can also be run by making a simple instrument1 such as is described in ,chapter ,,xv4 ,still another plan of drawing a line perpendicular to another line ,,ab1 from a point ,p1 consists in swinging a tape from ,p1 cutting ,,ab at ,x and ,y1 and then bisecting ,,xy by doubling the tape4 ,this fixes the foot of the perpendicular4 ,illustration #d4 ,it is now possible to find the area of a field of irregular shape by dividing it into triangles and trapezoids1 as shown in the figure4 ,pupils know from their work in arithmetic how to find the area of a triangle or a trapezoid1 so that the area of the field is easily found4 ,the work may be checked by comparing the results of different groups of pupils1 or by drawing another diagonal and dividing the field into other triangles and trapezoids4 ,these are about as many types of field work as there is any advantage in undertaking for the purpose of securing the interest of pupils as a preliminary to the work in geometry4 ,whether any of it is necessary1 and for what pupils it is necessary1 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 class4 ,illustration ,illustration ,illustration ,illustration ,illustration ,illustration ,illustration ,a second difficulty of the pupil is seen in his attitude of mind towards proofs in general4 ,he does not see why vertical angles should be proved equal when he knows that they are so by looking at the figure4 ,this difficulty should also be anticipated by giving him some opportunity to know the weakness of his judgment1 and for this purpose figures like the following should be placed before him4 ,he should be asked which of these lines is longer1 ,,ab or ,,xy4 ,two equal lines should then be arranged in the form of a letter ,t1 as here shown1 and he should be asked which is the longer1 ,,ab or ,,cd4 ,a figure that is very deceptive1 particularly if drawn larger and with heavy cross lines1 is this one in which ,,ab and ,,cd are really parallel1 but do not seem to be so4 ,other interesting deceptions have to do with producing lines1 as in these figures1 where it is quite difficult in advance to tell whether ,,ab and ,,cd are in the same line1 and similarly for ,,wx and ,,yz4 ,equally deceptive is this figure1 in which it is difficult to tell which line ,,ab will lie along when produced4 ,in the next figure ,,ab appears to be curved when in reality it is straight1 and ,,cd appears straight when in reality it is curved4 ,the first of the following circles seems to be slightly flattened at the points ,p1 ,q1 ,r1 ,s1 and in the second one the distance ,,bd seems greater than the distance ,,ac4 ,there are many equally deceptive figures1 and a few of them will convince the beginner that the proofs are necessary features of geometry4 ,it is interesting1 in connection with the tendency to feel that a statement is apparent without proof1 to recall an anecdote related by the ,french mathematician1 ,biot1 concerning the great scientist1 ,laplace3 ,once ,laplace1 having been asked about a certain point in his ,celestial ,mechanics1 spent nearly an hour in trying to recall the chain of reasoning which he had carelessly concealed by the words ,it is easy to see4 ,a third difficulty lies in the necessity for putting a considerable number of definitions at the beginning of geometry1 in order to get a working vocabulary4 ,although practically all writers scatter the definitions as much as possible1 there must necessarily be some vocabulary at the beginning4 ,in order to minimize the difficulty of remembering so many new terms1 it is helpful to mingle with them a considerable number of exercises in which these terms are employed1 so that they may become fixed in mind through actual use4 ,thus it is of value to have a class find the complements of #bg deg41 #cb deg4 #bj'1 #da deg4 #cb' #dh1 #bf4ge deg41 #cc #a_/c deg41 and #j deg44 ,it is true that into the pure geometry of ,euclid the measuring of angles in degrees does not enter1 but it has place in the practical applications1 and it serves at this juncture to fix the meaning of a new term like complement4 ,the teacher who thus anticipates the question as to the reason for studying geometry1 the mental opposition to proving statements1 and the forgetfulness of the meaning of common terms will find that much of the initial difficulty is avoided4 ,if1 now1 great care is given to the first half dozen propositions1 the pupil will be well on his way in geometry4 ,as to these propositions1 two plans of selection are employed4 ,the first takes a few preliminary propositions1 easily demonstrated1 and seeks thus to introduce the pupil to the nature of a proof4 ,this has the advantage of inspiring confidence and the disadvantage of appearing to prove the obvious4 ,the second plan discards all such apparently obvious propositions as those about the equality of right angles1 and the sum of two adjacent angles formed by one line meeting another1 and begins at once on things that seem to the pupil as worth the proving4 ,in this latter plan the introduction is usually made with the proposition concerning vertical angles1 and the two simplest cases of congruent triangles4 ,whichever plan of selection is taken1 it is important to introduce a considerable number of one-step exercises immediately1 that is1 exercises that require only one significant step in the proof4 ,in this way the pupil acquires confidence in his own powers1 he finds that geometry is not mere memorizing1 and he sees that each proposition makes him the master of a large field4 ,to delay the exercises to the end of each book1 or even to delay them for several lessons1 is to sow seeds that will result in the attempt to master geometry by the sheer process of memorizing4 ,as to the nature of these exercises1 however1 the mistake must not be made of feeling that only those have any value that relate to football or the laying out of a tennis court4 ,such exercises are valuable1 but such exercises alone are one-sided4 ,moreover1 any one who examines the hundreds of suggested exercises that are constantly appearing in various journals1 or who1 in the preparation of teachers1 looks through the thousands of exercises that come to him in the papers of his students1 comes very soon to see how hollow is the pretense of most of them4 ,as has already been said1 there are relatively few propositions in geometry that have any practical applications1 applications that are even honest in their pretense4 ,the principle that the writer has so often laid down in other works1 that whatever pretends to be practical should really be so1 applies with much force to these exercises4 ,when we can find the genuine application1 if it is within reasonable grasp of the pupil1 by all means let us use it4 ,but to put before a class of girls some technicality of the steam engine that only a skilled mechanic would be expected to know is not education1--it is mere sham4 ,there is a noble dignity to geometry1 a dignity that a large majority of any class comes to appreciate when guided by an earnest teacher2 but the best way to destroy this dignity1 to take away the appreciation of pure mathematics1 and to furnish weaker candidates than now for advance in this field is to deceive our pupils and ourselves into believing that the ultimate purpose of mathematics is to measure things in a way in which no one else measures them or has ever measured them4 ,in the proof of the early propositions of plane geometry1 and again at the beginning of solid geometry1 there is a little advantage in using colored crayon to bring out more distinctly the equal parts of two figures1 or the lines outside the plane1 or to differentiate one plane from another4 ,this device1 however1 like that of models in solid geometry1 can easily be abused1 and hence should be used sparingly1 and only until the purpose is accomplished4 ,the student of mathematics must learn to grasp the meaning of a figure drawn in black on white paper1 or1 more rarely1 in white on a blackboard1 and the sooner he is able to do this the better for him4 ,the same thing may be said of the constructing of models for any considerable number of figures in solid geometry2 enough work of this kind to enable a pupil clearly to visualize the solids is valuable1 but thereafter the value is usually more than offset by the time consumed and the weakened power to grasp the meaning of a geometric drawing4 ,there is often a tendency on the part of teachers in their first years of work to overestimate the logical powers of their pupils and to introduce forms of reasoning and technical terms that experience has proved to be unsuited to one who is beginning geometry4 ,usually but little harm is done1 because the enthusiasm of any teacher who would use this work would carry the pupils over the difficulties without much waste of energy on their part4 ,in the long run1 however1 the attempt is usually abandoned as not worth the effort4 ,such a term as contrapositive1 such distinctions as that between the logical and the geometric converse1 or between perfect and partial geometric conversion1 and such pronounced formalism as the syllogistic method1--all these are happily unknown to most teachers and might profitably be unknown to all pupils4 ,the modern ,american textbook in geometry does not begin to be as good a piece of logic as ,euclid's ,elements1 and yet it is to be observed that none of these terms is found in this classic work1 so that they cannot be thought to be necessary to a logical treatment of the subject4 ,we need the word converse1 and some reference to the law of converse is therefore permissible2 the meaning of the reductio ad absurdum1 of a necessary and sufficient condition1 and of the terms synthesis and analysis may properly form part of the pupil's equipment because of their universal use2 but any extended incursion into the domain of logic will be found unprofitable1 and it is liable to be positively harmful to a beginner in geometry4 ,a word should be said as to the lettering of the figures in the early stages of geometry4 ,in general1 it is a great aid to the eye if this is carried out with some system1 and the following suggestions are given as in accord with the best authors who have given any attention to the subject3 #a4 ,in general1 letter a figure counterclockwise1 for the reason that we read angles in this way in higher mathematics1 and it is as easy to form this habit now as to form one that may have to be changed4 ,where two triangles are congruent1 however1 but have their sides arranged in opposite order1 it is better to letter them so that their corresponding parts appear in the same order1 although this makes one read clockwise4 ,illustration #b4 ,for the same reason1 read angles counterclockwise4 ,thus ,l,a is read ,,bac1 the reflex angle on the outside of the triangle being read ,,cab4 ,of course this is not vital1 and many authors pay no attention to it2 but it is convenient1 and if the teacher habitually does it1 the pupils will also tend to do it4 ,it is helpful in trigonometry1 and it saves confusion in the case of a reflex angle in a polygon4 ,designate an angle by a single letter if this can conveniently be done4 #c4 ,designate the sides opposite angles ,a1 ,b1 ,c1 in a triangle1 by a1 b1 c1 and use these letters in writing proofs4 #d4 ,in the case of two congruent triangles use the letters ,a1 ,b1 ,c and ,a'1 ,b'1 ,c'1 or ,x1 ,y1 ,z1 instead of letters chosen at random1 like ,d1 ,k1 ,l4 ,it is easier to follow a proof where some system is shown in lettering the figures4 ,some teachers insist that a pupil at the blackboard should not use the letters given in the textbook1 hoping thereby to avoid memorizing4 ,while the danger is probably exaggerated1 it is easy to change with some system1 using ,p1 ,q1 ,r and ,p'1 ,q'1 ,r'1 for example4 #e4 ,use small letters for lines1 as above stated1 and also place them within angles1 it being easier to speak of and to see ,lm than ,l,,def4 ,the ,germans have a convenient system that some ,american teachers follow to advantage1 but that a textbook has no right to require4 ,they use1 as in the following figure1 ,a for the point1 a for the opposite side1 and the ,greek letter alpha 7alpha7 for the angle4 ,the learning of the first three ,greek letters1 alpha 7alpha71 beta 7beta71 and gamma 7gamma71 is not a hardship1 and they are worth using1 although ,greek is so little known in this country to-day that the alphabet cannot be demanded of teachers who do not care to use it4 ,illustration #f4 ,also use small letters to represent numerical values4 ,for example1 write c #bpir instead of ,c #bpi,r4 ,this is in accord with the usage in algebra to which the pupil is accustomed4 #g4 ,use initial letters whenever convenient1 as in the case of a for area1 b for base1 c for circumference1 d for diameter1 h for height 7altitude71 and so on4 ,many of these suggestions seem of slight importance in themselves1 and some teachers will be disposed to object to any attempt at lettering a figure with any regard to system4 ,if1 however1 they will notice how a class struggles to follow a demonstration given with reference to a figure on the blackboard1 they will see how helpful it is to have some simple standards of lettering4 ,it is hardly necessary to add that in demonstrating from a figure on a blackboard it is usually better to say this line1 or the red line1 than to say1 without pointing to it1 the line ,,ab4 ,it is by such simplicity of statement and by such efforts to help the class to follow demonstrations that pupils are led through many of the initial discouragements of the subject4 ,,footnotes3 #dj ,carson1 loc4 cit41 p4 #ac4 ,,chapter ,x ,,the ,,conduct ,,of ,a ,,class ,,in ,,geometry ,no definite rules can be given for the detailed conduct of a class in any subject4 ,if it were possible to formulate such rules1 all the personal magnetism of the teacher1 all the enthusiasm1 all the originality1 all the spirit of the class1 would depart1 and we should have a dull1 dry mechanism4 ,there is no one best method of teaching geometry or anything else4 ,the experience of the schools has shown that a few great principles stand out as generally accepted1 but as to the carrying out of these principles there can be no definite rules4 ,let us first consider the general question of the employment of time in a recitation in geometry4 ,we might all agree on certain general principles1 and yet no two teachers ever would or ever should divide the period even approximately in the same way4 ,first1 a class should have an opportunity to ask questions4 ,a teacher here shows his power at its best1 listening sympathetically to any good question1 quickly seeing the essential point1 and either answering it or restating it in such a way that the pupil can answer it for himself4 ,certain questions should be answered by the teacher2 he is there for that purpose4 ,others can at once be put in such a light that the pupil can himself answer them4 ,others may better be answered by the class4 ,occasionally1 but more rarely1 a pupil may be told to look that up for to-morrow1 a plan that is commonly considered by students as a confession of weakness on the part of the teacher1 as it probably is4 ,of course a class will waste time in questioning a weak teacher1 but a strong one need have no fear on this account4 ,five minutes given at the opening of a recitation to brisk1 pointed questions by the class1 with the same credit given to a good question as to a good answer1 will do a great deal to create a spirit of comradeship1 of frankness1 and of honesty1 and will reveal to a sympathetic teacher the difficulties of a class much better than the same amount of time devoted to blackboard work4 ,but there must be no dawdling1 and the class must feel that it has only a limited time1 say five minutes at the most1 to get the help it needs4 ,next in order of the division of the time may be the teacher's report on any papers that the class has handed in4 ,it is impossible to tell how much of this paper work should be demanded4 ,the local school conditions1 the mental condition of the class1 and the time at the disposal of the teacher are all factors in the case4 ,in general1 it may be said that enough of this kind of work is necessary to see that pupils are neat and accurate in setting down their demonstrations4 ,on the other hand1 paper work gives an opportunity for dishonesty1 and it consumes a great deal of the teacher's time that might better be given to reading good books on the subject that he is teaching4 ,if1 however1 any papers have been submitted1 about five minutes may well be given to a rapid review of the failures and the successes4 ,in general1 it is good educational policy to speak of the errors and failures impersonally1 but occasionally to mention by name any one who has done a piece of work that is worthy of special comment4 ,pupils may better be praised in public and blamed in private4 ,there is such a thing1 however1 as praising too much1 when nothing worthy of note has been done1 just as there is danger of blaming too much1 resulting in mere nagging4 ,the third division of the recitation period may profitably go to assigning the advance lesson4 ,the class questions and the teacher's report on written work have shown the mental status of the pupils1 so that the teacher now knows what he may expect for the next lesson4 ,if he assigns his lesson at the beginning of the period1 he does not have this information4 ,if he waits to the end1 he may be too hurried to give any development that the new lesson may require4 ,there can be no rule as to how to assign a new lesson2 it all depends upon what the lesson is1 upon the mental state of the class1 and not a little upon the idiosyncrasy of the teacher4 ,the ,german educator1 ,herbart1 laid down certain formal steps in developing a new lesson1 and his successors have elaborated these somewhat as follows3 #a4 ,aim4 ,always take a class into your confidence4 ,tell the members at the outset the goal4 ,no one likes to be led blindfolded4 #b4 ,preparation4 ,a few brief questions to bring the class to think of what is to be considered4 #c4 ,presentation of the new4 ,preferably this is done by questions1 the answers leading the members of the class to discover the new truth for themselves4 #d4 ,apperception4 ,calling attention to the fact that this new fact was known before1 in part1 and that it relates to a number of things already in the mind4 ,the more the new can be tied up to the old the more tenaciously it will be held4 #e4 ,generalization and application4 ,it is evident at once that a great deal of time may be wasted in always following such a plan1 perhaps in ever following it consciously4 ,but1 on the other hand1 probably every good teacher1 whether he has heard of ,herbart or not1 naturally covers these points in substantially this order4 ,for an inexperienced teacher it is helpful to be familiar with them1 that he may call to mind the steps1 arranged in a psychological sequence1 that he would do well to follow4 ,it must always be remembered that there is quite as much danger in developing too much as in taking the opposite extreme4 ,a mechanical teacher may develop a new lesson where there is need for only a question or two or a mere suggestion4 ,it should also be recognized that students need to learn to read mathematics for themselves1 and that always to take away every difficulty by explanations given in advance is weakening to any one4 ,therefore1 in assigning the new lesson we may say ,take the next two pages1 and thus discourage most of the class4 ,on the other hand1 we may spend an unnecessary amount of time and overdevelop the work of those same pages1 and have the whole lesson lose all its zest4 ,it is here that the genius of the teacher comes forth to find the happy mean4 ,the fourth division of the hour should be reached1 in general1 in about ten minutes4 ,this includes the recitation proper4 ,but as to the nature of this work no definite instructions should be attempted4 ,to a good teacher they would be unnecessary1 to a poor one they would be harmful4 ,part of the class may go to the board1 and as they are working1 the rest may be reciting4 ,those at the board should be limited as to time1 for otherwise a premium is placed on mere dawdling4 ,they should be so arranged as to prevent copying1 but the teacher's eye is the best preventive of this annoying feature4 ,those at their seats may be called upon one at a time to demonstrate at the blackboard1 the rest being called upon for quick responses1 as occasion demands4 ,the ,european plan of having small blackboards is in many respects better than ours1 since pupils cannot so easily waste time4 ,they have to work rapidly and talk rapidly1 or else take their seats4 ,what should be put on the board1 whether the figure alone1 or the figure and the proof1 depends upon the proposition4 ,in general1 there should be a certain number of figures put on the board for the sake of rapid work and as a basis for the proofs of the day4 ,there should also be a certain amount of written work for the sake of commending or of criticizing adversely the proof used4 ,there are some figures that are so complicated as to warrant being put upon sheets of paper and hung before the class4 ,thus there is no rule upon the subject1 and the teacher must use his judgment according to the circumstances and the propositions4 ,if the early originals are one-step exercises1 and a pupil is required to recite rapidly1 a habit of quick expression is easily acquired that leads to close attention on the part of all the class4 ,students as a rule recite slower than they need to1 from mere habit4 ,phlegmatic as we think the ,german is1 and nervous as is the ,american temperament1 a student in geometry in a ,german school will usually recite more quickly and with more vigor than one with us4 ,our extensive blackboards have something to do with this1 allowing so many pupils to be working at the board that a teacher cannot attend to them all4 ,the result is a habit of wasting the minutes that can only be overcome by the teacher setting a definite but reasonable time limit1 and holding the pupil responsible if the work is not done in the time specified4 ,if this matter is taken in hand the first day1 and special effort made in the early weeks of the year1 much of the difficulty can be overcome4 ,as to the nature of the recitation to be expected from the pupil1 no definite rule can be laid down1 since it varies so much with the work of the day4 ,in general1 however1 a pupil should state the theorem quickly1 state exactly what is given and what is to be proved1 with respect to the figure1 and then give the proof4 ,at first it is desirable that he should give the authorities in full1 and later give only the essential part in a few words4 ,it is better to avoid the expression by previous proposition1 for it soon comes to be abused1 and of course the learning of section numbers in a book is a barbarism4 ,it is only by continually stating the propositions used that a pupil comes to have well fixed in his memory the basal theorems of geometry1 and without these he cannot make progress in his subsequent mathematics4 ,in general1 it is better to allow a pupil to finish his proof before asking him any questions1 the constant interruptions indulged in by some teachers being the cause of no little confusion and hesitancy on the part of pupils4 ,sometimes it is well to have a figure drawn differently from the one in the book1 or lettered differently1 so as to make sure that the pupil has not memorized the proof1 but in general such devices are unnecessary1 for a teacher can easily discover whether the proof is thoroughly understood1 either by the manner of the pupil or by some slight questioning4 ,a good textbook has the figures systematically lettered in some helpful way that is easily followed by the class that is listening to the recitation1 and it is not advisable to abandon this for a random set of letters arranged in no proper order4 ,it is good educational policy for the teacher to commend at least as often as he finds fault when criticizing a recitation at the blackboard and when discussing the pupils' papers4 ,optimism1 encouragement1 sympathy1 the genuine desire to help1 the putting of one's self in the pupil's place1 the doing to the pupil as the teacher would that he should do in return1--these are educational policies that make for better geometry as they make for better life4 ,the prime failure in teaching geometry lies unquestionably in the lack of interest on the part of the pupil1 and this has been brought about by the ancient plan of simply reading and memorizing proofs4 ,it is to get away from this that teachers resort to some such development of the lesson in advance1 as has been suggested above4 ,it is usually a good plan to give the easier propositions as exercises before they are reached in the text1 where this can be done4 ,an ,english writer has recently contributed this further idea3 ,it might be more stimulating to encourage investigation than to demand proofs of stated facts2 that is to say1 ,here is a figure drawn in this way1 find out anything you can about it4 ,some such exercises having been performed jointly by teachers and pupils1 the lust of investigation and healthy competition which is present in every normal boy or girl might be awakened so far as to make such little researches really attractive2 moreover1 the training thus given is of far more value than that obtained by proving facts which are stated in advance1 for it is seldom1 if ever1 that the problems of adult life present themselves in this manner4 ,the spirit of the question1 ,what is true8 is positive and constructive1 but that involved in ,is this true8 is negative and destructive4#da ,when the question is asked1 ,how shall ,i teach8 or ,what is the ,method8 there is no answer such as the questioner expects4 ,a ,japanese writer1 ,motowori1 a great authority upon the ,shinto faith of his people1 once wrote these words3 ,to have learned that there is no way to be learned and practiced is really to have learned the way of the gods4 ,,footnotes3 #da ,carson1 loc4 cit41 p4 #ab4 ,,chapter ,,xi ,,the ,,axioms ,,and ,,postulates ,the interest as well as the value of geometry lies chiefly in the fact that from a small number of assumptions it is possible to deduce an unlimited number of conclusions4 ,with the truth of these assumptions we are not so much concerned as with the reasoning by which we draw the conclusions1 although it is manifestly desirable that the assumptions should not be false1 and that they should be as few as possible4 ,it would be natural1 and in some respects desirable1 to call these foundations of geometry by the name assumptions1 since they are simply statements that are assumed to be true4 ,the real foundation principles cannot be proved2 they are the means by which we prove other statements4 ,but as with most names of men or things1 they have received certain titles that are time-honored1 and that it is not worth the while to attempt to change4 ,in ,english we call them axioms and postulates1 and there is no more reason for attempting to change these terms than there is for attempting to change the names of geometry#db and of algebra4#dc ,since these terms are likely to continue1 it is necessary to distinguish between them more carefully than is often done1 and to consider what assumptions we are justified in including under each4 ,in the first place1 these names do not go back to ,euclid1 as is ordinarily supposed1 although the ideas and the statements are his4 ,postulate is a ,latin form of the ,greek ,greek3 aitema 7aitema71 and appears only in late translations4 ,euclid stated in substance1 ,let the following be assumed4 ,axiom 7,greek3 axioma1 axioma7 dates perhaps only from ,proclus 7fifth century ,a4,d471 ,euclid using the words common notions 7,greek3 koinai ennoiai1 koinai ennoiai7 for axioms1 as ,aristotle before him had used common things1 common principles1 and common opinions4 ,the distinction between axiom and postulate was not clearly made by ancient writers4 ,probably what was in ,euclid's mind was the ,aristotelian distinction that an axiom was a principle common to all sciences1 self-evident but incapable of proof1 while the postulates were the assumptions necessary for building up the particular science under consideration1 in this case geometry4#dd ,we thus come to the modern distinction between axiom and postulate1 and say that a general statement admitted to be true without proof is an axiom1 while a postulate in geometry is a geometric statement admitted to be true1 without proof4 ,for example1 when we say ,if equals are added to equals1 the sums are equal1 we state an assumption that is taken also as true in arithmetic1 in algebra1 and in elementary mathematics in general4 ,this is therefore an axiom4 ,at one time such a statement was defined as a self-evident truth1 but this has in recent years been abandoned1 since what is evident to one person is not necessarily evident to another1 and since all such statements are mere matters of assumption in any case4 ,on the other hand1 when we say1 ,a circle may be described with any given point as a center and any given line as a radius1 we state a special assumption of geometry1 and this assumption is therefore a geometric postulate4 ,some few writers have sought to distinguish between axiom and postulate by saying that the former was an assumed theorem and the latter an assumed problem1 but there is no standard authority for such a distinction1 and indeed the difference between a theorem and a problem is very slight4 ,if we say1 ,a circle may be passed through three points not in the same straight line1 we state a theorem2 but if we say1 ,required to pass a circle through three points1 we state a problem4 ,the mental process of handling the two propositions is1 however1 practically the same in spite of the minor detail of wording4 ,so with the statement1 ,a straight line may be produced to any required length4 ,this is stated in the form of a theorem1 but it might equally well be stated thus3 ,to produce a straight line to any required length4 ,it is unreasonable to call this an axiom in one case and a postulate in the other4 ,however stated1 it is a geometric postulate and should be so classed4 ,what1 now1 are the axioms and postulates that we are justified in assuming1 and what determines their number and character8 ,it seems reasonable to agree that they should be as few as possible1 and that for educational purposes they should be so clear as to be intelligible to beginners4 ,but here we encounter two conflicting ideas4 ,to get the irreducible minimum of assumptions is to get a set of statements quite unintelligible to students beginning geometry or any other branch of elementary mathematics4 ,such an effort is laudable when the results are intended for advanced students in the university1 but it is merely suggestive to teachers rather than usable with pupils when it touches upon the primary steps of any science4 ,in recent years several such attempts have been made4 ,in particular1 ,professor ,hilbert has given a system#de of congruence postulates1 but they are rather for the scientist than for the student of elementary geometry4 ,in view of these efforts it is well to go back to ,euclid and see what this great teacher of university men#df had to suggest4 ,the following are the five common notions that ,euclid deemed sufficient for the purposes of elementary geometry4 #a4 ,things equal to the same thing are also equal to each other4 ,this axiom has persisted in all elementary textbooks4 ,of course it is a simple matter to attempt criticism1--to say that -#b is the square root of #d1 and #b is also the square root of #d1 whence -#b #b2 but it is evident that the argument is not sound1 and that it does not invalidate the axiom4 ,proclus tells us that ,apollonius attempted to prove the axiom by saying1 ,let a equal b1 and b equal c4 ,i say that a equals c4 ,for1 since a equals b1 a occupies the same space as b4 ,therefore a occupies the same space as c4 ,therefore a equals c4 ,the proof is of no value1 however1 save as a curiosity4 #b4 ,and if to equals equals are added1 the wholes are equal4 #c4 ,if equals are subtracted from equals1 the remainders are equal4 ,axioms #b and #c are older than ,euclid's time1 and are the only ones given by him relating to the solution of the equation4 ,certain other axioms were added by later writers1 as1 ,things which are double of the same thing are equal to one another1 and ,things which are halves of the same thing are equal to one another4 ,these two illustrate the ancient use of duplatio 7doubling7 and mediatio 7halving71 the primitive forms of multiplication and division4 ,euclid would not admit the multiplication axiom1 since to him this meant merely repeated addition4 ,the partition 7halving7 axiom he did not need1 and if needed1 he would have inferred its truth4 ,there are also the axioms1 ,if equals are added to unequals1 the wholes are unequal1 and ,if equals are subtracted from unequals1 the remainders are unequal1 neither of which ,euclid would have used because he did not define unequals4 ,the modern arrangement of axioms1 covering addition1 subtraction1 multiplication1 division1 powers1 and roots1 sometimes of unequals as well as equals1 comes from the development of algebra4 ,they are not all needed for geometry1 but in so far as they show the relation of arithmetic1 algebra1 and geometry1 they serve a useful purpose4 ,there are also other axioms concerning unequals that are of advantage to beginners1 even though unnecessary from the standpoint of strict logic4 #d4 ,things that coincide with one another are equal to one another4 ,this is no longer included in the list of axioms4 ,it is rather a definition of equal1 or of congruent1 to take the modern term4 ,if not a definition1 it is certainly a postulate rather than an axiom1 being purely geometric in character4 ,it is probable that ,euclid included it to show that superposition is to be considered a legitimate form of proof1 but why it was not placed among the postulates is not easily seen4 ,at any rate it is unfortunately worded1 and modern writers generally insert the postulate of motion instead1--that a figure may be moved about in space without altering its size or shape4 ,the ,german philosopher1 ,schopenhauer 7#ahdd71 criticized ,euclid's axiom as follows3 ,coincidence is either mere tautology or something entirely empirical1 which belongs not to pure intuition but to external sensuous experience4 ,it presupposes1 in fact1 the mobility of figures4 #e4 ,the whole is greater than the part4 ,to this ,clavius 7#aegd7 added1 ,the whole is equal to the sum of its parts1 which may be taken to be a definition of whole1 but which is helpful to beginners1 even if not logically necessary4 ,some writers doubt the genuineness of this axiom4 ,having considered the axioms of ,euclid1 we shall now consider the axioms that are needed in the study of elementary geometry4 ,the following are suggested1 not from the standpoint of pure logic1 but from that of the needs of the teacher and pupil4 #a4 ,if equals are added to equals1 the sums are equal4 ,instead of this axiom1 the one numbered #h below is often given first4 ,for convenience in memorizing1 however1 it is better to give the axioms in the following order3 7#a7 addition1 7#b7 subtraction1 7#c7 multiplication1 7#d7 division1 7#e7 powers and roots1--all of equal quantities4 #b4 ,if equals are subtracted from equals1 the remainders are equal4 #c4 ,if equals are multiplied by equals1 the products are equal4 #d4 ,if equals are divided by equals1 the quotients are equal4 #e4 ,like powers or like positive roots of equals are equal4 ,formerly students of geometry knew nothing of algebra1 and in particular nothing of negative quantities4 ,now1 however1 in ,american schools a pupil usually studies algebra a year before he studies demonstrative geometry4 ,it is therefore better1 in speaking of roots1 to limit them to positive numbers1 since the two square roots of #d 7#b and -#b71 for example1 are not equal4 ,if the pupil had studied complex numbers before he began geometry1 it would have been advisable to limit the roots still further to real roots1 since the four fourth roots of #a 7#a1 -#a1 sqrt7-#a71 -sqrt7-#a771 for example1 are not equal save in absolute value4 ,it is well1 however1 to eliminate these fine distinctions as far as possible1 since their presence only clouds the vision of the beginner4 ,it should also be noted that these five axioms might be combined in one1 namely1 ,if equals are operated on by equals in the same way1 the results are equal4 ,in ,axiom #a this operation is addition1 in ,axiom #b it is subtraction1 and so on4 ,indeed1 in order to reduce the number of axioms two are already combined in ,axiom #e4 ,but there is a good reason for not combining the first four with the fifth1 and there is also a good reason for combining two in ,axiom #e4 ,the reason is that these are the axioms continually used in equations1 and to combine them all in one would be to encourage laxness of thought on the part of the pupil4 ,he would always say by ,axiom #a whatever he did to an equation1 and the teacher would not be certain whether the pupil was thinking definitely of dividing equals by equals1 or had a hazy idea that he was manipulating an equation in some other way that led to an answer4 ,on the other hand1 ,axiom #e is not used as often as the preceding four1 and the interchange of integral and fractional exponents is relatively common1 so that the joining of these two axioms in one for the purpose of reducing the total number is justifiable4 #f4 ,if unequals are operated on by positive equals in the same way1 the results are unequal in the same order4 ,this includes in a single statement the six operations mentioned in the preceding axioms2 that is1 if a b and if x y1 then a x b y1 a - x b - y1 ax by1 etc4 ,the reason for thus combining six axioms in one in the case of inequalities is apparent4 ,they are rarely used in geometry1 and if a teacher is in doubt as to the pupil's knowledge1 he can easily inquire in the few cases that arise1 whereas it would consume a great deal of time to do this for the many equations that are met4 ,the axiom is stated in such a way as to exclude multiplying or dividing by negative numbers1 this case not being needed4 #g4 ,if unequals are added to unequals in the same order1 the sums are unequal in the same order2 if unequals are subtracted from equals1 the remainders are unequal in the reverse order4 ,these are the only cases in which unequals are necessarily combined with unequals1 or operate upon equals in geometry1 and the axiom is easily explained to the class by the use of numbers4 #h4 ,quantities that are equal to the same quantity or to equal quantities are equal to each other4 ,in this axiom the word quantity is used1 in the common manner of the present time1 to include number and all geometric magnitudes 7length1 area1 volume74 #i4 ,a quantity may be substituted for its equal in an equation or in an inequality4 ,this axiom is tacitly assumed by all writers1 and is very useful in the proofs of geometry4 ,it is really the basis of several other axioms1 and if we were seeking the irreducible minimum1 it would replace them4 ,since1 however1 we are seeking only a reasonably abridged list of convenient assumptions that beginners will understand and use1 this axiom has much to commend it4 ,if we consider the equations 7#a7 a x and 7#b7 b x1 we see that for x in equation 7#a7 we may substitute b from equation 7#b7 and have a b2 in other words1 that ,axiom #h is included in ,axiom #i4 ,furthermore1 if 7#a7 a b and 7#b7 x y1 then since a x is the same as a x1 we may1 by substituting1 say that a x a x b x b y4 ,in other words1 ,axiom #a is included in ,axiom #i4 ,thus an axiom that includes others has a legitimate place1 because a beginner would be too much confused by seeing its entire scope1 and because he will make frequent use of it in his mathematical work4 #aj4 ,if the first of three quantities is greater than the second1 and the second is greater than the third1 then the first is greater than the third4 ,this axiom is needed several times in geometry4 ,the case in which a b and b c1 therefore a c1 is provided for in ,axiom #i4 #aa4 ,the whole is greater than any of its parts and is equal to the sum of all its parts4 ,the latter part of this axiom is really only the definition of whole1 and it would be legitimate to state a definition accordingly and refer to it where the word is employed4 ,where1 however1 we wish to speak of a polygon1 for example1 and wish to say that the area is equal to the combined areas of the triangles composing it1 it is more satisfactory to have this axiom to which to refer4 ,it will be noticed that two related axioms are here combined in one1 for a reason similar to the one stated under ,axiom #e4 ,in the case of the postulates we are met by a problem similar to the one confronting us in connection with the axioms1--the problem of the irreducible minimum as related to the question of teaching4 ,manifestly ,euclid used postulates that he did not state1 and proved some statements that he might have postulated4#dg ,the postulates given by ,euclid under the name ,greek3 aitemata7aitemata7 were requests made by the teacher to his pupil that certain things be conceded4 ,they were five in number1 as follows3 #a4 ,let the following be conceded3 to draw a straight line from any point to any point4 ,strictly speaking1 ,euclid might have been required to postulate that points and straight lines exist1 but he evidently considered this statement sufficient4 ,aristotle had1 however1 already called attention to the fact that a mere definition was sufficient only to show what a concept is1 and that this must be followed by a proof that the thing exists4 ,we might1 for example1 define x as a line that bisects an angle without meeting the vertex1 but this would not show that an x exists1 and indeed it does not exist4 ,euclid evidently intended the postulate to assert that this line joining two points is unique1 which is only another way of saying that two points determine a straight line1 and really includes the idea that two straight lines cannot inclose space4 ,for purposes of instruction1 the postulate would be clearer if it read1 ,one straight line1 and only one1 can be drawn through two given points4 #b4 ,to produce a finite straight line continuously in a straight line4 ,in this postulate ,euclid practically assumes that a straight line can be produced only in a straight line2 in other words1 that two different straight lines cannot have a common segment4 ,several attempts have been made to prove this fact1 but without any marked success4 #c4 ,to describe a circle with any center and radius4 #d4 ,that all right angles are equal to one another4 ,while this postulate asserts the essential truth that a right angle is a determinate magnitude so that it really serves as an invariable standard by which other 7acute and obtuse7 angles may be measured1 much more than this is implied1 as will easily be seen from the following consideration4 ,if the statement is to be proved1 it can only be proved by the method of applying one pair of right angles to another and so arguing their equality4 ,but this method would not be valid unless on the assumption of the invariability of figures1 which would have to be asserted as an antecedent postulate4 ,euclid preferred to assert as a postulate1 directly1 the fact that all right angles are equal2 and hence his postulate must be taken as equivalent to the principle of invariability of figures1 or its equivalent1 the homogeneity of space4#dh ,it is better educational policy1 however1 to assert this fact more definitely1 and to state the additional assumption that figures may be moved about in space without deformation4 ,the fourth of ,euclid's postulates is often given as an axiom1 following the idea of the ,greek philosopher ,geminus 7who flourished in the first century ,b4,c471 but this is because ,euclid's distinction between axiom and postulate is not always understood4 ,proclus 7#daj-dhe ,a4,d47 endeavored to prove the postulate1 and a later and more scientific effort was made by the ,italian geometrician ,saccheri 7#affg-agcc74 ,it is very commonly taken as a postulate that all straight angles are equal1 this being more evident to the senses1 and the equality of right angles is deduced as a corollary4 ,this method of procedure has the sanction of many of our best modern scholars4 #e4 ,that1 if a straight line falling on two straight lines make the interior angle on the same side less than