UNIVERSITY OF CALIFORNIA AT LOS ANGELES A NAERATIVE -ESSAY yikral ^trmatiott, CHIEFLY EMBODIED IN THE ACCOUNT OF AN ATTEMPT TO GIVE A LIBERAL EDUCATION CHILDKEN OF THE WORKING CLASSES. BY THE REV. S. HAWTEEY, A.M., ASSISTANT-MASTEE, ETON. LONDON" : HAMILTON, ADAMS, AND 0. 32, PATERNOSTER ROW. 1868. 5 2 4 2 8 LONnoN: UABT S. SIOKEBBT, PBINTBB, 4a, WALBBOQK, UANSION HOUSE. B. 0. O IGiV A NAREATIVE-ESSAY ITikral ^trntniim. S2 Many are speaking and writing aboiit Education, '^ — some with authority. The Education of every % class of the community, and its bearing on the ^ national character and the future prosperity of England, is under discussion. First, how to promote and extend the Educa- tion of the lower classes, and make it tell on °° the arts and manufactures of the country. ^ Secondly, how to supply to the middle class a CO . ^ better Education than has hitherto been within ^ their reach, and what is the character of the Education that shall be most serviceable to them. Thirdly, what changes and developments shall be made in the system of Education given to the wealthier classes of the country. These subjects are all now under discussion, ^ and the essays and speeches bearing upon them 5 are read and thought over by those who are ^ interested in the progress and well-being of their country. a2 As a coiitril)utioii towards the discussion of the foregoing questions, I give, in the following pages, not a theory as to what ought to he done, but a narrative of what has been done, — a narrative — somewhat detailed — of an attempt to give a liberal Education to boys of the working class. The School to which I allude is not unknown by name to friends among whom have been circu- lated, from time to time, narratives printed with a view of elucidating the moral system there pursued. This School, taking its rise from a very small beginning twenty-one years ago, has been devel- oping to the present time, and has been, and still is, a source of veiy great hope, interest, and happiness to those connected with it. This mode of approaching a subject, w^hich may be called the " topic of the day," will, I trust, commend itself to many minds. A valued col- league of the writer's (Hany Dupuis), when he was leaving Eton for a sphere of parochial work, said to him, " We don't want essays on Educa- tion — WT have plenty of them. Tell us what you have done, and how you have done it." Harry Dupuis is no more ; but others may share the feeling he expressed, and consider the narrative, at this juncture, not ill-timed. For even if it seems that the accidental circumstances surrounding St. Mark's School are too excej^tional to allow it to be a model, in every respect, for schools of the poorest class, I hope it may suggest some hmts to those engaged at this moment in the establishment of middle-class schools. I need not say a word about moral training here. I have written so fully on this subject in all the educational tracts and stories above alluded to, that it would be superfluous to say more. It is only necessary for me to premise that the power of sympathy — " the contact of human living soul with human living soul " — is pre- supposed in all that may be said about intellectual culture in the present Narrative-Essay. The first detail I shall go into is about church. By all means, where the opportunity allows, take the boys to a service in church daily. We have always begun the day by going to church. In fact, the church is the first place of meeting each morning for the boys coming from their various homes. And here we have taken a useful hint from the custom of public schools. Instead of the school children being formed two and two, and put into their seats in church like sheep into a pen, — that is, instead of the string of boys being passed into a seat till it is full, upon which the next seat is opened and filled in like manner, — each boy at the beginning of the school-time has his place in church allotted to him. This is the first instance I have to name of a principle that pervades the whole of our system, — the individualiziug each boy, and so creating a sense of individual responsibility. In each boy's place hangs a little bag con- taining his hymn-book, prayer-book, and Testa- ment, these remaining for his daily use in church ; and, as the united cost of the three books only amounts to the sum of sixpence, the expense of a separate set of books for use in church is not more than parents of the working class are willing to afford. Each boy, then, on coming independently from his own home, enters church and goes at once to his own place. We take the precaution in arranging the places in church to intermingle in order the elder and the younger boys. A number of veiy little boys together are apt to be fidgety, and the trust that is implied in placing a little boy on either side of an elder boy begets trust- w^orthiness. With this precaution, and a few distinct rules of conduct, such as would suggest themselves to any watchful parent, the boys are very much left to themselves ; we trust to the force of public opinion, general habit, and a prevalent wish to do right, for good and reverent behaviour. It does boys much more good to see the Masters trying to get good to themselves by following the prayers and lessons, than to be kept in order by a Master's eye looking about and resting on them, his own attention being manifestly distracted thereby. It is better even that instances of misconduct should occur and not be seen or noticed, than that the eye of the Master should be on the watch. And it is obvious, that a decent reverence of manner, produced by habit and the force of opinion and good feeling, is far more valuable than the most punctilious behaviour, the consequence of being closely watched. The daily morning service thus conducted is an incalculable blessing to the School : it has a gradual, quiet, but most sensible effect on the whole tone and bearing of the boys. The service is shortened ; it does not take more than from fifteen to seventeen minutes. We have only one lesson, that from the New Testament, and each boy having his prayer-book, hymn-book, and Testament, the burst of young fresh voices as they read the alternate verses with the officiating Minister, or sing the hymn (for the whole School learn to sing by note), is animating and helpful to one's own devotions. So is the stillness during the reading of the lesson, each boy following it on the pages of his own Testament. It is a comfort to think that these boys by ear and eye, and without the irreverent familiarity en- gendered by using the Holy Scriptures as a school- book, are getting day by day additional acquaint- ance with the Word of God, at least with that part of it most suitable for the reading of Christian boys. Three times a year in the course of the daily morning service do they go through the Gospels and the Acts of the Apostles. Now, as I am persuaded that most of what it is the present fashion to call distressing doubts about the very fundamentals of Christianity arises from want of reading reverently and regularly the Gospels, I prize this method, whereby an acquaint- ance with the facts, the words, the spirit of the Gospels is being very quietly and gradually, but steadily, imparted year by year to our boys. The reader of the morning lesson is one of themselves. Good readers with distinct articula- tion are chosen for the duty. Of course, if the boys were affected and read with self-conscious- ness, this would not answer. But in good Eng- lish schools of a manly character, and where hard work is done, boys are not affected or self-con- scious. It is delightful to observe how free they are from this fault ; I am certain all good School- masters will heartily agree in this statement. That being the case, there is a very great charm in the reading of the Gospel lesson simply and re- verently by a young boy with a clear fresh voice. The whole congregation seems to feel it. To the objection sometimes urged that a service in church takes so much out of the daily school- hours, it inay be replied, that it is an admirable and powerful incentive to good reading for the boys of a school to hear a chapter, which they are following with their eyes, well read by one of them- selves. Especially it is useful to boys in the lower divisions who do not often come in contact with those of the upper division, out of which the readers in church are naturally selected : they see the result of the rules and the method of teaching that are applied to themselves, and this makes them attend to those rules and do their best. I repeat emphatically what I said many years ago,* that the reading of the Psalms, by the boys of a school, verse about with the officiating Cler- gyman, and their following with their eyes on their books the lesson which is being read by a good reader from among themselves, will do more to teach good reading than the same amount of time spent in common reading lessons, or at least it admirably supplements those lessons. But this is altogether a minor consideration. It is desirable that the boys should have plenty of room allowed them. A little expense which may make them feel that they are thought of, and that their comfort is attended to, will not be thrown away. Each of our boys has a separate kneeling cushion provided for him. The silent effect of arrangements that make boys feel that in the house of God boys and Masters are all one — that, one beside the other, they are praying to and praising Him who is ^equally and infinitely above both — is to engender a tone and feeling in * See "Account of St. Mark's School, 1857," pages 4:2, 43. 