two right angles1 the two straight lines1 if produced indefinitely1 meet on that side on which are the angles less than the two right angles4 ,this famous postulate1 long since abandoned in teaching the beginner in geometry1 is a remarkable evidence of the clear vision of ,euclid4 ,for two thousand years mathematicians sought to prove it1 only to demonstrate the wisdom of its author in placing it among the assumptions4#di ,every proof adduced contains some assumption that practically conceals the postulate itself4 ,thus the great ,english mathematician ,john ,wallis 7#afaf-agjc7 gave a proof based upon the assumption that given a figure1 another figure is possible which is similar to the given one1 and of any size whatever4 ,legendre 7#ageb-ahcc7 did substantially the same at one time1 and offered several other proofs1 each depending upon some equally unprovable assumption4 ,the definite proof that the postulate cannot be demonstrated is due to the ,italian ,beltrami 7#ahfh74 ,of the alternative forms of the postulate1 that of ,proclus is generally considered the best suited to beginners4 ,as stated by ,playfair 7#agie71 this is1 ,through a given point only one parallel can be drawn to a given straight line2 and as stated by ,proclus1 ,if a straight line intersect one of two parallels1 it will intersect the other also4 ,playfair's form is now the common postulate of parallels1 and is the one that seems destined to endure4 ,posidonius and ,geminus1 both ,stoics of the first century ,b4,c41 gave as their alternative1 ,there exist straight lines everywhere equidistant from one another4 ,one of ,legendre's alternatives is1 ,there exists a triangle in which the sum of the three angles is equal to two right angles4 ,one of the latest attempts to suggest a substitute is that of the ,italian ,ingrami 7#aijd71 ,two parallel straight lines intercept1 on every transversal which passes through the mid-point of a segment included between them1 another segment the mid-point of which is the mid-point of the first4 ,of course it is entirely possible to assume that through a point more than one line can be drawn parallel to a given straight line1 in which case another type of geometry can be built up1 equally rigorous with ,euclid's4 ,this was done at the close of the first quarter of the nineteenth century by ,lobachevsky 7#agic-ahef7 and ,bolyai 7#ahjb-ahfj71 resulting in the first of several non-,euclidean geometries4#ej ,taking the problem to be that of stating a reasonably small number of geometric assumptions that may form a basis to supplement the general axioms1 that shall cover the most important matters to which the student must refer1 and that shall be so simple as easily to be understood by a beginner1 the following are recommended3 #a4 ,one straight line1 and only one1 can be drawn through two given points4 ,this should also be stated for convenience in the form1 ,two points determine a straight line4 ,from it may also be drawn this corollary1 ,two straight lines can intersect in only one point1 since two points would determine a straight line4 ,such obvious restatements of or corollaries to a postulate are to be commended1 since a beginner is often discouraged by having to prove what is so obvious that a demonstration fails to commend itself to his mind4 #b4 ,a straight line may be produced to any required length4 ,this1 like ,postulate #a1 requires the use of a straightedge for drawing the physical figure4 ,the required length is attained by using the compasses to measure the distance4 ,the straightedge and the compasses are the only two drawing instruments recognized in elementary geometry4#ea ,while this involves more than ,euclid's postulate1 it is a better working assumption for beginners4 #c4 ,a straight line is the shortest path between two points4 ,this is easily proved by the method of ,euclid#eb for the case where the paths are broken lines1 but it is needed as a postulate for the case of curve paths4 ,it is a better statement than the common one that a straight line is the shortest distance between two points2 for distance is measured on a line1 but it is not itself a line4 ,furthermore1 there are scientific objections to using the word distance any more than is necessary4 #d4 ,a circle may be described with any given point as a center and any given line as a radius4 ,this involves the use of the second of the two geometric instruments1 the compasses4 #e4 ,any figure may be moved from one place to another without altering the size or shape4 ,this is the postulate of the homogeneity of space1 and asserts that space is such that we may move a figure as we please without deformation of any kind4 ,it is the basis of all cases of superposition4 #f4 ,all straight angles are equal4 ,it is possible to prove this1 and therefore1 from the standpoint of strict logic1 it is unnecessary as a postulate4 ,on the other hand1 it is poor educational policy for a beginner to attempt to prove a thing that is so obvious4 ,the attempt leads to a loss of interest in the subject1 the proposition being 7to state a paradox7 hard because it is so easy4 ,it is1 of course1 possible to postulate that straight angles are equal1 and to draw the conclusion that their halves 7right angles7 are equal2 or to proceed in the opposite direction1 and postulate that all right angles are equal1 and draw the conclusion that their doubles 7straight angles7 are equal4 ,of the two the former has the advantage1 since it is probably more obvious that all straight angles are equal4 ,it is well to state the following definite corollaries to this postulate3 7#a7 ,all right angles are equal2 7#b7 ,from a point in a line only one perpendicular can be drawn to the line1 since two perpendiculars would make the whole 7right angle7 equal to its part2 7#c7 ,equal angles have equal complements1 equal supplements1 and equal conjugates2 7#d7 ,the greater of two angles has the less complement1 the less supplement1 and the less conjugate4 ,all of these four might appear as propositions1 but1 as already stated1 they are so obvious as to be more harmful than useful to beginners when given in such form4 ,the postulate of parallels may properly appear in connection with that topic in ,book ,i1 and it is accordingly treated in ,chapter ,,xiv4 ,there is also another assumption that some writers are now trying to formulate in a simple fashion4 ,we take1 for example1 a line segment ,,ab1 and describe circles with ,a and ,b respectively as centers1 and with a radius ,,ab4 ,we say that the circles will intersect as at ,c and ,d4 ,but how do we know that they intersect8 ,we assume it1 just as we assume that an indefinite straight line drawn from a point inclosed by a circle will1 if produced far enough1 cut the circle twice4 ,of course a pupil would not think of this if his attention was not called to it1 and the harm outweighs the good in doing this with one who is beginning the study of geometry4 ,with axioms and with postulates1 therefore1 the conclusion is the same3 from the standpoint of scientific geometry there is an irreducible minimum of assumptions1 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 beginner4 ,bibliography4 ,smith1 ,teaching of ,elementary ,mathematics1 ,new ,york1 #aijj2 ,young1 ,the ,teaching of ,mathematics1 ,new ,york1 #aija2 ,moore1 ,on the ,foundations of ,mathematics1 ,bulletin of the ,american ,mathematical ,society1 #aijc1 p4 #djb2 ,betz1 ,intuition and ,logic in ,geometry1 ,the ,mathematics ,teacher1 ,vol4 ,,ii1 p4 #c2 ,hilbert1 ,the ,foundations of ,geometry1 ,chicago1 #aijb2 ,veblen1 ,a ,system of ,axioms for ,geometry1 ,transactions of the ,american ,mathematical ,society1 #aijd1 p4 #cdc4 ,,footnotes3 #db ,from the ,greek ,greek3 ge1 ge 7earth71 ,greek3 metrein1 metrein 7to measure71 although the science has not had to do directly with the measure of the earth for over two thousand years4 #dc ,from the ,arabic al 7the7 jabr 7restoration71 referring to taking a quantity from one side of an equation and then restoring the balance by taking it from the other side 7see page #cg74 #dd ,one of the clearest discussions of the subject is in ,w4 ,b4 ,frankland1 ,the ,first ,book of ,euclid's ',elements1' p4 #bf1 ,cambridge1 #aije4 #de ,grundlagen der ,geometrie1 ,leipzig1 #ahii4 ,see ,heath's ,euclid1 ,vol4 ,i1 p4 #bbi1 for an ,english version2 also ,d4 ,e4 ,smith1 ,teaching of ,elementary ,mathematics1 p4 #bff1 ,new ,york1 #aijj4 #df ,we need frequently to recall the fact that ,euclid's ,elements was intended for advanced students who went to ,alexandria as young men now go to college1 and that the book was used only in university instruction in the ,middle ,ages and indeed until recent times4 #dg ,for example1 he moves figures without deformation1 but states no postulate on the subject2 and he proves that one side of a triangle is less than the sum of the other two sides1 when he might have postulated that a straight line is the shortest path between two points4 ,indeed1 his followers were laughed at for proving a fact so obvious as this one concerning the triangle4 #dh ,t4 ,l4 ,heath1 ,euclid1 ,vol4 ,i1 p4 #bjj4 #di ,for a resume of the best known attempts to prove this postulate1 see ,heath1 ,euclid1 ,vol4 ,i1 p4 #bjb2 ,w4 ,b4 ,frankland1 ,theories of ,parallelism1 ,cambridge1 #aiaj4 #ej ,for the early history of this movement see ,engel and ,staeckel1 ,die ,theorie der ,parallellinien von ,euklid bis auf ,gauss1 ,leipzig1 #ahie2 ,bonola1 ,sulla teoria delle parallele e sulle geometrie non-euclidee1 in his ,questioni riguardanti la geometria elementare1 #aijj2 ,karagiannides1 ,die nichteuklidische ,geometrie vom ,alterthum bis zur ,gegenwart1 ,berlin1 #ahic4 #ea ,this limitation upon elementary geometry was placed by ,plato 7died #cdg ,b4,c471 as already stated4 #eb ,book ,i1 ,proposition #bj4 ,,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 definitions than is the case in most of the other sciences4 ,the essence of all geometry worthy of serious study is not the knowledge of some fact1 but the proof of that fact2 and this proof is always based upon preceding proofs1 assumptions 7axioms or postulates71 or definitions4 ,if we are to prove that one line is perpendicular to another1 it is essential that we have an exact definition of perpendicular1 else we shall not know when we have reached the conclusion of the proof4 ,the essential features of a definition are that the term defined shall be described in terms that are simpler than1 or at least better known than1 the thing itself2 that this shall be done in such a way as to limit the term to the thing defined2 and that the description shall not be redundant4 ,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 angle1 because the concept perigon is not so simple1 and the term perigon is not so well known1 as the term and the concept right angle1 and because the definition is redundant1 containing more than is necessary4 ,it is evident that satisfactory definitions are not always possible2 for since the number of terms is limited1 there must be at least one that is at least as simple as any other1 and this cannot be described in terms simpler than itself4 ,such1 for example1 is the term angle4 ,we can easily explain the meaning of this word1 and we can make the concept clear1 but this must be done by a certain amount of circumlocution and explanation1 not by a concise and perfect definition4 ,unless a beginner in geometry knows what an angle is before he reads the definition in a textbook1 he will not know from the definition4 ,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 direction4 ,it should also be understood in this connection that a definition makes no assertion as to the existence of the thing defined4 ,if we say that a tangent to a circle is an unlimited straight line that touches the circle in one point1 and only one1 we do not assert that it is possible to have such a line2 that is a matter for proof4 ,not in all cases1 however1 can this proof be given1 as in the existence of the simplest concepts4 ,we cannot1 for example1 prove that a point or a straight line exists after we have defined these concepts4 ,we therefore tacitly or explicitly assume 7postulate7 the existence of these fundamentals of geometry4 ,on the other hand1 we can prove that a tangent exists1 and this may properly be considered a legitimate proposition or corollary of elementary geometry4 ,in relation to geometric proof it is necessary to bear in mind1 therefore1 that we are permitted to define any term we please2 for example1 a seven-edged polyhedron or ,leibnitz's ten-faced regular polyhedron1 neither of which exists2 but1 strictly speaking1 we have no right to make use of a definition in a proof until we have shown or postulated that the thing defined has an existence4 ,this is one of the strong features of ,euclid's textbook4 ,not being able to prove that a point1 a straight line1 and a circle exists1 he practically postulates these facts2 but he uses no other definition in a proof without showing that the thing defined exists1 and this is his reason for mingling his problems with his theorems4 ,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 logic4 ,it was pointed out by ,aristotle1 long before ,euclid1 that it is not a satisfactory procedure to define a thing by means of terms that are strictly not prior to it1 as when we attempt to define something by means of its opposite4 ,thus to define a curve as a line1 no part of which is straight1 would be a bad definition unless straight had already been explicitly defined2 and to define bad as not good is unsatisfactory for the reason that bad and good are concepts that are evolved simultaneously4 ,but all this is only a detail under the general principle that a definition must employ terms that are better understood than the one defined4 ,it should be understood that some definitions are much more important than others1 considered from the point of view of the logic of geometry4 ,those that enter into geometric proofs are basal2 those that form part of the conversational language of geometry are not4 ,euclid gave twenty-three definitions in ,book ,i1 and did not make use of even all of these terms4 ,other terms1 those not employed in his proofs1 he assumed to be known1 just as he assumed a knowledge of any other words in his language4 ,such procedure would not be satisfactory under modern conditions1 but it is of great importance that the teacher should recognize that certain definitions are basal1 while others are merely informational4 ,it is now proposed to consider the basal definitions of geometry1 first1 that the teacher may know what ones are to be emphasized and learned2 and second1 that he may know that the idea that the standard definitions can easily be improved is incorrect4 ,it is hoped that the result will be the bringing into prominence of the basal concepts1 and the discouraging of attempts to change in unimportant respects the definitions in the textbook used by the pupil4 ,in order to have a systematic basis for work1 the definitions of two books of ,euclid will first be considered4#ec #a4 ,,point4 ,a point is that which has no part4 ,this was incorrectly translated by ,capella in the fifth century1 ,punctum est cuius pars nihil est 7a point is that of which a part is nothing71 which is as much as to say that the point itself is nothing4 ,it generally appears1 however1 as in the ,campanus edition1#ed ,punctus est cuius pars non est1 which is substantially ,euclid's wording4 ,aristotle tells of the definitions of point1 line1 and surface that prevailed in his time1 saying that they all defined the prior by means of the posterior4#ee ,thus a point was defined as an extremity of a line1 a line as the extremity of a surface1 and a surface as the extremity of a solid1--definitions still in use and not without their value4 ,for it must not be assumed that scientific priority is necessarily priority in fact2 a child knows of solid before he knows of point1 so that it may be a very good way to explain1 if not to define1 by beginning with solid1 passing thence to surface1 thence to line1 and thence to point4 ,the first definition of point of which ,proclus could learn is attributed by him to the ,pythagoreans1 namely1 a monad having position1 the early form of our present popular definition of a point as position without magnitude4 ,plato defined it as the beginning of a line1 thus presupposing the definition of line2 and1 strangely enough1 he anticipated by two thousand years ,cavalieri1 the ,italian geometer1 by speaking of points as indivisible lines4 ,to ,aristotle1 who protested against ,plato's definitions1 is due the definition of a point as something indivisible but having position4 ,euclid's definition is essentially that of ,aristotle1 and is followed by most modern textbook writers1 except as to its omission of the reference to position4 ,it has been criticized as being negative1 which has no part2 but it is generally admitted that a negative definition is admissible in the case of the most elementary concepts4 ,for example1 blind must be defined in terms of a negation4 ,at present not much attention is given to the definition of point1 since the term is not used as the basis of a proof1 but every effort is made to have the concept clear4 ,it is the custom to start from a small solid1 conceive it to decrease in size1 and think of the point as the limit to which it is approaching1 using these terms in their usual sense without further explanation4 #b4 ,,line4 ,a line is breadthless length4 ,this is usually modified in modern textbooks by saying that a line is that which has length without breadth or thickness1 a statement that is better understood by beginners4 ,euclid's definition is thought to be due to ,plato1 and is only one of many definitions that have been suggested4 ,the ,pythagoreans having spoken of the point as a monad naturally were led to speak of the line as dyadic1 or related to two4 ,proclus speaks of another definition1 magnitude in one dimension1 and he gives an excellent illustration of line as the edge of a shadow1 thus making it real but not material4 ,aristotle speaks of a line as a magnitude divisible in one way only1 as contrasted with a surface which is divisible in two ways1 and with a solid which is divisible in three ways4 ,proclus also gives another definition as the flux of a point1 which is sometimes rendered as the path of a moving point4 ,aristotle had suggested the idea when he wrote1 ,they say that a line by its motion produces a surface1 and a point by its motion a line4 ,euclid did not deem it necessary to attempt a classification of lines1 contenting himself with defining only a straight line and a circle1 and these are really the only lines needed in elementary geometry4 ,his commentators1 however1 made the attempt4 ,for example4 ,heron 7first century ,a4,d47 probably followed his definition of line by this classification3 ,straight ,lines ,circular circumferences ,not straight ,spiral shaped ,curved 7generally7 ,proclus relates that both ,plato and ,aristotle divided lines into straight1 circular1 and a mixture of the two1 a statement which is not quite exact1 but which shows the origin of a classification not infrequently found in recent textbooks4 ,geminus 7ca4 #ej ,b4,c47 is said by ,proclus to have given two classifications1 of which one will suffice for our purposes3 ,composite 7broken line forming an angle7 ,lines ,forming a figure1 or determinate4 7,circle1 ellipse1 cissoid47 ,incomposite ,not forming a figure1 or indeterminate and extending without a limit4 7,straight line1 parabola1 hyperbola1 conchoid47 ,of course his view of the cissoid1 the curve represented by the equation y#b7;a x7 7a - x7#c1 is not the modern view4 #c4 ,the extremities of a line are points4 ,this is not a definition in the sense of its two predecessors4 ,a modern writer would put it as a note under the definition of line4 ,euclid did not wish to define a point as the extremity of a line1 for ,aristotle had asserted that this was not scientific2 so he defined point and line1 and then added this statement to show the relation of one to the other4 ,aristotle had improved upon this by stating that the division of a line1 as well as an extremity1 is a point1 as is also the intersection of two lines4 ,these statements1 if they had been made by ,euclid1 would have avoided the objection made by ,proclus1 that some lines have no extremities1 as1 for example1 a circle1 and also a straight line extending infinitely in both directions4 #d4 ,,straight ,,line4 ,a straight line is that which lies evenly with respect to the points on itself4 ,this is the least satisfactory of all of the definitions of ,euclid1 and emphasizes the fact that the straight line is the most difficult to define of the elementary concepts of geometry4 ,what is meant by lies evenly8 ,who would know what a straight line is1 from this definition1 if he did not know in advance8 ,the ancients suggested many definitions of straight line1 and it is well to consider a few in order to appreciate the difficulties involved4 ,plato spoke of it as that of which the middle covers the ends1 meaning that if looked at endways1 the middle would make it impossible to see the remote end4 ,this is often modified to read that a straight line when looked at endways appears as a point1--an idea that involves the postulate that our line of sight is straight4 ,archimedes made the statement that of all the lines which have the same extremities1 the straight line is the least1 and this has been modified by later writers into the statement that a straight line is the shortest distance between two points4 ,this is open to two objections as a definition3 7#a7 a line is not distance1 but distance is the length of a line1--it is measured on a line2 7#b7 it is merely stating a property of a straight line to say that a straight line is the shortest path between two points1--a proper postulate but not a good definition4 ,equally objectionable is one of the definitions suggested by both ,heron and ,proclus1 that a straight line is a line that is stretched to its uttermost2 for even then it is reasonable to think of it as a catenary1 although ,proclus doubtless had in mind the ,archimedes statement4 ,he also stated that a straight line is a line such that if any part of it is in a plane1 the whole of it is in the plane1--a definition that runs in a circle1 since plane is defined by means of straight line4 ,proclus also defines it as a uniform line1 capable of sliding along itself1 but this is also true of a circle4 ,of the various definitions two of the best go back to ,heron1 about the beginning of our era4 ,proclus gives one of them in this form1 ,that line which1 when its ends remain fixed1 itself remains fixed4 ,heron proposed to add1 when it is1 as it were1 turned round in the same plane4 ,this has been modified into that which does not change its position when it is turned about its extremities as poles1 and appears in substantially this form in the works of ,leibnitz and ,gauss4 ,the definition of a straight line as such a line as1 with another straight line1 does not inclose space1 is only a modification of this one4 ,the other definition of ,heron states that in a straight line all its parts fit on all in all ways1 and this in its modern form is perhaps the most satisfactory of all4 ,in this modern form it may be stated1 ,a line such that any part1 placed with its ends on any other part1 must lie wholly in the line1 is called a straight line1 in which the force of the word must should be noted4 ,this whole historical discussion goes to show how futile it is to attempt to define a straight line4 ,what is needed is that we should explain what is meant by a straight line1 that we should illustrate it1 and that pupils should then read the definition understandingly4 #e4 ,,surface4 ,a surface is that which has length and breadth4 ,this is substantially the common definition of our modern textbooks4 ,as with line1 so with surface1 the definition is not entirely satisfactory1 and the chief consideration is that the meaning of the term should be made clear by explanations and illustrations4 ,the shadow cast on a table top is a good illustration1 since all idea of thickness is wanting4 ,it adds to the understanding of the concept to introduce ,aristotle's statement that a surface is generated by a moving line1 modified by saying that it may be so generated1 since the line might slide along its own trace1 or1 as is commonly said in mathematics1 along itself4 #f4 ,the extremities of a surface are lines4 ,this is open to the same explanation and objection as definition #c1 and is not usually given in modern textbooks4 ,proclus calls attention to the fact that the statement is hardly true for a complete spherical surface4 #g4 ,,plane4 ,a plane surface is a surface which lies evenly with the straight lines on itself4 ,euclid here follows his definition of straight line1 with a result that is equally unsatisfactory4 ,for teaching purposes the translation from the ,greek is not clear to a beginner1 since lies evenly is a term not simpler than the one defined4 ,as with the definition of a straight line1 so with that of a plane1 numerous efforts at improvement have been made4 ,proclus1 following a hint of ,heron's1 defines it as the surface which is stretched to the utmost1 and also1 this time influenced by ,archimedes's assumption concerning a straight line1 as the least surface among all those which have the same extremities4 ,heron gave one of the best definitions1 ,a surface all the parts of which have the property of fitting on each other4 ,the definition that has met with the widest acceptance1 however1 is a modification of one due to ,proclus1 ,a surface such that a straight line fits on all parts of it4 ,proclus elsewhere says1 ,a plane surface is such that the straight line fits on it all ways1 and ,heron gives it in this form1 ,a plane surface is such that1 if a straight line pass through two points on it1 the line coincides with it at every spot1 all ways4 ,in modern form this appears as follows3 ,a surface such that a straight line joining any two of its points lies wholly in the surface is called a plane1 and for teaching purposes we have no better definition4 ,it is often known as ,simson's definition1 having been given by ,robert ,simson in #agef4 ,the ,french mathematician1 ,fourier1 proposed to define a plane as formed by the aggregate of all the straight lines which1 passing through one point on a straight line in space1 are perpendicular to that line4 ,this is clear1 but it is not so usable for beginners as ,simson's definition4 ,it appears as a theorem in many recent geometries4 ,the ,german mathematician1 ,crelle1 defined a plane as a surface containing all the straight lines 7throughout their whole length7 passing through a fixed point and also intersecting a straight line in space1 but of course this intersected straight line must not pass through the fixed point4 ,crelle's definition is occasionally seen in modern textbooks1 but it is not so clear to the pupil as ,simson's4 ,of the various ultrascientific definitions of a plane that have been suggested of late it is hardly of use to speak in a book concerned primarily with practical teaching4 ,no one of them is adapted to the needs and the comprehension of the beginner1 and it seems that we are not likely to improve upon the so-called ,simson form4 #h4 ,,plane ,,angle4 ,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 line4 ,this definition1 it will be noticed1 includes curvilinear angles1 and the expression and do not lie in a straight line states that the lines must not be continuous one with the other1 that is1 that zero and straight angles are excluded4 ,since ,euclid does not use the curvilinear angle1 and it is only the rectilinear angle with which we are concerned1 we will pass to the next definition and consider this one in connection therewith4 #i4 ,,rectilinear ,,angle4 ,when the lines containing the angle are straight1 the angle is called rectilinear4 ,this definition1 taken with the preceding one1 has always been a subject of criticism4 ,in the first place it expressly excludes the straight angle1 and1 indeed1 the angles of ,euclid are always less than #ahj deg41 contrary to our modern concept4 ,in the second place it defines angle by means of the word inclination1 which is itself as difficult to define as angle4 ,to remedy these defects many substitutes have been proposed4 ,apollonius defined angle as a contracting of a surface or a solid at one point under a broken line or surface4 ,another of the ,greeks defined it as a quantity1 namely1 a distance between the lines or surfaces containing it4 ,schotten#ef says that the definitions of angle generally fall into three groups3 a4 ,an angle is the difference of direction between two lines that meet4 ,this is no better than ,euclid's1 since difference of direction is as difficult to define as inclination4 b4 ,an angle is the amount of turning necessary to bring one side to the position of the other side4 c4 ,an angle is the portion of the plane included between its sides4 ,of these1 b is given by way of explanation in most modern textbooks4 ,indeed1 we cannot do better than simply to define an angle as the opening between two lines which meet1 and then explain what is meant by size1 through the bringing in of the idea of rotation4 ,this is a simple presentation1 it is easily understood1 and it is sufficiently accurate for the real purpose in mind1 namely1 the grasping of the concept4 ,we should frankly acknowledge that the concept of angle is such a simple one that a satisfactory definition is impossible1 and we should therefore confine our attention to having the concept understood4 #aj4 ,when a straight line set up on a straight line makes the adjacent angles equal to one another1 each of the equal angles is right1 and the straight line standing on the other is called a perpendicular to that on which it stands4 ,we at present separate these definitions and simplify the language4 #aa4 ,an obtuse angle is an angle greater than a right angle4 #ab4 ,an acute angle is an angle less than a right angle4 ,the question sometimes asked as to whether an angle of #bjj deg4 is obtuse1 and whether a negative angle1 say -#ij deg41 is acute1 is answered by saying that ,euclid did not conceive of angles equal to or greater than #ahj deg4 and had no notion of negative quantities4 ,generally to-day we define an obtuse angle as greater than one and less than two right angles4 ,an acute angle is defined as an angle less than a right angle1 and is considered as positive under the general understanding that all geometric magnitudes are positive unless the contrary is stated4 #ac4 ,a boundary is that which is an extremity of anything4 ,the definition is not exactly satisfactory1 for a circle is the boundary of the space inclosed1 but we hardly consider it as the extremity of that space4 ,euclid wishes the definition before ,no4 #ad4 #ad4 ,a figure is that which is contained by any boundary or boundaries4 ,the definition is not satisfactory1 since it excludes the unlimited straight line1 the angle1 an assemblage of points1 and other combinations of lines and points which we should now consider as figures4 #ae4 ,a circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another4 #af4 ,and the point is called the center of the circle4 ,some commentators add after one line1 definition #ae1 the words which is called the circumference1 but these are not in the oldest manuscripts4 ,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 circumference1 although ,aristotle used circle as synonymous with the bounding line4 ,with the growth of modern mathematics1 however1 and particularly as a result of the development of analytic geometry1 the word circle has come to mean the bounding line1 as it did with ,aristotle1 a century before ,euclid's time4 ,this has grown out of the equations of the various curves1 x#b y#b r#b representing the circle-line1 a#by#b b#bx#b a#b;b#b representing the ellipse-line1 and so on4 ,it is natural1 therefore1 that circle1 ellipse1 parabola1 and hyperbola should all be looked upon as lines4 ,since this is the modern use of circle in ,english1 it has naturally found its way into elementary geometry1 in order that students should not have to form an entirely different idea of circle on beginning analytic geometry4 ,the general body of ,american teachers1 therefore1 at present favors using circle to mean the bounding line and circumference to mean the length of that line4 ,this requires redefining area of a circle1 and this is done by saying that it is the area of the plane space inclosed4 ,the matter is not of greatest consequence1 but teachers will probably prefer to join in the modern ,american usage of the term4 #ag4 ,,diameter4 ,a diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle1 and such a straight line also bisects the circle4 ,the word diameter is from two ,greek words meaning a through measurer1 and it was also used by ,euclid for the diagonal of a square1 and more generally for the diagonal of any parallelogram4 ,the word diagonal is a later term and means the through angle4 ,it will be noticed that ,euclid adds to the usual definition the statement that a diameter bisects the circle4 ,he does this apparently to justify his definition 7#ah71 of a semicircle 7a half circle74 ,thales is said to have been the first to prove that a diameter bisects the circle1 this being one of three or four propositions definitely attributed to him1 and it is sometimes given as a proposition to be proved4 ,as a proposition1 however1 it is unsatisfactory1 since the proof of what is so evident usually instills more doubt than certainty in the minds of beginners4 #ah4 ,,semicircle4 ,a semicircle is the figure contained by the diameter and the circumference cut off by it4 ,and the center of the semicircle is the same as that of the circle4 ,proclus remarked that the semicircle is the only plane figure that has its center on its perimeter4 ,some writers object to defining a circle as a line and then speaking of the area of a circle1 showing minds that have at least one characteristic of that of ,proclus4 ,the modern definition of semicircle is half of a circle1 that is1 an arc of #ahj deg41 although the term is commonly used to mean both the arc and the segment4 #ai4 ,,rectilinear ,,figures4 ,rectilinear figures are those which are contained by straight lines1 trilateral figures being those contained by three1 quadrilateral those contained by four1 and multilateral those contained by more than four1 straight lines4 #bj4 ,of trilateral figures1 an equilateral triangle is that which has its three sides equal1 an isosceles triangle that which has two of its sides alone equal1 and a scalene triangle that which has its three sides unequal4 #ba4 ,further1 of trilateral figures1 a right-angled triangle is that which has a right angle1 an obtuse-angled triangle that which has an obtuse angle1 and an acute-angled triangle that which has its three angles acute4 ,these three definitions may properly be considered together4 ,rectilinear is from the ,latin translation of the ,greek euthygrammos1 and means right-lined1 or straight-lined4 ,euclid's idea of such a figure is that of the space inclosed1 while the modern idea is tending to become that of the inclosing lines4 ,in elementary geometry1 however1 the ,euclidean idea is still held4 ,trilateral is from the ,latin translation of the ,greek tripleuros 7three-sided74 ,in elementary geometry the word triangle is more commonly used1 although quadrilateral is more common than quadrangle4 ,the use of these two different forms is eccentric and is merely a matter of fashion4 ,thus we speak of a pentagon but not of a tetragon or a trigon1 although both words are correct in form4 ,the word multilateral 7many-sided7 is a translation of the ,greek polypleuros4 ,fashion has changed this to polygonal 7many-angled71 the word multilateral rarely being seen4 ,of the triangles1 equilateral means equal-sided2 isosceles is from the ,greek isoskeles1 meaning with equal legs1 and scalene from skalenos1 possibly from skazo 7to limp71 or from skolios 7crooked74 ,euclid's limitation of isosceles to a triangle with two1 and only two1 equal sides would not now be accepted4 ,we are at present more given to generalizing than he was1 and when we have proved a proposition relating to the isosceles triangle1 we wish to say that we have thereby proved it for the equilateral triangle4 ,we therefore say that an isosceles triangle has two sides equal1 leaving it possible that all three sides should be equal4 ,the expression equal legs is now being discarded on the score of inelegance4 ,in place of right-angled triangle modern writers speak of right triangle1 and so for the obtuse and acute triangles4 ,the terms are briefer and are as readily understood4 ,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 ,pluto1 ,bacchus1 and ,mars4 ,he also states that the ,pythagoreans called the equilateral triangle the head-born ,minerva and ,tritogeneia 7born of ,triton7 because it may be equally divided by the perpendicular lines drawn from each of its angles4 #bb4 ,of quadrilateral figures a square is that which is both equilateral and right-angled2 an oblong that which is right-angled but not equilateral2 a rhombus that which is equilateral and not right-angled2 and a rhomboid that which has its opposite sides and angles equal to one another1 but is neither equilateral nor right-angled4 ,and let all quadrilaterals other than these be called trapezia4 ,in this definition ,euclid also specializes in a manner not now generally approved4 ,thus we are more apt to-day to omit the oblong and rhomboid as unnecessary1 and to define rhombus in such a manner as to include a square4 ,we use parallelogram to cover rhomboid1 rhombus1 oblong1 and square4 ,for oblong we use rectangle1 letting it include square4 ,euclid's definition of square illustrates his freedom in stating more attributes than are necessary1 in order to make sure that the concept is clear2 for he might have said that it is that which is equilateral and has one right angle4 ,we may profit by his method1 sacrificing logic to educational necessity4 ,euclid does not use oblong1 rhombus1 rhomboid1 and trapezium 7plural1 trapezia7 in his proofs1 so that he might well have omitted the definitions1 as we often do4 #bc4 ,,parallels4 ,parallel straight lines are straight lines which1 being in the same plane and being produced indefinitely in both directions1 do not meet one another in either direction4 ,this definition of parallels1 simplified in its language1 is the one commonly used to-day4 ,other definitions have been suggested1 but none has been so generally used4 ,proclus states that ,posidonius gave the definition based upon the lines always being at the same distance apart4 ,geminus has the same idea in his definition4 ,there are1 as ,schotten has pointed out1 three general types of definitions of parallels1 namely3 a4 ,they have no point in common4 ,this may be expressed by saying that 7#a7 they do not intersect1 7#b7 they meet at infinity4 b4 ,they are equidistant from one another4 c4 ,they have the same direction4 ,of these1 the first is ,euclid's1 the idea of the point at infinity being suggested by ,kepler 7#afjd74 ,the second part of this definition is1 of course1 unusable for beginners4 ,dr4 7now ,sir ,thomas7 ,heath says1 ,it seems best1 therefore1 to leave to higher geometry the conception of infinitely distant points on a line and of two straight lines meeting at infinity1 like imaginary points of intersection1 and1 for the purposes of elementary geometry1 to rely on the plain distinction between 'parallel' and 'cutting1' which average human intelligence can readily grasp4 ,the direction definition seems to have originated with ,leibnitz4 ,it is open to the serious objection that direction is not easy of definition1 and that it is used very loosely4 ,if two people on different meridians travel due north1 do they travel in the same direction8 on parallel lines8 ,the definition is as objectionable as that of angle as the difference of direction of two intersecting lines4 ,from these definitions of the first book of ,euclid we see 7#a7 what a small number ,euclid considered as basal2 7#b7 what a change has taken place in the generalization of concepts2 7#c7 how the language has varied4 ,nevertheless we are not to be commended if we adhere to ,euclid's small number1 because geometry is now taught to pupils whose vocabulary is limited4 ,it is necessary to define more terms1 and to scatter the definitions through the work for use as they are needed1 instead of massing them at the beginning1 as in a dictionary4 ,the most important lesson to be learned from ,euclid's definitions is that only the basal ones1 relatively few in number1 need to be learned1 and these because they are used as the foundations upon which proofs are built4 ,it should also be noticed that ,euclid explains nothing in these definitions2 they are hard statements of fact1 massed at the beginning of his treatise4 ,not always as statements1 and not at all in their arrangement1 are they suited to the needs of our boys and girls at present4 ,having considered ,euclid's definitions of ,book ,i1 it is proper to turn to some of those terms that have been added from time to time to his list1 and are now usually incorporated in ,american textbooks4 ,it will be seen that most of these were assumed by ,euclid to be known by his mature readers4 ,they need to be defined for young people1 but most of them are not basal1 that is1 they are not used in the proofs of propositions4 ,some of these terms1 such as magnitudes1 curve line1 broken line1 curvilinear figure1 bisector1 adjacent angles1 reflex angles1 oblique angles and lines1 and vertical angles1 need merely a word of explanation so that they may be used intelligently4 ,if they were numerous enough to make it worth the while1 they could be classified in our textbooks as of minor importance1 but such a course would cause more trouble than it is worth4 ,other terms have come into use in modern times that are not common expressions with which students are familiar4 ,such a term is straight angle1 a concept not used by ,euclid1 but one that adds so materially to the interest and value of geometry as now to be generally recognized4 ,there is also the word perigon1 meaning the whole angular space about a point4 ,this was excluded by the ,greeks because their idea of angle required it to be less than a straight angle4 ,the word means around angle1 and is the best one that has been coined for the purpose4 ,flat angle and whole angle are among the names suggested for these two modern concepts4 ,the terms complement1 supplement1 and conjugate1 meaning the difference between a given angle and a right angle1 straight angle1 and perigon respectively1 have also entered our vocabulary and need defining4 ,there are also certain terms expressing relationship which ,euclid does not define1 and which have been so changed in recent times as to require careful definition at present4 ,chief among these are the words equal1 congruent1 and equivalent4 ,euclid used the single word equal for all three concepts1 although some of his recent editors have changed it to identically equal in the case of congruence4 ,in modern speech we use the word equal commonly to mean like-valued1 having the same measure1 as when we say the circumference of a circle equals a straight line whose length is #bpir1 although it could not coincide with it4 ,of late1 therefore1 in ,europe and ,america1 and wherever ,european influence reaches1 the word congruent is coming into use to mean identically equal in the sense of superposable4 ,we therefore speak of congruent triangles and congruent parallelograms as being those that are superposable4 ,it is a little unfortunate that equal has come to be so loosely used in ordinary conversation that we cannot keep it to mean congruent2 but our language will not permit it1 and we are forced to use the newer word4 ,whenever it can be used without misunderstanding1 however1 it should be retained1 as in the case of equal straight lines1 equal angles1 and equal arcs of the same circle4 ,the mathematical and educational world will never consent to use congruent straight lines1 or congruent angles1 for the reason that the terms are unnecessarily long1 no misunderstanding being possible when equal is used4 ,the word equivalent was introduced by ,legendre at the close of the eighteenth century to indicate equality of length1 or of area1 or of volume4 ,euclid had said1 ,parallelograms which are on the same base and in the same parallels are equal to one another1 while ,legendre and his followers would modify the wording somewhat and introduce equivalent for equal4 ,this usage has been retained4 ,congruent polygons are therefore necessarily equivalent1 but equivalent polygons are not in general congruent4 ,congruent polygons have mutually equal sides and mutually equal angles1 while equivalent polygons have no equality save that of area4 ,in general1 as already stated1 these and other terms should be defined just before they are used instead of at the beginning of geometry4 ,the reason for this1 from the educational standpoint and considering the present position of geometry in the curriculum1 is apparent4 ,we shall now consider the definitions of ,euclid's ,book ,,iii1 which is usually taken as ,book ,,ii in ,america4 #a4 ,,equal ,,circles4 ,equal circles are those the diameters of which are equal1 or the radii of which are equal4 ,manifestly this is a theorem1 for it asserts that if the radii of two circles are equal1 the circles may be made to coincide4 ,in some textbooks a proof is given by superposition1 and the proof is legitimate1 but ,euclid usually avoided superposition if possible4 ,nevertheless he might as well have proved this as that two triangles are congruent if two sides and the included angle of the one are respectively equal to the corresponding parts of the other1 and he might as well have postulated the latter as to have substantially postulated this fact4 ,for in reality this definition is a postulate1 and it was so considered by the great ,italian mathematician ,tartaglia 7ca4 #aejj-;ca4 #aeeg74 ,the plan usually followed in ,america to-day is to consider this as one of many unproved propositions1 too evident1 indeed1 for proof1 accepted by intuition4 ,the result is a loss in the logic of ,euclid1 but the method is thought to be better adapted to the mind of the youthful learner4 ,it is interesting to note in this connection that the ,greeks had no word for radius1 and were therefore compelled to use some such phrase as the straight line from the center1 or1 briefly1 the from the center1 as if from the center were one word4 #b4 ,,tangent4 ,a straight line is said to touch a circle which1 meeting the circle and being produced1 does not cut the circle4 ,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 used1--phrases that signify nothing unless circle is taken to mean the line4 ,so ,aristotle uses an expression meaning that the locus of a certain point is a circle1 and he speaks of a circle as passing through all the angles4 ,our word touch is from the ,latin tangere1 from which comes tangent1 and also tag1 an old touching game4 #c4 ,,tangent ,,circles4 ,circles are said to touch one another which1 meeting one another1 do not cut one another4 ,the definition has not been looked upon as entirely satisfactory1 even aside from its unfortunate phraseology4 ,it is not certain1 for instance1 whether ,euclid meant that the circles could not cut at some other point than that of tangency4 ,furthermore1 no distinction is made between external and internal contact1 although both forms are used in the propositions4 ,modern textbook makers find it convenient to define tangent circles as those that are tangent to the same straight line at the same point1 and to define external and internal tangency by reference to their position with respect to the line1 although this may be characterized as open to about the same objection as ,euclid's4 #d4 ,,distance4 ,in a circle straight lines are said to be equally distant from the center1 when the perpendiculars drawn to them from the center are equal4 ,it is now customary to define distance from a point to a line as the length of the perpendicular from the point to the line1 and to do this in ,book ,i4 ,in higher mathematics it is found that distance is not a satisfactory term to use1 but the objections to it have no particular significance in elementary geometry4 #e4 ,,greater ,,distance4 ,and that straight line is said to be at a greater distance on which the greater perpendicular falls4 ,such a definition is not thought essential at the present time4 #f4 ,,segment4 ,a segment of a circle is the figure contained by a straight line and the circumference of a circle4 ,the word segment is from the ,latin root sect1 meaning cut4 ,so we have sector 7a cutter71 section 7a cut71 intersect1 and so on4 ,the word is not limited to a circle2 we have long spoken of a spherical segment1 and it is common to-day to speak of a line segment1 to which some would apply a new name sect4 ,there is little confusion in the matter1 however1 for the context shows what kind of a segment is to be understood1 so that the word sect is rather pedantic than important4 ,it will be noticed that ,euclid here uses circumference to mean arc4 #g4 ,,angle ,,of ,a ,,segment4 ,an angle of a segment is that contained by a straight line and a circumference of a circle4 ,this term has entirely dropped out of geometry1 and few teachers would know what it meant if they should hear it used4 ,proclus called such angles mixed4 #h4 ,,angle ,,in ,a ,,segment4 ,an angle in a segment is the angle which1 when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment1 is contained by the straight lines so joined4 ,such an involved definition would not be usable to-day4 ,moreover1 the words circumference of the segment would not be used4 #i4 ,and when the straight lines containing the angle cut off a circumference1 the angle is said to stand upon that circumference4 #aj4 ,,sector4 ,a sector of a circle is the figure which1 when an angle is constructed at the center of the circle1 is contained by the straight lines containing the angle and the circumference cut off by them4 ,there is no reason for such an extended definition1 our modern phraseology being both more exact 7as seen in the above use of circumference for arc7 and more intelligible4 ,the ,greek word for sector is knife 7tomeus71 sector being the ,latin translation4 ,a sector is supposed to resemble a shoemaker's knife1 and hence the significance of the term4 ,euclid followed this by a definition of similar sectors1 a term now generally abandoned as unnecessary4 ,it will be noticed that ,euclid did not use or define the word polygon4 ,he uses rectilinear figure instead4 ,polygon may be defined to be a bounding line1 as a circle is now defined1 or as the space inclosed by a broken line1 or as a figure formed by a broken line1 thus including both the limited plane and its boundary4 ,it is not of any great consequence geometrically which of these ideas is adopted1 so that the usual definition of a portion of a plane bounded by a broken line may be taken as sufficient for elementary purposes4 ,it is proper to call attention1 however1 to the fact that we may have cross polygons of various types1 and that the line that bounds the polygon must be continuous1 as the definition states4 ,that is1 in the second of these figures the shaded portion is not considered a polygon4 ,such special cases are not liable to arise1 but if questions relating to them are suggested1 the teacher should be prepared to answer them4 ,if suggested to a class1 a note of this kind should come out only incidentally as a bit of interest1 and should not occupy much time nor be unduly emphasized4 ,illustration ,it may also be mentioned to a class at some convenient time that the old idea of a polygon was that of a convex figure1 and that the modern idea1 which is met in higher mathematics1 leads to a modification of earlier concepts4 ,for example1 here is a quadrilateral with one of its diagonals1 ,,bd1 outside the figure4 ,furthermore1 if we consider a quadrilateral as a figure formed by four intersecting lines1 ,,ac1 ,,cf1 ,,be1 and ,,ea1 it is apparent that this general quadrilateral has six vertices1 ,a1 ,b1 ,c1 ,d1 ,e1 ,f1 and three diagonals1 ,,ad1 ,,bf1 and ,,ce4 ,such broader ideas of geometry form the basis of what is called modern elementary geometry4 ,illustration ,the other definitions of plane geometry need not be discussed1 since all that have any historical interest have been considered4 ,on the whole it may be said that our definitions to-day are not in general so carefully considered as those of ,euclid1 who weighed each word with greatest skill1 but they are more teachable to beginners1 and are1 on the whole1 more satisfactory from the educational standpoint4 ,the greatest lesson to be learned from this discussion is that the number of basal definitions to be learned for subsequent use is very small4 ,since teachers are occasionally disturbed over the form in which definitions are stated1 it is well to say a few words upon this subject4 ,there are several standard types that may be used4 7#a7 ,we may use the dictionary form1 putting the word defined first1 thus3 ,right triangle4 ,a triangle that has one of its angles a right angle4 ,this is scientifically correct1 but it is not a complete sentence1 and hence it is not easily repeated when it has to be quoted as an authority4 7#b7 ,we may put the word defined at the end1 thus3 ,a triangle that has one of its angles a right angle is called a right triangle4 ,this is more satisfactory4 7#c7 ,we may combine 7#a7 and 7#b71 thus3 ,right triangle4 ,a triangle that has one of its angles a right angle is called a right triangle4 ,this is still better1 for it has the catchword at the beginning of the paragraph4 ,there is occasionally some mental agitation over the trivial things of a definition1 such as the use of the words is called4 ,it would not be a very serious matter if they were omitted1 but it is better to have them there4 ,the reason is that they mark the statement at once as a definition4 ,for example1 suppose we say that a triangle that has one of its angles a right angle is a right triangle4 ,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 triangle4 ,the style of statement is the same1 and we have nothing in the phraseology to show that the first is a definition and the second a theorem4 ,this may happen with most of the definitions1 and hence the most careful writers have not consented to omit the distinctive words in question4 ,apropos of the definitions of geometry1 the great ,french philosopher and mathematician1 ,pascal1 set forth certain rules relating to this subject1 as also to the axioms employed1 and these may properly sum up this chapter4 #a4 ,do not attempt to define terms so well known in themselves that there are no simpler terms by which to express them4 #b4 ,admit no obscure or equivocal terms without defining them4 #c4 ,use in the definitions only terms that are perfectly understood or are there explained4 #d4 ,omit no necessary principles without general agreement1 however clear and evident they may be4 #e4 ,set forth in the axioms only those things that are in themselves perfectly evident4 #f4 ,do not attempt to demonstrate anything that is so evident in itself that there is nothing more simple by which to prove it4 #g4 ,prove whatever is in the least obscure1 using in the demonstration only axioms that are perfectly evident in themselves1 or propositions already demonstrated or allowed4 #h4 ,in case of any uncertainty arising from a term employed1 always substitute mentally the definition for the term itself4 ,bibliography4 ,heath1 ,euclid1 as cited2 ,frankland1 ,the ,first ,book of ,euclid1 as cited2 ,smith1 ,teaching of ,elementary ,mathematics1 p4 #beg1 ,new ,york1 #aijj2 ,young1 ,teaching of ,mathematics1 p4 #ahi1 ,new ,york1 #aijg2 ,veblen1 ,on ,definitions1 in the ,monist1 #aijc1 p4 #cjc4 ,,footnotes3 #ec ,free use has been made of ,w4 ,b4 ,frankland1 ,the ,first ,book of ,euclid's ',elements1' ,cambridge1 #aije2 ,t4 ,l4 ,heath1 ,the ,thirteen ,books of ,euclid's ',elements1' ,cambridge1 #aijh2 ,h4 ,schotten1 ,inhalt und ,methode des planimetrischen ,unterrichts1 ,leipzig1 #ahic2 ,m4 ,simon1 ,euclid und die sechs planimetrischen ,buecher1 ,leipzig1 #aija4 #ed ,for a facsimile of a thirteenth-century ,,ms4 containing this definition1 see the author's ,rara ,arithmetica1 ,plate ,,iv1 ,boston1 #aiji4 #ee ,our slang expression ,the cart before the horse is suggestive of this procedure4 #ef ,loc4 cit41 ,vol4 ,,ii1 p4 #id4 ,,chapter ,,xiii ,,how ,,to ,,attack ,,the ,,exercises ,the old geometry1 say of a century ago1 usually consisted1 as has been stated1 of a series of theorems fully proved and of problems fully solved4 ,during the nineteenth century exercises were gradually introduced1 thus developing geometry from a science in which one learned by seeing things done1 into one in which he gained power by actually doing things4 ,of the nature of these exercises 7originals1 riders71 and of their gradual change in the past few years1 mention has been made in ,chapter ,,vii4 ,it now remains to consider the methods of attacking these exercises4 ,it is evident that there is no single method1 and this is a fortunate fact1 since if it were not so1 the attack would be too mechanical to be interesting4 ,there is no one rule for solving every problem nor even for seeing how to begin4 ,on the other hand1 a pupil is saved some time by having his attention called to a few rather definite lines of attack1 and he will undoubtedly fare the better by not wasting his energies over attempts that are in advance doomed to failure4 ,there are two general questions to be considered3 first1 as to the discovery of new truths1 and second1 as to the proof4 ,with the first the pupil will have little to do1 not having as yet arrived at this stage in his progress4 ,a bright student may take a little interest in seeing what he can find out that is new 7at least to him71 and if so1 he may be told that many new propositions have been discovered by the accurate drawing of figures2 that some have been found by actually weighing pieces of sheet metal of certain sizes2 and that still others have made themselves known through paper folding4 ,in all of these cases1 however1 the supposed proposition must be proved before it can be accepted4 ,as to the proof1 the pupil usually wanders about more or less until he strikes the right line1 and then he follows this to the conclusion4 ,he should not be blamed for doing this1 for he is pursuing the method that the world followed in the earliest times1 and one that has always been common and always will be4 ,this is the synthetic method1 the building up of the proof from propositions previously proved4 ,if the proposition is a theorem1 it is usually not difficult to recall propositions that may lead to the demonstration1 and to select the ones that are really needed4 ,if it is a problem1 it is usually easy to look ahead and see what is necessary for the solution and to select the preceding propositions accordingly4 ,but pupils should be told that if they do not rather easily find the necessary propositions for the construction or the proof1 they should not delay in resorting to another and more systematic method4 ,this is known as the method of analysis1 and it is applicable both to theorems and to problems4 ,it has several forms1 but it is of little service to a pupil to have these differentiated1 and it suffices that he be given the essential feature of all these forms1 a feature that goes back to ,plato and his school in the fifth century ,b4,c4 ,for a theorem1 the method of analysis consists in reasoning as follows3 ,i can prove this proposition if ,i can prove this thing2 ,i can prove this thing if ,i can prove that2 ,i can prove that if ,i can prove a third thing1 and so the reasoning runs until the pupil comes to the point where he is able to add1 but ,i can prove that4 ,this does not prove the proposition1 but it enables him to reverse the process1 beginning with the thing he can prove and going back1 step by step1 to the thing that he is to prove4 ,analysis is1 therefore1 his method of discovery of the way in which he may arrange his synthetic proof4 ,pupils often wonder how any one ever came to know how to arrange the proofs of geometry1 and this answers the question4 ,some one guessed that a statement was true2 he applied analysis and found that he could prove it2 he then applied synthesis and did prove it4 ,for a problem1 the method of analysis is much the same as in the case of a theorem4 ,two things are involved1 however1 instead of one1 for here we must make the construction and then prove that this construction is correct4 ,the pupil1 therefore1 first supposes the problem solved1 and sees what results follow4 ,he then reverses the process and sees if he can attain these results and thus effect the required construction4 ,if so1 he states the process and gives the resulting proof4 ,for example3 ,in a triangle ,,abc1 to draw ,,pq parallel to the base ,,ab1 cutting the sides in ,p and ,q1 so that ,,pq shall equal ,,ap ,,bq4 ,illustration ,analysis4 ,assume the problem solved4 ,then ,,ap must equal some part of ,,pq as ,,px1 and ,,bq must equal ,,qx4 ,but if ,,ap ,,px1 what must ,l,,pxa equal8 because ,,pq is ,,ab1 what does ,l,,pxa equal8 ,then why must ,l,,bax ,l,,xap8 ,similarly1 what about ,l,,qbx and ,l,,xba8 ,construction4 ,now reverse the process4 ,what may we do to ,ls ,a and ,b in order to fix ,x8 ,then how shall ,,pq be drawn8 ,now give the proof4 ,illustration ,illustration ,the third general method of attack applies chiefly to problems where some point is to be determined4 ,this is the method of the intersection of loci4 ,thus1 to locate an electric light at a point eighteen feet from the point of intersection of two streets and equidistant from them1 evidently one locus is a circle with a radius eighteen feet and the center at the vertex of the angle made by the streets1 and the other locus is the bisector of the angle4 ,the method is also occasionally applicable to theorems4 ,for example1 to prove that the perpendicular bisectors of the sides of a triangle are concurrent4 ,here the locus of points equidistant from ,a and ,b is ,,pp'1 and the locus of points equidistant from ,b and ,c is ,,qq'4 ,these can easily be shown to intersect1 as at ,o4 ,then ,o1 being equidistant from ,a1 ,b1 and ,c1 is also on the perpendicular bisector of ,,ac4 ,therefore these bisectors are concurrent in ,o4 ,these are the chief methods of attack1 and are all that should be given to an average class for practical use4 ,besides the methods of attack1 there are a few general directions that should be given to pupils4 #a4 ,in attacking either a theorem or a problem1 take the most general figure possible4 ,thus1 if a proposition relates to a quadrilateral1 take one with unequal sides and unequal angles rather than a square or even a rectangle4 ,the simpler figures often deceive a pupil into feeling that he has a proof1 when in reality he has one only for a special case4 #b4 ,set forth very exactly the thing that is given1 using letters relating to the figure that has been drawn4 ,then set forth with the same exactness the thing that is to be proved4 ,the neglect to do this is the cause of a large per cent of the failures4 ,the knowing of exactly what we have to do and exactly what we have with which to do it is half the battle4 #c4 ,if the proposition seems hazy1 the difficulty is probably with the wording4 ,in this case try substituting the definition for the name of the thing defined4 ,thus instead of thinking too long about proving that the median to the base of an isosceles triangle is perpendicular to the base1 draw the figure and think that there is given ,,ac ,,bc1 ,,ad ,,bd1 and that there is to be proved that ,l,,cda ,l,,bdc4 ,illustration ,here we have replaced median1 isosceles1 and perpendicular by statements that express the same idea in simpler language4 ,bibliography4 ,petersen1 ,methods and ,theories for the ,solution of ,geometric ,problems of ,construction1 ,copenhagen1 #ahgi1 a curious piece of ,english and an extreme view of the subject1 but well worth consulting2 ,alexandroff1 ,problemes de geometrie elementaire1 ,paris1 #ahii1 with a ,german translation in #aijc2 ,loomis1 ,original ,investigation2 or1 ,how to attack an ,exercise in ,geometry1 ,boston1 #aija2 ,sauvage1 ,les ,lieux geometriques en geometrie elementaire1 ,paris1 #ahic2 ,hadamard1 ,lecons de geometrie1 p4 #bfa1 ,paris1 #ahih2 ,duhamel1 ,des ,methodes dans les sciences de raisonnement1 #c;e ed41 ,paris1 #ahhe2 ,henrici and ,treutlein1 ,lehrbuch der ,elementar-,geometrie1 ,leipzig1 #c4 ,aufl41 #ahig2 ,henrici1 ,congruent ,figures1 ,london1 #ahgi4 ,,chapter ,,xiv ,,book ,i ,,and ,,its ,,propositions ,having considered the nature of the geometry that we have inherited1 and some of the opportunities for improving upon the methods of presenting it1 the next question that arises is the all-important one of the subject matter1 ,what shall geometry be in detail8 ,shall it be the text or the sequence of ,euclid8 ,few teachers have any such idea at the present time4 ,shall it be a mere dabbling with forms that are seen in mechanics or architecture1 with no serious logical sequence8 ,this is an equally dangerous extreme4 ,shall it be an entirely new style of geometry based upon groups of motions8 ,this may sometime be developed1 but as yet it exists in the future if it exists at all1 since the recent efforts in this respect are generally quite as ill suited to a young pupil as is ,euclid's ,elements itself4 ,no one can deny the truth of ,m4 ,bourlet's recent assertion that ,industry1 daughter of the science of the nineteenth century1 reigns to-day the mistress of the world2 she has transformed all ancient methods1 and she has absorbed in herself almost all human activity4#eg ,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 building1 so it demands another type of geometry4 ,but what does this mean8 ,that geometry is to exist merely as it touches industry1 or that bad architecture is to replace the good8 ,by no means4 ,a building should to-day have steam heat and elevators and electric lights1 but it should be constructed of just as enduring materials as the ,parthenon1 and it should have lines as pleasing as those of a ,gothic facade4 ,architecture should still be artistic and construction should still be substantial1 else a building can never endure4 ,so geometry must still exemplify good logic and must still bring to the pupil a feeling of exaltation1 or it will perish and become a mere relic in the museum of human culture4 ,what1 then1 shall the propositions of geometry be1 and in what manner shall they answer to the challenge of the industrial epoch in which we live8 ,in reply1 they must be better adapted to young minds and to all young minds than ,euclid ever intended his own propositions to be4 ,furthermore1 they must have a richness of application to pure geometry1 in the way of carefully chosen exercises1 that ,euclid never attempted4 ,and finally1 they must have application to this same life of industry of which we have spoken1 whenever this can really be found1 but there must be no sham and pretense about it1 else the very honesty that permeated the ancient geometry will seem to the pupil to be wanting in the whole subject4#eh ,until some geometry on a radically different basis shall appear1 and of this there is no very hopeful sign at present1 the propositions will be the essential ones of ,euclid1 excluding those that may be considered merely intuitive1 and excluding all that are too difficult for the pupil who to-day takes up their study4 ,the number will be limited in a reasonable way1 and every genuine type of application will be placed before the teacher to be used as necessity requires4 ,but a fair amount of logic will be retained1 and the effort to make of geometry an empty bauble of a listless mind will be rejected by every worthy teacher4 ,what the propositions should be is a matter upon which opinions may justly differ2 but in this chapter there is set forth a reasonable list for ,book ,i1 arranged in a workable sequence1 and this list may fairly be taken as typical of what the ,american school will probably use for many years to come4 ,with the list is given a set of typical applications1 and some of the general information that will add to the interest in the work and that should form part of the equipment of the teacher4 ,an ancient treatise was usually written on a kind of paper called papyrus1 made from the pith of a large reed formerly common in ,egypt1 but now growing luxuriantly only above ,khartum in ,upper ,egypt1 and near ,syracuse in ,sicily2 or else it was written on parchment1 so called from ,pergamos in ,asia ,minor1 where skins were first prepared in parchment form2 or occasionally they were written on ordinary leather4 ,in any case they were generally written on long strips of the material used1 and these were rolled up and tied4 ,hence we have such an expression as keeping the roll in school1 and such a word as volume1 which has in it the same root as involve 7to roll in71 and evolve 7to roll out74 ,several of these rolls were often necessary for a single treatise1 in which case each was tied1 and all were kept together in a receptacle resembling a pail1 or in a compartment on a shelf4 ,the ,greeks called each of the separate parts of a treatise biblion 7,greek3 biblion71 a word meaning book4 ,hence we have the books of the ,bible1 the books of ,homer1 and the books of ,euclid4 ,from the same root1 indeed1 comes ,bible1 bibliophile 7booklover71 bibliography 7list of books71 and kindred words4 ,thus the books of geometry are the large chapters of the subject1 chapter being from the ,latin caput 7head71 a section under a new heading4 ,there have been efforts to change books to chapters1 but they have not succeeded1 and there is no reason why they should succeed1 for the term is clear and has the sanction of long usage4 ,,theorem4 ,if two lines intersect1 the vertical angles are equal4 ,this was ,euclid's ,proposition #ae1 being put so late because he based the proof upon his ,proposition #ac1 now thought to be best taken without proof1 namely1 ,if a straight line set upon a straight line makes angles1 it will make either two right angles or angles equal to two right angles4 ,it is found to be better pedagogy to assume that this follows from the definition of straight angle1 with reference1 if necessary1 to the meaning of the sum of two angles4 ,this proposition on vertical angles is probably the best one with which to begin geometry1 since it is not so evident as to seem to need no proof1 although some prefer to rank it as semiobvious1 while the proof is so simple as easily to be understood4 ,eudemus1 a ,greek who wrote not long before ,euclid1 attributed the discovery of this proposition to ,thales of ,miletus 7ca4 #fdj-edh ,b4,c471 one of the ,seven ,wise ,men of ,greece1 of whom ,proclus wrote3 ,thales it was who visited ,egypt and first transferred to ,hellenic soil this theory of geometry4 ,he himself1 indeed1 discovered much1 but still more did he introduce to his successors the principles of the science4 ,the proposition is the only basal one relating to the intersection of two lines1 and hence there are no others with which it is necessarily grouped4 ,this is the reason for placing it by itself1 followed by the congruence theorems4 ,there are many familiar illustrations of this theorem4 ,indeed1 any two crossed lines1 as in a pair of shears or the legs of a camp stool1 bring it to mind4 ,the word straight is here omitted before lines in accordance with the modern convention that the word line unmodified means a straight line4 ,of course in cases of special emphasis the adjective should be used4 ,,theorem4 ,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 other4 ,this is ,euclid's ,proposition #d1 his first three propositions being problems of construction4 ,this would therefore have been his first proposition if he had placed his problems later1 as we do to-day4 ,the words congruent and equal are not used as in ,euclid1 for reasons already set forth on page #aea4 ,there have been many attempts to rearrange the propositions of ,book ,i1 putting in separate groups those concerning angles1 those concerning triangles1 and those concerning parallels1 but they have all failed1 and for the cogent reason that such a scheme destroys the logical sequence4 ,this proposition may properly follow the one on vertical angles simply because the latter is easier and does not involve superposition4 ,as far as possible1 ,euclid and all other good geometers avoid the proof by superposition4 ,as a practical test superposition is valuable1 but as a theoretical one it is open to numerous objections4 ,as ,peletier pointed out in his 7#aeeg7 edition of ,euclid1 if the superposition of lines and figures could freely be assumed as a method of demonstration1 geometry would be full of such proofs4 ,there would be no reason1 for example1 why an angle should not be constructed equal to a given angle by superposing the given angle on another part of the plane4 ,indeed1 it is possible that we might then assume to bisect an angle by imagining the plane folded like a piece of paper4 ,heath 7#aijh7 has pointed out a subtle defect in ,euclid's proof1 in that it is said that because two lines are equal1 they can be made to coincide4 ,euclid says1 practically1 that if two lines can be made to coincide1 they are equal1 but he does not say that if two straight lines are equal1 they can be made to coincide4 ,for the purposes of elementary geometry the matter is hardly worth bringing to the attention of a pupil1 but it shows that even ,euclid did not cover every point4 ,applications of this proposition are easily found1 but they are all very much alike4 ,there are dozens of measurements that can be made by simply constructing a triangle that shall be congruent to another triangle4 ,it seems hardly worth the while at this time to do more than mention one typical case1#ei leaving it to teachers who may find it desirable to suggest others to their pupils4 ,illustration ,wishing to measure the distance across a river1 some boys sighted from ,a to a point ,p4 ,they then turned and measured ,,ab at right angles to ,,ap4 ,they placed a stake at ,o1 halfway from ,a to ,b1 and drew a perpendicular to ,,ab at ,b4 ,they placed a stake at ,c1 on this perpendicular1 and in line with ,o and ,p4 ,they then found the width by measuring ,,bc4 ,prove that they were right4 ,this involves the ranging of a line1 and the running of a line at right angles to a given line1 both of which have been described in ,chapter ,,ix4 ,it is also fairly accurate to run a line at any angle to a given line by sighting along two pins stuck in a protractor4 ,,theorem4 ,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 other4 ,euclid combines this with his ,proposition #bf3 ,if two triangles have the two angles equal to two angles respectively1 and one side equal to one side1 namely1 either the side adjoining the equal angles1 or that subtending one of the equal angles1 they will also have the remaining sides equal to the remaining sides1 and the remaining angle to the remaining angle4 ,he proves this cumbersome statement without superposition1 desiring to avoid this method1 as already stated1 whenever possible4 ,the proof by superposition is old1 however1 for ,al-,nairizi#fj gives it and ascribes it to some earlier author whose name he did not know4 ,proclus tells us that ,eudemus in his geometrical history refers this theorem to ,thales4 ,for he says that in the method by which they say that ,thales proved the distance of ships in the sea1 it was necessary to make use of this theorem4 ,how ,thales did this is purely a matter of conjecture1 but he might have stood on the top of a tower rising from the level shore1 or of such headlands as abound near ,miletus1 and by some simple instrument sighted to the ship4 ,then1 turning1 he might have sighted along the shore to a point having the same angle of declination1 and then have measured the distance from the tower to this point4 ,this seems more reasonable than any of the various plans suggested1 and it is found in so many practical geometries of the first century of printing that it seems to have long been a common expedient4 ,the stone astrolabe from ,mesopotamia1 now preserved in the ,british ,museum1 shows that such instruments for the measuring of angles are very old1 and for the purposes of ,thales even a pair of large compasses would have answered very well4 ,an illustration of the method is seen in ,belli's work of #aefi1 as here shown4 ,at the top of the picture a man is getting the angle by means of the visor of his cap2 at the bottom of the picture a man is using a ruler screwed to a staff4#fa ,the story goes that one of ,napoleon's engineers won the imperial favor by quickly measuring the width of a stream that blocked the progress of the army1 using this very method4 ,illustration3 ,,sixteenth-,,century ,,mensuration ,belli's ,del ,misurar con la ,vista1 ,venice1 #aefi ,this proposition is the reciprocal or dual of the preceding one4 ,the relation between the two may be seen from the following arrangement3 ,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 other4 ,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 other4 ,in general1 to every proposition involving points and lines there is a reciprocal proposition involving lines and points respectively that is often true1--indeed1 that is always true in a certain line of propositions4 ,this relation is known as the ,principle of ,reciprocity or of ,duality4 ,instead of points and lines we have here angles 7suggested by the vertex points7 and lines4 ,it is interesting to a class to have attention called to such relations1 but it is not of sufficient importance in elementary geometry to justify more than a reference here and there4 ,there are other dual features that are seen in geometry besides those given above4 ,,theorem4 ,in an isosceles triangle the angles opposite the equal sides are equal4 ,this is ,euclid's ,proposition #e1 the second of his theorems1 but he adds1 and if the equal straight lines be produced further1 the angles under the base will be equal to one another4 ,since1 however1 he does not use this second part1 its genuineness is doubted4 ,he would not admit the common proof of to-day of supposing the vertical angle bisected1 because the problem about bisecting an angle does not precede this proposition1 and therefore his proof is much more involved than ours4 ,he makes ,,cx ,,cy1 and proves triangles,,xbc and ,,yac congruent1#fb and also triangles,,xba and ,,yab congruent4 ,then from ,l,,yac he takes ,l,,yab1 leaving ,l,,bac1 and so on the other side1 leaving ,l,,cba1 these therefore being equal4 ,illustration ,this proposition has long been called the pons asinorum1 or bridge of asses1 but no one knows where or when the name arose4 ,it is usually stated that it came from the fact that fools could not cross this bridge1 and it is a fact that in the ,middle ,ages this was often the limit of the student's progress in geometry4 ,it has however been suggested that the name came from ,euclid's figure1 which resembles the simplest type of a wooden truss bridge4 ,the name is applied by the ,french to the ,pythagorean ,theorem4 ,proclus attributes the discovery of this proposition to ,thales4 ,he also says that ,pappus 7third century ,a4,d471 a ,greek commentator on ,euclid1 proved the proposition as follows3 ,let ,,abc be the triangle1 with ,,ab ,,ac4 ,conceive of this as two triangles2 then ,,ab ,,ac1 ,,ac ,,ab1 and ,l,a is common2 hence the triangles,,abc and ,,acb are congruent1 and ,l,b of the one equals ,l,c of the other4 ,this is a better plan than that followed by some textbook writers of imagining triangle,,abc taken up and laid down on itself4 ,even to lay it down on its trace is more objectionable than the plan of ,pappus4 ,,theorem4 ,if two angles of a triangle are equal1 the sides opposite the equal angles are equal1 and the triangle is isosceles4 ,the statement is1 of course1 tautological1 the last five words being unnecessary from the mathematical standpoint1 but of value at this stage of the student's progress as emphasizing the nature of the triangle4 ,euclid stated the proposition thus1 ,if in a triangle two angles be equal to one another1 the sides which subtend the equal angles will also be equal to one another4 ,he did not define subtend1 supposing such words to be already understood4 ,this is the first case of a converse proposition in geometry4 ,heath distinguishes the logical from the geometric converse4 ,the logical converse of ,euclid ,i1 #e1 would be that some triangles with two angles equal are isosceles1 while the geometric converse is the proposition as stated4 ,proclus called attention to two forms of converse 7and in the course of the work1 but not at this time1 the teacher may have to do the same73 7#a7 the complete converse1 in which that which is given in one becomes that which is to be proved in the other1 and vice versa1 as in this and the preceding proposition2 7#b7 the partial converse1 in which two 7or even more7 things may be given1 and a certain thing is to be proved1 the converse being that one 7or more7 of the preceding things is now given1 together with what was to be proved1 and the other given thing is now to be proved4 ,symbolically1 if it is given that a b and c d1 to prove that x y1 the partial converse would have given a b and x y1 to prove that c d4 ,several proofs for the proposition have been suggested1 but a careful examination of all of them shows that the one given below is1 all things considered1 the best one for pupils beginning geometry and following the sequence laid down in this chapter4 ,it has the sanction of some of the most eminent mathematicians1 and while not as satisfactory in some respects as the reductio ad absurdum1 mentioned below1 it is more satisfactory in most particulars4 ,the proof is as follows3 ,illustration3 ,given the triangle ,,abc1 with the angle ,a equal to the angle ,b4 ,to prove that ,,ac ,,bc4 ,proof4 ,suppose the second triangle ,a',b',c' to be an exact reproduction of the given triangle ,,abc4 ,turn the triangle ,a',b',c' over and place it upon ,,abc so that ,b' shall fall on ,a and ,a' shall fall on ,b4 ,then ,b',a' will coincide with ,,ab4 ,since ,l,a' ,l,b'1 ,given and ,l,a ,l,a'1 ,hyp4 therefore,l,a ,l,b'4 therefore,b',c' will lie along ,,ac4 ,similarly1 ,a',c' will lie along ,,bc4 ,therefore ,c' will fall on both ,,ac and ,,bc1 and hence at their intersection4 therefore,b',c' ,,ac4 ,but ,b',c' was made equal to ,,bc4 therefore,,ac ,,bc4 ,q4,e4,d4 ,if the proposition should be postponed until after the one on the sum of the angles of a triangle1 the proof would be simpler1 but it is advantageous to couple it with its immediate predecessor4 ,this simpler proof consists in bisecting the vertical angle1 and then proving the two triangles congruent4 ,among the other proofs is that of the reductio ad absurdum1 which the student might now meet1 but which may better be postponed4 ,the phrase reductio ad absurdum seems likely to continue in spite of the efforts to find another one that is simpler4 ,such a proof is also called an indirect proof1 but this term is not altogether satisfactory4 ,probably both names should be used1 the ,latin to explain the nature of the ,english4 ,the ,latin name is merely a translation of one of several ,greek names used by ,aristotle1 a second being in ,english proof by the impossible1 and a third being proof leading to the impossible4 ,if teachers desire to introduce this form of proof here1 it must be borne in mind that only one supposition can be made if such a proof is to be valid1 for if two are made1 then an absurd conclusion simply shows that either or both must be false1 but we do not know which is false1 or if only one is false4 ,,theorem4 ,two triangles are congruent if the three sides of the one are equal respectively to the three sides of the other4 ,it would be desirable to place this after the fourth proposition mentioned in this list if it could be done1 so as to get the triangles in a group1 but we need the fourth one for proving this1 so that the arrangement cannot be made1 at least with this method of proof4 ,this proposition is a partial converse of the second proposition in this list2 for if the triangles are ,,abc and ,a',b',c'1 with sides a1 b1 c and a'1 b'1 c'1 then the second proposition asserts that if b b'1 c c'1 and ,l,a ,l,a'1 then a a' and the triangles are congruent1 while this proposition asserts that if a a'1 b b'1 and c c'1 then ,l,a ,l,a' and the triangles are congruent4 ,the proposition was known at least as early as ,aristotle's time4 ,euclid proved it by inserting a preliminary proposition to the effect that it is impossible to have on the same base ,,ab and the same side of it two different triangles ,,abc and ,,abc'1 with ,,ac ,,ac'1 and ,,bc ,,bc'4 ,the proof ordinarily given to-day1 wherein the two triangles are constructed on opposite sides of the base1 is due to ,philo of ,byzantium1 who lived after ,euclid's time but before the ,christian era1 and it is also given by ,proclus4 ,there are really three cases1 if one wishes to be overparticular1 corresponding to the three pairs of equal sides4 ,but if we are allowed to take the longest side for the common base1 only one case need be considered4 ,of the applications of the proposition one of the most important relates to making a figure rigid by means of diagonals4 ,for example1 how many diagonals must be drawn in order to make a quadrilateral rigid8 to make a pentagon rigid8 a hexagon8 a polygon of n sides4 ,in particular1 the following questions may be asked of a class3 ,illustration #a4 ,three iron rods are hinged at the extremities1 as shown in this figure4 ,is the figure rigid8 ,why8 #b4 ,four iron rods are hinged1 as shown in this figure4 ,is the figure rigid8 ,if not1 where would you put in the fifth rod to make it rigid8 ,prove that this would accomplish the result4 ,illustration ,another interesting application relates to the most ancient form of leveling instrument known to us4 ,this kind of level is pictured on very ancient monuments1 and it is still used in many parts of the world4 ,pupils in manual training may make such an instrument1 and indeed one is easily made out of cardboard4 ,if the plumb line passes through the mid-point of the base1 the two triangles are congruent and the plumb line is then perpendicular to the base4 ,in other words1 the base is level4 ,with such simple primitive instruments1 easily made by pupils1 a good deal of practical mathematical work can be performed4 ,the interesting old illustration here given shows how this form of level was used three hundred years ago4 ,illustration3 ,,early ,,methods ,,of ,,leveling ,pomodoro's ,la geometria prattica1 ,rome1 #afbd ,illustration ,teachers who seek for geometric figures in practical mechanics will find this proposition illustrated in the ordinary hoisting apparatus of the kind here shown4 ,from the study of such forms and of simple roof and bridge trusses1 a number of the usual properties of the isosceles triangle may be derived4 ,,theorem4 ,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 drawn1 but included by them4 ,it should be noted that the words the extremities of are necessary1 for it is possible to draw from a certain point within a certain triangle two lines to the base such that their sum is greater than the sum of the other two sides4 ,illustration ,thus1 in the right triangle ,,abc draw any line ,,cx from ,c to the base4 ,make ,,xy ,,ac1 and ,,cp ,,py4 ,then it is easily shown that ,,pb ,,px ,,cb ,,ca4 ,illustration ,it is interesting to a class to have a teacher point out that1 in this figure1 ,,ap ,,pb ,,ac ,,cb1 and ,,ap' ,p',b ,,ap ,,pb1 and that the nearer ,p gets to ,,ab1 the shorter ,,ap ,,pb becomes1 the limit being the line ,,ab4 ,from this we may infer 7although we have not proved7 that a straight line 7,,ab7 is the shortest path between two points4 ,,theorem4 ,only one perpendicular can be drawn to a given line from a given external point4 ,,theorem4 ,two lines drawn from a point in a perpendicular to a given line1 cutting off on the given line equal segments from the foot of the perpendicular1 are equal and make equal angles with the perpendicular4 ,,theorem4 ,of two lines drawn from the same point in a perpendicular to a given line1 cutting off on the line unequal segments from the foot of the perpendicular1 the more remote is the greater4 ,,theorem4 ,the perpendicular is the shortest line that can be drawn to a straight line from a given external point4 ,these four propositions1 while known to the ancients and incidentally used1 are not explicitly stated by ,euclid4 ,the reason seems to be that he interspersed his problems with his theorems1 and in his ,propositions #aa and #ab1 which treat of drawing a perpendicular to a line1 the essential features of these theorems are proved4 ,further