10 reference to Divine worship just and true, and of the highest value to the boys. When I think of the number of newly -built or restored churches which there are throughout the country, many of them close to the school and parsonage, I cannot but regret that they are not thus utilized on week-days, and that the Clergy- man of the parish has not the opportunity of meeting each morning the boys and girls of his Parochial School in his church. In our church the seats on one side of the centre aisle are appropriated during the week-days to the boys ; the opposite side is sufficient for the girls and the other attendants at the daily service. I now leave this, the most important part of my whole subject, with one concluding remark : I have heard the heads of establishments say of different social arrangements, that such or such {e. g., the appointments of their dinner-table or the like) were the key-stone of the discipline and good order of their house. I say that the daily quarter of an hour in church is the key-stone of the moral tone of our School. On leaving church for school the boys fall into rank in different divisions under their respective Masters. For it is a feature of our system, de- rived immediately from the Public School system, that each Master has a separate division of the School, for which he is responsible. It cannot 11 be expected that a Master will work with energy and hope, if he has not full credit for the success of his work. The system, therefore, of having a set of Assistants or Monitors, with a Head Master exercising a general superintendence over the whole School, I have long given up as unsatis- factory. Each Master is wholly responsible for the tuition and discipline of his division. Parents look to him and communicate with him, and the credit for the good discipline and progress of his division is wholly his own. I most strongly recommend that at any school where there is more than one Master this system should be adopted. I am happy to say that the Committee of the Windsor National School, in- stead of having one Head Master to superintend a staff of Junior Assistants and paid Monitors, have appointed two Assistant Masters in addition to the Head Master, each to be responsible for a division of the School ; each Master probably to have two " standards " out of the six under his tuition and superintendence ; and the school-room is being divided that each Master may have a separate compartment for his division. It may be mentioned here that the last week of each school-time is devoted to an examination in the work of the preceding half. As soon as the examination is finished the boys separate for the holidays ; and when the result of the exami- 12 nation is made out, the new school-list is printed, in which the meritorious of each division are placed in three classes according to merit, followed by the unclassed, if any — those who have taken a high class entitling themselves thereby to be placed in a higher division the following school- time. This system is very efficacious in giving tone to a school. The assurance that industry and pro- gress will be rewarded by accelerated promotion gives a healthy stimulus to exertion ; indeed, what stimulus can be more healthy than the wish on the part of a boy for an advancement, (which does not necessitate the keeping back another,) into a higher division of the School, where he will do harder lessons and associate with more advanced boys. This, of course, is none other than the system pursued (with perhaps certain varieties of detail) in most public, and good Grammar Schools. My reason for naming it here is that it has been found capable of application in a day-school for children of tlie working class, and has been of incalculable benefit to the school. An observation of our former Inspector, Mr. Bellairs, is very much in point. It is, indeed, so very flattering, that I should hesitate to repeat it here, if it did not give a most valuable testimony in favour of the system of divisions which I advocate. He had begun his inspection at the bottom of 13 the School, and worked up, taking division after division, to the top ; when it was over, he said that the School reminded him of nothing so much as of a paper mill, where one sees the rough rags put in at one end ; and each step of the process refines and purifies the material, till the clean white paper comes out at the other end. The sense of trust and responsibility will make a good Master out of an imperfect one. To call upon him to take now this class and now that, at the bidding of another, without his having either responsibility or credit, will go far to spoil a good one. While we have been talking over these matters, we will suppose the boys have been walking from church to school. When they get to school, what do they find there ? In the first place, plenty of oxygen. The Government requires eighty cubic feet of pure air for each child. We give them four hundred. So that we never, even at the end of a long day, see any symptom of the presence of carbonic acid in the pale face, drawn features, and languid eye. As they enter, each boy hangs his cap and over-coat on his own peg ; those who come from a distance, and do not return to their homes till the evening, drop their dinner-bags into the large basket which the matron holds to receive them, giving her a kindly greeting as they pass on to their covered playground — (a grand refuge in wet 14 weather). Here a short inspection takes place to see they are all tidy ; after which they ascend by the diiferent staircases to their several compart- ments of the great School. Here again we seek to educate them to a sense of individual responsibility by giving each boy a locker (which, by the way, does not lock ) ; in this he keeps and takes care of all his books, paper, pens, and school requisites : this plan we adopt even with the little boys, that they may early learn to take care of their own property and to respect that of their neighbours. Dr. Keate used to say that a boy's bureau was his castle ; so we say of the locker. It is poor training to take charge of a boy's slate, book, pen, &c., to give them out for the lesson, and then take them back. It may look very orderly, but it is not good education. Let the boys be treated as moral agents, not as so many parts of a great educational mill which is to be kept in good working order. Another provision that works well is to have a large tablet suspended in the compartment occu- pied by each division, on which there is a weekly register kept of the industry, conduct, and progress of each boy ; a box of large printed letters and figures, with a pot of paste and a brush, and the help of one or two of his division, enables each Master weekly to add on to the previous record the result of the current week ; this register the boys always have before their eyes. Any person inter- 15 ested in them, such as the Clergyman of the parish, or the promoters of the School, can, at a glance, tell how the boys are getting on. Every week that they continue free from a blemish for inattention, irregularity, and the like, makes them all the more anxious to keep free from blemish the next week. And, as they see their marks for school and home lessons groiv weekly, they are all the more anxious they should keep on growing. It is pleasant, when a half holiday at Eton allows me to do so, to go to the School and see who deserves a shake of the hand or a pat on the back for his sustained good conduct and industry through the school-time. Weekly reports are also sent home, and much valued ; and the periodical meeting of parents, when the final re- sult of the weekly register and of the examina- tion at the end of the school-time is given out, is a most festive and happy day with us. At the end of the covered playground we have a row of warm baths, and all boys must either be washed all over at home at least weekly, or have a bath at school. Practically, the baths at school are a real delight and enjoyment to all, and a great source of health. I am afraid to say how much the rate of mortality was decreased at the Hanwell District School by the introduction of soap and warm water. But of all the arrangements at St. Mark's, 16 none is more valuable, morally or socially, than the provisions for the boys' retiring closets. So valuable are they, that I hope no reader of these pages who is building a school, or altering school premises, will omit to come and see, or make inquiries about what Mr. Street has done for us in this respect. We now come to the vexata questio, the chief subject of discussion in all the daily speeches and essays one meets with on Education. By what studies shall we inform boys' minds, and culti- vate their intellects ? Of course, all agree that the foundation of Eeading, Writing, and Arith- metic must be first laid. Of these, I have not much to say. But, as I wish to be useful, I will say a few words on each of these heads. I. Beading. — The rule for attaining the art of clear, articulate Beading is a very short one — FINISH YOUR WORDS. Do not let the last consonant be divided from its own word and tacked on to the next following, — for instance, do not let the words " dwelt in " be read as though they were written " dwel tin." Expression, force, and propriety in Beading must be gained by a general quickening of the faculties. — (See Appendix A.) II. For Writing, a short, good rule, which we 17 owe to Mr Harris, the old Writing Master at Eton, is, 23oint the knuckle of your fore-finger to the ceiling. A pleasant occasional exercise is to ba- lance a halfpenny on the knuckle of the fore-finger ; if it does not slip off till the end of the copy, let it then slip into the writer's pocket. — (See Appendix B.) III. Arithmetic. — Prolonged experience teaches me that explanation of principles in teaching Arithmetic may he easily overdone. I have erred myself in overdoing explanation again and again. G-ive out the rule ; let them write it out, learn it, and do a number of examples by it. An occa- sional pithy and pointed explanation of a principle is all very good, especially when accompanied by some practical illustration. For instance, — a square of ginger-bread or a plate of apples is admirably useful to illustrate the definitions and fundamental rules of vulgar fractions, especially if the " parts into which the unit is divided" are at the close of the lesson shared amongst the members of the class; but lengthened explanations children cannot follow. — (See Appendix C.) Any account of the Education given at St. Mark's School would be very incomplete without a reference to Music. I will here, then, intro- duce parenthetically what I have to say. Singing by note is taught as part of the school business to 18 every boy. All that I said in the *' Account of St. Mark's School," pubhshcd m 1856, I endorse now. As to the intellectual, moral, and social benefit of the study, I am fortunate in being able to leave to others less prejudiced than myself to say if they think that the experiment has been a success. The CoUege of Eton, in instituting a choir of their own for their chapel services, have done me the honour to take their choristers from St. Mark's, and have endowed the School with twelve choral scholarships. Simultaneously with this step, they have appointed as their organist, and as the head of the music at Eton, the Rev. G. L. Hayne, Mus. D., Oxon., himself an old Etonian. Now, Dr. Ha}Tie is in daily intercourse with these choristers, and he can therefore, better than any one else, measure them musically, mentally, and morally. To him with confidence I leave it to say if he thinks that the result of our having made music at St. Mark's a study, not a pastime, is good. To resume : — The foundation of reading, writ- ing, and arithmetic being well laid, what next ? Is it to be History ? Geography ? the knowledge of common things ? an insight into the interesting discoveries of our age ? My experience tells me, No. These are not mental cultivators. These are not what dig, plough, harrow the brain. These are rather of the nature of seeds which, when 19 thrown into ground previously prepared, take root, grow, and bear fruit. Then what is the mental cultivator that is to prepare the ground ? Lan- guage, I should say undoubtedly. Is it to be English or Latin ? I answer, both. Our Schoolmasters having been trained at St. Mark's and Culham Training Colleges, there is naturally far more attention paid to the study of English with us than at regular classical schools. English words are analysed, their formation and meaning traced, synonymous words are examined and distinguished, metaphorical sentences are put into common language : the analysis of sentences also is a favourite exercise, and the boys seem to get a very intelligent knowledge of the structure of language thereby. Eeading lessons, as in schools for their class and age, form a prominent item in the work of each day, and through their reading lessons they get an extensive acquaintance with good English au- thors ; for not only are School Beading Books now very good, and improving every year, but AUman, (by his circulating library for schools,) and other enterprising publishers, enable us to put standard authors into our boys' hands, such as " The Tales of a Grandfather," "Bobinson Crusoe," "The Fool of Quality," (that grand old book full of generosity and chivalry, without a shadow of modern flip- pancy,) Shakespear, of course, and Sir Walter B 2 20 Scott's Poems and Novels,* the sentiment and romance of which form a good antidote to the slang and flippancy in which much of the modern writing which comes within the reach of school- boys is steeped. Speaking of these matters, I cannot omit to record the great benefit which Mr. Warbnrton conferred upon the School when he was our In- spector. He not only told us that we should find the learning — with perfect accuracy — good and choice pieces of English to be spoken with great care, good articulation, and proper emphasis, a valuable educational exercise — but he further found time, amidst his avocations as Queen's Inspector, to gather for us and copy with his own hand a long manuscript of most choice pieces of English,— " The Battle of the Nile," "The Mining Adventure in Peru," " Physical Geogra- phy," " St. Peter's Chapel in the Tower," " Ma- terial Progress," "The Earl of Warwick," &c., culled from such writers as Macaulay, Dr. Arnold, S. T. * It may not be amiss to remind my readers that Sir Walter Scott's Novels may be liad for sixpence each complete work ; of course, they are in paper covers ; but if before the book is given out as a reading book a few holes are stabbed in it with a bookbinder's awl, as near the back as possible, (for there is very little margin,) and a bit of strong carpet-thread or wax-end be drawn in and out through the holes tightly, the book will never fall to pieces, and will last, as a reading book, as long as if it were bound. 21 Coleridge, Soutliey, and his own poor brother, will fix him for a long time in our memories. We found his words most true. His pieces of English, in style most beautiful, and breathing manly and heroic sentiments, have been a precious possession to the School. I can testify that this most kind act of his has done the School the greatest good both intellectual and moral ; and I would venture to suggest that if he would add other pieces to those which he has selected for us, and were to publish them in a book, he would be conferring a benefit on other schools which is now confined to ourselves. I trust all this will satisfy Professor Seeley that English is not overlooked, but neither am I dis- posed to forego Latin. At least, we have begun it with our lads at St. Mark's School, three years earlier than the age which he names (fourteen) for beginning Latin ; and, I cannot help seeing, with the best result. Shall I be called a " Philistine " if I urge that Latin is a fine thing for " evening lessons ? " There is something very definite in committing to memory a page of accidence, and in a Latin exercise or translation, VN^hich has to be done, written out fair, examined, have its faults marked and corrected. I have never been able to make English such a good old-fashioned lesson as this — boys cannot always be learning choice bits of poetry, and to be told to look over so many pages 22 of s}Tionymes, or of history, geography, &c., is not detiiiitc Hkc a Latin exorcise. But, besides this, I am persuaded that the in- flections of Latin give quite a new view of the reahty of concords. And, as an instrument for cultivating the mental faculties, what can be better than an exercise that requires a simul- taneous effort to get number, gender, case, tense, mood, person, all right together. To an educated person all this seems to come intuitively, but it should not be forgotten that to get each of them right requires a separate mental effort when a boy is constructing a Latin sentence. The boys like their Latin lesson ; they are interested and animated by it ; a little know- ledge of Latin they find a great help in pulling English words to pieces ; and they analyse the structure of sentences in the English fashion all the better from parsing their Latin lesson. With regard to grammar. "Whether they first read the progressive sentences in their elementary Latin-books and build up for themselves (under the teacher's guidance) laws, rules of construction, or take the law at the hands of those who have travelled over the ground before them, and trace its action in the structure of the sentences framed for them, and subsequently in those authors whose writings they may read, is a question which I do not think worth fighting about. What is wanted is that their knowledge of 23 language slionld be systematic, drawn up in orderly array, under well-defined rules wliicli they can apply. And it is not less valuable surely, as an intellectual exercise, than the arrangement of stones, or flowers, or insects, in classified order. It is but a little way that we go in Latin, but every step seems to be valuable, per se. Of course, we do not touch verses, that is beyond us. But finding, as we do in familiar intercourse with these boys, that ready discernment, quickness of ap- prehension and thought, the power of taking in and estimating what is said, of reading a glance of the eye, seeing the point of a joke, or detecting covert irony : — in a word, a general mental acumen is developed by their study of language, I feel disposed to say, if Latin, so far as we go at St. Mark's, gives this mental activity, why should not the good effect of the exercise go on if the boys were able to advance yet further ? What if in addition to getting mood, tense, person, number, gender, case of a sentence, all right, they were still further required to bring out the sentence subject to metrical rules and the rhythm and poiiit of verse ? Are we quite sure that the Remove and Fourth- form boy is not deriving intellectual profit from that "worry and torture of brain" which makes him knit his brow, bite his pen, and turn from Gradus to Dictionary and back again from Dictionary to Gradus ? Especially as it is an 24 exercise that is enlivened, I contend, from time to time, with pleasurable emotion. Who cannot call to mind his looking back on a couplet which he hoped was rather good, and the thought arising, " I wonder what my tutor will say to that verse ? " How long this work should continue, at what time, if a boy does not become a good composer, verse -making should be given up, I do not pretend to say. But this I do say, before we give up verses as mental cultivation and training, let us see that we have something to put in their place that will do intellectually the same or more good. Mr. Farrar, I suppose, will answer with an un- hesitating voice, " Science will do it." And very beautiful is the posy of scientific flowerets which he gathered from the fields of modern discovery, and presented to his charmed audience at the British Institution. With such art and grace were they entwined, that I could not but ask myself, How much of his power in running down verses does Mr. Farrar owe to his having worked at verses ? Is he not somewhat in the position of a Swiss mountaineer who, having by hard labour reached some Alpine eminence, now luxuriates in the charms of his high position, and despises, flings away the good staff which helped him to sur- mount the rugged difficulties of the ascent, and enabled him to crown the height ? 