mention will be made of them when we come to consider the problems in question4 ,many textbook writers put the second and third of the four before the first1 forgetting that the first is assumed in the other two1 and hence should precede them4 ,,theorem4 ,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 other4 ,,theorem4 ,two right triangles are congruent if the hypotenuse and an adjacent angle of the one are equal respectively to the hypotenuse and an adjacent angle of the other4 ,as stated in the notes on the third proposition in this sequence1 ,euclid's cumbersome ,proposition #bf covers several cases1 and these two among them4 ,of course this present proposition could more easily be proved after the one concerning the sum of the angles of a triangle1 but the proof is so simple that it is better to leave the proposition here in connection with others concerning triangles4 ,,theorem4 ,two lines in the same plane perpendicular to the same line cannot meet1 however far they are produced4 ,this proposition is not in ,euclid1 and it is introduced for educational rather than for mathematical reasons4 ,euclid introduced the subject by the proposition that1 if alternate angles are equal1 the lines are parallel4 ,it is1 however1 simpler to begin with this proposition1 and there is some advantage in stating it in such a way as to prove that parallels exist before they are defined4 ,the proposition is properly followed by the definition of parallels and by the postulate that has been discussed on page #abg4 ,a good application of this proposition is the one concerning a method of drawing parallel lines by the use of a carpenter's square4 ,here two lines are drawn perpendicular to the edge of a board or a ruler1 and these are parallel4 ,,theorem4 ,if a line is perpendicular to one of two parallel lines1 it is perpendicular to the other also4 ,this1 like the preceding proposition1 is a special case under a later theorem4 ,it simplifies the treatment of parallels1 however1 and the beginner finds it easier to approach the difficulties gradually1 through these two cases of perpendiculars4 ,it should be noticed that this is an example of a partial converse1 as explained on page #age4 ,the preceding proposition may be stated thus3 ,if a is perp to x and b is perp to x1 then a is to b4 ,this proposition may be stated thus3 ,if a is perp to x and a is to b1 then b is perp to x4 ,this is1 therefore1 a partial converse4 ,these two propositions having been proved1 the usual definitions of the angles made by a transversal of two parallels may be given4 ,it is unfortunate that we have no name for each of the two groups of four equal angles1 and the name of transverse angles has been suggested4 ,this would simplify the statements of certain other propositions2 thus3 ,if two parallel lines are cut by a transversal1 the transverse angles are equal1 and this includes two propositions as usually given4 ,there is not as yet1 however1 any general sanction for the term4 ,,theorem4 ,if two parallel lines are cut by a transversal1 the alternate-interior angles are equal4 ,euclid gave this as half of his ,proposition #bi4 ,indeed1 he gives only four theorems on parallels1 as against five propositions and several corollaries in most of our ,american textbooks4 ,the reason for increasing the number is that each proposition may be less involved4 ,thus1 instead of having one proposition for both exterior and interior angles1 modern authors usually have one for the exterior and one for the interior1 so as to make the difficult subject of parallels easier for beginners4 ,,theorem4 ,when two straight lines in the same plane are cut by a transversal1 if the alternate-interior angles are equal1 the two straight lines are parallel4 ,this is the converse of the preceding theorem1 and is half of ,euclid ,i1 #bh1 his theorem being divided for the reason above stated4 ,there are several typical pairs of equal or supplemental angles that would lead to parallel lines1 of which ,euclid uses only part1 leaving the other cases to be inferred4 ,this accounts for the number of corollaries in this connection in later textbooks4 ,surveyors make use of this proposition when they wish1 without using a transit instrument1 to run one line parallel to another4 ,illustration ,for example1 suppose two boys are laying out a tennis court and they wish to run a line through ,p parallel to ,,ab4 ,take a #fj-;foot tape and swing it around ,p until the other end rests on ,,ab1 as at ,m4 ,put a stake at ,o1 #cj feet from ,p and ,m4 ,then take any convenient point ,n on ,,ab1 and measure ,,on4 ,suppose it equals #bj feet4 ,then sight from ,n through ,o1 and put a stake at ,q just #bj feet from ,o4 ,then ,p and ,q determine the parallel1 according to the proposition just mentioned4 ,,theorem4 ,if two parallel lines are cut by a transversal1 the exterior-interior angles are equal4 ,this is also a part of ,euclid ,i1 #bi4 ,it is usually followed by several corollaries1 covering the minor and obvious cases omitted by the older writers4 ,while it would be possible to dispense with these corollaries1 they are helpful for definite reference in later propositions4 ,,theorem4 ,the sum of the three angles of a triangle is equal to two right angles4 ,euclid stated this as follows3 ,in any triangle1 if one of the sides be produced1 the exterior angle is equal to the two interior and opposite angles1 and the three interior angles of the triangle are equal to two right angles4 ,this states more than is necessary for the basal fact of the proposition1 which is the constancy of the sum of the angles4 ,the theorem is one of the three most important propositions in plane geometry1 the other two being the so-called ,pythagorean ,theorem1 and a proposition relating to the proportionality of the sides of two triangles4 ,these three form the foundation of trigonometry and of the mensuration of plane figures4 ,the history of the proposition is extensive4 ,eutocius 7ca4 #eaj ,a4,d471 in his commentary on ,apollonius1 says that ,geminus 7first century ,b4,c47 testified that the ancients investigated the theorem of the two right angles in each individual species of triangle1 first in the equilateral1 again in the isosceles1 and afterwards in the scalene triangle4 ,this1 indeed1 was the ancient plan1 to proceed from the particular to the general4 ,it is the natural order1 it is the world's order1 and it is well to follow it in all cases of difficulty in the classroom4 ,proclus 7#daj-dhe ,a4,d47 tells us that ,eudemus1 who lived just before ,euclid 7or probably about #cbe ,b4,c471 affirmed that the theorem was due to the ,pythagoreans1 although this does not necessarily mean to the actual pupils of ,pythagoras4 ,the proof as he gives it consists in showing that a a'1 b b'1 and a' c b' two right angles4 ,since the proposition about the exterior angle of a triangle is attributed to ,philippus of ,mende 7ca4 #chj ,b4,c471 the figure given by ,eudemus is probably the one used by the ,pythagoreans4 ,illustration ,there is also some reason for believing that ,thales 7ca4 #fjj ,b4,c47 knew the theorem1 for ,diogenes ,laertius 7ca4 #bjj ,a4,d47 quotes ,pamphilius 7first century ,a4,d47 as saying that he1 having learned geometry from the ,egyptians1 was the first to inscribe a right triangle in a circle1 and sacrificed an ox4 ,the proof of this proposition requires the knowledge that the sum of the angles1 at least in a right triangle1 is two right angles4 ,the proposition is frequently referred to by ,aristotle4 ,there have been numerous attempts to prove the proposition without the use of parallel lines4 ,of these a ,german one1 first given by ,thibaut in the early part of the eighteenth century1 is among the most interesting4 ,this1 in simplified form1 is as follows3 ,illustration ,suppose an indefinite line ,,xy to lie on ,,ab4 ,let it swing about ,a1 counterclockwise1 through ,l,a1 so as to lie on ,,ac1 as ,x',y'4 ,then let it swing about ,c1 through ,l,c1 so as to lie on ,,cb1 as ,x'',y''4 ,then let it swing about ,b1 through ,l,b1 so as to lie on ,,ba1 as ,x''',y'''4 ,it now lies on ,,ab1 but it is turned over1 ,x''' being where ,y was1 and ,y''' where ,x was4 ,in turning through ,ls,a1 ,b1 and ,c it has therefore turned through two right angles4 ,one trouble with the proof is that the rotation has not been about the same point1 so that it has never been looked upon as other than an interesting illustration4 ,proclus tried to prove the theorem by saying that1 if we have two perpendiculars to the same line1 and suppose them to revolve about their feet so as to make a triangle1 then the amount taken from the right angles is added to the vertical angle of the triangle1 and therefore the sum of the angles continues to be two right angles4 ,but1 of course1 to prove his statement requires a perpendicular to be drawn from the vertex to the base1 and the theorem of parallels to be applied4 ,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 angle4 ,this theorem furnishes an opportunity for many interesting exercises1 and in particular for determining the third angle when two angles of a triangle are given1 or the second acute angle of a right triangle when one acute angle is given4 ,of the simple outdoor applications of the proposition1 one of the best is illustrated in this figure4 ,illustration ,to ascertain the height of a tree or of the school building1 fold a piece of paper so as to make an angle of #de deg44 ,then walk back from the tree until the top is seen at an angle of #de deg4 with the ground 7being therefore careful to have the base of the triangle level74 ,then the height ,,ac will equal the base ,,ab1 since ,,abc is isosceles4 ,a paper protractor may be used for the same purpose4 ,distances can easily be measured by constructing a large equilateral triangle of heavy pasteboard1 and standing pins at the vertices for the purpose of sighting4 ,illustration ,to measure ,,pc1 stand at some convenient point ,a and sight along ,,apc and also along ,,ab4 ,then walk along ,,ab until a point ,b is reached from which ,,bc makes with ,,ba an angle of the triangle 7#fj deg474 ,then ,,ac ,,ab1 and since ,,ap can be measured1 we can find ,,pc4 ,another simple method of measuring a distance ,,ac across a stream is shown in this figure4 ,illustration ,measure the angle ,,cax1 either in degrees1 with a protractor1 or by sighting along a piece of paper and marking down the angle4 ,then go along ,,xa produced until a point ,b is reached from which ,,bc makes with ,a an angle equal to half of angle ,,cax4 ,then it is easily shown that ,,ab ,,ac4 ,a navigator uses the same principle when he doubles the angle on the bow to find his distance from a lighthouse or other object4 ,illustration ,if he is sailing on the course ,,abc and notes a lighthouse ,l when he is at ,a1 and takes the angle ,a1 and if he notices when the angle that the lighthouse makes with his course is just twice the angle noted at ,a1 then ,,bl ,,ab4 ,he has ,,ab from his log 7an instrument that tells how far a ship goes in a given time71 so he knows ,,bl4 ,he has doubled the angle on the bow to get this distance4 ,it would have been possible for ,thales1 if he knew this proposition1 to have measured the distance of the ship at sea by some such device as this3 ,illustration ,make a large isosceles triangle out of wood1 and1 standing at ,t1 sight to the ship and along the shore on a line ,,ta1 using the vertical angle of the triangle4 ,then go along ,,ta until a point ,p is reached1 from which ,t and ,s can be seen along the sides of a base angle of the triangle4 ,then ,,tp ,,ts4 ,by measuring ,,tb1 ,,bs can then be found4 ,,theorem4 ,the sum of two sides of a triangle is greater than the third side1 and their difference is less than the third side4 ,if the postulate is assumed that a straight line is the shortest path between two points1 then the first part of this theorem requires no further proof1 and the second part follows at once from the axiom of inequalities4 ,this seems the better plan for beginners1 and the proposition may be considered as semiobvious4 ,euclid proved the first part1 not having assumed the postulate4 ,proclus tells us that the ,epicureans 7the followers of ,epicurus1 the ,greek philosopher1 #cdb-bgj ,b4,c47 used to ridicule this theorem1 saying that even an ass knew it1 for if he wished to get food1 he walked in a straight line and not along two sides of a triangle4 ,proclus replied that it was one thing to know the truth and another thing to prove it1 meaning that the value of geometry lay in the proof rather than in the mere facts1 a thing that all who seek to reform the teaching of geometry would do well to keep in mind4 ,the theorem might simply appear as a corollary under the postulate if it were of any importance to reduce the number of propositions one more4 ,if the proposition is postponed until after those concerning the inequalities of angles and sides of a triangle1 there are several good proofs4 ,illustration ,for example1 produce ,,ac to ,x1 making ,,cx ,,cb4 ,then ,l,x ,l,,xbc4 therefore ,l,,xba ,l,x4 therefore ,,ax ,,ab4 therefore ,,ac ,,cb ,,ab4 ,the above proof is due to ,euclid4 ,heron of ,alexandria 7first century ,a4,d47 is said by ,proclus to have given the following3 ,illustration ,let ,,cx bisect ,l,c4 ,then ,l,,bxc ,l,,acx4 therefore ,l,,bxc ,l,,xcb4 therefore ,,cb ,,xb4 ,similarly1 ,,ac ,,ax4 ,adding1 ,,ac ,,cb ,,ab4 ,,theorem4 ,if two sides of a triangle are unequal1 the angles opposite these sides are unequal1 and the angle opposite the greater side is the greater4 ,euclid stated this more briefly by saying1 ,in any triangle the greater side subtends the greater angle4 ,this is not so satisfactory1 for there may be no greater side4 ,,theorem4 ,if two angles of a triangle are unequal1 the sides opposite these angles are unequal1 and the side opposite the greater angle is the greater4 ,euclid also stated this more briefly1 but less satisfactorily1 thus1 ,in any triangle the greater angle is subtended by the greater side4 ,students should have their attention called to the fact that these two theorems are reciprocal or dual theorems1 the words sides and angles of the one corresponding to the words angles and sides respectively of the other4 ,it may also be noticed that the proof of this proposition involves what is known as the ,law of ,converse2 for 7#a7 if b c1 then ,l,b ,l,c2 7#b7 if b c1 then ,l,b ,l,c2 7#c7 if b c1 then ,l,b ,l,c2 therefore the converses must necessarily be true as a matter of logic2 for if ,l,b ,l,c1 then b cannot be greater than c without violating 7#b71 and b cannot be less than c without violating 7#c71 therefore b c2 and if ,l,b ,l,c1 then b cannot equal c without violating 7#a71 and b cannot be less than c without violating 7#c71 therefore b c2 similarly1 if ,l,b ,l,c1 then b c4 ,this ,law of ,converse may readily be taught to pupils1 and it has several applications in geometry4 ,,theorem4 ,if two triangles have two sides of the one equal respectively to two sides of the other1 but the included angle of the first triangle greater than the included angle of the second1 then the third side of the first is greater than the third side of the second1 and conversely4 ,illustration ,in this proposition there are three possible cases3 the point ,y may fall below ,,ab1 as here shown1 or on ,,ab1 or above ,,ab4 ,as an exercise for pupils all three may be considered if desired4 ,following ,euclid and most early writers1 however1 only one case really need be proved1 provided that is the most difficult one1 and is typical4 ,proclus gave the proofs of the other two cases1 and it is interesting to pupils to work them out for themselves4 ,in such work it constantly appears that every proposition suggests abundant opportunity for originality1 and that the complete form of proof in a textbook is not a bar to independent thought4 ,the ,law of ,converse1 mentioned on page #aij1 may be applied to the converse case if desired4 ,,theorem4 ,two angles whose sides are parallel1 each to each1 are either equal or supplementary4 ,this is not an ancient proposition1 although the ,greeks were well aware of the principle4 ,it may be stated so as to include the case of the sides being perpendicular1 each to each1 but this is better left as an exercise4 ,it is possible1 by some circumlocution1 to so state the theorem as to tell in what cases the angles are equal and in what cases supplementary4 ,it cannot be tersely stated1 however1 and it seems better to leave this point as a subject for questioning by the teacher4 ,,theorem4 ,the opposite sides of a parallelogram are equal4 ,,theorem4 ,if the opposite sides of a quadrilateral are equal1 the figure is a parallelogram4 ,illustration ,this proposition is a very simple test for a parallelogram4 ,it is the principle involved in the case of the common folding parallel ruler1 an instrument that has long been recognized as one of the valuable tools of practical geometry4 ,it will be of some interest to teachers to see one of the early forms of this parallel ruler1 as shown in the illustration4#fc ,if such an instrument is not available in the school1 one suitable for illustrative purposes can easily be made from cardboard4 ,illustration3 ,,parallel ,,ruler ,,of ,,the ,,seventeenth ,,century ,san ,giovanni's ,seconda squara mobile1 ,vicenza1 #afhf ,a somewhat more complicated form of this instrument may also be made by pupils in manual training1 as is shown in this illustration from ,bion's great treatise4 ,the principle involved may be taken up in class1 even if the instrument is not used4 ,it is evident that1 unless the workmanship is unusually good1 this form of parallel ruler is not as accurate as the common one illustrated above4 ,the principle is sometimes used in iron gates4 ,illustration3 ,,parallel ,,ruler ,,of ,,the ,,eighteenth ,,century ,n4 ,bion's ,traite de la construction 444 des instrumens de mathematique1 ,the ,hague1 #agbc ,,theorem4 ,two parallelograms are congruent if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other4 ,this proposition is discussed in connection with the one that follows4 ,,theorem4 ,if three or more parallels intercept equal segments on one transversal1 they intercept equal segments on every transversal4 ,these two propositions are not given in ,euclid1 although generally required by ,american syllabi of the present time4 ,the last one is particularly useful in subsequent work4 ,neither one offers any difficulty1 and neither has any interesting history4 ,there are1 however1 numerous interesting applications to the last one4 ,one that is used in mechanical drawing is here illustrated4 ,illustration ,if it is desired to divide a line ,,ab into five equal parts1 we may take a piece of ruled tracing paper and lay it over the given line so that line #j passes through ,a1 and line #e through ,b4 ,we may then prick through the paper and thus determine the points on ,,ab4 ,similarly1 we may divide ,,ab into any other number of equal parts4 ,among the applications of these propositions is an interesting one due to the ,arab ,al-,nairizi 7ca4 #ijj ,a4,d474 ,the problem is to divide a line into any number of equal parts1 and he begins with the case of trisecting ,,ab4 ,it may be given as a case of practical drawing even before the problems are reached1 particularly if some preliminary work with the compasses and straightedge has been given4 ,make ,,bq and ,,aq' perpendicular to ,,ab1 and make ,,bp ,,pq ,,ap' ,p',q'4 ,then triangle,,xyz is congruent to triangle,,ybp1 and also to triangle,,xap'4 ,therefore ,,ax ,,xy ,,yb4 ,in the same way we might continue to produce ,,bq until it is made up of n - #a lengths ,,bp1 and so for ,,aq'1 and by properly joining points we could divide ,,ab into n equal parts4 ,in particular1 if we join ,p and ,p'1 we bisect the line ,,ab4 ,illustration ,,theorem4 ,if two sides of a quadrilateral are equal and parallel1 then the other two sides are equal and parallel1 and the figure is a parallelogram4 ,this was ,euclid's first proposition on parallelograms1 and ,proclus speaks of it as the connecting link between the theory of parallels and that of parallelograms4 ,the ancients1 writing for mature students1 did not add the words and the figure is a parallelogram1 because that follows at once from the first part and from the definition of parallelogram1 but it is helpful to younger students because it emphasizes the fact that here is a test for this kind of figure4 ,,theorem4 ,the diagonals of a parallelogram bisect each other4 ,this proposition was not given in ,euclid1 but it is usually required in ,american syllabi4 ,there is often given in connection with it the exercise in which it is proved that the diagonals of a rectangle are equal4 ,when this is taken1 it is well to state to the class that carpenters and builders find this one of the best checks in laying out floors and other rectangles4 ,it is frequently applied also in laying out tennis courts4 ,if the class is doing any work in mensuration1 such as finding the area of the school grounds1 it is a good plan to check a few rectangles by this method4 ,an interesting outdoor application of the theory of parallelograms is the following3 ,illustration ,suppose you are on the side of this stream opposite to ,,xy1 and wish to measure the length of ,,xy4 ,run a line ,,ab along the bank4 ,then take a carpenter's square1 or even a large book1 and walk along ,,ab until you reach ,p1 a point from which you can just see ,x and ,b along two sides of the square4 ,do the same for ,y1 thus fixing ,p and ,q4 ,using the tape1 bisect ,,pq at ,m4 ,then walk along ,,ym produced until you reach a point ,y' that is exactly in line with ,m and ,y1 and also with ,p and ,x4 ,then walk along ,,xm produced until you reach a point ,x' that is exactly in line with ,m and ,x1 and also with ,q and ,y4 ,then measure ,y',x' and you have the length of ,,xy4 ,for since ,,yx' is perp to ,,pq1 and ,,xy' is also perp to ,,pq1 ,,yx' is to ,,xy'4 ,and since ,,pm ,,mq1 therefore ,,xm ,,mx' and ,y',m ,,my4 ,therefore ,y',x',,yx is a parallelogram4 ,the properties of the parallelogram are often applied to proving figures of various kinds congruent1 or to constructing them so that they will be congruent4 ,illustration ,for example1 if we draw ,a',b' equal and parallel to ,,ab1 ,b',c' equal and parallel to ,,bc1 and so on1 it is easily proved that ,,abcd and ,a',b',c',d' are congruent4 ,this may be done by ordinary superposition1 or by sliding ,,abcd along the dotted parallels4 ,there are many applications of this principle of parallel translation in practical construction work4 ,the principle is more far-reaching than here intimated1 however1 and a few words as to its significance will now be in place4 ,the efforts usually made to improve the spirit of ,euclid are trivial4 ,they ordinarily relate to some commonplace change of sequence1 to some slight change in language1 or to some narrow line of applications4 ,such attempts require no particular thought and yield no very noticeable result4 ,but there is a possibility1 remote though it may be at present1 that a geometry will be developed that will be as serious as ,euclid's and as effective in the education of the thinking individual4 ,if so1 it seems probable that it will not be based upon the congruence of triangles1 by which so many propositions of ,euclid are proved1 but upon certain postulates of motion1 of which one is involved in the above illustration1--the postulate of parallel translation4 ,if to this we join the two postulates of rotation about an axis1#fd leading to axial symmetry2 and rotation about a point1#fe leading to symmetry with respect to a center1 we have a group of three motions upon which it is possible to base an extensive and rigid geometry4#ff ,it will be through some such effort as this1 rather than through the weakening of the ,euclid-,legendre style of geometry1 that any improvement is likely to come4 ,at present1 in ,america1 the important work for teachers is to vitalize the geometry they have1--an effort in which there are great possibilities1--seeing to it that geometry is not reduced to mere froth1 and recognizing the possibility of another geometry that may sometime replace it1--a geometry as rigid1 as thought-compelling1 as logical1 and as truly educational4 ,,theorem4 ,the sum of the interior angles of a polygon is equal to two right angles1 taken as many times less two as the figure has sides4 ,this interesting generalization of the proposition about the sum of the angles of a triangle is given by ,proclus4 ,there are several proofs1 but all are based upon the possibility of dissecting the polygon into triangles4 ,the point from which lines are drawn to the vertices is usually taken at a vertex1 so that there are n - #b triangles4 ,it may however be taken within the figure1 making n triangles1 from the sum of the angles of which the four right angles about the point must be subtracted4 ,the point may even be taken on one side1 or outside the polygon1 but the proof is not so simple4 ,teachers who desire to do so may suggest to particularly good students the proving of the theorem for a concave polygon1 or even for a cross polygon1 although the latter requires negative angles4 ,some schools have transit instruments for the use of their classes in trigonometry4 ,in such a case it is a good plan to measure the angles in some piece of land so as to verify the proposition1 as well as show the care that must be taken in reading angles4 ,in the absence of this exercise it is well to take any irregular polygon and measure the angles by the help of a protractor1 and thus accomplish the same results4 ,,theorem4 ,the sum of the exterior angles of a polygon1 made by producing each of its sides in succession1 is equal to four right angles4 ,this is also a proposition not given by the ancient writers4 ,we have1 however1 no more valuable theorem for the purpose of showing the nature and significance of the negative angle2 and teachers may arouse a great deal of interest in the negative quantity by showing to a class that when an interior angle becomes #ahj deg4 the exterior angle becomes #j1 and when the polygon becomes concave the exterior angle becomes negative1 the theorem holding for all these cases4 ,we have few better illustrations of the significance of the negative quantity1 and few better opportunities to use the knowledge of this kind of quantity already acquired in algebra4 ,illustration ,in the hilly and mountainous parts of ,america1 where irregular-shaped fields are more common than in the more level portions1 a common test for a survey is that of finding the exterior angles when the transit instrument is set at the corners4 ,in this field these angles are given1 and it will be seen that the sum is #cfj deg44 ,in the absence of any outdoor work a protractor may be used to measure the exterior angles of a polygon drawn on paper4 ,if there is an irregular piece of land near the school1 the exterior angles can be fairly well measured by an ordinary paper protractor4 ,the idea of locus is usually introduced at the end of ,book ,i4 ,it is too abstract to be introduced successfully any earlier1 although authors repeat the attempt from time to time1 unmindful of the fact that all experience is opposed to it4 ,the loci propositions are not ancient4 ,the ,greeks used the word locus 7in ,greek1 topos71 however4 ,proclus1 for example1 says1 ,i call those locus theorems in which the same property is found to exist on the whole of some locus4 ,teachers should be careful to have the pupils recognize the necessity for proving two things with respect to any locus3 7#a7 that any point on the supposed locus satisfies the condition2 7#b7 that any point outside the supposed locus does not satisfy the given condition4 ,the first of these is called the sufficient condition1 and the second the necessary condition4 ,thus in the case of the locus of points in a plane equidistant from two given points1 it is sufficient that the point be on the perpendicular bisector of the line joining the given points1 and this is the first part of the proof2 it is also necessary that it be on this line1 i4e4 it cannot be outside this line1 and this is the second part of the proof4 ,the proof of loci cases1 therefore1 involves a consideration of the necessary and sufficient condition that is so often spoken of in higher mathematics4 ,this expression might well be incorporated into elementary geometry1 and when it becomes better understood by teachers1 it probably will be more often used4 ,in teaching loci it is helpful to call attention to loci in space 7meaning thereby the space of three dimensions71 without stopping to prove the proposition involved4 ,indeed1 it is desirable all through plane geometry to refer incidentally to solid geometry4 ,in the mensuration of plane figures1 which may be boundaries of solid figures1 this is particularly true4 ,it is a great defect in most school courses in geometry that they are entirely confined to two dimensions4 ,even if solid geometry in the usual sense is not attempted1 every occasion should be taken to liberate boys' minds from what becomes the tyranny of paper4 ,thus the questions3 ,what is the locus of a point equidistant from two given points2 at a constant distance from a given straight line or from a given point8 should be extended to space4#fg ,the two loci problems usually given at this time1 referring to a point equidistant from the extremities of a given line1 and to a point equidistant from two intersecting lines1 both permit of an interesting extension to three dimensions without any formal proof4 ,it is possible to give other loci at this point1 but it is preferable merely to introduce the subject in ,book ,i1 reserving the further discussion until after the circle has been studied4 ,it is well1 in speaking of loci1 to remember that it is entirely proper to speak of the locus of a point or the locus of points4 ,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 circle4 ,in analytic geometry we usually speak of the locus of a point1 thinking of the point as being anywhere on the locus4 ,some teachers of elementary geometry1 however1 prefer to speak of the locus of points1 or the locus of all points1 thus tending to make the language of elementary geometry differ from that of analytic geometry4 ,since it is a trivial matter of phraseology1 it is better to recognize both forms of expression and to let pupils use the two interchangeably4 ,,footnotes3 #eg ,address at ,brussels1 ,august1 #aiaj4 #eh ,for a recent discussion of this general subject1 see ,professor ,hobson on ,the ,tendencies of ,modern ,mathematics1 in the ,educational ,review1 ,new ,york1 #aiaj1 ,vol4 ,,xl1 p4 #ebd4 #ei ,a more extended list of applications is given later in this work4 #fj ,abu'l-',abbas al-,fadl ibn ,hatim al-,nairizi1 so called from his birthplace1 ,nairiz1 was a well-known ,arab writer4 ,he died about #ibb ,a4,d4 ,he wrote a commentary on ,euclid4 #fa ,this illustration1 taken from a book in the author's library1 appeared in a valuable monograph by ,w4 ,e4 ,stark1 ,measuring ,instruments of ,long ,ago1 published in ,school ,science and ,mathematics1 ,vol4 ,x1 pp4 #dh1 #abf4 ,with others of the same nature it is here reproduced by the courtesy of ,principal ,stark and of the editors of the journal in which it appeared4 #fb ,in speaking of two congruent triangles it is somewhat easier to follow the congruence if the two are read in the same order1 even though the relatively unimportant counterclockwise reading is neglected4 ,no one should be a slave to such a formalism1 but should follow the plan when convenient4 #fc ,stark1 loc4 cit4 #fd ,of which so much was made by ,professor ,olaus ,henrici in his ,congruent ,figures1 ,london1 #ahgi1--;a book that every teacher of geometry should own4 #fe ,much is made of this in the excellent work by ,henrici and ,treutlein1 ,lehrbuch der ,geometrie1 ,leipzig1 #ahha4 #ff ,meray did much for this movement in ,france1 and the recent works of ,bourlet and ,borel have brought it to the front in that country4 #fg ,w4 ,n4 ,bruce1 ,teaching of ,geometry and ,graphic ,algebra in ,secondary ,schools1 ,board of ,education circular 7,no4 #gaa71 p4 #h1 ,london1 #aiji4 ,,chapter ,,xv ,,the ,,leading ,,propositions ,,of ,,book ,,ii ,having taken up all of the propositions usually given in ,book ,i1 it seems unnecessary to consider as specifically all those in subsequent books4 ,it is therefore proposed to select certain ones that have some special interest1 either from the standpoint of mathematics or from that of history or application1 and to discuss them as fully as the circumstances seem to warrant4 ,,theorems4 ,in the same circle or in equal circles equal central angles intercept equal arcs2 and of two unequal central angles the greater intercepts the greater arc1 and conversely for both of these cases4 ,euclid made these the twenty-sixth and twenty-seventh propositions of his ,book ,,iii1 but he limited them as follows3 ,in equal circles equal angles stand on equal circumferences1 whether they stand at the centers or at the circumferences1 and conversely4 ,he therefore included two of our present theorems in one1 thus making the proposition doubly hard for a beginner4 ,after these two propositions the ,law of ,converse1 already mentioned on page #aij1 may properly be introduced4 ,,theorems4 ,in the same circle or in equal circles1 if two arcs are equal1 they are subtended by equal chords2 and if two arcs are unequal1 the greater is subtended by the greater chord1 and conversely4 ,euclid dismisses all this with the simple theorem1 ,in equal circles equal circumferences are subtended by equal straight lines4 ,it will therefore be noticed that he has no special word for chord and none for arc1 and that the word circumference1 which some teachers are so anxious to retain1 is used to mean both the whole circle and any arc4 ,it cannot be doubted that later writers have greatly improved the language of geometry by the use of these modern terms4 ,the word arc is the same1 etymologically1 as arch1 each being derived from the ,latin arcus 7a bow74 ,chord is from the ,greek1 meaning the string of a musical instrument4 ,subtend is from the ,latin sub 7under71 and tendere 7to stretch74 ,it should be noticed that ,euclid speaks of equal circles1 while we speak of the same circle or equal circles1 confining our proofs to the latter1 on the supposition that this sufficiently covers the former4 ,,theorem4 ,a line through the center of a circle perpendicular to a chord bisects the chord and the arcs subtended by it4 ,this is an improvement on ,euclid1 ,,iii1 #c3 ,if in a circle a straight line through the center bisects a straight line not through the center1 it also cuts it at right angles2 and if it cuts it at right angles1 it also bisects it4 ,it is a very important proposition1 theoretically and practically1 for it enables us to find the center of a circle if we know any part of its arc4 ,a civil engineer1 for example1 who wishes to find the center of the circle of which some curve 7like that on a running track1 on a railroad1 or in a park7 is an arc1 takes two chords1 say of one hundred feet each1 and erects perpendicular bisectors4 ,it is well to ask a class why1 in practice1 it is better to take these chords some distance apart4 ,engineers often check their work by taking three chords1 the perpendicular bisectors of the three passing through a single point4 ,illustrations of this kind of work are given later in this chapter4 ,,theorem4 ,in the same circle or in equal circles equal chords are equidistant from the center1 and chords equidistant from the center are equal4 ,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 tape1 or that cannot easily be reached from the center on account of obstructions4 ,illustration ,if part of the curve ,,apb is known1 take ,p as the mid-point4 ,then stretch the tape from ,a to ,b and draw ,,pm perpendicular to it4 ,then swing the length ,,am about ,p1 and ,,pm about ,b1 until they meet at ,l1 and stretch the length ,,ab along ,,pl to ,q4 ,this fixes the point ,q4 ,in the same way fix the point ,c4 ,points on the curve can thus be fixed as near together as we wish4 ,the chords ,,ab1 ,,pq1 ,,bc1 and so on1 are equal and are equally distant from the center4 ,,theorem4 ,a line perpendicular to a radius at its extremity is tangent to the circle4 ,the enunciation of this proposition by ,euclid is very interesting4 ,it is as follows3 ,the straight line drawn at right angles to the diameter of a circle at its extremity will fall outside the circle1 and into the space between the straight line and the circumference another straight line cannot be interposed2 further1 the angle of the semicircle is greater and the remaining angle less than any acute rectilineal angle4 ,the first assertion is practically that of tangency1--will fall outside the circle4 ,the second one states1 substantially1 that there is only one such tangent1 or1 as we say in modern mathematics1 the tangent is unique4 ,the third statement relates to the angle formed by the diameter and the circumference1--a mixed angle1 as ,proclus called it1 and a kind of angle no longer used in elementary geometry4 ,the fourth statement practically asserts that the angle between the tangent and circumference is less than any assignable quantity4 ,this gives rise to a difficulty that seems to have puzzled many of ,euclid's commentators1 and that will interest a pupil3 ,as the circle diminishes this angle apparently increases1 while as the circle increases the angle decreases1 and yet the angle is always stated to be zero4 ,vieta 7#aedj-afjc71 who did much to improve the science of algebra1 attempted to explain away the difficulty by adopting a notion of circle that was prevalent in his time4 ,he said that a circle was a polygon of an infinite number of sides 7which it cannot be1 by definition71 and that1 a tangent simply coincided with one of the sides1 and therefore made no angle with it2 and this view was also held by ,galileo 7#aefd-afdb71 the great physicist and mathematician who first stated the law of the pendulum4 ,,theorem4 ,parallel lines intercept equal arcs on a circle4 ,the converse of this proposition has an interesting application in outdoor work4 ,illustration ,suppose we wish to run a line through ,p parallel to a given line ,,ab4 ,with any convenient point ,o as a center1 and ,,op as a radius1 describe a circle cutting ,,ab in ,x and ,y4 ,draw ,,px4 ,then with ,y as a center and ,,px as a radius draw an arc cutting the circle in ,q4 ,then run the line from ,p to ,q4 ,,pq is parallel to ,,ab by the converse of the above theorem1 which is easily shown to be true for this figure4 ,,theorem4 ,if two circles are tangent to each other1 the line of centers passes through the point of contact4 ,there are many illustrations of this theorem in practical work1 as in the case of cogwheels4 ,an interesting application to engineering is seen in the case of two parallel streets or lines of track which are to be connected by a reversed curve4 ,illustration ,if the lines are ,,ab and ,,cd1 and the connection is to be made1 as shown1 from ,b to ,c1 we may proceed as follows3 ,draw ,,bc and bisect it at ,m4 ,erect ,,po1 the perpendicular bisector of ,,bm2 and ,,bo1 perpendicular to ,,ab4 ,then ,o is one center of curvature4 ,in the same way fix ,o'4 ,then to check the work apply this theorem1 ,m being in the line of centers ,,oo'4 ,the curves may now be drawn1 and they will be tangent to ,,ab1 to ,,cd1 and to each other4 ,at this point in the ,american textbooks it is the custom to insert a brief treatment of measurement1 explaining what is meant by ratio1 commensurable and incommensurable quantities1 constant and variable1 and limit1 and introducing one or more propositions relating to limits4 ,the object of this departure from the ancient sequence1 which postponed this subject to the book on ratio and proportion1 is to treat the circle more completely in ,book ,,iii4 ,it must be confessed that the treatment is not as scientific as that of ,euclid1 as will be explained under ,book ,,iii1 but it is far better suited to the mind of a boy or girl4 ,it begins by defining measurement in a practical way1 as the finding of the number of times a quantity of any kind contains a known quantity of the same kind4 ,of course this gives a number1 but this number may be a surd1 like sqrt#b4 ,in other words1 the magnitude measured may be incommensurable with the unit of measure1 a seeming paradox4 ,with this difficulty1 however1 the pupil should not be called upon to contend at this stage in his progress4 ,the whole subject of incommensurables might safely be postponed1 although it may be treated in an elementary fashion at this time4 ,the fact that the measure of the diagonal of a square1 of which a side is unity1 is sqrt#b1 and that this measure is an incommensurable number1 is not so paradoxical as it seems1 the paradox being verbal rather than actual4 ,it is then customary to define ratio as the quotient of the numerical measures of two quantities in terms of a common unit4 ,this brings all ratios to the basis of numerical fractions1 and while it is not scientifically so satisfactory as the ancient concept which considered the terms as lines1 surfaces1 angles1 or solids1 it is more practical1 and it suffices for the needs of elementary pupils4 ,commensurable1 incommensurable1 constant1 and variable are then defined1 and these definitions are followed by a brief discussion of limit4 ,it simplifies the treatment of this subject to state at once that there are two classes of limits1--those which the variable actually reaches1 and those which it can only approach indefinitely near4 ,we find the one as frequently as we find the other1 although it is the latter that is referred to in geometry4 ,for example1 the superior limit of a chord is a diameter1 and this limit the chord may reach4 ,the inferior limit is zero1 but we do not consider the chord as reaching this limit4 ,it is also well to call the attention of pupils to the fact that a quantity may decrease towards its limit as well as increase towards it4 ,such further definitions as are needed in the theory of limits are now introduced4 ,among these is area of a circle4 ,it might occur to some pupil that since a circle is a line 7as used in modern mathematics71 it can have no area4 ,this is1 however1 a mere quibble over words4 ,it is not pretended that the line has area1 but that area of a circle is merely a shortened form of the expression area inclosed by a circle4 ,the ,principle of ,limits is now usually given as follows3 ,if1 while approaching their respective limits1 two variables are always equal1 their limits are equal4 ,this was expressed by ,d',alembert in the eighteenth century as ,magnitudes which are the limits of equal magnitudes are equal1 or this in substance4 ,it would easily be possible to elaborate this theory1 proving1 for example1 that if x approaches y as its limit1 then ax approaches ay as its limit1 and x_/a approaches y_/a as its limit1 and so on4 ,very much of this theory1 however1 wearies a pupil so that the entire meaning of the subject is lost1 and at best the treatment in elementary geometry is not rigorous4 ,it is another case of having to sacrifice a strictly scientific treatment to the educational abilities of the pupil4 ,teachers wishing to find a scientific treatment of the subject should consult a good work on the calculus4 ,,theorem4 ,in the same circle or in equal circles two central angles have the same ratio as their intercepted arcs4 ,this is usually proved first for the commensurable case and then for the incommensurable one4 ,the latter is rarely understood by all of the class1 and it may very properly be required only of those who show some aptitude in geometry4 ,it is better to have the others understand fully the commensurable case and see the nature of its applications1 possibly reading the incommensurable proof with the teacher1 than to stumble about in the darkness of the incommensurable case and never reach the goal4 ,in ,euclid there was no distinction between the two because his definition of ratio covered both2 but1 as we shall see in ,book ,,iii1 this definition is too difficult for our pupils4 ,theon of ,alexandria 7fourth century ,a4,d471 the father of the ,hypatia who is the heroine of ,kingsley's well-known novel1 wrote a commentary on ,euclid1 and he adds that sectors also have the same ratio as the arcs1 a fact very easily proved4 ,in propositions of this type1 referring to the same circle or to equal circles1 it is not worth while to ask pupils to take up both cases1 the proof for either being obviously a proof for the other4 ,many writers state this proposition so that it reads that central angles are measured by their intercepted arcs4 ,this1 of course1 is not literally true1 since we can measure anything only by some thing1 of the same kind4 ,thus we measure a volume by finding how many times it contains another volume which we take as a unit1 and we measure a length by taking some other length as a unit2 but we cannot measure a given length in quarts nor a given weight in feet1 and it is equally impossible to measure an arc by an angle1 and vice versa4 ,nevertheless it is often found convenient to define some brief expression that has no meaning if taken literally1 in such way that it shall acquire a meaning4 ,thus we define area of a circle1 even when we use circle to mean a line2 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 arcs4 ,this is done by most writers1 and is legitimate as explaining an abbreviated expression4 ,,theorem4 ,an inscribed angle is measured by half the intercepted arc4 ,in ,euclid this proposition is combined with the preceding one in his ,book ,,vi1 ,proposition #cc4 ,such a procedure is not adapted to the needs of students to-day4 ,euclid gave in ,book ,,iii1 however1 the proposition 7,no4 #bj7 that a central angle is twice an inscribed angle standing on the same arc4 ,since ,euclid never considered an angle greater than #ahj deg41 his inscribed angle was necessarily less than a right angle4 ,the first one who is known to have given the general case1 taking the central angle as being also greater than #ahj deg41 was ,heron of ,alexandria1 probably of the first century ,a4,d4#fh ,in this he was followed by various later commentators1 including ,tartaglia and ,clavius in the sixteenth century4 ,one of the many interesting exercises that may be derived from this theorem is seen in the case of the horizontal danger angle observed by ships4 ,illustration ,if some dangerous rocks lie off the shore1 and ,l and ,l' are two lighthouses1 the angle ,a is determined by observation1 so that ,a will lie on a circle inclosing the dangerous area4 ,angle ,a is called the horizontal danger angle4 ,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 ,a4 ,to this proposition there are several important corollaries1 including the following3 #a4 ,an angle inscribed in a semicircle is a right angle4 ,this corollary is mentioned by ,aristotle and is attributed to ,thales1 being one of the few propositions with which his name is connected4 ,it enables us to describe a circle by letting the arms of a carpenter's square slide along two nails driven in a board1 a pencil being held at the vertex4 ,illustration ,a more practical use for it is made by machinists to determine whether a casting is a true semicircle4 ,taking a carpenter's square as here shown1 if the vertex touches the curve at every point as the square slides around1 it is a true semicircle4 ,by a similar method a circle may be described by sliding a draftsman's triangle so that two sides touch two tacks driven in a board4 ,illustration ,another interesting application of this corollary may be seen by taking an ordinary paper protractor ,,acb1 and fastening a plumb line at ,b4 ,if the protractor is so held that the plumb line cuts the semicircle at ,c1 then ,,ac is level because it is perpendicular to the vertical line ,,bc4 ,thus1 if a class wishes to determine the horizontal line ,,ac1 while sighting up a hill in the direction ,,ab1 this is easily determined without a spirit level4 ,it follows from this corollary1 as the pupil has already found1 that the mid-point of the hypotenuse of a right triangle is equidistant from the three vertices4 ,this is useful in outdoor measuring1 forming the basis of one of the best methods of letting fall a perpendicular from an external point to a line4 ,illustration ,suppose ,,xy to be the edge of a sidewalk1 and ,p a point in the street from which we wish to lay a gas pipe perpendicular to the walk4 ,from ,p swing a cord or tape1 say #fj feet long1 until it meets ,,xy at ,a4 ,then take ,m1 the mid-point of ,,pa1 and swing ,,mp about ,m1 to meet ,,xy at ,b4 ,then ,b is the foot of the perpendicular1 since ,l,,pba can be inscribed in a semicircle4 #b4 ,angles inscribed in the same segment are equal4 ,illustration ,by driving two nails in a board1 at ,a and ,b1 and taking an angle ,p made of rigid material 7in particular1 as already stated1 a carpenter's square71 a pencil placed at ,p will generate an arc of a circle if the arms slide along ,a and ,b4 ,this is an interesting exercise for pupils4 ,,theorem4 ,an angle formed by two chords intersecting within the circle is measured by half the sum of the intercepted arcs4 ,,theorem4 ,an angle formed by a tangent and a chord drawn from the point of tangency is measured by half the intercepted arc4 ,,theorem4 ,an angle formed by two secants1 a secant and a tangent1 or two tangents1 drawn to a circle from an external point1 is measured by half the difference of the intercepted arcs4 ,these three theorems are all special cases of the general proposition that the angle included between two lines that cut 7or touch7 a circle is measured by half the sum of the intercepted arcs4 ,if the point passes from within the circle to the circle itself1 one arc becomes zero and the angle becomes an inscribed angle4 ,if the point passes outside the circle1 the smaller arc becomes negative1 having passed through zero4 ,the point may even go to infinity1 as is said in higher mathematics1 the lines then becoming parallel1 and the angle becoming zero1 being measured by half the sum of one arc and a negative arc of the same absolute value4 ,this is one of the best illustrations of the ,principle of ,continuity to be found in geometry4 ,,problem4 ,to let fall a perpendicular upon a given line from a given external point4 ,this is the first problem that a student meets in most ,american geometries4 ,the reason for treating the problems by themselves instead of mingling them with the theorems has already been discussed4#fi ,the student now has a sufficient body of theorems1 by which he can prove that his constructions are correct1 and the advantage of treating these constructions together is greater than that of following ,euclid's plan of introducing them whenever needed4 ,proclus tells us that this problem was first investigated by ,oenopides1#gj who thought it useful for astronomy4 ,proclus speaks of such a line as a gnomon1 a common name for the perpendicular on a sundial1 which casts the shadow by which the time of day is known4 ,he also speaks of two kinds of perpendiculars1 the plane and solid1 the former being a line perpendicular to a line1 and the latter a line perpendicular to a plane4 ,it is interesting to notice that the solution tacitly assumes that a certain arc is going to cut the given line in two points1 and only two4 ,strictly speaking1 why may it not cut it in only one point1 or even in three points8 ,we really assume that if a straight line is drawn through a point within a circle1 this line must get out of the circle on each of two sides of the given point1 and in getting out it must cut the circle twice4 ,proclus noticed this assumption and endeavored to prove it4 ,it is better1 however1 not to raise the question with beginners1 since it seems to them like hair-splitting to no purpose4 ,the problem is of much value in surveying1 and teachers would do well to ask a class to let fall a perpendicular to the edge of a sidewalk from a point #bj feet from the walk1 using an ordinary #ff-;foot or #ej-;foot tape4 ,practically1 the best plan is to swing #cj feet of the tape about the point and mark the two points of intersection with the edge of the walk4 ,then measure the distance between the points and take half of this distance1 thus fixing the foot of the perpendicular4 ,,problem4 ,at a given point in a line1 to erect a perpendicular to that line4 ,this might be postponed until after the problem to bisect an angle1 since it merely requires the bisection of a straight angle2 but considering the immaturity of the average pupil1 it is better given independently4 ,the usual case considers the point not at the extremity of the line1 and the solution is essentially that of ,euclid4 ,in practice1 however1 as for example in surveying1 the point may be at the extremity1 and it may not be convenient to produce the line4 ,illustration ,surveyors sometimes measure ,,pb #c ft41 and then take #i ft4 of tape1 the ends being held at ,b and ,p1 and the tape being stretched to ,a1 so that ,,pa #d ft4 and ,,ab #e ft4 ,then ,p is a right angle by the ,pythagorean ,theorem4 ,this theorem not having yet been proved1 it cannot be used at this time4 ,a solution for the problem of erecting a perpendicular from the extremity of a line that cannot be produced1 depending1 however1 on the problem of bisecting an angle1 and therefore to be given after that problem1 is attributed by ,al-,nairizi 7tenth century ,a4,d47 to ,heron of ,alexandria4 ,it is also given by ,proclus4 ,illustration ,required to draw from ,p a perpendicular to ,,ap4 ,take ,x anywhere on the line and erect ,,xy perp to ,,ap in the usual manner4 ,bisect ,l,,pxy by the line ,,xm4 ,on ,,xy take ,,xn ,,xp1 and draw ,,nm perp to ,,xy4 ,then draw ,,pm4 ,the proof is evident4 ,these may at the proper time be given as interesting variants of the usual solution4 ,,problem4 ,to bisect a given line4 ,euclid said finite straight line1 but this wording is not commonly followed1 because it will be inferred that the line is finite if it is to be bisected1 and we use line alone to mean a straight line4 ,euclid's plan was to construct an equilateral triangle 7by his ,proposition #a of ,book ,i7 on the line as a base1 and then to bisect the vertical angle4 ,proclus tells us that ,apollonius of ,perga1 who wrote the first great work on conic sections1 used a plan which is substantially that which is commonly found in textbooks to-day1--constructing two isosceles triangles upon the line as a common base1 and connecting their vertices4 ,,problem4 ,to bisect a given angle4 ,it should be noticed that in the usual solution two arcs intersect1 and the point thus determined is connected with the vertex4 ,now these two arcs intersect twice1 and since one of the points of intersection may be the vertex itself1 the other point of intersection must be taken4 ,it is not1 however1 worth while to make much of this matter with pupils4 ,proclus calls attention to the possible suggestion that the point of intersection may be imagined to lie outside the angle1 and he proceeds to show the absurdity2 but here1 again1 the subject is not one of value to beginners4 ,he also contributes to the history of the trisection of an angle4 ,any angle is easily trisected by means of certain higher curves1 such as the conchoid of ,nicomedes 7ca4 #ahj ,b4,c471 the quadratrix of ,hippias of ,elis 7ca4 #dbj ,b4,c471 or the spiral of ,archimedes 7ca4 #bej ,b4,c474 ,but since this problem1 stated algebraically1 requires the solution of a cubic equation1 and this involves1 geometrically1 finding three points1 we cannot solve the problem by means of straight lines and circles alone4 ,in other words1 the trisection of any angle1 by the use of the straightedge and compasses alone1 is impossible4 ,special angles may however be trisected4 ,thus1 to trisect an angle of #ij deg4 we need only to construct an angle of #fj deg41 and this can be done by constructing an equilateral triangle4 ,but while we cannot trisect the angle1 we may easily approximate trisection4 ,for since1 in the infinite geometric series #a_/b #a_/h #a_/cb #a_/abh 4441 s a _/ 7#a - r71 we have s #a_/b _/ #c_/d #b_/c4 ,in other words1 if we add #a_/b of the angle1 #a_/h of the angle1 #a_/cb of the angle1 and so on1 we approach as a limit #b_/c of the angle2 but all of these fractions can be obtained by repeated bisections1 and hence by bisections we may approximate the trisection4 ,the approximate bisection 7or any other division7 of an angle may of course be effected by the help of the protractor and a straightedge4 ,the geometric method is1 however1 usually more accurate1 and it is advantageous to have the pupils try both plans1 say for bisecting an angle of about #di #a_/b deg44 ,illustration ,applications of this problem are numerous4 ,it may be desired1 for example1 to set a lamp-post on a line bisecting the angle formed by two streets that come together a little unsymmetrically1 as here shown1 in which case the bisecting line can easily be run by the use of a measuring tape1 or even of a stout cord4 ,a more interesting illustration is1 however1 the following3 ,illustration ,let the pupils set a stake1 say about #e feet high1 at a point ,n on the school grounds about #i ,a4,m41 and carefully measure the length of the shadow1 ,,nw1 placing a small wooden pin at ,w4 ,then about #c ,p4,m4 let them watch until the shadow ,,ne is exactly the same length that it was when ,w was fixed1 and then place a small wooden pin at ,e4 ,if the work has been very carefully done1 and they take the tape and bisect the line ,,we1 thus fixing the line ,,ns1 they will have a north and south line4 ,if this is marked out for a short distance from ,n1 then when the shadow falls on ,,ns1 it will be noon by sun time 7not standard time7 at the school4 ,,problem4 ,from a given point in a given line1 to draw a line making an angle equal to a given angle4 ,proclus says that ,eudemus attributed to ,oenopides the discovery of the solution which ,euclid gave1 and which is substantially the one now commonly seen in textbooks4 ,the problem was probably solved in some fashion before the time of ,oenopides1 however4 ,the object of the problem is primarily to enable us to draw a line parallel to a given line4 ,practically1 the drawing of one line parallel to another is usually effected by means of a parallel ruler 7see page #aia71 or by the use of draftsmen's triangles1 as here shown1 or even more commonly by the use of a ,t-square1 such as is here seen4 ,this illustration shows two ,t-squares used for drawing lines parallel to the sides of a board upon which the drawing paper is fastened4#ga ,illustration ,illustration ,an ingenious instrument described by ,baron ,dupin is illustrated below4 ,illustration ,to the bar ,a is fastened the sliding check ,b4 ,a movable check ,d may be fastened by a screw ,c4 ,a sharp point is fixed in ,b1 so that as ,d slides along the edge of a board1 the point marks a line parallel to the edge4 ,moreover1 ,f and ,g are two brass arms of equal length joined by a pointed screw ,h that marks a line midway between ,b and ,d4 ,furthermore1 it is evident that ,h will draw a line bisecting any irregular board if the checks ,b and ,d are kept in contact with the irregular edges4 ,book ,,ii offers two general lines of application that may be introduced to advantage1 preferably as additions to the textbook work4 ,one of these has reference to topographical drawing and related subjects1 and the other to geometric design4 ,as long as these can be introduced to the pupil with an air of reality1 they serve a good purpose1 but if made a part of textbook work1 they soon come to have less interest than the exercises of a more abstract character4 ,if a teacher can relate the problems in topographical drawing to the pupil's home town1 and can occasionally set some outdoor work of the nature here suggested1 the results are usually salutary2 but if he reiterates only a half-dozen simple propositions time after time1 with only slight changes in the nature of the application1 then the results will not lead to a cultivation of power in geometry1--a point which the writers on applied geometry usually fail to recognize4 ,illustration ,one of the simple applications 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 corner4 ,it is also used in laying out park walks and drives4 ,illustration ,the figure in the middle of the page represents two streets1 ,,ap and ,,bq1 that would1 if prolonged1 intersect at ,c4 ,it is required to construct an arc so that they shall begin to curve at ,p and ,q1 where ,,cp ,,cq1 and hence the center of curvature ,o must be found4 ,the problem is a common one in railroad work1 only here ,,ap is usually oblique to ,,bq if they are produced to meet at ,c1 as in the second figure on page #bah4 ,it is required to construct an arc so that the tracks shall begin to curve at ,p and ,q1 where ,,cp ,,cq4 ,illustration ,the problem becomes a little more complicated1 and correspondingly more interesting1 when we have to find the center of curvature for a street railway track that must turn a corner in such a way as to allow1 say1 exactly #e feet from the point ,p1 on account of a sidewalk4 ,illustration ,the problem becomes still more difficult if we have two roads of different widths that we wish to join on a curve4 ,here the two centers of curvature are not the same1 and the one road narrows to the other on the curve4 ,the solutions will be understood from a study of the figures4 ,the number of problems of this kind that can easily be made is limitless1 and it is well to avoid the danger of hobby riding on this or any similar topic4 ,therefore a single one will suffice to close this group4 ,illustration ,if a road ,,ab on an arc described about ,o1 is to be joined to road ,,cd1 described about ,o'1 the arc ,,bc should evidently be internally tangent to ,,ab and externally tangent to ,,cd4 ,hence the center is on ,,box and ,o',,cy1 and is therefore at ,p4 ,the problem becomes more real if we give some width to the roads in making the drawing1 and imagine them in a park that is being laid out with drives4 ,it will be noticed that the above problems require the erecting of perpendiculars1 the bisecting of angles1 and the application of the propositions on tangents4 ,a somewhat different line of problems is that relating to the passing of a circle through three given points4 ,it is very easy to manufacture problems of this kind that have a semblance of reality4 ,illustration ,for example1 let it be required to plan a driveway from the gate ,g to the porch ,p so as to avoid a mass of rocks ,r1 an arc of a circle to be taken4 ,of course1 if we allow pupils to use the ,pythagorean ,theorem at this time 7and for metrical purposes this is entirely proper1 because they have long been familiar with it71 then we may ask not only for the drawing1 but we may1 for example1 give the length from ,g to the point on ,r 7which we may also call ,r71 and the angle ,,rgo as #fj deg41 to find the radius4 ,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 geometry4 ,the copying or the making of designs requiring the describing of circles1 their inscription in or circumscription about triangles1 and their construction in various positions of tangency1 has some value as applying the various problems studied in this book4 ,for a number of years past1 several enthusiastic teachers have made much of the designs found in ,gothic windows1 having their pupils make the outline drawings by the help of compasses and straightedge4 ,while such work has its value1 it is liable soon to degenerate into purposeless formalism1 and hence to lose interest by taking the vigorous mind of youth from the strong study of geometry to the weak manipulation of instruments4 ,nevertheless its value should be appreciated and conserved1 and a few illustrations of these forms are given in order that the teacher may have examples from which to select4 ,the best way of using this material is to offer it as supplementary work1 using much or little1 as may seem best1 thus giving to it a freshness and interest that some have trouble in imparting to the regular book work4 ,the best plan is to sketch rapidly the outline of a window on the blackboard1 asking the pupils to make a rough drawing1 and to bring in a mathematical drawing on the following day4 ,illustration ,it might be said1 for example1 that in planning a ,gothic window this drawing is needed4 ,the arc ,,bc is drawn with ,a as a center and ,,ab as a radius4 ,the small arches are described with ,a1 ,d1 and ,b as centers and ,,ad as a radius4 ,the center ,p is found by taking ,a and ,b as centers and ,,ae as a radius4 ,how may the points ,d1 ,e1 and ,f be found8 ,draw the figure4 ,from the study of the rectilinear figures suggested by such a simple pattern the properties of the equilateral triangle may be inferred4 ,the ,gothic window also offers some interesting possibilities in connection with the study of the square4 ,for example1 the illustration given on page #bbc shows a number of traceries involving the construction of a square1 the bisecting of angles1 and the describing of circles4#gb ,illustration ,the properties of the square1 a figure now easily constructed by the pupils1 are not numerous4 ,what few there are may be brought out through the study of art forms1 if desired4 ,in case these forms are shown to a class1 it is important that they should be selected from good designs4 ,we have enough poor art in the world1 so that geometry should not contribute any more4 ,this illustration is a type of the best medieval ,gothic parquetry4#gc ,illustration3 ,,gothic ,,designs ,,employing ,,circles ,,and ,,bisected ,,angles ,even simple designs of a semipuzzling nature have their advantage in this connection4 ,in the following example the inner square contains all of the triangles1 the letters showing where they may be fitted4#gd ,still more elaborate designs1 based chiefly upon the square and circle1 are shown in the window traceries on page #bbe1 and others will be given in connection with the study of the regular polygons4 ,illustration ,designs like the figure below are typical of the simple forms1 based on the square and circle1 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 perpendiculars and squares4 ,illustration ,among the applications of the problem to draw a tangent to a given circle is the case of the common tangents to two given circles4 ,some authors give this as a basal problem1 although it is more commonly given as an exercise or a corollary4 ,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 wheels1#ge ,a and ,b1 as shown on page #bbf4 ,illustration3 ,,gothic ,,designs ,,employing ,,circles ,,and ,,bisected ,,angles ,the band may either not be crossed 7the case of the two exterior tangents71 or be crossed 7the interior tangents71 the latter allowing the wheels to turn in opposite directions4 ,in case the band is liable to change its length1 on account of stretching or variation in heat or moisture1 a third wheel1 ,d1 is used4 ,we then have the case of tangents to three pairs of circles4 ,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 work4 ,illustration ,,footnotes3 #fh ,this is the latest opinion4 ,he is usually assigned to the first century ,b4,c4 #fi ,see page #ed4 #gj ,a ,greek philosopher and mathematician of the fifth century ,b4,c4 #ga ,this illustration and the following two are from ,c4 ,dupin1 ,mathematics ,practically ,applied1 translated from the ,french by ,g4 ,birkbeck1 ,halifax1 #ahed4 ,this is probably the most scholarly attempt ever made at constructing a practical geometry4 #gb ,this illustration and others of the same type used in this work are from the excellent drawings by ,r4 ,w4 ,billings1 in ,the ,infinity of ,geometric ,design ,exemplified1 ,london1 #ahdi4 #gc ,from ,h4 ,kolb1 ,der ,ornamentenschatz 444 aus allen ,kunst-,epochen1 ,stuttgart1 #ahhc4 ,the original is in the ,church of ,saint ,anastasia in ,verona4 #gd ,from ,j4 ,bennett1 ,the ,arcanum 444 ,a ,concise ,theory of ,practicable ,geometry1 ,london1 #ahch1 one of the many books that have assumed to revolutionize geometry by making it practical4 #ge ,the figures are from ,dupin1 loc4 cit4 ,,chapter ,,xvi ,,the ,,leading ,,propositions ,,of ,,book ,,iii ,in the ,american textbooks ,book ,,iii is usually assigned to proportion4 ,it is therefore necessary at the beginning of this discussion to consider what is meant by ratio and proportion1 and to compare the ancient and the modern theories4 ,the subject is treated by ,euclid in his ,book ,v1 and an anonymous commentator has told us that it is the discovery of ,eudoxus1 the teacher of ,plato4 ,now proportion had been known long before the time of ,eudoxus 7#djh-cee ,b4,c471 but it was numerical proportion1 and as such it had been studied by the ,pythagoreans4 ,they were also the first to study seriously the incommensurable number1 and with this study the treatment of proportion from the standpoint of rational numbers lost its scientific position with respect to geometry4 ,it was because of this that ,eudoxus worked out a theory of geometric proportion that was independent of number as an expression of ratio4 ,the following four definitions from ,euclid are the basal ones of the ancient theory3 ,a ratio is a sort of relation in respect of size between two magnitudes of the same kind4 ,magnitudes are said to have a ratio to one another which are capable1 when multiplied1 of exceeding one another4 ,magnitudes are said to be in the same ratio1 the first to the second and the third to the fourth1 when1 if any equimultiples whatever be taken of the first and third1 and any equimultiples whatever of the second and fourth1 the former equimultiples alike exceed1 are alike equal to1 or alike fall short of1 the latter equimultiples respectively taken in corresponding order4 ,let magnitudes which have the same ratio be called proportional4#gf ,of these1 the first is so loose in statement as often to have been thought to be an interpolation of some later writer4 ,it was probably1 however1 put into the original for the sake of completeness1 to have some kind of statement concerning ratio as a preliminary to the important definition of quantities in the same ratio4 ,like the definition of straight line1 it was not intended to be taken seriously as a mathematical statement4 ,the second definition is intended to exclude zero and infinite magnitudes1 and to show that incommensurable magnitudes are included4 ,the third definition is the essential one of the ancient theory4 ,it defines what is meant by saying that magnitudes are in the same ratio2 in other words1 it defines a proportion4 ,into the merits of the definition it is not proposed to enter1 for the reason that it is no longer met in teaching in ,america1 and is practically abandoned even where the rest of ,euclid's work is in use4 ,it should be said1 however1 that it is scientifically correct1 that it covers the case of incommensurable magnitudes as well as that of commensurable ones1 and that it is the ,greek forerunner of the modern theories of irrational numbers4 ,as compared with the above treatment1 the one now given in textbooks is unscientific4 ,we define ratio as the quotient of the numerical measures of two quantities of the same kind1 and proportion as an equality of ratios4 ,but what do we mean by the quotient1 say of sqrt#b by sqrt#c8 ,and when we multiply a ratio by sqrt#e1 what is the meaning of this operation8 ,if we say that sqrt#b 3 sqrt#c means a quotient1 what meaning shall we assign to quotient8 ,if it is the number that shows how many times one number is contained in another1 how many times is sqrt#c contained in sqrt#b8 ,if to multiply is to take a number a certain number of times1 how many times do we take it when we multiply by sqrt#e8 ,we certainly take it more than #b times and less than #c times1 but what meaning can we assign to sqrt#e times8 ,it will thus be seen that our treatment of proportion assumes that we already know the theory of irrationals and can apply it to geometric magnitudes1 while the ancient treatment is independent of this theory4 ,educationally1 however1 we are forced to proceed as we do4 ,just as ,dedekind's theory of numbers is a simple one for college students1 so is the ancient theory of proportion2 but as the former is not suited to pupils in the high school1 so the latter must be relegated to the college classes4 ,and in this we merely harmonize educational progress with world progress1 for the numerical theory of proportion long preceded the theory of ,eudoxus4 ,the ancients made much of such terms as duplicate1 triplicate1 alternate1 and inverse ratio1 and also such as composition1 separation1 and conversion of ratio4 ,these entered into such propositions as1 ,if four magnitudes are proportional1 they will also be proportional alternately4 ,in later works they appear in the form of proportion by composition1 by division1 and by composition and division4 ,none of these is to-day of much importance1 since modern symbolism has greatly simplified the ancient expressions1 and in particular the proposition concerning composition and division is no longer a basal theorem in geometry4 ,indeed1 if our course of study were properly arranged1 we might well relegate the whole theory of proportion to algebra1 allowing this to precede the work in geometry4 ,we shall now consider a few of the principal propositions of ,book ,,iii4 ,,theorem4 ,if a line is drawn through two sides of a triangle parallel to the third side1 it divides those sides proportionally4 ,in addition to the usual proof it is instructive to consider in class the cases in which the parallel is drawn through the two sides produced1 either below the base or above the vertex1 and also in which the parallel is drawn through the vertex4 ,,theorem4 ,the bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides4 ,the proposition relating to the bisector of an exterior angle may be considered as a part of this one1 but it is usually treated separately in order that the proof shall appear less involved1 although the two are discussed together at this time4 ,the proposition relating to the exterior angle was recognized by ,pappus of ,alexandria4 ,if ,,abc is the given triangle1 and ,,cp#a1 ,,cp#b are respectively the internal and external bisectors1 then ,,ab is divided harmonically by ,p#a and ,p#b4 therefore,,ap#a 3 ,p#a;,b ,,ap#b 3 ,p#b;,b4 therefore,,ap#b 3 ,p#b;,b ,,ap#b - ,p#a,p#b 3 ,p#a,p#b - ,p#b;,b1 and this is the criterion for the harmonic progression still seen in many algebras4 ,for1 letting ,,ap#b a1 ,p#a,p#b b1 ,p#b;,b c1 we have a_/c 7a - b7_/7b - c71 which is also derived from taking the reciprocals of a1 b1 c1 and placing them in an arithmetical progression1 thus3 #a_/;b - #a_/;a #a_/;c - #a_/;b1 whence 7a - b7_/ab 7b - c7_/bc1 or 7a - b7_/7b - c7 ab_/bc a_/c4 ,this is the reason why the line ,,ab is said to be divided harmonically4 ,the line ,p#a,p#b is also called the harmonic mean between ,,ap#b and ,p#b;,b1 and the points ,a1 ,p#a1 ,b1 ,p#b are said to form an harmonic range4 ,illustration ,it may be noted that ,l,p#b;,,cp#a1 being made up of halves of two supplementary angles1 is a right angle4 ,furthermore1 if the ratio ,,ca 3 ,,cb is given1 and ,,ab is given1 then ,p#a and ,p#b are both fixed4 ,hence ,c must lie on a semicircle with ,p#a,p#b as a diameter1 and therefore the locus of a point such that its distances from two given points are in a given ratio is a circle4 ,this fact1 ,pappus tells us1 was known to ,apollonius4 ,at this point it is customary to define similar polygons as such as have their corresponding angles equal and their corresponding sides proportional4 ,aristotle gave substantially this definition1 saying that such figures have their sides proportional and their angles equal4 ,euclid improved upon this by saying that they must have their angles severally equal and the sides about the equal angles proportional4 ,our present phraseology seems clearer4 ,instead of corresponding angles we may say homologous angles1 but there seems to be no reason for using the less familiar word4 ,illustration ,illustration ,illustration ,it is more general to proceed by first considering similar figures instead of similar polygons1 thus including the most obviously similar of all figures1--two circles2 but such a procedure is felt to be too difficult by many teachers4 ,by this plan we first define similar sets of points1 ,a#a1 ,a#b1 ,a#c1 4441 and ,b#a1 ,b#b1 ,b#c1 4441 as such that ,a#a;,a#b1 ,b#a;,b#b1 ,c#a;,c#b1 444 are concurrent in ,o1 and ,a#a,o 3 ,a#b,o ,b#a,o 3 ,b#b,o ,c#a,o 3 ,c#b,o 444 ,here the constant ratio ,a#a,o 3 ,a#b,o is called the ratio of similitude1 and ,o is called the center of similitude4 ,having defined similar sets of points1 we then define similar figures as those figures whose points form similar sets4 ,then the two circles1 the four triangles1 and the three quadrilaterals respectively are similar figures4 ,if the ratio of similitude is #a1 the similar figures become symmetric figures1 and they are therefore congruent4 ,all of the propositions relating to similar figures can be proved from this definition1 but it is customary to use the ,greek one instead4 ,illustration ,among the interesting applications of similarity is the case of a shadow1 as here shown1 where the light is the center of similitude4 ,it is also well known to most high school pupils that in a camera the lens reverses the image4 ,the mathematical arrangement is here shown1 the lens inclosing the center of similitude4 ,the proposition may also be applied to the enlargement of maps and working drawings4 ,the propositions concerning similar figures have no particularly interesting history1 nor do they present any difficulties that call for discussion4 ,in schools where there is a little time for trigonometry1 teachers sometimes find it helpful to begin such work at this time1 since all of the trigonometric functions depend upon the properties of similar triangles1 and a brief explanation of the simplest trigonometric functions may add a little interest to the work4 ,in the present state of our curriculum we cannot do more than mention the matter as a topic of general interest in this connection4 ,it is a mistaken idea that geometry is a prerequisite to trigonometry4 ,we can get along very well in teaching trigonometry if we have three propositions3 7#a7 the one about the sum of the angles of a triangle2 7#b7 the ,pythagorean ,theorem2 7#c7 the one that asserts that two right triangles are similar if an acute angle of the one equals an acute angle of the other4 ,for teachers who may care to make a little digression at this time1 the following brief statement of a few of the facts of trigonometry may be of value3 ,illustration ,in the right triangle ,,oab we shall let ,,ab y1 ,,oa x1 ,,ob r1 thus adopting the letters of higher mathematics4 ,then1 so long as ,l,o remains the same1 such ratios as y_/x1 y_/r1 etc41 will remain the same1 whatever is the size of the triangle4 ,some of these ratios have special names4 ,for example1 we call y_/r the sine of ,o1 and we write sin ,o y_/r2 x_/r the cosine of ,o1 and we write cos ,o x_/r2 y_/x the tangent of ,o1 and we write tan ,o y_/x4 ,now because sin ,o y_/r1 therefore r sin ,o y2 and because cos ,o x_/r1 therefore r cos ,o x2 and because tan ,o y_/x1 therefore x tan ,o y4 ,hence1 if we knew the values of sin ,o1 cos ,o1 and tan ,o for the various angles1 we could find x1 y1 or r if we knew any one of them4 ,now the values of the sine1 cosine1 and tangent 7functions of the angles1 as they are called7 have been computed for the various angles1 and some interest may be developed by obtaining them by actual measurement1 using the protractor and squared paper4 ,some of those needed for such angles as a pupil in geometry is likely to use are as follows3 ,,angle ,,sine ,,cosine,,tangent ,,angle ,,sine ,,cosine,,tangent ---------------------------------------- ------------ #e deg4 4#jhg 4#iif 4#jhg #ej deg4 4#gff 4#fdc #a4aib ---------------------------------------- ------------ #aj deg4 4#agd 4#ihe 4#agf #ee deg4 4#hai 4#egd #a4dbh ---------------------------------------- ------------ #ae deg4 4#bei 4#iff 4#bfh #fj deg4 4#hff 4#ejj #a4gcb ---------------------------------------- ------------ #bj deg4 4#cdb 4#idj 4#cfd #fe deg4 4#ijf 4#dbc #b4ade ---------------------------------------- ------------ #be deg4 4#dbc 4#ijf 4#dff #gj deg4 4#idj 4#cdb #b4gdh ---------------------------------------- ------------ #cj deg4 4#ejj 4#hff 4#egg #ge deg4 4#iff 4#bei #c4gcb ---------------------------------------- ------------ #ce deg4 4#egd 4#hai 4#gjj #hj deg4 4#ihe 4#agd #e4fga ---------------------------------------- ------------ #dj deg4 4#fdc 4#gff 4#hci #he deg4 4#iif 4#jhg #aa4dcj ---------------------------------------- ------------ #de deg4 4#gjg 4#gjg #a4jjj #ij deg4 #a4jj 4#jjj infinity ,it will of course be understood that the values are correct only to the nearest thousandth4 ,thus the cosine of #e deg4 is #j4iifai1 and the sine of #he deg4 is #j4iifai4 ,the entire table can be copied by a class in five minutes if a teacher wishes to introduce this phase of the work1 and the author has frequently assigned the computing of a simpler table as a class exercise4 ,referring to the figure1 if we know that r #cj and ,l,o #dj deg41 then since y r sin ,o1 we have y #cj x #j4fdc #ai4bi4 ,if we know that x #fj and ,l,o #ce deg41 then since y x tan ,o1 we have y #fj x #j4g #db4 ,we may also find r1 for cos ,o x_/r1 whence r x_/7cos ,o7 #fj_/j4hai #gc4bf4 ,therefore1 if we could easily measure ,l,o and could measure the distance x1 we could find the height of a building y4 ,in trigonometry we use a transit for measuring angles1 but it is easy to measure them with sufficient accuracy for illustrative purposes by placing an ordinary paper protractor upon something level1 so that the center comes at the edge1 and then sighting along a ruler held against it1 so as to find the angle of elevation of a building4 ,we may then measure the distance to the building and apply the formula y x tan ,o4 ,illustration3 ,a ,,quadrant ,,of ,,the ,,sixteenth ,,century ,finaeus's ,de re et praxi geometrica1 ,paris1 #aeef ,it should always be understood that expensive apparatus is not necessary for such illustrative work4 ,the telescope used on the transit is only three hundred years old1 and the world got along very well with its trigonometry before that was invented4 ,so a little ingenuity will enable any one to make from cheap protractors about as satisfactory instruments as the world used before #afjj4 ,in order that this may be the more fully appreciated1 a few illustrations are here given1 showing the old instruments and methods used in practical surveying before the eighteenth century4 ,illustration3 ,a ,,quadrant ,,of ,,the ,,seventeenth ,,century ,the illustration on page #bcf shows a simple form of the quadrant1 an instrument easily made by a pupil who may be interested in outdoor work4 ,it was the common surveying instrument of the early days4 ,a more elaborate example is seen in the illustration1 on page #bcg1 of a seventeenth-century brass specimen in the author's collection4#gg ,illustration3 ,a ,,quadrant ,,of ,,the ,,seventeenth ,,century ,bartoli's ,del modo di misurare1 ,venice1 #afhi ,another type1 easily made by pupils1 is shown in the above illustration from ,bartoli1 #afhi4 ,such instruments were usually made of wood1 brass1 or ivory4#gh ,instruments for the running of lines perpendicular to other lines were formerly common1 and are easily made4 ,they suffice1 as the following illustration shows1 for surveying an ordinary field4 ,illustration3 ,,surveying ,,instrument ,,of ,,the ,,eighteenth ,,century ,n4 ,bion's ,traite de la construction 444 des instrumens de mathematique1 ,the ,hague1 #agbc ,illustration3 ,,the ,,quadrant ,,used ,,for ,,altitudes ,finaeus's ,de re et praxi geometrica1 ,paris1 #aeef ,the quadrant was practically used for all sorts of outdoor measuring4 ,for example1 the illustration from ,finaeus1 on this page1 shows how it was used for altitudes1 and the one reproduced on page #bdj shows how it was used for measuring depths4 ,a similar instrument from the work of ,bettinus is given on page #bda1 the distance of a ship being found by constructing an isosceles triangle4 ,a more elaborate form1 with a pendulum attachment1 is seen in the illustration from ,de ,judaeis1 which also appears on page #bda4 ,illustration3 ,,the ,,quadrant ,,used ,,for ,,depths ,finaeus's ,protomathesis1 ,paris1 #aecb ,illustration3 ,a ,,quadrant ,,of ,,the ,,sixteenth ,,century ,de ,judaeis's ,de quadrante geometrico1 ,nuernberg1 #aeid ,illustration3 ,,the ,,quadrant ,,used ,,for ,,distances ,bettinus's ,apiaria universae philosophiae mathematicae1 ,bologna1 #afde ,the quadrant finally developed into the octant1 as shown in the following illustration from ,hoffmann1 and this in turn developed into the sextant1 which is now used by all navigators4 ,illustration3 ,,the ,,octant ,hoffmann's ,de ,octantis1 ,jena1 #afab ,in connection with this general subject the use of the speculum 7mirror7 in measuring heights should be mentioned4 ,the illustration given on page #bdc shows how in early days a simple device was used for this purpose4 ,two similar triangles are formed in this way1 and we have only to measure the height of the eye above the ground1 and the distances of the mirror from the tower and the observer1 to have three terms of a proportion4 ,all of these instruments are easily made4 ,the mirror is always at hand1 and a paper protractor on a piece of board1 with a plumb line attached1 serves as a quadrant4 ,for a few cents1 and by the expenditure of an hour or so1 a school can have almost as good instruments as the ordinary surveyor had before the nineteenth century4 ,illustration3 ,,the ,,speculum ,finaeus's ,de re et praxi geometrica1 ,paris1 #aeef ,a well-known method of measuring the distance across a stream is illustrated in the figure below1 where the distance from ,a to some point ,p is required4 ,illustration ,run a line from ,a to ,c by standing at ,c in line with ,a and ,p4 ,then run two perpendiculars from ,a and ,c by any of the methods already given1--sighting on a protractor or along the edge of a book if no better means are at hand4 ,then sight from some point ,d1 on ,,cd1 to ,p1 putting a stake at ,b4 ,then run the perpendicular ,,be4 ,since ,,de 3 ,,eb ,,ba 3 ,,ap1 and since we can measure ,,de1 ,,eb1 and ,,ba with the tape1 we can compute the distance ,,ap4 ,there are many variations of this scheme of measuring distances by means of similar triangles1 and pupils may be encouraged to try some of them4 ,other figures are suggested on page #bdd1 and the triangles need not be confined to those having a right angle4 ,a very simple illustration of the use of similar triangles is found in one of the stories told of ,thales4 ,it is related that he found the height of the pyramids by measuring their shadow at the instant when his own shadow just equaled his height4 ,he thus had the case of two similar isosceles triangles4 ,this is an interesting exercise which may be tried about the time that pupils are leaving school in the afternoon4 ,illustration ,another application of the same principle is seen in a method often taken for measuring the height of a tree4 ,illustration ,the observer has a large right triangle made of wood4 ,such a triangle is shown in the picture1 in which ,,ab ,,bc4 ,he holds ,,ab level and walks toward the tree until he just sees the top along ,,ac4 ,then because ,,ab ,,bc1 and ,,ab 3 ,,bc ,,ad 3 ,,de1 the height above ,d will equal the distance ,,ad4 ,questions like the following may be given to the class3 #a4 ,what is the height of the tree in the picture if the triangle is #e ft4 #d in4 from the ground1 and ,,ad is #bc ft4 #h in48 #b4 ,suppose a triangle is used which has ,,ab twice ,,bc4 ,what is the height if ,,ad #ge ft48 ,there are many variations of this principle4 ,one consists in measuring the shadows of a tree and a staff at the same time4 ,the height of the staff being known1 the height of the tree is found by proportion4 ,another consists in sighting from the ground1 across a mark on an upright staff1 to the top of the tree4 ,the height of the mark being known1 and the distances from the eye to the staff and to the tree being measured1 the height of the tree is found4 ,illustration ,an instrument sold by dealers for the measuring of heights is known as the hypsometer4 ,it is made of brass1 and is of the form here shown4 ,the base is graduated in equal divisions1 say #ej1 and the upright bar is similarly divided4 ,at the ends of the hinged radius are two sights4 ,if the observer stands #ej feet from a tree and sights at the top1 so that the hinged radius cuts the upright bar at #bg1 then he knows at once that the tree is #bg feet high4 ,it is easy for a class to make a fairly good instrument of this kind out of stiff pasteboard4 ,an interesting application of the theorem relating to similar triangles is this3 ,extend your arm and point to a distant object1 closing your left eye and sighting across your finger tip with your right eye4 ,now keep the finger in the same position and sight with your left eye4 ,the finger will