25 In like manner those brilliant speeches of Mr. Lowe, with which he is shaking the country's traditional opinions about Education, are not their wit, vivacity, and power due to that intellectual training which he would bring into disrepute ? His enthusiastic audiences, who are exulting in his slashing cuts and telling hits, do not reflect that his sword was sharpened and his arrow pointed by that very engine the cumbrous inu- tility of which they are laughing at. My reason for not using science, I ought rather to say scientific informatioup as an instrument of intellectual Education, I have stated above. In- ordinary hands it is not a good instrument for cultivating the mental faculties. I have on all occasions" expressed my contempt and abhorrence of that species of mental training which was in vogue in our advanced, sharp National Schools some fifteen to twenty-five years ago, — a training which consisted in putting into minds not prepared to receive them — and wholly incapable of digesting and assimilating them — the facts of geography and history, and crude morsels of scientific and miscellaneous knowledge. I got a lesson when I was a very young school- boy, which I have never forgotten. We had taken a wood-pidgeon's nest, and brought home in pride a fine, vigorous, young bird nearly fledged, * See " Account of St. Mark's School, 1857," pages 27, 28, and " Eeminiscences of a French Eton," page 74. 26 It was our ambition to rear it. We went out and bought some beans, dry liorse beans, opened its beak, and slipped one and another and another of the beans down its gullet. Thinking that we had given it a good meal, we left it for the night to digest its food. The next morning we hurried to see our bird ; the result may be anticipated. It was lying at the bottom of its cage with an ex- tended crop, dead. We had given it food which it was not able to digest, and so had killed it. When I have listened to a National School- master giving to school-children crude morsels of scientific and miscellaneous Imowledge, which they gulped down, indeed, but were clearly unable to digest, "It is hcans," I have said to myself, " that he is cramming down their throats : instead of nourishing them, he is intellectually killing them." If, during these last twenty and more years, St. Mark's has sent out into the world young men able and intelligent, fitted to be good and useful men, it is because this practice I reprobate is there eschewed ; the aim of the School being not so much to impart knowledge as to develope the boys' faculties, and to teach them how to learn. Now, in answer to the cry for science, which resounds in the present day, I fear a recurrence in schools of a higher rank to the bad system which has been of late years wisely abandoned in schools for the lower class. 27 Another danger to be apprehended is this, that boys, finding that they are to be released from severer study on the ground of its being dis- tasteful, more and more will find severe study distasteful, and that thus our English boys will lose some of that mental vigour which they now have, and take with them into the affairs of life the weakness which belongs to discursive and superficial men. To follow out one subject in which the learner feels interest is a splendid Education. " One subject thoroughly," says Professor Tulloch, most wisely. What man is there of any class more worthy of respect than the mechanic, who, after his hours of toil, takes up a study or pursuit by wiiich he may educate himself — be it a branch of mathema- tics or philology, — engineering, drawing, flowers, stones, or beetles. And the inquiring mind and persevering spirit which the artisan shows in thus educating himself is met with in other classes and in different conditions of life. I recollect being once for some days wind-bound in one of the least interesting spots I remember to have visited, — the island of Alderney. Among the few families that were to be found on the island, I had the pleasure of meeting a lady who had turned the dull island into a para- dise of beauty and interest. Folio after folio she showed me containing treasures of beauty in form 28 and colour, such as I had no idea existed on earth. In fact, she hved in a world of beauty and order; her knowledge was extensive and accurate, her conversation was as instructive as it was charming and animated. I found she was in correspondence with men of European cele- brity. How was all this brought about ? Why, she had gone to the bottom of sea-iceed ! Another vision of past days now comes up before my mind. I was in abookseller's shop inPaternoster Row. An old pupil accosted me. We kept together a good part of the day, talking of old days and many things. I found he was in quest of books, learned, rare, old, to assist him in disentangling some intricate point of historical geography. He accompanied me to the station, and then left me to return to his quarters. He was in the Grenadier Guards and was stationed at the Tower. My old pupil has since risen to an exalted position in the country, deservedly honoured, admired, I might add, beloved by a large circle of friends. When I hear him extolled by those who know him, I think of the day when I watched the young officer of the Guards returning to the Tower to meet the learned books he had ordered, which were to help him in his careful search after truth. Here are instances of the highest and most valuable kind of Education ; but the Alderney lady, the young Guardsman, the Manchester artisan, 29 are not common types of their several classes. They are exceptional cases. They re-appear, at intervals, as Eton or other public schools-hoys, hut these are just the boys who give us no trouble or anxiety ; they would make themselves under any system. "We get great credit, indeed, and praise for their success, but it is credit which we only partially deserve : on the other hand, we deserve a great deal of credit in cases where we get little or none. I mean in the case of boys of no intellectual or moral power, whom, by care, and patience, and fellow-feeling, we have led to use self- control ; whose weak or dormant faculties we have called out, and whom we send out into the world not indeed to shine, but to stand on a higher intellectual and moral level than they would have reached had they not passed through our hands. Now, it is these last that form the bulk of humanity ; it is for them, and not for exceptional cases, that we must legislate and form our plans. And it is for the mass that I fear, when I hear the cry that boys should be freed from the severer labour of studying language if it is distasteful, and therefore, it is said, unprofitable, and should learn, instead, something about the wonders which science has achieved within the present century. Suppose them freed from verse-making, and from learning about ye and civ and allowed, instead, to learn about the electric telegraph, the light- ning conductor, the electric light, the Davy 30 Safety Lamp, chloroform, and vaccination, (for these — deprived of the poetic imagery with which he surrounds them — are the subjects whicli Mr. Farrar puts in the place of the idol worshipped in the shape of classics,) would there not be a great danger of the boys becoming less vigorous- minded than they are ? Certain of them I grant might, and doubtless would, educate themselves by botany, chemistry, electricity, and the like. But it will be those who have an inquiring mind, and a decided love for the science which they take in hand. But how will it be with the mass ? Will their becoming acquainted with a string of scientific results stand them instead of the mental training they now get ? Am I satisfied with the training we now give to boys through our own subjects ? By no means. But do not let us therefore begin on another tack ; let us try to do, what we do, better. And notwithstanding the imperfection of our teaching and the idleness of the boys, what are the intellectual characteristics of the pupils which the public schools of England turn out ? Is not their apprehension quick, and discernment ready and correct ? Do they not take in, w^eigh, and set at its true value what is said, with a perfectly marvellous quickness and unerring instinct ? If an observation is just, in good taste, well-timed, do they not feel that it is so, and respond ? If not, do they not detect the weakness or fallacy 31 instantly, and . eschew it ? Will they tolerate verbiage ? Have they not a fine sense of the ridiculous ? When Mr. Hullah was giving singing