then seem to be pointing to an object some distance to the right of the one at which you were pointing4 ,if you can estimate the distance between these two objects1 which can often be done with a fair degree of accuracy when there are houses intervening1 then you will be able to tell approximately your distance from the objects1 for it will be ten times the estimated distance between them4 ,the finding of the reason for this by measuring the distance between the pupils of the two eyes1 and the distance from the eye to the finger tip1 and then drawing the figure1 is an interesting exercise4 ,perhaps some pupil who has read ,thoreau's descriptions of outdoor life may be interested in what he says of his crude mathematics4 ,he writes1 ,i borrowed the plane and square1 level and dividers1 of a carpenter1 and with a shingle contrived a rude sort of a quadrant1 with pins for sights and pivots4 ,with this he measured the heights of a cliff on the ,massachusetts coast1 and with similar home-made or school-made instruments a pupil in geometry can measure most of the heights and distances in which he is interested4 ,,theorem4 ,if in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse3 #a4 ,the triangles thus formed are similar to the given triangle1 and are similar to each other4 #b4 ,the perpendicular is the mean proportional between the segments of the hypotenuse4 #c4 ,each of the other sides is the mean proportional between the hypotenuse and the segment of the hypotenuse adjacent to that side4 ,to this important proposition there is one corollary of particular interest1 namely1 ,the perpendicular from any point on a circle to a diameter is the mean proportional between the segments of the diameter4 ,by means of this corollary we can easily construct a line whose numerical value is the square root of any number we please4 ,thus we may make ,,ad #b in41 ,,db #c in41 and erect ,,dc perp to ,,ab4 ,then the length of ,,dc will be sqrt#f in41 and we may find sqrt#f approximately by measuring ,,dc4 ,illustration ,furthermore1 if we introduce negative magnitudes into geometry1 and let ,,db #c and ,,da -#b1 then ,,dc will equal sqrt7-#f74 ,in other words1 we have a justification for representing imaginary quantities by lines perpendicular to the line on which we represent real quantities1 as is done in the graphic treatment of imaginaries in algebra4 ,it is an interesting exercise to have a class find1 to one decimal place1 by measuring as above1 the value of sqrt#b1 sqrt#c1 sqrt#e1 and sqrt#i1 the last being integral4 ,if1 as is not usually the case1 the class has studied the complex number1 the absolute value of sqrt7-#f71 sqrt7-#g71 4441 may be found in the same way4 ,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 books4 ,it seems to have come from the ,roman surveyors4 ,illustration ,if a carpenter's square is put on top of an upright stick1 as here shown1 and an observer sights along the arms to a distant point ,b and a point ,a near the stick1 then the two triangles are similar4 ,hence ,,ad 3 ,,dc ,,dc 3 ,,db4 ,hence1 if ,,ad and ,,dc are measured1 ,,db can be found4 ,the experiment is an interesting and instructive one for a class1 especially as the square can easily be made out of heavy pasteboard4 ,,theorem4 ,if two chords intersect within a circle1 the product of the segments of the one is equal to the product of the segments of the other4 ,,theorem4 ,if from a point without a circle a secant and a tangent are drawn1 the tangent is the mean proportional between the secant and its external segment4 ,,corollary4 ,if from a point without a circle a secant is drawn1 the product of the secant and its external segment is constant in whatever direction the secant is drawn4 ,these two propositions and the corollary are all parts of one general proposition3 ,if through a point a line is drawn cutting a circle1 the product of the segments of the line is constant4 ,illustration ,if ,p is within the circle1 then xx' yy'2 if ,p is on the circle1 then x and y become #j1 and #j 4 x' #j 4 y' #j2 if ,p is at ,p#c1 then x and y1 having passed through #j1 may be considered negative if we wish1 although the two negative signs would cancel out in the equation2 if ,p is at ,p#d1 then y y' and we have xx' y#b1 or x 3 y y 3 x'1 as stated in the proposition4 ,we thus have an excellent example of the ,principle of ,continuity1 and classes are always interested to consider the result of letting ,p assume various positions4 ,among the possible cases is the one of two tangents from an external point1 and the one where ,p is at the center of the circle4 ,students should frequently be questioned as to the meaning of product of lines4 ,the ,greeks always used rectangle of lines1 but it is entirely legitimate to speak of product of lines1 provided we define the expression consistently4 ,most writers do this1 saying that by the product of lines is meant the product of their numerical values1 a subject already discussed at the beginning of this chapter4 ,,theorem4 ,the square on the bisector of an angle of a triangle is equal to the product of the sides of this angle diminished by the product of the segments made by the bisector upon the third side of the triangle4 ,this proposition enables us to compute the length of a bisector of a triangle if the lengths of the sides are known4 ,illustration ,for1 in this figure1 let a #c1 b #e1 and c #f4 ,then because x 3 y b 3 a1 and y #f - x1 we have x_/7#f - x7 #e_/c4 therefore #cx #cj - #ex4 therefore x #c #c_/d1 y #b #a_/d4 ,by the theorem1 z#b ab - xy #ae - 7#h #g_/af7 #f #i_/af4 therefore z sqrt7#f #i_/af7 #a_/d sqrt#aje #b4e4 ,,theorem4 ,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 side4 ,this enables us1 after the ,pythagorean ,theorem has been studied1 to compute the length of the diameter of the circumscribed circle in terms of the three sides4 ,illustration ,for if we designate the sides by a1 b1 and c1 as usual1 and let ,,cd d and ,,pb x1 then 7,,cp7#b a#b - x#b b#b - 7c - x7#b4 therefore a#b - x#b b#b - c#b #b;cx - x#b4 therefore x 7a#b - b#b c#b7 _/ #b;c4 therefore 7,,cp7#b a#b - 77a#b - b#b c#b7 _/ #b;c7#b4 ,but ,,cp 4 d ab4 therefore d #b;abc _/ sqrt7#d;a#b;c#b - 7a#b - b#b c#b7b74 ,this is not available at this time1 however1 because the ,pythagorean ,theorem has not been proved4 ,these two propositions are merely special cases of the following general theorem1 which may be given as an interesting exercise3 ,if ,,abc is an inscribed triangle1 and through ,c there are drawn two straight lines ,,cd1 meeting ,,ab in ,d1 and ,,cp1 meeting the circle in ,p1 with angles ,,acd and ,,pcb equal1 then ,,ac x ,,bc will equal ,,cd x ,,cp4 ,illustration3 ,,fig4 #a ,illustration3 ,,fig4 #b ,illustration3 ,,fig4 #c ,illustration3 ,,fig4 #d ,fig4 #a is the general case where ,d falls between ,a and ,b4 ,if ,,cp is a diameter1 it reduces to the second figure given on page #bdi4 ,if ,,cp bisects ,l,,acb1 we have ,fig4 #c1 from which may be proved the proposition given at the foot of page #bdh4 ,if ,d lies on ,,ba produced1 we have ,fig4 #b4 ,if ,d lies on ,,ab produced1 we have ,fig4 #d4 ,this general proposition is proved by showing that triangles,,adc and ,,pbc are similar1 exactly as in the second proposition given on page #bdi4 ,these theorems are usually followed by problems of construction1 of which only one has great interest1 namely1 ,to divide a given line in extreme and mean ratio4 ,the purpose of this problem is to prepare for the construction of the regular decagon and pentagon4 ,the division of a line in extreme and mean ratio is called the golden section1 and is probably the section mentioned by ,proclus when he says that ,eudoxus greatly added to the number of the theorems which ,plato originated regarding the section4 ,the expression golden section is not old1 however1 and its origin is uncertain4 ,if a line ,,ab is divided in golden section at ,p1 we have ,,ab x ,,pb 7,,ap7#b4 ,therefore1 if ,,ab a1 and ,,ap x1 we have a7a - x7 x#b1 or x#b ax - a#b #j2 whence x - a_/#b - a_/#bsqrt#e a7#a4aah - #j4e7 #j4fah;a1 the other root representing the external point4 ,that is1 x about #j4f;a1 and a - x about #j4d;a1 and a is therefore divided in about the ratio of #b 3 #c4 ,there has been a great deal written upon the aesthetic features of the golden section4 ,it is claimed that a line is most harmoniously divided when it is either bisected or divided in extreme and mean ratio4 ,a painting has the strong feature in the center1 or more often at a point about #j4d of the distance from one side1 that is1 at the golden section of the width of the picture4 ,it is said that in nature this same harmony is found1 as in the division of the veins of such leaves as the ivy and fern4 ,,footnotes3 #gf ,for a very full discussion of these four definitions see ,heath's ,euclid1 ,vol4 ,,ii1 p4 #aaf1 and authorities there cited4 #gg ,these two and several which follow are from ,stark1 loc4 cit4 #gh ,the author has a beautiful ivory specimen of the ,sixteenth century4 ,,chapter ,,xvii ,,the ,,leading ,,propositions ,,of ,,book ,,iv ,book ,,iv treats of the area of polygons1 and offers a large number of practical applications4 ,since the number of applications to the measuring of areas of various kinds of polygons is unlimited1 while in the first three books these applications are not so obvious1 less effort is made in this chapter to suggest practical problems to the teachers4 ,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 important4 ,,theorem4 ,two rectangles having equal altitudes are to each other as their bases4 ,euclid's statement 7,book ,,vi1 ,proposition #a7 was as follows3 ,triangles and parallelograms which are under the same height are to one another as their bases4 ,our plan of treating the two figures separately is manifestly better from the educational standpoint4 ,in the modern treatment by limits the proof is divided into two parts3 first1 for commensurable bases2 and second1 for incommensurable ones4 ,of these the second may well be omitted1 or merely be read over by the teacher and class and the reasons explained4 ,in general1 it is doubtful if the majority of an ,american class in geometry get much out of the incommensurable case4 ,of course1 with a bright class a teacher may well afford to take it as it is given in the textbook1 but the important thing is that the commensurable case should be proved and the incommensurable one recognized4 ,euclid's treatment of proportion was so rigorous that no special treatment of the incommensurable was necessary4 ,the ,french geometer1 ,legendre1 gave a rigorous proof by reductio ad absurdum4 ,in ,america the pupils are hardly ready for these proofs1 and so our treatment by limits is less rigorous than these earlier ones4 ,,theorem4 ,the area of a rectangle is equal to the product of its base by its altitude4 ,the easiest way to introduce this is to mark a rectangle1 with commensurable sides1 on squared paper1 and count up the squares2 or1 what is more convenient1 to draw the rectangle and mark the area off in squares4 ,it is interesting and valuable to a class to have its attention called to the fact that the perimeter of a rectangle is no criterion as to the area4 ,thus1 if a rectangle has an area of #a square foot and is only #a_/ddj of an inch high1 the perimeter is over #b miles4 ,the story of how ,indians were induced to sell their land by measuring the perimeter is a very old one4 ,proclus speaks of travelers who described the size of cities by the perimeters1 and of men who cheated others by pretending to give them as much land as they themselves had1 when really they made only the perimeters equal4 ,thucydides estimated the size of ,sicily by the time it took to sail round it4 ,pupils will be interested to know in this connection that of polygons having the same perimeter and the same number of sides1 the one having equal sides and equal angles is the greatest1 and that of plane figures having the same perimeter1 the circle is the greatest4 ,these facts were known to the ,greek writers1 ,zenodorus 7ca4 #aej ,b4,c47 and ,proclus 7#daj-dhe ,a4,d474 ,the surfaces of rectangular solids may now be found1 there being an advantage in thus incidentally connecting plane and solid geometry wherever it is natural to do so4 ,,theorem4 ,the area of a parallelogram is equal to the product of its base by its altitude4 ,the best way to introduce this theorem is to cut a parallelogram from paper1 and then1 with the class1 separate it into two parts by a cut perpendicular to the base4 ,the two parts may then be fitted together to make a rectangle4 ,in particular1 if we cut off a triangle from one end and fit it on the other1 we have the basis for the proof of the textbooks4 ,the use of squared paper for such a proposition is not wise1 since it makes the measurement appear to be merely an approximation4 ,the cutting of the paper is in every way more satisfactory4 ,,theorem4 ,the area of a triangle is equal to half the product of its base by its altitude4 ,of course1 the ,greeks would never have used the wording of either of these two propositions4 ,euclid1 for example1 gives this one as follows3 ,if a parallelogram have the same base with a triangle and be in the same parallels1 the parallelogram is double of the triangle4 ,as to the parallelogram1 he simply says it is equal to a parallelogram of equal base and in the same parallels1 which makes it equal to a rectangle of the same base and the same altitude4 ,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 class4 ,teachers may now introduce pyramids1 requiring the areas of the triangular faces to be found4 ,the ,ahmes papyrus 7ca4 #agjj ,b4,c47 gives the area of an isosceles triangle as #a_/b bs1 where s is one of the equal sides1 thus taking s for the altitude4 ,this shows the primitive state of geometry at that time4 ,,theorem4 ,the area of a trapezoid is equal to half the sum of its bases multiplied by the altitude4 ,illustration ,an interesting variation of the ordinary proof is made by placing a trapezoid ,t'1 congruent to ,t1 in the position here shown4 ,the parallelogram formed equals a7b b'71 and therefore ,t a 4 7b b'7_/#b4 ,the proposition should be discussed for the case b b'1 when it reduces to the one about the area of a parallelogram4 ,if b' #j1 the trapezoid reduces to a triangle1 and ,t a 4 b_/#b4 ,this proposition is the basis of the theory of land surveying1 a piece of land being1 for purposes of measurement1 divided into trapezoids and triangles1 the latter being1 as we have seen1 a kind of special trapezoid4 ,the proposition is not in ,euclid1 but is given by ,proclus in the fifth century4 ,the term isosceles trapezoid is used to mean a trapezoid with two opposite sides equal1 but not parallel4 ,the area of such a figure was incorrectly given by the ,ahmes papyrus as #a_/b7;b b'7s1 where s is one of the equal sides4 ,this amounts to taking s a4 ,the proposition is particularly important in the surveying of an irregular field such as is found in hilly districts4 ,it is customary to consider the field as a polygon1 and to draw a meridian line1 letting fall perpendiculars upon it from the vertices1 thus forming triangles and trapezoids that can easily be measured4 ,an older plan1 but one better suited to the use of pupils who may be working only with the tape1 is given on page #ii4 ,,theorem4 ,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 angles4 ,this proposition may be omitted as far as its use in plane geometry is concerned1 for we can prove the next proposition here given without using it4 ,in solid geometry it is used only in a proposition relating to the volumes of two triangular pyramids having a common trihedral angle1 and this is usually omitted4 ,but the theorem is so simple that it takes but little time1 and it adds greatly to the student's appreciation of similar triangles4 ,it not only simplifies the next one here given1 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 triangle is also equal to a second angle of the other triangle4 ,it is helpful to give numerical values to the sides of a few triangles having such equal angles1 and to find the numerical ratio of the areas4 ,,theorem4 ,the areas of two similar triangles are to each other as the squares on any two corresponding sides4 ,illustration ,this may be proved independently of the preceding proposition by drawing the altitudes p and p'4 ,then triangle,,abc_/triangle,a',b',c' cp_/c'p'4 ,but c_/c' p_/p'1 by similar triangles4 therefore triangle,,abc_/triangle,a',b',c' c#b_/;c'#b1 and so for other sides4 ,this proof is unnecessarily long1 however1 because of the introduction of the altitudes4 ,in this and several other propositions in ,book ,,iv occurs the expression the square on a line4 ,we have1 in our departure from ,euclid1 treated a line either as a geometric figure or as a number 7the length of the line71 as was the more convenient4 ,of course if we are speaking of a line1 the preferable expression is square on the line1 whereas if we speak of a number1 we say square of the number4 ,in the case of a rectangle of two lines we have come to speak of the product of the lines1 meaning the product of their numerical values4 ,we are therefore not as accurate in our phraseology as ,euclid1 and we do not pretend to be1 for reasons already given4 ,but when it comes to square on a line or square of a line1 the former is the one demanding no explanation or apology1 and it is even better understood than the latter4 ,,theorem4 ,the areas of two similar polygons are to each other as the squares on any two corresponding sides4 ,this is a proposition of great importance1 and in due time the pupil sees that it applies to circles1 with the necessary change of the word sides to lines4 ,it is well to ask a few questions like the following3 ,if one square is twice as high as another1 how do the areas compare8 ,if the side of one equilateral triangle is three times as long as that of another1 how do the perimeters compare8 how do the areas compare8 ,if the area of one square is twenty-five times the area of another square1 the side of the first is how many times as long as the side of the second8 ,if a photograph is enlarged so that a tree is four times as high as it was before1 what is the ratio of corresponding dimensions8 ,the area of the enlarged photograph is how many times as great as the area of the original8 ,,theorem4 ,the square on the hypotenuse of a right triangle is equivalent to the sum of the squares on the other two sides4 ,of all the propositions of geometry this is the most famous and perhaps the most valuable4 ,trigonometry is based chiefly upon two facts of plane geometry3 7#a7 in similar triangles the corresponding sides are proportional1 and 7#b7 this proposition4 ,in mensuration1 in general1 this proposition enters more often than any others1 except those on the measuring of the rectangle and triangle4 ,it is proposed1 therefore1 to devote considerable space to speaking of the history of the theorem1 and to certain proofs that may profitably be suggested from time to time to different classes for the purpose of adding interest to the work4 ,proclus1 the old ,greek commentator on ,euclid1 has this to say of the history3 ,if we listen to those who wish to recount ancient history1 we may find some of them referring this theorem to ,pythagoras and saying that he sacrificed an ox in honor of his discovery4 ,but for my part1 while ,i admire those who first observed the truth of this theorem1 ,i marvel more at the writer of the ',elements' 7,euclid71 not only because he made it fast by a most lucid demonstration1 but because he compelled assent to the still more general theorem by the irrefragable arguments of science in ,book ,,vi4 ,for in that book he proves1 generally1 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 angle4 ,now it appears from this that ,proclus1 in the fifth century ,a4,d41 thought that ,pythagoras discovered the proposition in the sixth century ,b4,c41 that the usual proof1 as given in most of our ,american textbooks1 was due to ,euclid1 and that the generalized form was also due to the latter4 ,for it should be made known to students that the proposition is true not only for squares1 but for any similar figures1 such as equilateral triangles1 parallelograms1 semicircles1 and irregular figures1 provided they are similarly placed on the three sides of the right triangle4 ,besides ,proclus1 ,plutarch testifies to the fact that ,pythagoras was the discoverer1 saying that ,pythagoras sacrificed an ox on the strength of his proposition as ,apollodotus says1 but saying that there were two possible propositions to which this refers4 ,this ,apollodotus was probably ,apollodorus1 surnamed ,logisticus 7the ,calculator71 whose date is quite uncertain1 and who speaks in some verses of a famous proposition discovered by ,pythagoras1 and all tradition makes this the one4 ,cicero1 who comments upon these verses1 does not question the discovery1 but doubts the story of the sacrifice of the ox4 ,of other early writers1 ,diogenes ,laertius1 whose date is entirely uncertain 7perhaps the second century ,a4,d471 and ,athenaeus 7third century ,a4,d47 may be mentioned as attributing the theorem to ,pythagoras1 while ,heron 7first century ,a4,d47 says that he gave a rule for forming right triangles with rational integers for the sides1 like #c1 #d1 #e1 where #cb #db #eb4 ,it should be said1 however1 that the ,pythagorean origin has been doubted1 notably in an article by ,h4 ,vogt1 published in the ,bibliotheca ,mathematica in #aijh 7,vol4 ,,ix 7#c71 p4 #ae71 entitled ,die ,geometrie des ,pythagoras1 and by ,g4 ,junge1 in his work entitled ,wann haben die ,griechen das ,irrationale entdeckt8 7,halle1 #aijg74 ,these writers claim that all the authorities attributing the proposition to ,pythagoras are centuries later than his time1 and are open to grave suspicion4 ,nevertheless it is hardly possible that such a general tradition1 and one so universally accepted1 should have arisen without good foundation4 ,the evidence has been carefully studied by ,heath in his ,euclid1 who concludes with these words3 ,on the whole1 therefore1 ,i see no sufficient reason to question the tradition that1 so far as ,greek geometry is concerned 4441 ,pythagoras was the first to introduce the theorem 444 and to give a general proof of it4 ,that the fact was known earlier1 probably without the general proof1 is recognized by all modern writers4 ,illustration ,pythagoras had studied in ,egypt and possibly in the ,east before he established his school at ,crotona1 in southern ,italy4 ,in ,egypt1 at any rate1 he could easily have found that a triangle with the sides #c1 #d1 #e1 is a right triangle1 and ,vitruvius 7first century ,b4,c47 tells us that he taught this fact4 ,the ,egyptian harpedonaptae 7rope stretchers7 stretched ropes about pegs so as to make such a triangle for the purpose of laying out a right angle in their surveying1 just as our surveyors do to-day4 ,the great pyramids have an angle of slope such as is given by this triangle4 ,indeed1 a papyrus of the twelfth dynasty1 lately discovered at ,kahun1 in ,egypt1 refers to four of these triangles1 such as #ab 7#c_/d7b 7#a #a_/d7b4 ,this property seems to have been a matter of common knowledge long before ,pythagoras1 even as far east as ,china4 ,he was1 therefore1 naturally led to attempt to prove the general property which had already been recognized for special cases1 and in particular for the isosceles right triangle4 ,how ,pythagoras proved the proposition is not known4 ,it has been thought that he used a proof by proportion1 because ,proclus says that ,euclid gave a new style of proof1 and ,euclid does not use proportion for this purpose1 while the subject1 in incomplete form1 was highly esteemed by the ,pythagoreans4 ,heath suggests that this is among the possibilities3 ,illustration triangles,,abc and ,,apc are similar4 therefore ,,ab x ,,ap 7,,ac7#b4 ,similarly1 ,,ab x ,,pb 7,,bc7#b4 therefore ,,ab7,,ap ,,pb7 7,,ac7#b 7,,bc7#b1 or 7,,ab7#b 7,,ac7#b 7,,bc7#b4 ,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 sides1 in some such manner as this3 ,illustration3 ,,fig4 #a ,illustration3 ,,fig4 #b ,here ,fig4 #a is evidently h#b #d triangles4 ,fig4 #b is evidently a#b b#b #d triangles4 therefore h#b #d triangles a#b b#b #d triangles1 the triangles all being congruent4 therefore h#b a#b b#b4 ,the great ,hindu mathematician1 ,bhaskara 7born #aaad ,a4,d471 proceeds in a somewhat similar manner4 ,he draws this figure1 but gives no proof4 ,it is evident that he had in mind this relation3 ,illustration h#b #d 4 ab_/#b 7b - a7#b a#b b#b4 ,a somewhat similar proof can be based upon the following figure3 ,illustration ,if the four triangles1 #a #b #c #d1 are taken away1 there remains the square on the hypotenuse4 ,but if we take away the two shaded rectangles1 which equal the four triangles1 there remain the squares on the two sides4 ,therefore the square on the hypotenuse must equal the sum of these two squares4 ,illustration ,it has long been thought that the truth of the proposition was first observed by seeing the tiles on the floors of ancient temples4 ,if they were arranged as here shown1 the proposition would be evident for the special case of an isosceles right triangle4 ,the ,hindus knew the proposition long before ,bhaskara1 however1 and possibly before ,pythagoras4 ,it is referred to in the old religious poems of the ,brahmans1 the ,sulvasutras1 but the date of these poems is so uncertain that it is impossible to state that they preceded the sixth century ,b4,c41#gi in which ,pythagoras lived4 ,the ,sulvasutra of ,apastamba has a collection of rules1 without proofs1 for constructing various figures4 ,among these is one for constructing right angles by stretching cords of the following lengths3 #c1 #d1 #e2 #ab1 #af1 #bj2 #ae1 #bj1 #be 7the two latter being multiples of the first72 #e1 #ab1 #ac2 #ae1 #cf1 #ci2 #h1 #ae1 #ag2 #ab1 #ce1 #cg4 ,whatever the date of these ,sulvasutras1 there is no evidence that the ,indians had a definite proof of the theorem1 even though they1 like the early ,egyptians1 recognized the general fact4 ,it is always interesting to a class to see more than one proof of a famous theorem1 and many teachers find it profitable to ask their pupils to work out proofs that are 7to them7 original1 often suggesting the figure4 ,two of the best known historic proofs are here given4 ,the first makes the ,pythagorean ,theorem a special case of a proposition due to ,pappus 7fourth century ,a4,d471 relating to any kind of a triangle4 ,illustration ,somewhat simplified1 this proposition asserts that if ,,abc is any kind of triangle1 and ,,mc1 ,,nc are parallelograms on ,,ac1 ,,bc1 the opposite sides being produced to meet at ,p2 and if ,,pc is produced making ,,qr ,,pc2 and if the parallelogram ,,at is constructed1 then ,,at ,,mc ,,nc4 ,for ,,mc ,,ap ,,ar1 having equal bases and equal altitudes4 ,similarly1 ,,nc ,,qt4 ,adding1 ,,mc ,,nc ,,at4 ,if1 now1 ,,abc is a right triangle1 and if ,,mc and ,,nc are squares1 it is easy to show that ,,at is a square1 and the proposition reduces to the ,pythagorean ,theorem4 ,the ,arab writer1 ,al-,nairizi 7died about #ibb ,a4,d471 attributes to ,thabit ben ,qurra 7#hbf-ija ,a4,d47 a proof substantially as follows3 ,illustration ,the four triangles ,t can be proved congruent4 ,then if we take from the whole figure ,t and ,t'1 we have left the squares on the two sides of the right angle4 ,if we take away the other two triangles instead1 we have left the square on the hypotenuse4 ,therefore the former is equivalent to the latter4 ,a proof attributed to the great artist1 ,leonardo da ,vinci 7#adeb-aeai71 is as follows3 ,illustration ,the construction of the following figure is evident4 ,it is easily shown that the four quadrilaterals ,,abmx1 ,,xnca1 ,,sbcp1 and ,,srqp are congruent4 therefore ,,abmxnca equals ,,sbcpqrs but is not congruent to it1 the congruent quadrilaterals being differently arranged4 ,subtract the congruent triangles ,,mxn1 ,,abc1 ,,raq1 and the proposition is proved4#hj ,the following is an interesting proof of the proposition3 ,let ,,abc be the original triangle1 with ,,ab ,,bc4 ,turn the triangle about ,b1 through #ij deg41 until it comes into the position ,a',,bc'4 ,then because it has been turned through #ij deg41 ,c',a',p will be perpendicular to ,,ac4 ,then #a_/b7;,,ab7#b triangle,,aba'1 and #a_/b7;,,bc'7#b triangle,,bc',c1 because ,,bc ,,bc'4 therefore #a_/b77;,,ab7#b 7,,bc7#b7 triangle,,aba' triangle,,bc',c4 therefore #a_/b77;,,ab7#b 7,,bc7#b7 triangle,,ac',a' triangle,a',c',c ,illustration 7,for triangle,,aba' triangle,,bc',a' triangle,a',c',c is the second member of both equations47 #a_/b;,a',c' 4 ,,ap #a_/b;,a',c' 4 ,,pc #a_/b;,a',c' 4 ,,ac #a_/b7;,,ac7#b4 therefore 7,,ab7#b 7,,bc7#b 7,,ac7#b4 ,the ,pythagorean ,theorem1 as it is generally called1 has had other names4 ,it is not uncommonly called the pons asinorum 7see page #agd7 in ,france4 ,the ,arab writers called it the ,figure of the ,bride1 although the reason for this name is unknown2 possibly two being joined in one has something to do with it4 ,it has also been called the ,bride's ,chair1 and the shape of the ,euclid figure is not unlike the chair that a slave carries on his back1 in which the ,eastern bride is sometimes transported to the wedding ceremony4 ,schopenhauer1 the ,german philosopher1 referring to the figure1 speaks of it as a proof walking on stilts1 and as a mouse-trap proof4 ,an interesting theory suggested by the proposition is that of computing the sides of right triangles so that they shall be represented by rational numbers4 ,pythagoras seems to have been the first to take up this theory1 although such numbers were applied to the right triangle before his time1 and ,proclus tells us that ,plato also contributed to it4 ,the rule of ,pythagoras1 put in modern symbols1 was as follows3 n#b 77n#b - #a7_/b7b 77n#b #a7_/b7b1 the sides being n1 7n#b - #a7_/b1 and 7n#b #a7_/b4 ,if for n we put #c1 we have #c1 #d1 #e4 ,if we take the various odd numbers1 we have n #a1 #c1 #e1 #g1 #i1 4441 7n#b - #a7_/b #j1 #d1 #ab1 #bd1 #dj1 4441 7n#b #a7_/b #a1 #e1 #ac1 #be1 #da1 4444 ,of course n may be even1 giving fractional values4 ,thus1 for n #b we have for the three sides1 #b1 #a #a_/b1 #b #a_/b4 ,other formulas are also known4 ,plato's1 for example1 is as follows3 7#bn7#b 7n#b - #a7b 7n#b #a7b4 ,if #bn #b1 #d1 #f1 #h1 #aj1 4441 then n#b - #a #j1 #c1 #h1 #ae1 #bd1 4441 and n#b #a #b1 #e1 #aj1 #ag1 #bf1 4444 ,this formula evidently comes from that of ,pythagoras by doubling the sides of the squares4#ha ,,theorem4 ,in any triangle the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides by the projection of the other upon that side4 ,,theorem4 ,a similar statement for the obtuse triangle4 ,these two propositions are usually proved by the help of the ,pythagorean ,theorem4 ,some writers1 however1 actually construct the squares and give a proof similar to the one in that proposition4 ,this plan goes back at least to ,gregoire de ,st4 ,vincent 7#afdg74 ,illustration ,it should be observed that a#b b#b c#b - #b;b'c4 ,if ,l,a #ij deg41 then b' #j1 and this becomes a#b b#b c#b4 ,if ,l,a is obtuse1 then b' passes through #j and becomes negative1 and a#b b#b c#b #b;b'c4 ,thus we have three propositions in one4 ,illustration ,at the close of ,book ,,iv many geometries give as an exercise1 and some give as a regular proposition1 the celebrated problem that bears the name of ,heron of ,alexandria1 namely1 to compute the area of a triangle in terms of its sides4 ,the result is the important formula ,area sqrt7s7s - a77s - b77s - c771 where a1 b1 and c are the sides1 and s is the semiperimeter #a_/b7;a b c74 ,as a practical application the class may be able to find a triangular piece of land1 as here shown1 and to measure the sides4 ,if the piece is clear1 the result may be checked by measuring the altitude and applying the formula a #a_/b;bh4 ,it may be stated to the class that ,heron's formula is only a special case of the more general one developed about #fdj ,a4,d41 by a famous ,hindu mathematician1 ,brahmagupta4 ,this formula gives the area of an inscribed quadrilateral as sqrt77s - a77s - b77s - c77s - d771 where a1 b1 c1 and d are the sides and s is the semiperimeter4 ,if d #j1 the quadrilateral becomes a triangle and we have ,heron's formula4#hb ,at the close of ,book ,,iv1 also1 the geometric equivalents of the algebraic formulas for 7a b7#b1 7a - b7#b1 and 7a b77a - b7 are given4 ,the class may like to know that ,euclid had no algebra and was compelled to prove such relations as these by geometry1 while we do it now much more easily by algebraic multiplication4 ,,footnotes3 #gi ,see1 for example1 ,g4 ,b4 ,kaye1 ,the ,source of ,hindu ,mathematics1 in the ,journal of the ,royal ,asiatic ,society1 ,july1 #aiaj4 #hj ,an interesting ,japanese proof of this general character may be seen in ,y4 ,mikami1 ,mathematical ,papers from the ,far ,east1 p4 #abg1 ,leipzig1 #aiaj4 #ha ,special recognition of indebtedness to ,h4 ,a4 ,naber's ,das ,theorem des ,pythagoras 7,haarlem1 #aijh71 ,heath's ,euclid1 ,gow's ,history of ,greek ,mathematics1 and ,cantor's ,geschichte is due in connection with the ,pythagorean ,theorem4 #hb ,the rule was so ill understood that ,bhaskara 7twelfth century7 said that ,brahmagupta was a blundering devil for giving it 7,lilavati1 ,sec4 #agb74 ,,chapter ,,xviii ,,the ,,leading ,,propositions ,,of ,,book ,v ,illustration ,book ,v treats of regular polygons and circles1 and includes the computation of the approximate value of pi4 ,it opens with a definition of a regular polygon as one that is both equilateral and equiangular4 ,while in elementary geometry the only regular polygons studied are convex1 it is interesting to a class to see that there are also regular cross polygons4 ,indeed1 the regular cross pentagon was the badge of the ,pythagoreans1 as ,lucian 7ca4 #ajj ,b4,c47 and an unknown commentator on ,aristophanes 7ca4 #djj ,b4,c47 tell us4 ,at the vertices of this polygon the ,pythagoreans placed the ,greek letters signifying health4 ,euclid was not interested in the measure of the circle1 and there is nothing in his ,elements on the value of pi4 ,indeed1 he expressly avoided numerical measures of all kinds in his geometry1 wishing the science to be kept distinct from that form of arithmetic known to the ,greeks as logistic1 or calculation4 ,his ,book ,,iv is devoted to the construction of certain regular polygons1 and his propositions on this subject are now embodied in ,book ,v as it is usually taught in ,america4 ,if we consider ,book ,v as a whole1 we are struck by three features4 ,of these the first is the pure geometry involved1 and this is the essential feature to be emphasized4 ,the second is the mensuration of the circle1 a relatively unimportant piece of theory in view of the fact that the pupil is not ready for incommensurables1 and a feature that imparts no information that the pupil did not find in arithmetic4 ,the third is the somewhat interesting but mathematically unimportant application of the regular polygons to geometric design4 ,as to the mensuration of the circle it is well for us to take a broad view before coming down to details4 ,there are only four leading propositions necessary for the mensuration of the circle and the determination of the value of pi4 ,these are as follows3 7#a7 ,the inscribing of a regular hexagon1 or any other regular polygon of which the side is easily computed in terms of the radius4 ,we may start with a square1 for example1 but this is not so good as the hexagon because its side is incommensurable with the radius1 and its perimeter is not as near the circumference4 7#b7 ,the perimeters of similar regular polygons are proportional to their radii1 and their areas to the squares of the radii4 ,it is now necessary to state1 in the form of a postulate if desired1 that the circle is the limit of regular inscribed and circumscribed polygons as the number of sides increases indefinitely1 and hence that 7#b7 holds for circles4 7#c7 ,the proposition relating to the area of a regular polygon1 and the resulting proposition relating to the circle4 7#d7 ,given the side of a regular inscribed polygon1 to find the side of a regular inscribed polygon of double the number of sides4 ,it will thus be seen that if we were merely desirous of approximating the value of pi1 and of finding the two formulas c #bpir and a pir#b1 we should need only four propositions in this book upon which to base our work4 ,it is also apparent that even if the incommensurable cases are generally omitted1 the notion of limit is needed at this time1 and that it must briefly be reviewed before proceeding further4 ,there is1 however1 a much more worthy interest than the mere mensuration of the circle1 namely1 the construction of such polygons as can readily be formed by the use of compasses and straightedge alone4 ,the pleasure of constructing such figures and of proving that the construction is correct is of itself sufficient justification for the work4 ,as to the use of such figures in geometric design1 some discussion will be offered at the close of this chapter4 ,the first few propositions include those that lead up to the mensuration of the circle4 ,after they are proved it is assumed that the circle is the limit of the regular inscribed and circumscribed polygons as the number of sides increases indefinitely4 ,this may often be proved with some approach to rigor by a few members of an elementary class1 but it is the experience of teachers that the proof is too difficult for most beginners1 and so the assumption is usually made in the form of an unproved theorem4 ,the following are some of the leading propositions of this book3 ,,theorem4 ,two circumferences have the same ratio as their radii4 ,this leads to defining the ratio of the circumference to the diameter as pi4 ,although this is a ,greek letter1 it was not used by the ,greeks to represent this ratio4 ,indeed1 it was not until #agjf that an ,english writer1 ,william ,jones1 in his ,synopsis ,palmariorum ,matheseos1 used it in this way1 it being the initial letter of the ,greek word for periphery4 ,after establishing the properties that c #bpir1 and a pir#b1 the textbooks follow the ,greek custom and proceed to show how to inscribe and circumscribe various regular polygons1 the purpose being to use these in computing the approximate numerical value of pi4 ,of these regular polygons two are of special interest1 and these will now be considered4 ,,problem4 ,to inscribe a regular hexagon in a circle4 ,that the side of a regular inscribed hexagon equals the radius must have been recognized very early4 ,the common divisions of the circle in ancient art are into four1 six1 and eight equal parts4 ,no draftsman could have worked with a pair of compasses without quickly learning how to effect these divisions1 and that compasses were early used is attested by the specimens of these instruments often seen in museums4 ,there is a tradition that the ancient ,babylonians considered the circle of the year as made up of #cfj days1 whence they took the circle as composed of #cfj steps or grades 7degrees74 ,this tradition is without historic foundation1 however1 there being no authority in the inscriptions for this assumption of the #cfj-;division by the ,babylonians1 who seem rather to have preferred #h1 #ab1 #abj1 #bdj1 and #dhj as their division numbers4 ,the story of #cfj deg4 in the ,babylonian circle seems to start with ,achilles ,tatius1 an ,alexandrian grammarian of the second or third century ,a4,d4 ,it is possible1 however1 that the ,babylonians got their favorite number #fj 7as in #fj seconds make a minute1 #fj minutes make an hour or degree7 from the hexagon in a circle 7#a_/f of #cfj deg4 #fj deg471 although the probabilities seem to be that there is no such connection4#hc ,the applications of this problem to mensuration are numerous4 ,the fact that we may use for tiles on a floor three regular polygons--the triangle1 square1 and hexagon--is noteworthy1 a fact that ,proclus tells us was recognized by ,pythagoras4 ,the measurement of the regular hexagon1 given one side1 may be used in computing sections of hexagonal columns1 in finding areas of flower beds1 and in other similar cases4 ,this review of the names of the polygons offers an opportunity to impress their etymology again on the mind4 ,in this case1 for example1 we have hexagon from the ,greek words for six and angle4 ,,problem4 ,to inscribe a regular decagon in a given circle4 ,euclid states the problem thus3 ,to construct an isosceles triangle having each of the angles at the base double of the remaining one4 ,this makes each base angle #gb deg4 and the vertical angle #cf deg41 the latter being the central angle of a regular decagon1--essentially our present method4 ,this proposition seems undoubtedly due to the ,pythagoreans1 as tradition has always asserted4 ,proclus tells us that ,pythagoras discovered the construction of the cosmic figures1 or the five regular polyhedrons1 and one of these 7the dodecahedron7 involves the construction of the regular pentagon4 ,iamblichus 7ca4 #cbe ,a4,d47 tells us that ,hippasus1 a ,pythagorean1 was said to have been drowned for daring to claim credit for the construction of the regular dodecahedron1 when by the rules of the brotherhood all credit should have been assigned to ,pythagoras4 ,if a regular polygon of s sides can be inscribed1 we may bisect the central angles1 and therefore inscribe one of #bs sides1 and then of #ds sides1 and then of #hs sides1 and in general of #bns sides4 ,this includes the case of s #b and n #j1 for we can inscribe a regular polygon of two sides1 the angles being1 by the usual formula1 #b7b - #b7_/b #j1 although1 of course1 we never think of two equal and coincident lines as forming what we might call a digon4 ,we therefore have the following regular polygons3 ,from the equilateral triangle1 regular polygons of #bn 4 #c sides2 ,from the square1 regular polygons of #bn sides2 ,from the regular pentagon1 regular polygons of #bn 4 #e sides2 ,from the regular pentedecagon1 regular polygons of #bn 4 #ae sides4 ,this gives us1 for successive values of n1 the following regular polygons of less than #ajj sides3 ,from #bn 4 #c1 #c1 #f1 #ab1 #bd1 #dh1 #if2 ,from #bn1 #b1 #d1 #h1 #af1 #cb1 #fd2 ,from #bn 4 #e1 #e1 #aj1 #bj1 #dj1 #hj2 ,from #bn 4 #ae1 #ae1 #cj1 #fj4 ,illustration3 ,,roman ,,mosaic ,,found ,,at ,,pompeii ,gauss 7#aggg-ahee71 a celebrated ,german mathematician1 proved 7in #agif7 that it is possible also to inscribe a regular polygon of #ag sides1 and hence polygons of #bn 4 #ag sides1 or #ag1 #cd1 #fh1 4441 sides1 and also #c 4 #ag #ea and #e 4 #ag #he sides1 by the use of the compasses and straightedge1 but the proof is not adapted to elementary geometry4 ,in connection with the study of the regular polygons some interest attaches to the reference to various forms of decorative design4 ,the mosaic floor1 parquetry1 ,gothic windows1 and patterns of various kinds often involve the regular figures4 ,if the teacher uses such material1 care should be taken to exemplify good art4 ,for example1 the equilateral triangle and its relation to the regular hexagon is shown in the picture of an ancient ,roman mosaic floor on page #bgd4hd ,in the next illustration some characteristic ,moorish mosaic work appears1 in which it will be seen that the basal figure is the square1 although at first sight this would not seem to be the case4#he ,this is followed by a beautiful ,byzantine mosaic1 the original of which was in five colors of marble4 ,here it will be seen that the equilateral triangle and the regular hexagon are the basal figures1 and a few of the properties of these polygons might be derived from the study of such a design4 ,in the ,arabic pattern on page #bgf the dodecagon appears as the basis1 and the remarkable powers of the ,arab designer are shown in the use of symmetry without employing regular figures4 ,illustration3 ,,mosaic ,,from ,,damascus ,illustration3 ,,mosaic ,,from ,,an ,,ancient ,,byzantine ,,church ,,problem4 ,given the side and the radius of a regular inscribed polygon1 to find the side of the regular inscribed polygon of double the number of sides4 ,illustration3 ,,arabic ,,pattern ,the object of this proposition is1 of course1 to prepare the way for finding the perimeter of a polygon of #bn sides1 knowing that of n sides4 ,the ,greek plan was generally to use both an inscribed and a circumscribed polygon1 thus approaching the circle as a limit both from without and within4 ,this is more conclusive from the ultrascientific point of view1 but it is1 if anything1 less conclusive to a beginner1 because he does not so readily follow the