lessons at Eton, he made two observations on the charac- teristics of his Eton class which showed his penetration. Eirst, that they seemed all one, bound together as by a three-fold cord ; compared to them, his other classes appeared encircled by a rope of sand. His second observation was, that, in teaching them, he was baffled by the quickness of their ap- prehension. Before he had finished his sentence, he felt that they had hold of it, and had time to become distracted and inattentive, while he was adding such further explanation as he commonly gave to his classes. And as we trace them on in life, I cannot but think that we see these intellectual characteristics of public school-boys developing themselves. Who shall say how much the quick appreciation of practical good sense v/hicli characterizes the House of Commons, its intolerance of humbug, its admirable instincts, and the other kindred qualities in which the country feels both pride and confidence, are due to the large number of public school-men which it contains ? Look, on the other hand, at those wiio have not been under the training of our public schools, nor have formed themselves, by some study or pur- S2 suit, of a solid, vigorous character, but, instead, have dabbled in science, — men who have left the beaten track of classics and mathematics for the flowery meads of scientific discovery. They are not a numerous class, and are still rather un- English. But here and there they may be found. I dare say most people have one or two specimens of the class before their mind's eye. If so, they can say whether I overdraw their characteristics in describing them as self-complacent and self- conscious ; wanting in manliness, in vigour of mind, and in sound judgment ; mouthy-talkers, without the sense to know that they are dull ; even in their own subjects apt to be hazy, and when talking of one scientific subject, to confound and jumble it with another : a public school-man may very possibly be profoundly ignorant of both ; but he honestly says so, while the sciolist indulges in grandiloquent twaddle about what he does not understand. I leave it to English parents to say into which of these two types of men they would like to see their sons grow up. But are public school-men to be profoundly ignorant about the various interesting and marvel- lous discoveries which modern science is daily making, and their application to the comforts and conveniences of life ? Is the choice of necessity to be between ignorance and sciolism ? By no means. S3 In advocating the study of language as the best means of educing the mental powers, — of doing the work of digging and harrowing the brain, I have said only half my say. A mind educated by classics alone is only half educated. There is a book the study and understanding of which will make the education complete. No one who is master of that book can ever be a sciolist, and he will be mentally prepared by it to survey and firmly to grasp all branches of natural science : he will find it a master-key that unlocks aU such knowledge. His mind, imbued with that book, will deal with scientific subjects in a manner in which, without the knowledge of that book, it never would have done : — and that book is Euclid. But are we met in limine with a difficulty ? Is this first step, the mastery of Euclid, so difiicult a step, that it cannot be taken by the average of our school-boys ? Is it necessary that they shall give up the attempt ? must they be satisfied with a humbler and more incomplete and unsatisfactory acquaintance with natural science ? Mr. Wilson inserts, in his very interesting essay on a Liberal Education, a note, over which I have thought much since reading it. He writes as follows : — " The extreme repulsiveness of Euclid to almost every boy is a complete proof, if, indeed, other proofs were wanting, that the ordinary methods of studying geometry at pre- 34 paratory and public schools are wholly erroneous." Mr. Wilson speaks with deserved weight and authority. But I must say that the extreme re- pulsiveness of Euclid to almost every boy is not a fact which has come under my own observation. And holding as I do most firmly that a man is not perfectly educated unless to classical study he adds the mastering of Euclid ; — and that any one who has mastered Euclid has hereby the best of all preparations for a profitable study of all branches of natural science, I feel almost bound to speak, — to say what my experience is, — and to add some hints that I hope may be useful. It was in the year 1836 that I began to teach Euclid at Eton, and I have been teaching it ever since. Among the brightest and most interesting hours of my life have been those that I have spent in initiating boys, even boys of very weak mental power, into the knowledge of Euclid. I make this statement because I think it stands to reason that this would not have been the case if the study had proved repulsive to our pupils. The first steps to many minds, I grant, are difficult, — to some, very difficult. But the teacher is cheered on by the thought that the more difficult the appreciation of geometrical reasoning is to a boy's mind, the more does that mind stand in need of Euclid ; the more good is he doing to the boy in trying to make him understand it. 35 In looking back through years of experience, I feel borne out in saying that at Eton we do not find Euclid, as a rule, a repulsive study. Boys of very low mental calibre have taken to it, have been interested in it, attracted by it, and have said to me again and again, "It's the best thing we do at Eton." The Euclid hour is always a bright one. Whether we are taking the first steps with feeble pupils ; or breaking fresh ground with a more ad- vanced class ; or, recapitulating, we are linking on proposition to proposition ; or, reversing the opera- tion, we are tracing a proposition backward, through those on which it depends, to lirst principles ; or lastly, are setting the class to work their brains on riders and exercises, — every phase of Euclid teaching is pleasant in the retro- spect. Of course, I have had my disappointments — what Master has not ? — in boys not using the personal effort necessary to reproduce what they have learned. Many have done so, and many a time I have had to thank them for giving me a satis- faction, one of the greatest a teacher can have, that of finding that pupils retain and have made their own what he has taught them. But, whether the boys of a class be studiously disposed or otherwise, I cannot call to mind any to whom Euclid could rightly be said to be a repul- sive study. If ever I say to a class, ''What shall c 2 36 we have to-day ? " I am sure the cry will be, "Euclid."- And turning from my experience of Etonians to boys of a totally different class and station. I have now fresh in my recollection the heart- felt pleasure of a weekly evening meeting through last school-time with a party of St. Mark's school- boys, whom Mr. Blythe (the Head Master of the School) sent to me for a weekly lesson in Euclid. They were boys of ages ranging from 12 to 15. I took them through the third book of Euclid : we only used enunciation books. I set before them what had to be proved, and left them to find out the demonstration, helping them if necessary by giving the figure, and referring them to the propositions on which the proof depended. I do not know that there was a demonstration that they did not make out for themselves. Each would look keenly at the black board where the figure was drawn, and give me a nod when he had made out the demonstration. Then came the comparing notes, and great was the fun and hearty the laughter if one made a slip, or assumed what he had no right to do. The school-time was cro^vned by as beautiful a Euclid-paper from * May I take the observation that has been made to me more than once by a Cambridge tutor, " that in looking over Examination papers, it is refreshing to come on an Eton man's Euclid," as a corroboration of my assertion that Euclid is not found repulsive by our Eton pxipils 1 37 almost every one of them as I ever looked over. It was only last night that one of them, who is leaving school, came with his father, to give me (according to om* custom) his photograph and take leave. His manly workman father stood beside him, and the tears found a channel for themselves down his furrowed cheeks, tears of thankfulness at the thought of what St. Mark's had done for his son. Nor were the boys who formed this evening class more than good boys of average ability. The very boy who took leave yesterday (he will not be angry with me for saying so), Avlien he be- gan Euclid, used to jumble the data and qucesita in admired confusion in his brain. I thought at one time (good boy as he always was) he would never master Euclid, but I found him last school- time as sharp and clear-headed as any of the party. He would " spot " the false step in the reasoning as quickly as any one of them. And not only is the Euclid lesson an interest full of charm, whether with my Etonian pupils, or with mechanics' children at St. Mark's. It forms a bond of firm friendship in after life. How often, amidst the mountains and valleys of Switzerland, do I find my hand grasped, and, looking up, I see an animated face, radiant with pleasant memories ! " What, don't you know me ? I am one of your old fMva-Tat," — alluding to a long and 38 growing list that used to hang up in the mathe- matical school of " the Initiated," that is, of those who understood and could apply the fourth pro- position of the first book. These were my iJ^varat,, The Platonic motto " /i^^Set? dyecofMerp'nTo^ elairco" stand- ing at the head of the list, shut out the inofanum vuJgus ; and hearty, nay, vehement sometimes, were the struggles to get into it. All this I feel to be very outspoken ; but the matter in discussion is too interesting and im- portant for me to be squeamish about speaking out. And if we do interest boys in Euclid, and make them like it, only think what we are doing for them. Lund, in his Preface to his Geometry, says : — " Next to the Bible, Euclid is the most wonderful book in the world." I prefer to place the Bible in a category by itself, and, drawing no comparison between it and any other book, simply to say, Euclid is the most wonderful book ever written by man. Undoubtedly it is so. It is the book which has moulded European thought from generation to generation. Even in those nations who have given up Euclid as a text book, the mind has been trained to habits of thought by the rigour and precision of geometrical reasoning of which Euclid is the father. But we Englishmen, with our stability of taste, and our love of linking ourselves on with the traditions of the past, stiU use the very text 39 of Euclid for mental discipline and training. We value its calmness, its dignity, its permanence. We find Euclid to be a book the tendency of which is eminently to give what St. James calls "the meekness of wisdom:" there is no self- consciousness, or conceit, or affectation in the quickened intelligence which its study imparts. For myself, I can safely say that I can re- member few hours of greater enjoyment than those in which — shutting out the noise, eager- ness, and contention of the outward world, and taking up a book which has held its ground for more than two thousand years — I have endea- voured, by its help, to open the minds of the young, to bring out those faculties which God has given them, by the calm and unimpassioned inves- tigation of truth, for truth's sake, to which its pages guide us. It was a happy day for me, and a proud day, when a party of my old H'VaTaL who had entered into the Indian Civil Service, presented to me a copy of a Euclid which they had translated into Hindustanee, as the surest and best way of bring- ing the Asiatic mind and mode of thought into conformity with the European. Again, as I have said, Euclid is the true en- trance to the natural sciences, — and a noble en- trance it is, — how much better to enter on them thus, than to scramble up in some other way ! What, for instance, is the greatest and most 40 elevating of all the natural sciences ? It is that, incontestably, which issues in the researches and discoveries of Physical Astronomy. When we have made a pupil appreciate Euclid, we have placed his foot firmly on the first step of that great intellectual ladder which leads by natural and continuous sequence, step by step, up to the highest researches of celestial mechanics. The highest steps of this ladder no man has ever reached ; but the most highly gifted among man- kind are those that have climbed highest upon it, and are climbing still. Terrestrial and celestial mechanics are one harmonious whole, one continuous, unbroken rise. All matter obeys in its moiivement the same laws. *' La courbe," says La Place, — in the Preface to his Third book of the Systeme du Monde, — " decrite par I'atome leger que les vents semblent emporter au hasard est reglee d'une maniere aussi certaine que les orbes planetaires." Surely, for this branch of natural science may be claimed precedence before all others. Euclid, as we have said, is the first step on this intellectual ladder, and, moreover, each successive step upon it is so much clear gain. There is no room here for Professor Seeley's amusing simile: — "If a man is not quite up to going to Cambridge, it is no use for him to go to Eoyston and stop there ! " Though we may never reach the heights of 41 celestial meclianics, Euclid is a clear gain. The first principles of mechanics, the composition and resolution of forces, are clear gains. So is each successive step in advance, up to Newton's giant step, — " The moon is retained in her orbit by the force of gravity," (happy the man who gets thus far,) and so on. The know- ledge acquired and the mental training, no matter where we stop, are both of the highest benefit. I ask, then, with an outline of physical science so great, so pre-eminent before us, — Dr. Whewell truly calls it classical, — why should we divert our students' minds from it ; and propose in its place something so shaky as experimental lectures on branches of science not yet fixed ? It is not very long ago that a chemical lecturer told me that he had been obliged to modify his printed programme in consequence of some modi- fication of hypothesis which it had been found necessary to introduce during the preceding week ! Are branches of science which give, as yet, such uncertain sounds, to be named as educational studies, with those which are based on rigorously established laws, and have acquired all the firmness, authority, and permanence of classical science ? If boys have laid the sound foundation of classics and mathematics, by all means let them 42 have scientific lectures* and teaching.* Their intellectual training will prepare them to turn such lectures to a profitable use. What I deprecate is putting the scientific lectures in the place of classics and mathematics. — {See Appendix C.) To revert again to Euclid. All that I have been * The following is a list of the Lectures given at Eton, on scientific subjects, and on the discoveries of modern science, which I furnished to the Public School Commissioners at their request: — "I find thiit, during the last fourteen years, there have been courses of Lectures on the following subjects : — Natural Philosojihy ; Popular Astronomy ; Mechanical Philo- sophy ; the Forces of Attraction ; Light ; Heat ; Frictionai and Voltaic Electricity ; Chemistry (at various times and in its various branches and applications). Organic and Inorganic ■ Chemistry of Gases, and of Metals ; Chemistry of Common Things ; Water in its Three Conditions of Ice, Water, and Steam ; the Chemical and Physical Properties of the Atmo- sjihere ; Application of Scientific Discoveries to the Useful Arts and Manufactures ; on Coal and Coal Mines ; the Manufacture of Iron, &.C. Also X'emarkable discoveries or processes, which have from time to time attracted attention, have been illus- trated and explained by Lectures, such as the Electric Tele- graph ; Photography ; Electro-Plating ; the Construction of the Westminster Clock ; the Bessemer Process for Making Steel ; Rifling Cannons, and the Science of the Armstrong and Whitwortli Guns ; the Method of producing Brilliant Colours from Coal Tar ; Professor Bunsen and KirchhoflTs Analysis of the Spectrum, &c. " The following courses have also been given : — " Comi^arative Anatomy and Geology ; Physical Geography, including the Theory of the Winds, Rain, and Climate ; Mo- dern. History ; Ancient and Modern Assyria ; China and Japan from Personal Observation." — Report of Commisdon on Public Schools. 43 led on to say in these last paragraphs is based on the assumption that EncHd may be made an interesting, instead of a repulsive, study. I have said that at Eton the boys like Euclid, and take an interest in it. I ought to do more than this : I ought to try to show how this is brought about. Among the most charming pages of the essays on a Liberal Education (second indeed in interest only to the blunt and manly semi- autobiography given by Mr. Johnson, which they who remember his first coming to Eton Imow to be affectingly true) are those in which Mr. Wilson gives his graphic picture of the mode in which his colleague initiated a class of boys into the study of botany. One cannot help regretting that the poor boy who made a mistake in spelling was not privately corrected ; otherwise it is a beautiful picture, valuable and suggestive in the highest degree. I can well imagine that the French Commis- sioner would have been charmed with the intelli- gence and vivacity of a class which had been so taught. Now, I must confess to my readers that, to the juxta position of the above graphic account of the botanical lesson, and the note about the repulsive- ness of Euclid, they owe the pleasure and profit, such as they are, of this Narrative-Essay, which they are reading ; for the thought occurred to me that Euclid as well as botany may be made a bright and pleasant lesson ; at least, I thought it 44 might bo useful to sliow how we try to make it so at Eton.