proof4 ,the plan of using the two polygons was carried out by ,archimedes of ,syracuse 7#bhg-bab ,b4,c47 in his famous method of approximating the value of pi1 although before him ,antiphon 7fifth century ,b4,c47 had inscribed a square 7or equilateral triangle7 as a basis for the work1 and ,bryson 7his contemporary7 had attacked the problem by circumscribing as well as inscribing a regular polygon4 ,,problem4 ,to find the numerical value of the ratio of the circumference of a circle to its diameter4 ,as already stated1 the usual plan of the textbooks is in part the method followed by ,archimedes4 ,it is possible to start with any regular polygon of which the side can conveniently be found in terms of the radius4 ,in particular we might begin with an inscribed square instead of a regular hexagon4 ,in this case we should have ,length of ,side ,perimeter s#d #a4dad444 #a4da #e4ff s#h sqrt7#b - sqrt7#d - #a4dadb77 #j4gb #e4gf and so on4 ,it is a little easier to start with the hexagon1 however1 for we are already nearer the circle1 and the side and perimeter are both commensurable with the radius4 ,it is not1 of course1 intended that pupils should make the long numerical calculations4 ,they may be required to compute s#ab and possibly s#bd1 but aside from this they are expected merely to know the process4 ,if it were possible to find the value of pi exactly1 we could find the circumference exactly in terms of the radius1 since c #bpir4 ,if we could find the circumference exactly1 we could find the area exactly1 since a pir#b4 ,if we could find the area exactly in this form1 pi times a square1 we should have a rectangle1 and it is easy to construct a square equivalent to any rectangle4 ,therefore1 if we could find the value of pi exactly1 we could construct a square with area equivalent to the area of the circle2 in other words1 we could square the circle4 ,we could also1 as already stated1 construct a straight line equivalent to the circumference2 in other words1 we could rectify the circumference4 ,these two problems have attracted the attention of the world for over two thousand years1 but on account of their interrelation they are usually spoken of as a single problem1 to square the circle4 ,since we can construct sqrta by means of the straightedge and compasses1 it would be possible for us to square the circle if we could express pi by a finite number of square roots4 ,conversely1 every geometric construction reduces to the intersection of two straight lines1 of a straight line and a circle1 or of two circles1 and is therefore equivalent to a rational operation or to the extracting of a square root4 ,hence a geometric construction cannot be effected by the straightedge and compasses unless it is equivalent to a series of rational operations or to the extracting of a finite number of square roots4 ,it was proved by a ,german professor1 ,lindemann1 in #ahhb1 that pi cannot be expressed as an algebraic number1 that is1 as the root of an equation with rational coefficients1 and hence it cannot be found by the above operations1 and1 furthermore1 that the solution of this famous problem is impossible by elementary geometry4#hf ,it should also be pointed out to the student that for many practical purposes one of the limits of pi stated by ,archimedes1 namely1 #c #a_/g1 is sufficient4 ,for more accurate work #c4adaf is usually a satisfactory approximation4 ,indeed1 the late ,professor ,newcomb stated that ten decimal places are sufficient to give the circumference of the earth to the fraction of an inch1 and thirty decimal places would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope4 ,probably the earliest approximation of the value of pi was #c4 ,this appears very commonly in antiquity1 as in ,i ,kings vii1 #bc1 and #b ,chronicles iv1 #b4 ,in the ,ahmes papyrus 7ca4 #agjj ,b4,c47 there is a rule for finding the area of the circle1 expressed in modern symbols as 7#h_/i7b;d#b1 which makes pi #bef_/ha or #c4afjd4444 ,archimedes1 using a plan somewhat similar to ours1 found that pi lay between #c #a_/g and #c #aj_/ga4 ,ptolemy1 the great ,greek astronomer1 expressed the value as #c #ag_/abj1 or #c4adaff4444 ,the fact that ,ptolemy divided his diameter into #abj units and his circumference into #cfj units probably shows1 however1 the influence of the ancient value #c4 ,in ,india an approximate value appears in a certain poem written before the ,christian era1 but the date is uncertain4 ,about #ejj ,a4,d4 ,aryabhatta 7or possibly a later writer of the same name7 gave the value #fbhcb_/bjjjj1 or #c4adaf4 ,brahmagupta1 another ,hindu 7born #eih ,a4,d471 gave sqrt7#aj71 and this also appears in the writings of the ,chinese mathematician ,chang ,heng 7#gh-aci ,a4,d474 ,a little later in ,china1 ,wang ,fan 7#bbi-bfg7 gave #adb _/ #de1 or #c4aeee4442 and one of his contemporaries1 ,lui ,hui1 gave #aeg _/ #ej1 or #c4ad4 ,in the fifth century ,ch'ung-chih gave as the limits of pi1 #c4adaeibg and #c4adaeibf1 from which he inferred that #bb_/g and #cee_/aac were good approximations1 although he does not state how he came to this conclusion4 ,in the ,middle ,ages the greatest mathematician of ,italy1 ,leonardo ,fibonacci1 or ,leonardo of ,pisa 7about #abjj ,a4,d471 found as limits #c4adbg444 and #c4adaj4444 ,about #afjj the ,chinese value #cee_/aac was rediscovered by ,adriaen ,anthonisz 7#aebg-afjg71 being published by his son1 who is known as ,metius 7#aega-afce71 in the year #afbe4 ,about the same period the ,french mathematician ,vieta 7#aedj-afjc7 found the value of pi to #i decimal places1 and ,adriaen van ,rooman 7#aefa-afae7 carried it to #ag decimal places1 and ,ludolph van ,ceulen 7#aedj-afaj7 to #ce decimal places4 ,it was carried to #adj decimal places by ,georg ,vega 7died in #agic71 to #bjj by ,zacharias ,dase 7died in #ahdd71 to #ejj by ,richter 7died in #ahed71 and more recently by ,shanks to #gjg decimal places4 ,there have been many interesting formulas for pi1 among them being the following3 pi_/#b #b_/a 4 #b_/c 4 #d_/c 4 #d_/e 4 #f_/e 4 #f_/g 4 #h_/g 4 #h_/i 4 4444 7,wallis1 #afaf-agjc7 #d_/pi #a #a_/b #i_/b #be_/b #di_/b 4444 7,brouncker1 #afbj-afhd7 pi_/#d #a - #a_/c #a_/e - #a_/g 4444 7,gregory1 #afch-afge7 pi_/#f sqrt7#a_/c7 4 7#a - #a_/7c 4 #c7 #a_/7cb 4 #e7 - #a_/7cc 4 #g7 44474 pi_/#b 7log i7 _/ i4 7,bernoulli7 pi_/7#bsqrt7#c77 #a - #a_/e #a_/g - #a_/aa #a_/ac - #a_/ag #a_/ai4441 thus connecting the primes4 pi#b_/af #a - #a_/bb - #a_/cb #a_/db - #a_/eb #a_/fb - #a_/gb - #a_/hb #a_/ib 4444 pi_/#b x_/#b sin x 7sin#b x7 _/ #b 7sin#c x7 _/ #c 4444 7#j x #bpi7 pi_/#d #c_/d #a_/7b 4 #c 4 #d7 - #a_/7d 4 #e 4 #f7 #a_/7f 4 #g 4 #h7 - 4444 #bpi#b_/c #g - 7#a_/7a 4 #c7 #a_/7c 4 #f7 #a_/7f 4 #aj7 44474 pi #bnsqrt7#b - sqrt7#b sqrt7#b sqrt7#b sqrt7#b444777774 ,students of elementary geometry are not prepared to appreciate it1 but teachers will be interested in the remarkable formula discovered by ,euler 7#agjg-aghc71 the great ,swiss mathematician1 namely1 #a eipi #j4 ,in this relation are included the five most interesting quantities in mathematics1--zero1 the unit1 the base of the so-called ,napierian logarithms1 i sqrt7-#a71 and pi4 ,it was by means of this relation that the transcendence of e was proved by the ,french mathematician ,hermite1 and the transcendence of pi by the ,german ,lindemann4 ,illustration ,there should be introduced at this time1 if it has not already been done1 the proposition of the lunes of ,hippocrates 7ca4 #dgj ,b4,c471 who proved a theorem that asserts1 in somewhat more general form1 that if three semicircles be described on the sides of a right triangle as diameters1 as shown1 the lunes ,l ,l' are together equivalent to the triangle ,t4 ,illustration ,in the use of the circle in design one of the simplest forms suggested by ,book ,v is the trefoil 7three-leaf71 as here shown1 with the necessary construction lines4 ,this is a very common ornament in architecture1 both with rounded ends and with the ends slightly pointed4 ,the trefoil is closely connected with hexagonal designs1 since the regular hexagon is formed from the inscribed equilateral triangle by doubling the number of sides4 ,the following are designs that are easily made3 ,illustration ,it is not very profitable1 because it is manifestly unreal1 to measure the parts of such figures1 but it offers plenty of practice in numerical work4 ,illustration3 ,,choir ,,of ,,lincoln ,,cathedral ,illustration3 ,,porch ,,of ,,lincoln ,,cathedral ,in the illustrations of the ,gothic windows given in ,chapter ,,xv only the square and circle were generally involved4 ,teachers who feel it necessary or advisable to go outside the regular work of geometry for the purpose of increasing the pupil's interest or of training his hand in the drawing of figures will find plenty of designs given in any pictures of ,gothic cathedrals4 ,for example1 this picture of the noble window in the choir of ,lincoln ,cathedral shows the use of the square1 hexagon1 and pentagon4 ,in the porch of the same cathedral1 shown in the next illustration1 the architect has made use of the triangle1 square1 and pentagon in planning his ornamental stonework4 ,it is possible to add to the work in pure geometry some work in the mensuration of the curvilinear figures shown in these designs4 ,this form of mensuration is not of much value1 however1 since it places before the pupil a problem that he sees at once is fictitious1 and that has no human interest4 ,illustration3 ,,gothic ,,designs ,,employing ,,circles ,,and ,,bisected ,,angles ,illustration3 ,,gothic ,,designs ,,employing ,,circles ,,and ,,squares ,illustration3 ,,gothic ,,designs ,,employing ,,circles ,,and ,,the ,,equilateral ,,triangle ,illustration3 ,,gothic ,,designs ,,employing ,,circles ,,and ,,the ,,regular ,,hexagon ,the designs given on page #bhc involve chiefly the square as a basis1 but it will be seen from one of the figures that the equilateral triangle and the hexagon also enter4 ,the possibilities of endless variation of a single design are shown in the illustration on page #bhd1 the basis in this case being the square4 ,the variations in the use of the triangle and hexagon have been the object of study of many designers of ,gothic windows1 and some examples of these forms are shown on page #bhe4 ,in more simple form this ringing of the changes on elementary figures is shown on page #bhf4 ,some teachers have used color work with such designs for the purpose of increasing the interest of their pupils1 but the danger of thus using the time with no serious end in view will be apparent4 ,illustration ,in the matter of the mensuration of the circle the annexed design has some interest4 ,the figure is not uncommon in decoration1 and it is interesting to show1 as a matter of pure geometry1 that the area of the circle is divided into three equal portions by means of the four interior semicircles4 ,illustration ,an important application of the formula a pir#b is seen in the area of the annulus1 or ring1 the formula being a pir#b - pir'#b pi7r#b - r'#b7 pi7r r'77r - r'74 ,it is used in finding the area of the cross section of pipes1 and this is needed when we wish to compute the volume of the iron used4 ,another excellent application is that of finding the area of the surface of a cylinder1 there being no reason why such simple cases from solid geometry should not furnish working material for plane geometry1 particularly as they have already been met by the pupils in arithmetic4 ,a little problem that always has some interest for pupils is one that ,napoleon is said to have suggested to his staff on his voyage to ,egypt3 ,to divide a circle into four equal parts by the use of circles alone4 ,illustration ,here the circles ,b are tangent to the circle ,a at the points of division4 ,furthermore1 considering areas1 and taking r as the radius of ,a1 we have ,a pir#b1 and ,b pi7r_/#b7b4 ,hence ,b #a_/d;,a1 or the sum of the areas of the four circles ,b equals the area of ,a4 ,hence the four ,d's must equal the four ,c's1 and ,d ,c4 ,the rest of the argument is evident4 ,the problem has some interest to pupils aside from the original question suggested by ,napoleon4 ,at the close of plane geometry teachers may find it helpful to have the class make a list of the propositions that are actually used in proving other propositions1 and to have it appear what ones are proved by them4 ,this forms a kind of genealogical tree that serves to fix the parent propositions in mind4 ,such a work may also be carried on at the close of each book1 if desired4 ,it should be understood1 however1 that certain propositions are used in the exercises1 even though they are not referred to in subsequent propositions1 so that their omission must not be construed to mean that they are not important4 ,an exercise of distinctly less value is the classification of the definitions4 ,for example1 the classification of polygons or of quadrilaterals1 once so popular in textbook making1 has generally been abandoned as tending to create or perpetuate unnecessary terms4 ,such work is therefore not recommended4 ,,footnotes3 #hc ,bosanquet and ,sayre1 ,the ,babylonian ,astronomy1 ,monthly ,notices of the ,royal ,asiatic ,society1 ,vol4 ,,xl1 p4 #ajh4 #hd ,this and the three illustrations following are from ,kolb1 loc4 cit4 #he ,this was in five colors of marble4 #hf ,the proof is too involved to be given here4 ,the writer has set it forth in a chapter on the transcendency of pi in a work soon to be published by ,professor ,young of ,the ,university of ,chicago4 ,,chapter ,,xix ,,the ,,leading ,,propositions ,,of ,,book ,,vi ,there have been numerous suggestions with respect to solid geometry1 to the effect that it should be more closely connected with plane geometry4 ,the attempt has been made1 notably by ,meray in ,france and de ,paolis in ,italy1 to treat the corresponding propositions of plane and solid geometry together2 as1 for example1 those relating to parallelograms and parallelepipeds1 and those relating to plane and spherical triangles4 ,whatever the merits of this plan1 it is not feasible in ,america at present1 partly because of the nature of the college-entrance requirements4 ,while it is true that to a boy or girl a solid is more concrete than a plane1 it is not true that a geometric solid is more concrete than a geometric plane4 ,just as the world developed its solid geometry1 as a science1 long after it had developed its plane geometry1 so the human mind grasps the ideas of plane figures earlier than those of the geometric solid4 ,there is1 however1 every reason for referring to the corresponding proposition of plane geometry when any given proposition of solid geometry is under consideration1 and frequent references of this kind will be made in speaking of the propositions in this and the two succeeding chapters4 ,such reference has value in the apperception of the various laws of solid geometry1 and it also adds an interest to the subject and creates some approach to power in the discovery of new facts in relation to figures of three dimensions4 ,the introduction to solid geometry should be made slowly4 ,the pupil has been accustomed to seeing only plane figures1 and therefore the drawing of a solid figure in the flat is confusing4 ,the best way for the teacher to anticipate this difficulty is to have a few pieces of cardboard1 a few knitting needles filed to sharp points1 a pine board about a foot square1 and some small corks4 ,with the cardboard he can illustrate planes1 whether alone1 intersecting obliquely or at right angles1 or parallel1 and he can easily illustrate the figures given in the textbook in use4 ,there are models of this kind for sale1 but the simple ones made in a few seconds by the teacher or the pupil have much more meaning4 ,the knitting needles may be stuck in the board to illustrate perpendicular or oblique lines1 and if two or more are to meet in a point1 they may be held together by sticking them in one of the small corks4 ,such homely apparatus1 costing almost nothing1 to be put together in class1 seems much more real and is much more satisfactory than the ,german models4#hg ,an extensive use of models is1 however1 unwise4 ,the pupil must learn very early how to visualize a solid from the flat outline picture1 just as a builder or a mechanic learns to read his working drawings4 ,to have a model for each proposition1 or even to have a photograph or a stereoscopic picture1 is a very poor educational policy4 ,a textbook may properly illustrate a few propositions by photographic aids1 but after that the pupil should use the kind of figures that he must meet in his mathematical work4 ,a child should not be kept in a perambulator all his life1--he must learn to walk if he is to be strong and grow to maturity2 and it is so with a pupil in the use of models in solid geometry4#hh ,the case is somewhat similar with respect to colored crayons4 ,they have their value and their proper place1 but they also have their strict limitations4 ,it is difficult to keep their use within bounds2 pupils come to use them to make pleasing pictures1 and teachers unconsciously fall into the same habit4 ,the value of colored crayons is two-fold3 7#a7 they sometimes make two planes stand out more clearly1 or they serve to differentiate some line that is under consideration from others that are not2 7#b7 they enable a class to follow a demonstration more easily by hearing of the red plane perpendicular to the blue one1 instead of the plane ,,mn perpendicular to the plane ,,pq4 ,but it should always be borne in mind that in practical work we do not have colored ink or colored pencils commonly at hand1 nor do we generally have colored crayons4 ,pupils should therefore become accustomed to the pencil and the white crayon as the regulation tools1 and in general they should use them4 ,the figures may not be as striking1 but they are more quickly made and they are more practical4 ,the definition of plane has already been discussed in ,chapter ,,xii1 and the other definitions of ,book ,,vi are not of enough interest to call for special remark4 ,the axioms are the same as in plane geometry1 but there is at least one postulate that needs to be added1 although it would be possible to state various analogues of the postulates of plane geometry if we cared unnecessarily to enlarge the number4 ,the most important postulate of solid geometry is as follows3 ,one plane1 and only one1 can be passed through two intersecting straight lines4 ,this is easily illustrated1 as in most textbooks1 as also are three important corollaries derived from it3 #a4 ,a straight line and a point not in the line determine a plane4 ,of course this may be made the postulate1 as may also the next one1 the postulate being placed among the corollaries1 but the arrangement here adopted is probably the most satisfactory for educational purposes4 #b4 ,three points not in a straight line determine a plane4 ,the common question as to why a three-legged stool stands firmly1 while a four-legged table often does not1 will add some interest at this point4 #c4 ,two parallel lines determine a plane4 ,this requires a slight but informal proof to show that it properly follows as a corollary from the postulate1 but a single sentence suffices4 ,while studying this book questions of the following nature may arise with an advanced class1 or may be suggested to those who have had higher algebra3 ,how many straight lines are in general 7that is1 at the most7 determined by n points in space8 ,two points determine #a line1 a third point adds 7in general1 in all these cases7 #b more1 a fourth point adds #c more1 and an nth point n - #a more4 ,hence the maximum is #a #b #c 444 7n - #a71 or n7n-#a7_/b1 which the pupil will understand if he has studied arithmetical progression4 ,the maximum number of intersection points of n straight lines in the same plane is also n7n - #a7_/b4 ,how many straight lines are in general determined by n planes8 ,the answer is the same1 n7n - #a7_/b4 ,how many planes are in general determined by n points in space8 ,here the answer is #a #c #f #aj 444 7n - #b77n - #a7_/b1 or n7n - #a77n - #b7_/7a x #b x #c74 ,the same number of points is determined by n planes4 ,,theorem4 ,if two planes cut each other1 their intersection is a straight line4 ,among the simple illustrations are the back edges of the pages of a book1 the corners of the room1 and the simple test as to whether the edge of a card is straight by testing it on a plane4 ,it is well to call attention to the fact that if two intersecting straight lines move parallel to their original position1 and so that their intersection rests on a straight line not in the plane of those lines1 the figure generated will be that of this proposition4 ,in general1 if we cut through any figure of solid geometry in some particular way1 we are liable to get the figure of a proposition in plane geometry1 as will frequently be seen4 ,,theorem4 ,if a straight line is perpendicular to each of two other straight lines at their point of intersection1 it is perpendicular to the plane of the two lines4 ,if students have trouble in visualizing the figure in three dimensions1 some knitting needles through a piece of cardboard will make it clear4 ,teachers should call attention to the simple device for determining if a rod is perpendicular to a board 7or a pipe to a floor1 ceiling1 or wall71 by testing it twice1 only1 with a carpenter's square4 ,similarly1 it may be asked of a class1 ,how shall we test to see if the corner 7line7 of a room is perpendicular to the floor1 or if the edge of a box is perpendicular to one of the sides8 ,in some elementary and in most higher geometries the perpendicular is called a normal to the plane4 ,,theorem4 ,all the perpendiculars that can be drawn to a straight line at a given point lie in a plane which is perpendicular to the line at the given point4 ,thus the hands of a clock pass through a plane as the hands revolve1 if they are1 as is usual1 perpendicular to the axis2 and the same is true of the spokes of a wheel1 and of a string with a stone attached1 swung as rapidly as possible about a boy's arm as an axis4 ,a clock pendulum too swings in a plane1 as does the lever in a pair of scales4 ,,theorem4 ,through a given point within or without a plane there can be one perpendicular to a given plane1 and only one4 ,this theorem is better stated to a class as two theorems4 ,thus a plumb line hanging from a point in the ceiling1 without swinging1 determines one definite point in the floor2 and1 conversely1 if it touches a given point in the floor1 it must hang from one definite point in the ceiling4 ,it should be noticed that if we cut through this figure1 on the perpendicular line1 we shall have the figure of the corresponding proposition in plane geometry1 namely1 that there can be1 under similar circumstances1 only one perpendicular to a line4 ,,theorem4 ,oblique lines drawn from a point to a plane1 meeting the plane at equal distances from the foot of the perpendicular1 are equal1 etc4 ,there is no objection to speaking of a right circular cone in connection with this proposition1 and saying that the slant height is thus proved to be constant4 ,the usual corollary1 that if the obliques are equal they meet the plane in a circle1 offers a new plan of drawing a circle4 ,a plumb line that is a little too long to reach the floor will1 if swung so as just to touch the floor1 describe a circle4 ,a #aj-;foot pole standing in a #i-;foot room will1 if it moves so as to touch constantly a fixed point on either the floor or the ceiling1 describe a circle on the ceiling or floor respectively4 ,one of the corollaries states that the locus of points in space equidistant from the extremities of a straight line is the plane perpendicular to this line at its middle point4 ,this has been taken by some writers as the definition of a plane1 but it is too abstract to be usable4 ,it is advisable to cut through the figure along the given straight line1 and see that we come back to the corresponding proposition in plane geometry4 ,a good many ships have been saved from being wrecked by the principle involved in this proposition4 ,illustration ,if a dangerous shoal ,a is near a headland ,h1 the angle ,,hax is measured and is put down upon the charts as the vertical danger angle4 ,ships coming near the headland are careful to keep far enough away1 say at ,s1 so that the angle ,,hsx shall be less than this danger angle4 ,they are then sure that they will avoid the dangerous shoal4 ,related to this proposition is the problem of supporting a tall iron smokestack by wire stays4 ,evidently three stays are needed1 and they are preferably placed at the vertices of an equilateral triangle1 the smokestack being in the center4 ,the practical problem may be given of locating the vertices of the triangle and of finding the length of each stay4 ,,theorem4 ,two straight lines perpendicular to the same plane are parallel4 ,here again we may cut through the figure by the plane of the two parallels1 and we get the figure of plane geometry relating to lines that are perpendicular to the same line4 ,the proposition shows that the opposite corners of a room are parallel1 and that therefore they lie in the same plane1 or are coplanar1 as is said in higher geometry4 ,it is interesting to a class to have attention called to the corollary that if two straight lines are parallel to a third straight line1 they are parallel to each other2 and to have the question asked why it is necessary to prove this when the same thing was proved in plane geometry4 ,in case the reason is not clear1 let some student try to apply the proof used in plane geometry4 ,,theorem4 ,two planes perpendicular to the same straight line are parallel4 ,besides calling attention to the corresponding proposition of plane geometry1 it is well now to speak of the fact that in propositions involving planes and lines we may often interchange these words4 ,for example1 using line for straight line1 for brevity1 we have3 ,one line does not determine ,one plane does not determine a plane4 a line4 ,two intersecting lines ,two intersecting planes determine determine a plane4 a line4 ,two lines perpendicular to ,two planes perpendicular to a plane are parallel4 a line are parallel4 ,if one of two parallel lines ,if one of two parallel planes is perpendicular to a plane1 the is perpendicular to a line1 the other is also perpendicular to other is also perpendicular to the plane4 the line4 ,if two lines are parallel1 every ,if two planes are parallel1 plane containing one of the every line in one of the planes lines is parallel to the other is parallel to the other plane4 line4 ,,theorem4 ,the intersections of two parallel planes by a third plane are parallel lines4 ,thus one of the edges of a box is parallel to the next succeeding edge if the opposite faces are parallel1 and in sawing diagonally through an ordinary board 7with rectangular cross section7 the section is a parallelogram4 ,,theorem4 ,a straight line perpendicular to one of two parallel planes is perpendicular to the other also4 ,notice 7#a7 the corresponding proposition in plane geometry2 7#b7 the proposition that results from interchanging plane and 7straight7 line4 ,,theorem4 ,if two intersecting straight lines are each parallel to a plane1 the plane of these lines is parallel to that plane4 ,interchanging plane and 7straight7 line1 we have3 ,if two intersecting planes are each parallel to a line1 the line of 7intersection of7 these planes is parallel to that line4 ,is this true8 ,,theorem4 ,if two angles not in the same plane have their sides respectively parallel and lying on the same side of the straight line joining their vertices1 they are equal and their planes are parallel4 ,questions like the following may be asked in connection with the proposition3 ,what is the corresponding proposition in plane geometry8 ,why do we need another proof here8 ,try the plane-geometry proof here4 ,,theorem4 ,if two straight lines are cut by three parallel planes1 their corresponding segments are proportional4 ,here1 again1 it is desirable to ask for the corresponding proposition of plane geometry1 and to ask why the proof of that proposition will not suffice for this one4 ,the usual figure may be varied in an interesting manner by having the two lines meet on one of the planes1 or outside the planes1 or by having them parallel1 in which cases the proof of the plane-geometry proposition holds here4 ,this proposition is not of great importance from the practical standpoint1 and it is omitted from some of the standard syllabi at present1 although included in certain others4 ,it is easy1 however1 to frame some interesting questions depending upon it for their answers1 such as the following3 ,in a gymnasium swimming tank the water is #d feet deep and the ceiling is #h feet above the surface of the water4 ,a pole #ae feet long touches the ceiling and the bottom of the tank4 ,required to know what length of the pole is in the water4 ,at this point in ,book ,,vi it is customary to introduce the dihedral angle4 ,the word dihedral is from the ,greek1 di- meaning two1 and hedra meaning seat4 ,we have the root hedra also in trihedral 7three-seated71 polyhedral 7many-seated71 and cathedral 7a church having a bishop's seat74 ,the word is also1 but less properly1 spelled without the h1 diedral1 a spelling not favored by modern usage4 ,it is not necessary to dwell at length upon the dihedral angle1 except to show the analogy between it and the plane angle4 ,a few illustrations1 as of an open book1 the wall and floor of a room1 and a swinging door1 serve to make the concept clear1 while a plane at right angles to the edge shows the measuring plane angle4 ,so manifest is this relationship between the dihedral angle and its measuring plane angle that some teachers omit the proposition that two dihedral angles have the same ratio as their plane angles4 ,,theorem4 ,if two planes are perpendicular to each other1 a straight line drawn in one of them perpendicular to their intersection is perpendicular to the other4 ,this and the related propositions allow of numerous illustrations taken from the schoolroom1 as of door edges being perpendicular to the floor4 ,the pretended applications of these propositions are usually fictitious1 and the propositions are of value chiefly for their own interest and because they are needed in subsequent proofs4 ,,theorem4 ,the locus of a point equidistant from the faces of a dihedral angle is the plane bisecting the angle4 ,by changing plane to line1 and by making other obvious changes to correspond1 this reduces to the analogous proposition of plane geometry4 ,the figure formed by the plane perpendicular to the edge is also the figure of that analogous proposition4 ,this at once suggests that there are two planes in the locus1 provided the planes of the dihedral angle are taken as indefinite in extent1 and that these planes are perpendicular to each other4 ,it may interest some of the pupils to draw this general figure1 analogous to the one in plane geometry4 ,,theorem4 ,the projection of a straight line not perpendicular to a plane upon that plane is a straight line4 ,in higher mathematics it would simply be said that the projection is a straight line1 the special case of the projection of a perpendicular being considered as a line-segment of zero length4 ,there is no advantage1 however1 of bringing in zero and infinity in the course in elementary geometry4 ,the legitimate reason for the modern use of these terms is seldom understood by beginners4 ,this subject of projection 7,latin pro-1 forth1 and jacere1 to throw7 is extensively used in modern mathematics and also in the elementary work of the draftsman1 and it will be referred to a little later4 ,at this time1 however1 it is well to call attention to the fact that the projection of a straight line on a plane is a straight line or a point2 the projection of a curve may be a curve or it may be straight2 the projection of a point is a point2 and the projection of a plane 7which is easily understood without defining it7 may be a surface or it may be a straight line4 ,an artisan represents a solid by drawing its projection upon two planes at right angles to each other1 and a map maker 7cartographer7 represents the surface of the earth by projecting it upon a plane4 ,a photograph of the class is merely the projection of the class upon a photographic plate 7plane71 and when we draw a figure in solid geometry1 we merely project the solid upon the plane of the paper4 ,there are other projections than those formed by lines that are perpendicular to the plane4 ,the lines may be oblique to the plane1 and this is the case with most projections4 ,a photograph1 for example1 is not formed by lines perpendicular to a plane1 for they all converge in the camera4 ,if the lines of projection are all perpendicular to the plane1 the projection is said to be orthographic1 from the ,greek ortho- 7straight7 and graphein 7to draw74 ,a good example of orthographic projection may be seen in the shadow cast by an object upon a piece of paper that is held perpendicular to the sun's rays4 ,a good example of oblique projection is a shadow on the floor of the schoolroom4 ,,theorem4 ,between two straight lines not in the same plane there can be one common perpendicular1 and only one4 ,the usual corollary states that this perpendicular is the shortest line joining them4 ,it is interesting to compare this with the case of two lines in the same plane4 ,if they are parallel1 there may be any number of common perpendiculars4 ,if they intersect1 there is still a common perpendicular1 but this can hardly be said to be between them1 except for its zero segment4 ,there are many simple illustrations of this case4 ,for example1 what is the shortest line between any given edge of the ceiling and the various edges of the floor of the schoolroom8 ,if two galleries in a mine are to be connected by an air shaft1 how shall it be planned so as to save labor8 ,make a drawing of the plan4 ,at this point the polyhedral angle is introduced4 ,the word is from the ,greek polys 7many7 and hedra 7seat74 ,students have more difficulty in grasping the meaning of the size of a polyhedral angle than is the case with dihedral and plane angles4 ,for this reason it is not good policy to dwell much upon this subject unless the question arises1 since it is better understood when the relation of the polyhedral angle and the spherical polygon is met4 ,teachers will naturally see that just as we may measure the plane angle by taking the ratio of an arc to the whole circle1 and of a dihedral angle by taking the ratio of that part of the cylindric surface that is cut out by the planes to the whole surface1 so we may measure a polyhedral angle by taking the ratio of the spherical polygon to the whole spherical surface4 ,it should also be observed that just as we may have cross polygons in a plane1 so we may have spherical polygons that are similarly tangled1 and that to these will correspond polyhedral angles that are also cross1 their representation by drawings being too complicated for class use4 ,the idea of symmetric solids may be illustrated by a pair of gloves1 all their parts being mutually equal but arranged in opposite order4 ,our hands1 feet1 and ears afford other illustrations of symmetric solids4 ,,theorem4 ,the sum of the face angles of any convex polyhedral angle is less than four right angles4 ,there are several interesting points of discussion in connection with this proposition4 ,for example1 suppose the vertex ,v to approach the plane that cuts the edges in ,a1 ,b1 ,c1 ,d1 4441 the edges continuing to pass through these as fixed points4 ,the sum of the angles about ,v approaches what limit8 ,on the other hand1 suppose ,v recedes indefinitely2 then the sum approaches what limit8 ,then what are the two limits of this sum8 ,suppose the polyhedral angle were concave1 why would the proof not hold8 ,,footnotes3 #hg ,these may be purchased through the ,leipziger ,lehrmittelanstalt1 ,leipzig1 ,germany1 which will send catalogues to intending buyers4 #hh ,an excellent set of stereoscopic views of the figures of solid geometry1 prepared by ,e4 ,m4 ,langley of ,bedford1 ,england1 is published by ,underwood ,underwood1 ,new ,york4 ,such a set may properly have place in a school library or in a classroom in geometry1 to be used when it seems advantageous4 ,,chapter ,,xx ,,the ,,leading ,,propositions ,,of ,,book ,,vii ,book ,,vii relates to polyhedrons1 cylinders1 and cones4 ,it opens with the necessary definitions relating to polyhedrons1 the etymology of the terms often proving interesting and valuable when brought into the work incidentally by the teacher4 ,polyhedron is from the ,greek polys 7many7 and hedra 7seat74 ,the ,greek plural1 polyhedra1 is used in early ,english works1 but polyhedrons is the form now more commonly seen in ,america4 ,prism is from the ,greek prisma 7something sawed1 like a piece of wood sawed from a beam74 ,lateral is from the ,latin latus 7side74 ,parallelepiped is from the ,greek parallelos 7parallel7 and epipedon 7a plane surface71 from epi 7on7 and pedon 7ground74 ,by analogy to parallelogram the word is often spelled parallelopiped1 but the best mathematical works now adopt the etymological spelling above given4 ,truncate is from the ,latin truncare 7to cut off74 ,a few of the leading propositions are now considered4 ,,theorem4 ,the lateral area of a prism is equal to the product of a lateral edge by the perimeter of the right section4 ,it should be noted that although some syllabi do not give the proposition that parallel sections are congruent1 this is necessary for this proposition1 because it shows that the right sections are all congruent and hence that any one of them may be taken4 ,it is1 of course1 possible to construct a prism so oblique and so low that a right section1 that is1 a section cutting all the lateral edges at right angles1 is impossible4 ,in this case the lateral faces must be extended1 thus forming what is called a prismatic space4 ,this term may or may not be introduced1 depending upon the nature of the class4 ,this proposition is one of the most important in ,book ,,vii1 because it is the basis of the mensuration of the cylinder as well as the prism4 ,practical applications are easily suggested in connection with beams1 corridors1 and prismatic columns1 such as are often seen in school buildings4 ,most geometries supply sufficient material in this line1 however4 ,,theorem4 ,an oblique prism is equivalent to a right prism whose base is equal to a right section of the oblique prism1 and whose altitude is equal to a lateral edge of the oblique prism4 ,this is a fundamental theorem leading up to the mensuration of the prism4 ,attention should be called to the analogous proposition in plane geometry relating to the area of the parallelogram and rectangle1 and to the fact that if we cut through the solid figure by a plane parallel to one of the lateral edges1 the resulting figure will be that of the proposition mentioned4 ,as in the preceding proposition1 so in this case1 there may be a question raised that will make it helpful to introduce the idea of prismatic space4 ,,theorem4 ,the opposite lateral faces of a parallelepiped are congruent and parallel4 ,it is desirable to refer to the corresponding case in plane geometry1 and to note again that the figure is obtained by passing a plane through the parallelepiped parallel to a lateral edge4 ,the same may be said for the proposition about the diagonal plane of a parallelepiped4 ,these two propositions are fundamental in the mensuration of the prism4 ,,theorem4 ,two rectangular parallelepipeds are to each other as the products of their three dimensions4 ,this leads at once to the corollary that the volume of a rectangular parallelepiped equals the product of its three dimensions1 the fundamental law in the mensuration of all solids4 ,it is preceded by the proposition asserting that rectangular parallelepipeds having congruent bases are proportional to their altitudes4 ,this includes the incommensurable case1 but this case may be omitted4 ,the number of simple applications of this proposition is practically unlimited4 ,in all such cases it is advisable to take a considerable number of numerical exercises in order to fix in mind the real nature of the proposition4 ,any good geometry furnishes a certain number of these exercises4 ,the following is an interesting property of the rectangular parallelepiped1 often called the rectangular solid3 ,if the edges are a1 b1 and c1 and the diagonal is d1 then 7a_/d7#b 7b_/d7#b 7c_/d7#b #a4 ,this property is easily proved by the ,pythagorean ,theorem1 for d#b a#b b#b c#b1 whence 7a#b b#b c#b7 _/ d#b #a4 ,in case c #j1 this reduces to the ,pythagorean ,theorem4 ,the property is the fundamental one of solid analytic geometry4 ,,theorem4 ,the volume of any parallelepiped is equal to the product of its base by its altitude4 ,this is one of the few propositions in ,book ,,vii where a model is of any advantage4 ,it is easy to make one out of pasteboard1 or to cut one from wood4 ,if a wooden one is made1 it is advisable to take an oblique parallelepiped and1 by properly sawing it1 to transform it into a rectangular one instead of using three different solids4 ,on account of its awkward form1 this figure is sometimes called the ,devil's ,coffin1 but it is a name that it would be well not to perpetuate4 ,,theorem4 ,the volume of any prism is equal to the product of its base by its altitude4 ,this is also one of the basal propositions of solid geometry1 and it has many applications in practical mensuration4 ,a first-class textbook will give a sufficient list of problems involving numerical measurement1 to fix the law in mind4 ,for outdoor work1 involving measurements near the school or within the knowledge of the pupils1 the following problem is a type3 ,illustration ,if this represents the cross section of a railway embankment that is l feet long1 h feet high1 b feet wide at the bottom1 and b' feet wide at the top1 find the number of cubic feet in the embankment4 ,find the volume if l #cjj1 h #h1 b #fj1 and b' #bh4 ,the mensuration of the volume of the prism1 including the rectangular parallelepiped and cube1 was known to the ancients4 ,euclid was not concerned with practical measurement1 so that none of this part of geometry appears in his ,elements4 ,we find1 however1 in the papyrus of ,ahmes1 directions for the measuring of bins1 and the ,egyptian builders1 long before his time1 must have known the mensuration of the rectangular parallelepiped4 ,among the ,hindus1 long before the ,christian era1 rules were known for the construction of altars1 and among the ,greeks the problem of constructing a cube with twice the volume of a given cube 7the duplication of the cube7 was attacked by many mathematicians4 ,the solution of this problem is impossible by elementary geometry4 ,if e equals the edge of the given cube1 then e#c is its volume and #b;e#c is the volume of the required cube4 ,therefore the edge of the required cube is e#croot#b4 ,now if e is given1 it is not possible with the straightedge and compasses to construct a line equal to e#croot#b1 although it is easy to construct one equal to esqrt#b4 ,the study of the pyramid begins at this point4 ,in practical measurement we usually meet the regular pyramid4 ,it is1 however1 a simple matter to consider the oblique pyramid as well1 and in measuring volumes we sometimes find these forms4 ,,theorem4 ,the lateral area of a regular pyramid is equal to half the product of its slant height by the perimeter of its base4 ,this leads to the corollary concerning the lateral area of the frustum of a regular pyramid4 ,it should be noticed that the regular pyramid may be considered as a frustum with the upper base zero1 and the proposition as a special case under the corollary4 ,it is also possible1 if we choose1 to let the upper base of the frustum pass through the vertex and cut the lateral edges above that point1 although this is too complicated for most pupils4 ,if this case is considered1 it is well to bring in the general idea of pyramidal space1 the infinite space bounded on several sides by the lateral faces1 of the pyramid4 ,this pyramidal space is double1 extending on two sides of the vertex4 ,,theorem4 ,if a pyramid is cut by a plane parallel to the base3 #a4 ,the edges and altitude are divided proportionally4 #b4 ,the section is a polygon similar to the base4 ,to get the analogous proposition of plane geometry1 pass a plane through the vertex so as to cut the base4 ,we shall then have the sides and altitude of the triangle divided proportionally1 and of course the section will merely be a line-segment1 and therefore it is