* So now, as Mr. Wilson set his botanical master face to face with his forty pupils, we will suppose the mathematical master and his class are assembled for the first time for a lesson of Euclid. We wiU suppose them the weakest twenty of a division which has lately been advanced to that part of the school where the study of Euclid is begun. Some few may have had some lessons in Euclid at the last school : only perhaps to hate it. These are ready to ask the question, " What is the use of Euclid ? " The master deprecates the question, " Don't ask me now. We should not be talking on fair terms. I should know what I was talking about ; you would be talking about — you don't know what. Wait one month. If at the end of My readers must judge how far I have succeeded in my sketch. It has grown very long — is necessarily, in some degree, technical, and cannot be so interesting as the sketch of the lesson on botany, to the general reader. Still I hope the perusal of it will not be confined to those engaged in teaching Euclid, but that the general reader will try if he can extract some profit or, at any rate, some amusement from it. I ought also to premise that I do not make my colleagues responsible for a mode of expounding Euclid, which some grave teachers might think was trifling with the subject. They have their own methods, and I can add emphatically that some of them succeed equally with myself in making boys take an interest in Euclid. The St. Mark's Masters do teach it after my method. They learned Euclid from me, and as they were taught it, so they teach it. 45 that time you are a /u-uo-t???, that is, if you under- stand the meaning of EucHd — and then repeat your question, I will answer it ; hut I never met an instance of anyone asking the question after he understood Euclid." Now, open your books. Euclid begins with definitions ; of course, every subject has its own language, its technical words, as they are called. We must be agreed about the words we are going to use, and their meaning. Look at the first definition, it is a curious one, — "A point is that which hath no parts, no magnitude." You will better see its meaning if you consider how Euclid may have come to it. Here is a wooden brick I have got in my hand. You see that it has length, breadth, and thick- ness. Each of these is called a dimension. The brick, then, has three. Suppose I plane away its thickness till it is all gone, then there will remain two dimensions, length and breadth — that is, the surface of one of the faces of the brick ; again, in like manner, suppose the breadth be taken away, then only one dimension remains ; length — that is, the edge of one of the faces. Now, suppose the length to shrivel up from one end till all the length is gone, what would remain ? — no dimension at all. We should have just the dot of the corner of the brick. This is Euclid's "point : " it is deprived of all dimensions ; it has no parts, no magnitude. From this he starts, and, travelling the opposite way to that we have gone, he gets first to a line, that which has one dimension, length ; then to sur- face, that which has two, length and breadth. By adding on thickness to the surface he would come to the solid or full brick from which we started. He does not do so, because he treats, in the first six books of plane geometry only, that part of the subject in which the lines are drawn on a plane or flat surface. And here let us pause for a minute. I wish to say something to you, in a sort of parenthesis, which I hope may help you hereafter to under- stand two words that you will hear used to describe two different ways of reasoning. We began with the full brick, and went back undoing it, taking it to pieces till we got to the " point." Euclid, on the other hand, begins with a point and goes forward, putting things together, adding on length, &c., till he builds up the full brick, that is, arrives where we started from. Here you perceive are two different ways of considering the same question — the way of undoing, and the way of putting together. What do you think would be a good name for the way of undoing, of taking to pieces ? does dva-\u(o de- scribe it ? and for the other way, the way of putting together, do you think (rw-rlOrnML would be a good word for that ? Well, if hereafter you hear two rather long 47 words, which yon will easily trace to avaXvw and a-vvrcOTjfMi for two jdnds of reasoning, think of the brick; perhaps it will help you to understand them. Now, let us go back. Euchd wants a flat surface to write on, and how does he find out if his table is flat ? I will tell you. He takes a straight line ; we all know what a straight line is : — The edge of this ruler — carpenters call it a straight edge — is meant to be a straight line. If wherever this straight edge is placed along the table its edge lies exactly in the surface of the table (if there is no day-light between), then the table is flat, it is a plane surface. Here is a bit of warped board. My straight edge, though in some positions it may lie in the surface of the board, in others it does not do so ; so the face of this board is not a flat surface, and therefore will not do for drawing the lines oi plane geometry on. ' But supposing our table flat and our rulers straight, draw now any two straight lines from the same point, in any direction you like, and tell me what have we got here ? — An angle. — Yes, a flat straight lined, i. e., plane rectilineal, angle. Now, what is the meaning of this word ''Angle?" It is the inclination, the amount of opening be- tween those lines. You will see plenty of angles if you look round the room [an octogan roomj ; all the different corners are angles ; some are blunt, some are sharp ; in fact, an angle is a corner, the 48 opening at the corner. What does Horace say of that cosy corner of his Tiburtine farm where he hoped to sit in his old age, and drink his Falernian ? — Angulus ille milii ridet. Eemember, the size of an angle in no way depends on the length of the lines containing it. Here, as I open and close up the legs of my big wooden compasses, I make angles greater or less with the same straight lines ; and here, give me one of your little brass compasses, — if I open it equally, with the large compasses, the angles are equal, though the legs are much shorter. Plenty of time should be given to this matter — the meaning of an angle ; and when the boys have learned how, by means of three letters, an angle is indicated in Euclid, a number of inter- secting lines should be di'a's^Ti on the black board, and while the Master puts his finger into the angles, the boys should indicate them by naming the letters ; and conversely he should name the letters, and the boys should come up one by one to the board and put their finger into the angle named. A Master will often be surprised how long it will take before the whole of a class of weak boys can do this ; but he will be greatly helped in stimulating the slowest and weakest, by the hearty and not ill-natured laugh elicited by their bad attempts at hitting the angle named. All teachers should assure themselves that 49 every individual boy has got over this difficulty. I have met boys professedly reading the third book of Euclid, who actually did not know the angle they were naming. No wonder Euclid is repulsive to such. The same figure on the board will supply another exercise which should not be omitted, — the adding two contiguous angles, and pointing out and naming the single angle which they make when added together, also the taking one angle from another, and naming the angle that remains. This is a greater difficulty with many boys than would be supposed, but it is a great gain when they can do it promptly ; it helps them much in the winding up of the fifth proposition, and when they come to the seventh. The three classes into which angles are divided, according as they are acute, right, or obtuse, are easily set before them, by the Master putting the end of his long ruler to the edge of his table, and making it slant more or less ; they soon get to appreciate Euclid's test of a right angle. Before drawing a circle with a pair of com- passes, it is well to do it by the primitive mode of a tack in the black board, and a bit of string with a loop at one end to go round the tack, and a piece of chalk tied to the other. The rest of the definitions offer no difiiculty. It is better to pass over ah reference to parallel lines at present. 50 With regard to the Postulates, it may be enough to observe you are allowed the use of a straight niler and a pair of compasses. They are allowed you only that you may draw, and pro- duce, straight lines, and describe circles, and not to mecmLre tcith. It is better at present not to distract the boys by alluding to any deeper mean- ing of the Postulates. The Axioms are readily taken in. Let them know that Euclid asks that they may be accepted without proof, hccause he has no proof to offer, that . he proves everything he can. Of course, they will be told not to think about the twelfth axiom at present. And now we have arrived at the first pro- position. Let the figures of the first three pro- positions be drawn very carefully and deliberately ; the boys will enjoy doing it. Let the equilateral Fig. 1. triangle in the first proposition be drawn on each side of the line. In the second proposition let •^«'r/- 2. them join the point with each end of the given line. Vary also the relative positions of the point and the straight line, bring the point nearer and nearer to the straight line, let them see that the triangle thus becomes smaller and smaller, and guide them to feel what it becomes when the point actually coincides 'with an end of the line. Also place the point in the given line. Fifj. 3. In the third proposition, by all means let them do all the figure of the second proposition re- 61 quired for drawing AD equal to C ; they had better do it in j)€ncil, inking in the Hne AD ; then let them efface all the pencil figure, and they have the figure as given in their Euclid books, and can proceed with the third proposition. Boys like all this ; they take a great pleasure and pride in it : they like to bring out neat and good figures, and it is all very instructive. The demonstrations of the first three propo- sitions are very simple, and there is this good about them, that the boys have the advantage of going three times over the same argument, with a little variety in details. But even here boys think there is something mysterious about lines, so that it is desirable to throw them off the rut of Euclid's phraseology and ])ack on their own common sense. And on Horace's principle that FdcUculmn acri fortius et melius, secat res, we proceed in some such way as the following : — At one side of the black board a few strokes represent the Tower of London. Near it