similar to the base line4 ,the cutting plane may pass through the vertex1 or it may cut the pyramidal space above the vertex4 ,in either case the proof is essentially the same4 ,,theorem4 ,the volume of a triangular pyramid is equal to one third of the product of its base by its altitude1 and this is also true of any pyramid4 ,this is stated as two theorems in all textbooks1 and properly so4 ,it is explained to children who are studying arithmetic by means of a hollow pyramid and a hollow prism of equal base and equal altitude4 ,the pyramid is filled with sand or grain1 and the contents is poured into the prism4 ,this is repeated1 and again repeated1 showing that the volume of the prism is three times the volume of the pyramid4 ,it sometimes varies the work to show this to a class in geometry4 ,this proposition was first proved1 so ,archimedes asserts1 by ,eudoxus of ,cnidus1 famous as an astronomer1 geometer1 physician1 and lawgiver1 born in humble circumstances about #djg ,b4,c4 ,he studied at ,athens and in ,egypt1 and founded a famous school of geometry at ,cyzicus4 ,his discovery also extended to the volume of the cone1 and it was his work that gave the beginning to the science of stereometry1 the mensuration part of solid geometry4 ,,theorem4 ,the volume of the frustum of any pyramid is equal to the sum of the volumes of three pyramids whose common altitude is the altitude of the frustum1 and whose bases are the lower base1 the upper base1 and the mean proportional between the bases of the frustum4 ,attention should be called to the fact that this formula v #a_/c a7b b' sqrt7bb'77 applies to the pyramid by letting b' #j1 to the prism by letting b b'1 and also to the parallelepiped and cube1 these being special forms of the prism4 ,this formula is1 therefore1 a very general one1 relating to all the polyhedrons that are commonly met in mensuration4 ,,theorem4 ,there cannot be more than five regular convex polyhedrons4 ,eudemus of ,rhodes1 one of the principal pupils of ,aristotle1 in his history of geometry of which ,proclus preserves some fragments1 tells us that ,pythagoras discovered the construction of the mundane figures1 meaning the five regular polyhedrons4 ,iamblichus speaks of the discovery of the dodecahedron in these words3 ,as to ,hippasus1 who was a ,pythagorean1 they say that he perished in the sea on account of his impiety1 inasmuch as he boasted that he first divulged the knowledge of the sphere with the twelve pentagons4 ,hippasus assumed the glory of the discovery to himself1 whereas everything belongs to ,him1 for thus they designate ,pythagoras1 and do not call ,him by name4 ,iamblichus here refers to the dodecahedron inscribed in the sphere4 ,the ,pythagoreans looked upon these five solids as fundamental forms in the structure of the universe4 ,in particular ,plato tells us that they asserted that the four elements of the real world were the tetrahedron1 octahedron1 icosahedron1 and cube1 and ,plutarch ascribes this doctrine to ,pythagoras himself4 ,philolaus1 who lived in the fifth century ,b4,c41 held that the elementary nature of bodies depended on their form4 ,the tetrahedron was assigned to fire1 the octahedron to air1 the icosahedron to water1 and the cube to earth1 it being asserted that the smallest constituent part of each of these substances had the form here assigned to it4 ,although ,eudemus attributes all five to ,pythagoras1 it is certain that the tetrahedron1 cube1 and octahedron were known to the ,egyptians1 since they appear in their architectural decorations4 ,these solids were studied so extensively in the school of ,plato that ,proclus also speaks of them as the ,platonic bodies1 saying that ,euclid proposed to himself the construction of the so-called ,platonic bodies as the final aim of his arrangement of the ',elements4' ,aristaeus1 probably a little older than ,euclid1 wrote a book upon these solids4 ,as an interesting amplification of this proposition1 the centers of the faces 7squares7 of a cube may be connected1 an inscribed octahedron being thereby formed4 ,furthermore1 if the vertices of the cube are ,a1 ,b1 ,c1 ,d1 ,a'1 ,b'1 ,c'1 ,d'1 then by drawing ,,ac1 ,,cd'1 ,d',a1 ,d',b'1 ,b',a1 and ,b',c1 a regular tetrahedron will be formed4 ,since the construction of the cube is a simple matter1 this shows how three of the five regular solids may be constructed4 ,the actual construction of the solids is not suited to elementary geometry4#hi ,it is not difficult for a class to find the relative areas of the cube and the inscribed tetrahedron and octahedron4 ,if s is the side of the cube1 these areas are #fs#b1 7#a_/b7s#bsqrt#c1 and s#bsqrt#c2 that is1 the area of the octahedron is twice that of the tetrahedron inscribed in the cube4 ,somewhat related to the preceding paragraph is the fact that the edges of the five regular solids are incommensurable with the radius of the circumscribed sphere4 ,this fact seems to have been known to the ,greeks1 perhaps to ,theaetetus 7ca4 #djj ,b4,c47 and ,aristaeus 7ca4 #cjj ,b4,c471 both of whom wrote on incommensurables4 ,just as we may produce the sides of a regular polygon and form a regular cross polygon or stellar polygon1 so we may have stellar polyhedrons4 ,kepler1 the great astronomer1 constructed some of these solids in #afai1 and ,poinsot1 a ,french mathematician1 carried the constructions so far in #ahja that several of these stellar polyhedrons are known as ,poinsot solids4 ,there is a very extensive literature upon this subject4 ,the following table may be of some service in assigning problems in mensuration in connection with the regular polyhedrons1 although some of the formulas are too difficult for beginners to prove4 ,in the table e edge of the polyhedron1 r radius of circumscribed sphere1 r' radius of inscribed sphere1 a total area1 v volume4 ,,number ,,of ,,faces #d #f #h ---------------------------------------- --------------- r esqrt7#c_/h7 7e_/#b7sqrt#c esqrt7#a_/b7 r' esqrt7#a_/bd7 e_/#b esqrt7#a_/f7 a e#bsqrt#c #f;e#b #b;e#bsqrt#c v 7e#c_/ab7sqrt#b e#c 7e#c_/c7sqrt#b ---------------------------------------- ------------------ ,,number ,,of ,,faces #ab #bj ---------------------------------------- ------------------------------ r 7e_/#d7sqrt#c7sqrt#e #a7 esqrt77#e sqrt#e7_/h7 r' 7e_/#b7sqrt77#be #aasqrt#e7_/aj77;esqrt#c7_/ab7sqrt#e #c7 a #c;e#bsqrt7#e7e #bsqrt#e77 7#e;e#b7sqrt#c v 77e#c7_/d77ae #gsqrt#e7 77#e;e#c7_/ab77sqrt#e #c7 ---------------------------------------- -------------------------------- ,some interest is added to the study of polyhedrons by calling attention to their occurrence in nature1 in the form of crystals4 ,the computation of the surfaces and volumes of these forms offers an opportunity for applying the rules of mensuration1 and the construction of the solids by paper folding or by the cutting of crayon or some other substance often arouses a considerable interest4 ,the following are forms of crystals that are occasionally found3 ,illustration ,they show how the cube is modified by having its corners cut off4 ,a cube may be inscribed in an octahedron1 its vertices being at the centers of the faces of the octahedron4 ,if we think of the cube as expanding1 the faces of the octahedron will cut off the corners of the cube as seen in the first figure1 leaving the cube as shown in the second figure4 ,if the corners are cut off still more1 we have the third figure4 ,similarly1 an octahedron may be inscribed in a cube1 and by letting it expand a little1 the faces of the cube will cut off the corners of the octahedron4 ,this is seen in the following figures3 ,illustration ,this is a form that is found in crystals1 and the computation of the surface and volume is an interesting exercise4 ,the quartz crystal1 an hexagonal pyramid on an hexagonal prism1 is found in many parts of the country1 or is to be seen in the school museum1 and this also forms an interesting object of study in this connection4 ,the properties of the cylinder are next studied4 ,the word is from the ,greek kylindros1 from kyliein 7to roll74 ,in ancient mathematics circular cylinders were the only ones studied1 but since some of the properties are as easily proved for the case of a noncircular directrix1 it is not now customary to limit them in this way4 ,it is convenient to begin by a study of the cylindric surface1 and a piece of paper may be curved or rolled up to illustrate this concept4 ,if the paper is brought around so that the edges meet1 whatever curve may form a cross section the surface is said to inclose a cylindric space4 ,this concept is sometimes convenient1 but it need be introduced only as necessity for using it arises4 ,the other definitions concerning the cylinder are so simple as to require no comment4 ,the mensuration of the volume of a cylinder depends upon the assumption that the cylinder is the limit of a certain inscribed or circumscribed prism as the number of sides of the base is indefinitely increased4 ,it is possible to give a fairly satisfactory and simple proof of this fact1 but for pupils of the age of beginners in geometry in ,america it is better to make the assumption outright4 ,this is one of several cases in geometry where a proof is less convincing than the assumed statement4 ,,theorem4 ,the lateral area of a circular cylinder is equal to the product of the perimeter of a right section of the cylinder by an element4 ,for practical purposes the cylinder of revolution 7right circular cylinder7 is the one most frequently used1 and the important formula is therefore l #bpirh where l the lateral area1 r the radius1 and h the altitude4 ,applications of this formula are easily found4 ,,theorem4 ,the volume of a circular cylinder is equal to the product of its base by its altitude4 ,here again the important case is that of the cylinder of revolution1 where v pir#b;h4 ,the number of applications of this proposition is1 of course1 very great4 ,in architecture and in mechanics the cylinder is constantly seen1 and the mensuration of the surface and the volume is important4 ,a single illustration of this type of problem will suffice4 ,a machinist is making a crank pin 7a kind of bolt7 for an engine1 according to this drawing4 ,he considers it as weighing the same as three steel cylinders having the diameters and lengths in inches as here shown1 where #g #c_/d means #g #c_/d inches4 ,he has this formula for the weight 7w7 of a steel cylinder where d is the diameter and l is the length3 w #j4jgpid#bl4 ,taking pi #c #a_/g1 find the weight of the pin4 ,the most elaborate study of the cylinder1 cone1 and sphere 7the three round bodies7 in the ,greek literature is that of ,archimedes of ,syracuse 7on the island of ,sicily71 who lived in the third century ,b4,c4 ,archimedes tells us1 however1 that ,eudoxus 7born ca4 #djg ,b4,c47 discovered that any cone is one third of a cylinder of the same base and the same altitude4 ,tradition says that ,archimedes requested that a sphere and a cylinder be carved upon his tomb1 and that this was done4 ,cicero relates that he discovered the tomb by means of these symbols4 ,the tomb now shown to visitors in ancient ,syracuse as that of ,archimedes cannot be his1 for it bears no such figures1 and is not outside the gate of ,agrigentum1 as ,cicero describes4 ,the cone is now introduced4 ,a conic surface is easily illustrated to a class by taking a piece of paper and rolling it up into a cornucopia1 the space inclosed being a conic space1 a term that is sometimes convenient4 ,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 curve4 ,if we consider a straight line as the limit of a curve1 then the pointer may swing in a plane1 and so a plane is the limit of a conic surface4 ,if we swing the pointer about a point in the middle1 we shall generate the two nappes of the cone1 the conic space now being double4 ,in practice the right circular cone1 or cone of revolution1 is the important type1 and special attention should be given to this form4 ,,theorem4 ,every section of a cone made by a plane passing through its vertex is a triangle4 ,at this time1 or in speaking of the preliminary definitions1 reference should be made to the conic sections4 ,of these there are three great types3 7#a7 the ellipse1 where the cutting plane intersects all the elements on one side of the vertex2 a circle is a special form of the ellipse2 7#b7 the parabola1 where the plane is parallel to an element2 7#c7 the hyperbola1 where the plane cuts some of the elements on one side of the vertex1 and the rest on the other side2 that is1 where it cuts both nappes4 ,it is to be observed that the ellipse may vary greatly in shape1 from a circle to a very long ellipse1 as the cutting plane changes from being perpendicular to the axis to being nearly parallel to an element4 ,the instant it becomes parallel to an element the ellipse changes suddenly to a parabola4 ,if the plane tips the slightest amount more1 the section becomes an hyperbola4 ,while these conic sections are not studied in elementary geometry1 the terms should be known for general information1 particularly the ellipse and parabola4 ,the study of the conic sections forms a large part of the work of analytic geometry1 a subject in which the figures resemble the graphic work in algebra1 this having been taken from analytics1 as the higher subject is commonly called4 ,the planets move about the sun in elliptic orbits1 and ,halley's comet that returned to view in #aiji-aiaj has for its path an enormous ellipse4 ,most comets seem to move in parabolas1 and a body thrown into the air would take a parabolic path if it were not for the resistance of the atmosphere4 ,two of the sides of the triangle in this proposition constitute a special form of the hyperbola4 ,the study of conic sections was brought to a high state by the ,greeks4 ,they were not known to the ,pythagoreans1 but were discovered by ,menaechmus in the fourth century ,b4,c4 ,this discovery is mentioned by ,proclus1 who says1 ,further1 as to these sections1 the conics were conceived by ,menaechmus4 ,since if the cutting plane is perpendicular to the axis the section is a circle1 and if oblique it is an ellipse1 a parabola1 or an hyperbola1 it follows that if light proceeds from a point1 the shadow of a circle is a circle1 an ellipse1 a parabola1 or an hyperbola1 depending on the position of the plane on which the shadow falls4 ,it is interesting and instructive to a class to see these shadows1 but of course not much time can be allowed for such work4 ,at this point the chief thing is to have the names ellipse and parabola1 so often met in reading1 understood4 ,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 paper4 ,this is a practical method1 and is familiar to all teachers who have studied analytic geometry4 ,in designing elliptic arches1 however1 three circular arcs are often joined1 as here shown1 the result being approximately an elliptic arc4 ,illustration ,here ,o is the center of arc ,,bc1 ,m of arc ,,ab1 and ,n of arc ,,cd4 ,since ,,xy is perpendicular to ,,bm and ,,bo1 it is tangent to arcs ,,ab and ,,bc1 so there is no abrupt turning at ,b1 and similarly for ,c4#ij ,,theorem4 ,the volume of a circular cone is equal to one third the product of its base by its altitude4 ,it is easy to prove this for noncircular cones as well1 but since they are not met commonly in practice1 they may be omitted in elementary geometry4 ,the important formula at this time is v #a_/cpir#b;h4 ,as already stated1 this proposition was discovered by ,eudoxus of ,cnidus 7born ca4 #djg ,b4,c41 died ca4 #ced ,b4,c471 a man who1 as already stated1 was born poor1 but who became one of the most illustrious and most highly esteemed of all the ,greeks of his time4 ,,theorem4 ,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 height4 ,an interesting case for a class to notice is that in which the upper base becomes zero and the frustum becomes a cone1 the proposition being still true4 ,if the upper base is equal to the lower base1 the frustum becomes a cylinder1 and still the proposition remains true4 ,the proposition thus offers an excellent illustration of the elementary ,principle of ,continuity4 ,then follows1 in most textbooks1 a theorem relating to the volume of a frustum4 ,in the case of a cone of revolution v 7#a_/c7pih7r#b r'#b rr'74 ,here if r' #j1 we have v 7#a_/c7pir#b;h1 the volume of a cone4 ,if r' r1 we have v 7#a_/c7pih7r#b r#b r#b7 pihr#b1 the volume of a cylinder4 ,if one needs examples in mensuration beyond those given in a first-class textbook1 they are easily found4 ,the monument to ,sir ,christopher ,wren1 the professor of geometry in ,cambridge ,university1 who became the great architect of ,st4 ,paul's ,cathedral in ,london1 has a ,latin inscription which means1 ,reader1 if you would see his monument1 look about you4 ,so it is with practical examples in ,book ,,vii4 ,appended to this ,book1 or more often to the course in solid geometry1 is frequently found a proposition known as ,euler's ,theorem4 ,this is often considered too difficult for the average pupil and is therefore omitted4 ,on account of its importance1 however1 in the theory of polyhedrons1 some reference to it at this time may be helpful to the teacher4 ,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 faces4 ,in other words1 that e #b v f4 ,on account of its importance a proof will be given that differs from the one ordinarily found in textbooks4 ,let s#a1 s#b1 4441 sn be the number of sides of the various faces1 and f the number of faces4 ,now since the sum of the angles of a polygon of s sides is 7s - #b7ahj deg41 therefore the sum of the angles of all the faces is 7s#a s#b s#c 444 sn - #b;f7#ahj deg44 ,but s#a s#b s#c 444 sn is twice the number of edges1 because each edge belongs to two faces4 therefore the sum of the angles of all the faces is 7#b;e - #b;f7#ahj deg41 or 7e - f7#cfj deg44 ,since the polyhedron is convex1 it is possible to find some outside point of view1 ,p1 from which some face1 as ,,abcde1 covers up the whole figure1 as in this illustration4 ,if we think of all the vertices projected on ,,abcde1 by lines through ,p1 the sum of the angles of all the faces will be the same as the sum of the angles of all their projections on ,,abcde4 ,calling ,,abcde s#a1 and thinking of the projections as traced by dotted lines on the opposite side of s#a1 this sum is evidently equal to 7#a7 the sum of the angles in s#a1 or 7s#a - #b7 #ahj deg41 plus 7#b7 the sum of the angles on the other side of s#a1 or 7s#a - #b7ahj deg41 plus 7#c7 the sum of the angles about the various points shown as inside of s#a1 of which there are v - s#a points1 about each of which the sum of the angles is #cfj deg41 making 7v - s#a7cfj deg4 in all4 ,illustration ,adding1 we have 7s#a - #b7ahj deg4 7s#a - #b7ahj deg4 7v - s#a7cfj deg4 7s#a - #b7 7v - s#a7cfj deg4 7v - #b7cfj deg44 ,equating the two sums already found1 we have 7e - f7#cfj deg4 7v - #b7cfj deg41 or e - f v - #b1 or e #b v f4 ,this proof is too abstract for most pupils in the high school1 but it is more scientific than those found in any of the elementary textbooks1 and teachers will find it of service in relieving their own minds of any question as to the legitimacy of the theorem4 ,although this proposition is generally attributed to ,euler1 and was1 indeed1 rediscovered by him and published in #ageb1 it was known to the great ,french geometer ,descartes1 a fact that ,leibnitz mentions4#ia ,this theorem has a very practical application in the study of crystals1 since it offers a convenient check on the count of faces1 edges1 and vertices4 ,some use of crystals1 or even of polyhedrons cut from a piece of crayon1 is desirable when studying ,euler's proposition4 ,the following illustrations of common forms of crystals may be used in this connection3 ,illustration ,the first represents two truncated pyramids placed base to base4 ,here e #bj1 f #aj1 v #ab1 so that e #b f v4 ,the second represents a crystal formed by replacing each edge of a cube by a plane1 with the result that e #dj1 f #ah1 and v #bd4 ,the third represents a crystal formed by replacing each edge of an octahedron by a plane1 it being easy to see that ,euler's law still holds true4 ,,footnotes3 #hi ,the actual construction of these solids is given by ,pappus4 ,see his ,mathematicae ,collectiones1 p4 #dh1 ,bologna1 #affj4 #ij ,the illustration is from ,dupin1 loc4 cit4 #ia ,for the historical bibliography consult ,g4 ,holzmueller1 ,elemente der ,stereometrie1 ,vol4 ,i1 p4 #aha1 ,leipzig1 #aijj4 ,,chapter ,,xxi ,,the ,,leading ,,propositions ,,of ,,book ,,viii ,book ,,viii treats of the sphere4 ,just as the circle may be defined either as a plane surface or as the bounding line which is the locus of a point in a plane at a given distance from a fixed point1 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 point4 ,in higher mathematics the circle is defined as the bounding line and the sphere as the bounding surface2 that is1 each is defined as a locus4 ,this view of the circle as a line is becoming quite general in elementary geometry1 it being the desire that students may not have to change definitions in passing from elementary to higher mathematics4 ,the sphere is less frequently looked upon in geometry as a surface1 and in popular usage it is always taken as a solid4 ,analogous to the postulate that a circle may be described with any given point as a center and any given line as a radius1 is the postulate for constructing a sphere with any given center and any given radius4 ,this postulate is not so essential1 however1 as the one about the circle1 because we are not so concerned with constructions here as we are in plane geometry4 ,a good opportunity is offered for illustrating several of the definitions connected with the study of the sphere1 such as great circle1 axis1 small circle1 and pole1 by referring to geography4 ,indeed1 the first three propositions usually given in ,book ,,viii have a direct bearing upon the study of the earth4 ,,theorem4 ,a plane perpendicular to a radius at its extremity is tangent to the sphere4 ,the student should always have his attention called to the analogue in plane geometry1 where there is one4 ,if here we pass a plane through the radius in question1 the figure formed on the plane will be that of a line tangent to a circle4 ,if we revolve this about the line of the radius in question1 as an axis1 the circle will generate the sphere again1 and the tangent line will generate the tangent plane4 ,,theorem4 ,a sphere may be inscribed in any given tetrahedron4 ,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 tetrahedron4 ,we shall then form the figure of a circle inscribed in a triangle4 ,and just as in the case of the triangle we may have escribed circles by producing the sides1 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 sphere4 ,the figure is difficult to draw1 but it is not difficult to understand1 particularly if we construct the tetrahedron out of pasteboard4 ,,theorem4 ,a sphere may be circumscribed about any given tetrahedron4 ,by producing one of the faces indefinitely it will cut the sphere in a circle1 and the resulting figure1 on the plane1 will be that of the analogous proposition of plane geometry1 the circle circumscribed about a triangle4 ,it is easily proved from the proposition that the four perpendiculars erected at the centers of the faces of a tetrahedron meet in a point 7are concurrent71 the analogue of the proposition about the perpendicular bisectors of the sides of a triangle4 ,,theorem4 ,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 line4 ,the figure suggests the case of two circles in plane geometry4 ,in the case of two circles that do not intersect or touch1 one not being within the other1 there are four common tangents4 ,if the circles touch1 two close up into one4 ,if one circle is wholly within the other1 this last tangent disappears4 ,the same thing exists in relation to two spheres1 and the analogous cases are formed by revolving the circles and tangents about the line through their centers4 ,in plane geometry it is easily proved that if two circles intersect1 the tangents from any point on their common chord produced are equal4 ,for if the common chord is ,,ab and the point ,p is taken on ,,ab produced1 then the square on any tangent from ,p is equal to ,,pb x ,,pa4 ,the line ,,pba is sometimes called the radical axis4 ,similarly in this proposition concerning spheres1 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 sphere1 these tangents are equal4 ,for it is easily proved that all tangents to the same sphere from an external point are equal1 and it can be proved as in plane geometry that two tangents to the two spheres are equal4 ,among the interesting analogies between plane and solid geometry is the one relating to the four common tangents to two circles4 ,if the figure be revolved about the line of centers1 the circles generate spheres and the tangents generate conical surfaces4 ,to study this case for various sizes and positions of the two spheres is one of the most interesting generalizations of solid geometry4 ,an application of the proposition is seen in the case of an eclipse1 where the sphere ,o' represents the moon1 ,o the earth1 and ,s the sun4 ,it is also seen in the case of the full moon1 when ,s is on the other side of the earth4 ,in this case the part ,,min is fully illuminated by the moon1 but the zone ,,abnm is only partly illuminated1 as the figure shows4#ib ,illustration ,,theorem4 ,the sum of the sides of a spherical polygon is less than #cfj deg44 ,in all such cases the relation to the polyhedral angle should be made clear4 ,this is done in the proofs usually given in the textbooks4 ,it is easily seen that this is true only with the limitation set forth in most textbooks1 that the spherical polygons considered are convex4 ,thus we might have a spherical triangle that is concave1 with its base #cei deg41 and its other two sides each #ij deg41 the sum of the sides being #eci deg44 ,,theorem4 ,the sum of the angles of a spherical triangle is greater than #ahj deg4 and less than #edj deg44 ,it is for the purpose of proving this important fact that polar triangles are introduced4 ,this proposition shows the relation of the spherical to the plane triangle4 ,if our planes were in reality slightly curved1 being small portions of enormous spherical surfaces1 then the sum of the angles of a triangle would not be exactly #ahj deg41 but would exceed #ahj deg4 by some amount depending on the curvature of the surface4 ,just as a being may be imagined as having only two dimensions1 and living always on a plane surface 7in a space of two dimensions71 and having no conception of a space of three dimensions1 so we may think of ourselves as living in a space of three dimensions but surrounded by a space of four dimensions4 ,the flat being could not point to a third dimension because he could not get out of his plane1 and we cannot point to the fourth dimension because we cannot get out of our space4 ,now what the flat being thinks is his plane may be the surface of an enormous sphere in our three dimensions2 in other words1 the space he lives in may curve through some higher space without his being conscious of it4 ,so our space may also curve through some higher space without our being conscious of it4 ,if our planes have really some curvature1 then the sum of the angles of our triangles has a slight excess over #ahj deg44 ,all this is mere speculation1 but it may interest some student to know that the idea of fourth and higher dimensions enters largely into mathematical investigation to-day4 ,,theorem4 ,two symmetric spherical triangles are equivalent4 ,while it is not a subject that has any place in a school1 save perhaps for incidental conversation with some group of enthusiastic students1 it may interest the teacher to consider this proposition in connection with the fourth dimension just mentioned4 ,consider these triangles1 where ,l,a ,l,a'1 ,,ab ,a',b'1 ,,ac ,a',c'4 ,we prove them congruent by superposition1 turning one over and placing it upon the other4 ,but suppose we were beings in ,flatland1 beings with only two dimensions and without the power to point in any direction except in the plane we lived in4 ,we should then be unable to turn triangle,a',b',c' over so that it could coincide with triangle,,abc1 and we should have to prove these triangles equivalent in some other way1 probably by dividing them into isosceles triangles that could be superposed4 ,illustration ,illustration ,now it is the same thing with symmetric spherical triangles2 we cannot superpose them4 ,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 dimension8 ,it is interesting to think about this possibility even though we carry it no further1 and in these side lights on mathematics lies much of the fascination of the subject4 ,,theorem4 ,the shortest line that can be drawn on the surface of a sphere between two points is the minor arc of a great circle joining the two points4 ,it is always interesting to a class to apply this practically4 ,by taking a terrestrial globe and drawing a great circle between the southern point of ,ireland and ,new ,york ,city1 we represent the shortest route for ships crossing to ,england4 ,now if we notice where this great-circle arc cuts the various meridians and mark this on an ordinary ,mercator's projection map1 such as is found in any schoolroom1 we shall find that the path of the ship does not make a straight line4 ,passengers at sea often do not understand why the ship's course on the map is not a straight line2 but the chief reason is that the ship is taking a great-circle arc1 and this is not1 in general1 a straight line on a ,mercator projection4 ,the small circles of latitude are straight lines1 and so are the meridians and the equator1 but other great circles are represented by curved lines4 ,,theorem4 ,the area of the surface of a sphere is equal to the product of its diameter by the circumference of a great circle4 ,this leads to the remarkable formula1 a #dpir#b4 ,that the area of the sphere1 a curved surface1 should exactly equal the sum of the areas of four great circles1 plane surfaces1 is the remarkable feature4 ,this was one of the greatest discoveries of ,archimedes 7ca4 #bhg-bab ,b4,c471 who gives it as the thirty-fifth proposition of his treatise on the ,sphere and the ,cylinder1 and who mentions it specially in a letter to his friend ,dositheus1 a mathematician of some prominence4 ,archimedes also states that the surface of a sphere is two thirds that of the circumscribed cylinder1 or the same as the curved surface of this cylinder4 ,this is evident1 since the cylindric surface of the cylinder is #bpir x #br1 or #dpir#b1 and the two bases have an area pir#b pir#b1 making the total area #fpir#b4 ,,theorem4 ,the area of a spherical triangle is equal to the area of a lune whose angle is half the triangle's spherical excess4 ,this theorem1 so important in finding areas on the earth's surface1 should be followed by a considerable amount of computation of triangular areas1 else it will be rather meaningless4 ,students tend to memorize a proof of this character1 and in order to have the proposition mean what it should to them1 they should at once apply it4 ,the same is true of the following proposition on the area of a spherical polygon4 ,it is probable that neither of these propositions is very old2 at any rate1 they do not seem to have been known to the writers on elementary mathematics among the ,greeks4 ,,theorem4 ,the volume of a sphere is equal to the product of the area of its surface by one third of its radius4 ,this gives the formula v 7#d_/c7pir#c4 ,this is one of the greatest discoveries of ,archimedes4 ,he also found as a result that the volume of a sphere is two thirds the volume of the circumscribed cylinder4 ,this is easily seen1 since the volume of the cylinder is pir#b x #br1 or #bpir#c1 and 7#d_/c7pir#c is #b_/c of #bpir#c4 ,it was because of these discoveries on the sphere and cylinder that ,archimedes wished these figures engraved upon his tomb1 as has already been stated4 ,the ,roman general ,marcellus conquered ,syracuse in #bab ,b4,c41 and at the sack of the city ,archimedes was killed by an ignorant soldier4 ,marcellus carried out the wishes of ,archimedes with respect to the figures on his tomb4 ,the volume of a sphere can also be very elegantly found by means of a proposition known as ,cavalieri's ,theorem4 ,this asserts that if two solids lie between parallel planes1 and are such that the two sections made by any plane parallel to the given planes are equal in area1 the solids are themselves equal in volume4 ,thus1 if these solids have the same altitude1 a1 and if ,s and ,s' are equal sections made by a plane parallel to ,,mn1 then the solids have the same volume4 ,the proof is simple1 since prisms of the same altitude1 say a_/n1 and on the bases ,s and ,s' are equivalent1 and the sums of n such prisms are the given solids2 and as n increases1 the sums of the prisms approach the solids as their limits2 hence the volumes are equal4 ,illustration ,this proposition1 which will now be applied to finding the volume of the sphere1 was discovered by ,bonaventura ,cavalieri 7#aeia or #aeih-afdg74 ,he was a ,jesuit professor in the ,university of ,bologna1 and his best known work is his ,geometria ,indivisilibus1 which he wrote in #afbf1 at least in part1 and published in #afce 7second edition1 #afdg74 ,by means of the proposition it is also possible to prove several other theorems1 as that the volumes of triangular pyramids of equivalent bases and equal altitudes are equal4 ,illustration ,to find the volume of a sphere1 take the quadrant ,,opq1 in the square ,,oprq4 ,then if this figure is revolved about ,,op1 ,,opq will generate a hemisphere1 ,,opr will generate a cone of volume 7#a_/c7pir#c1 and ,,oprq will generate a cylinder of volume pir#c4 ,hence the figure generated by ,,orq will have a volume pir#c - 7#a_/c7pir#c1 or 7#b_/c7pir#c1 which we will call x4 ,now ,,oa ,,ab1 and ,,oc ,,ad2 also 7,,oc7#b - 7,,oa7#b 7,,ac7#b1 so that 7,,ad7#b - 7,,ab7#b 7,,ac7#b1 and pi7,,ad7#b - pi7,,ab7#b pi7,,ac7#b4 ,but pi7,,ad7#b - pi7,,ab7#b is the area of the ring generated by ,,bd1 a section of x1 and pi7,,ac7#b is the corresponding section of the hemisphere4 ,hence1 by ,cavalieri's ,theorem1 7#b_/c7pir#c the volume of the hemisphere4 therefore 7#d_/c7pir#c the volume of the sphere4 ,in connection with the sphere some easy work in quadratics may be introduced even if the class has had only a year in algebra4 ,for example1 suppose a cube is inscribed in a hemisphere of radius r and we wish to find its edge1 and thereby its surface and its volume4 ,if x the edge of the cube1 the diagonal of the base must be xsqrt#b1 and the projection of r 7drawn from the center of the base to one of the vertices7 on the base is half of this diagonal1 or 7xsqrt#b7_/b4 ,hence1 by the ,pythagorean ,theorem1 r#b x#b 77xsqrt#b7_/b7b 7#c_/b7x#b therefore x rsqrt7#b_/c71 and the total surface is #fx#b #dr#b1 and the volume is x#c 7#b_/c7r#csqrt7#b_/c74 ,,footnotes3 #ib ,the illustration is from ,dupin1 loc4 cit4 ,l',,envoi ,in the ,valley of ,youth1 through which all wayfarers must pass on their journey from the ,land of ,mystery to the ,land of the ,infinite1 there is a village where the pilgrim rests and indulges in various excursions for which the valley is celebrated4 ,there also gather many guides in this spot1 some of whom show the stranger all the various points of common interest1 and others of whom take visitors to special points from which the views are of peculiar significance4 ,as time has gone on new paths have opened1 and new resting places have been made from which these views are best obtained4 ,some of the mountain peaks have been neglected in the past1 but of late they too have been scaled1 and paths have been hewn out that approach the summits1 and many pilgrims ascend them and find that the result is abundantly worth the effort and the time4 ,the effect of these several improvements has been a natural and usually friendly rivalry in the body of guides that show the way4 ,the mountains have not changed1 and the views are what they have always been4 ,but there are not wanting those who say1 ,my mountain may not be as lofty as yours1 but it is easier to ascend2 or ,there are quarries on my peak1 and points of view from which a building may be seen in process of erection1 or a mill in operation1 or a canal1 while your mountain shows only a stretch of hills and valleys1 and thus you will see that mine is the more profitable to visit4 ,then there are guides who are themselves often weak of limb1 and who are attached to numerous sand dunes1 and these say to the weaker pilgrims1 ,why tire yourselves 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 precipice8 ,the result is not wholly disadvantageous1 for many who pass through the valley are able to approach the summits of the sand dunes only1 and would make progress with greatest difficulty should they attempt to scale a real mountain1 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 dunes4 ,then1 too1 there have of late come guides who have shown much ingenuity by digging tunnels into some of the greatest mountains4 ,these they have paved with smooth concrete1 and have arranged for rubber-tired cars that run without jar to the heart of some mountain4 ,arrived there the pilgrim has a glance1 as the car swiftly turns in a blaze of electric light1 at a roughly painted panorama of the view from the summit1 and he is assured by the guide that he has accomplished all that he would have done1 had he laboriously climbed the peak itself4 ,in the midst of all the advocacy of sand-dune climbing1 and of rubber-tired cars to see a painted view1 the great body of guides still climb their mountains with their little groups of followers1 and the vigor of the ascent and the magnificence of the view still attract all who are strong and earnest1 during their sojourn in the ,valley of ,youth4 ,among the mountains that have for ages attracted the pilgrims is ,mons ,latinus1 usually called in the valley by the more pleasing name ,latina4 ,mathematica1 and ,rhetorica1 and ,grammatica are also among the best known4 ,a group known as ,montes ,naturales comprises ,physica1 ,biologica1 and ,chemica1 and one great peak with minor peaks about it is called by the people ,philosophia4 ,there are those who claim that these great masses of rock are too old to be climbed1 as if that affected the view2 while others claim that the ascent is too difficult and that all who do not favor the sand dunes are reactionary4 ,but this affects only a few who belong to the real mountains1 and the others labor diligently to improve the paths and to lessen unnecessary toil1 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 work4 99 99 99 99 99 ,geometry is a mountain4 ,vigor is needed for its ascent4 ,the views all along the paths are magnificent4 ,the effort of climbing is stimulating4 ,a guide who points out the beauties1 the grandeur1 and the special places of interest commands the admiration of his group of pilgrims4 ,one who fails to do this1 who does not know the paths1 who puts unnecessary burdens upon the pilgrim1 or who blindfolds him in his progress1 is unworthy of his position4 ,the pretended guide who says that the painted panorama1 seen from the rubber-tired car1 is as good as the view from the summit is simply a fakir and is generally recognized as such4 ,the mountain will stand2 it will not be used as a mere commercial quarry for building stone2 it will not be affected by pellets thrown from the little hillocks about2 but its paths will be freed from unnecessary flints1 they will be straightened where this can advantageously be done1 and new paths on entirely novel plans will be made as time goes on1 but these paths will be hewed out of rock1 not made out of the dreams of a day4 ,every worthy guide will assist in all these efforts at betterment1 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 peaks2 but he will not take seriously the efforts of the fakir1 nor will he listen with more than passing interest to him who proclaims the sand heap to be a ,matterhorn4 ,,index ,ahmes1 #bg1 #bed1 #bgh1 #cjf ,alexandroff1 #afd ,algebra1 #cg1 #hd ,al-,khowarazmi1 #cg ,allman1 ,g4 ,j41 #bi ,almagest1 #ce ,al-,nairizi1 #aga1 #aic1 #bad1 #bfd ,al-,qift4i1 #di ,analysis1 #da1 #afa ,angle1 #adb1 #aee2 trisection of1 #ca1 #bae ,anthonisz1 ,adriaen1 #bgi ,antiphon1 #ca1 #cb1 #bgf ,apollodotus 7,apollodorus71 #bei ,apollonius1 #cd1 #bad1 #bca ,applied problems1 #ge1 #ajc1 #agh1 #ahf1 #aib1 #aie1 #bjc1 #bjd1 #bji1 #bae1 #bag1 #bdb1 #bfg1 #bie1 #cag ,appreciation of geometry1 #ai ,arab geometry1 #cg1 #ea ,archimedes1 #cd1 #db1 #dh1 #aci1 #ada1 #bae1 #bgf1 #bgh1 #cad1 #cbg1 #cbh ,aristaeus1 #caj ,aristotle1 #cc1 #db1 #acd1 #ace1 #acg1 #ade1 #aed1 #agg1 #bji ,aryabhatta1 #cf1 #bgi ,associations1 syllabi of1 #eh1 #fj1 #fd ,assumptions1 #aaf ,astrolabe1 #agb ,athelhard of ,bath1 #cg1 #ea ,athenaeus1 #bei ,axioms1 #ca1 #da1 #aaf ,babylon1 #bf1 #bgb ,bartoli1 #aj1 #dd1 #bch ,belli1 #aj1 #dd1 #agb ,beltinus1 #bci1 #bda 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,thibaut1 #ahe ,thoreau1 #bdf ,trapezium1 #adh ,treutlein1 #fh1 #afd ,triangle1 #adg ,trigonometry1 #bcd ,trisection problem1 #ca1 #bae ,trivium1 #db ,universities1 geometry in the1 #dc ,uselessness of mathematics1 #ac ,veblen1 #aca1 #aei ,vega1 #bgi ,veronese1 #fh ,vieta1 #bgi ,vogt1 #bei ,wallis1 #abg1 #bhj ,young1 ,j4 ,w4 ,a41 #be1 #aca1 #aei1 #bgg ,zamberti1 #eb ,zenodorus1 #cd1 #bec ,,announcements ,,books ,,for ,,teachers ,list price ,allen3 ,civics and ,health #a4be ,brigham3 ,geographic ,influences in ,american ,history #a4be ,channing and ,hart3 ,guide to the ,study of ,american ,history #b4jj ,hall3 ,aspects of ,child ,life and ,education #a4ej ,hodge3 ,nature ,study and ,life #a4ej ,johnson3 ,education by ,plays and ,games 4#ij ,johnson3 ,what to ,do at ,recess 4#be ,kern3 ,among ,country ,schools #a4be ,mace3 ,method in ,history #a4jj ,mac,vicar3 ,the ,principles of ,education 4#fj ,moral ,training in the ,public ,schools #a4be ,prince3 ,courses of ,studies and ,methods of ,teaching 4#ge ,scott3 ,social ,education #a4be ,tompkins3 ,philosophy of ,school ,management 4#ge ,tompkins3 ,philosophy of ,teaching 4#ge ,wiltse3 ,place of the ,story in ,early ,education1 and ,other ,essays4 ,a ,manual for ,teachers 4#ej ,,for ,,class ,,records ,comings3 ,complete ,record--,attendance and ,scholarship ,graded-,school ,edition 4#cj ,high-,school ,edition 4#cj ,ginn and ,company3 ,teacher's ,class ,books ,no4 ,i 4#cj ,no4 ,,ii 4#dj ,twenty ,weeks' ,class ,book 4#cj ,,civics ,,and ,,health ,by ,,william ,h4 ,,allen1 ,secretary of the ,bureau of ,municipal ,research1 ,new ,york ,city4 ,with an ,introduction by ,professor ,,william ,t4 ,,sedgwick1 ,professor of ,biology in the ,massachusetts ,institute of ,technology ,list price1 #a4be ,adopted by the ,teachers' ,reading ,circles of ,maryland1 ,kentucky1 ,north ,dakota1 ,south ,dakota1 ,oklahoma1 ,new ,mexico1 ,south ,carolina1 ,alabama1 ,arizona1 ,illinois1 ,michigan1 ,colorado1 ,texas1 ,virginia1 ,iowa1 ,arkansas1 ,wyoming1 ,missouri1 ,indiana1 ,nebraska1 and ,washington 99 99 99 99 99 ,for ,dr4 ,allen prevention is a text and the making of sound citizens a sermon4 ,in ,civics and ,health he sounds a slogan which should awaken every community in this country to its opportunities in municipal reform4 ,every teacher who reads this book will gain a new sense of duty in matters of hygiene and sanitation4 ,civics and ,health is enthrallingly interesting4 ,it is humanized sociology4 ,cleaning up children by scientific illumination will appeal to every father and mother1 every child lover who has any patriotism or desire to learn how we as a people are to make moral-reform agitations fruitful through health of ,american children1 and so establish health of national life4--,boston ,transcript4 ,this is one of the books we wish the law required every citizen to have in his house and to know by heart4 ,then1 indeed1 mankind would have made an immense stride forward4--,chicago ,medical ,recorder4 ,the book is alive from cover to cover4 ,it breathes reform but not of the platform variety4 ,it abounds in ugly facts but superabounds in the statement of best methods of getting rid of this ugliness4 ,as claimed by the publishers1 it is preeminently a book on getting things done4--,hygiene and ,physical ,education1 ,springfield1 ,mass4 ,,ginn ,,and ,,company ,publishers ,transcribers notes ,on page #cj3 ,pythagoras fled to ,megapontum has been left as printed1 though the author probably meant ,metapontum4 ,on page #bfi3 #ajj ,b4,c4 has been left as it was printed1 though it is probably a typo for #ajj ,a4,d4