we draw a soldier on guard, — soldier, musket, and all is done in a few strokes. At the other side of the board is drawn a round tower and a flag flying ; this is Windsor Castle, and we place a sentry on duty there too. Now tell them these are two soldiers of the Grenadier Guards, one battalion of which is stationed at Windsor, the other at D 2 52 the Tower, and there is a bet pending between these two men as to which is the taller. Un- fortunately they are both on duty, one at Windsor, the other in London, and cannot therefore settle the question by the primitive method of standing back to back and measuring. What shall they do ? The Windsor man has a comrade who measures with him ; they find they are exactly of equal height. This comrade is not on guard, and gets leave to go to London. Then, midway between Windsor Castle and the Tower, a few strokes exhibit the train and the '' comrade " inside. By the time we have drawn the scare- crow figure of the long-legged guardsman in the train, all are in a roar of laughter. Well, he gets to London, and measures back to back with the man at the Tower, and they find they are also of exactly equal height. Now, the London man and the Windsor man know, u-ithout measuring ivitli each other, that one is exactly of equal height with the other. By what axiom it is asked : they answer, doubt- less rightly, the first. Now, then, says the Master, who of you can pick out, in each of the first three figures, the London man and Windsor man, and, above all, the man that went by the train ? In the best humour they all turn to their figures and look and think till, in a little while, I 63 one after another, they all find out; and when they have all caught the man that went by the train, it is a great victory ! This done ; — turn, says the Master, to the thir- teenth proposition. I tell you beforehand, that the grand difficulty of that proposition is to see which is the man that went by the train. You are con- fused with the multitude of angles named, until you have discovered the man that went by the train, then it is all perfectly simple ; and I here give you notice that when you come to that pro- position, whoever finds out which is the man that went by the train, may claim an exemption. Make a note of that ! Now comes the tug of war : the fourth proposi- tion, which is the key of Euclid. We tell them that the first three propositions are put in first, because in the fifth and sixth propositions we have to cut off, from a greater line, a part equal to the less ; but that the real beginning of Euclid is the fourth proposition, the first theorem, and that on this proposition the battle with Euclid is to be fought, and the victory won. From the beginning one cannot be too parti- cular in pointing out the three distinct portions of each proposition — the enunciation, construction, and demonstration. In drawing the figure of the fourth proposition. Fig. 4. it is wise to be very deliberate, and make a great to do about making the two given lines and the 64 iucludcd angle in each triangle equal, in order that the fact of their equality may be strongly impressed on the learners' minds. Let each boy draw the AB and DE, and care- fully, by his boxwood ruler, make them (say) two inches long. [If any sharp boy retorts, *' You told us we were not to use our rulers to measure with," explain — this is only a datum; that it is for the qucesita they are not to measure.] For the included angle, if a boy has learned the use of the protractor, and done a little practi- cal geometry, (as he ought to,) he draws two lines equally inclined to the lines AB and DE ; if not, let him bend a bit of paper or cardboard to any angle, and with its help draw the second lines AC and BE equaUy inclined to the first ; and, by the boxwood ruler, let him make them carefully (say) two inches and a-half long. Does this seem long and useless ? I think it is highly important. It must be borne in mind that I have supposed a set of very weak boys before the Master. If he were to draw quickly the figures of two triangles, and say (as he might if he had more advanced pupils) we will suppose such lines and angles equal in each, and expect them to take in his words, he will be disappointed. I venture to say, if he names different lines and angles, and says was this and this given equal, or have they to be proved equal, he will find the boys again and again making bad shots ; but all this care- 55 fill deliberation which I advocate helps them gradually to take in the idea, and till they know accurately what is giveu, and what has to be proved, they can make no progress. It would be well at this point to take up the eighth axiom, and dwell on it and elucidate it : let them know that though in common parlance the words '' same " and " equal " are used as if they had the same meaning, (for instance, we speak of two boys being of the same height, of two masses being of the same weight, when we mean they are of equal height and equal weight,) the words are not confounded in Euclid : when he says the same, he means the same identical. He says, we know two magnitudes are equal, if we know, if we can assure ourselves, that they fill the very same identical space. Elucidate, by sup- posing the boy you are addressing to be casting bullets : he casts one out of the mould, then he casts another out of the same mould ; there they are lying side by side ; is he sure that they are equal ? yes, why ? because he knows them to ha.ve filled the same space, the same cavity of the mould in which they were cast. It is good, here, to get a promise from each boy that they will not say that any two things which they want to prove equal are so, until they have first shown that they fill the same space. We then begin with Euclid's ideal super- position : place A on D and AB on DE. Where, 66 then, will B naturally and of necessity fall, and why ? Some wall have difficulty in answering the question ; they arc frightened at the phraseology of geometry. Draw their minds, then, to some homely illustration. Here are a couple of walking- sticks, a long one and a short one. I lay them alongside of each other. I bring the handles together ; say, wdll the points come together ? No. Why? Because one is longer than the other. Quite right. Here are a couple of equal length ; I do the same with them. Now, do the points come together ? Yes. Why ? Because one stick is as long as the other. Quite right again. Now, says the Master, let us go back to the lines ; consider them to represent walking-sticks ; and remember, AB was given equal to DE. Now, then, if you place A on D and AB along DE, will B come on E ? Yes. Why ? Because they are equal. Quite right, admirably said. You have got over one-third of the difficulty of the main part of the proposition. Now, these lines AB and DE lying together, where will AC naturally and of necessity lie, and why ? Here it is well to have three triangles at hand in cardboard, w^hich have two sides in each respectively equal, the included angle being in two cases equal, in the third case unequal. Place them together as directed, bringing together alternately the triangles which have unequal and equal in- cluded angles. Thus the pupil will be brought to 57 feel the reason why AC will lie along DF. Two pair of compasses will do as well. And AC now lying along DF, will the point C naturally and of necessity fall on F ? Yes. It is hoped this time he will say firmly, " Because the lines are equal." He should now be called upon to give the three consecutive steps of the reasoning. If he does it correctly, he may deservedly be patted on the back; for out of a hundred boys that come to Eton professing to have read Euclid, not one can do as much. Now, take every boy consecutively, and make him try to say the same three steps of the proof. A Master will find comparatively few able to do so, even after the foregoing explanation. He must be ready to repeat and vary his illustrations ; also by going round and round till each boy can say it, one will teach the other, and the desire to be classed with the successful will be a stimulus that will help the poor weak fellows to fix their thoughts. At last, we may suppose they have all succeeded in saying this much, right. Now, call a pupil up to the board, and ask him where B now lies. It is ten chances to one that he wiU point to the letter B on the board. No, the Master cries energetically — and he dashes a couple of chalk marks across the triangle ABC, saying, ABC is not there ; it's lying on DEF. Now, try again. Where is B '? He puts his finger on E ; 58 and Where is C ? He puts his finger on F. Quite right now, observes the Master ; you have shown that the two hues BC and EF begin to- gether and end together. Now, answer me this. Beginning together and ending together, as they do, can they separate at all between the beginning and the end '? If they do not take in the meaning of this question, and possibly they will not, they might be helped thus : — The Master might take up a couple of pens, and bringing the points of the pens together with one finger and thumb, and the feather ends to- gether with the other, he might say. Here these two pens begin and end together, but they don't keep together ; they separate. In fact, they en- close a space, don't they ? Put your finger into the space they enclose. How is it that ' they en- close a space ? Because they are not straight. Exactly so, and if they w^ere straight lines, they could not enclose a space. Is not that just wdiat Euclid says ? Here are two straight pen-holders ; if I bring the beginning and end of these together as I did the quill pens just now, they do not enclose a space, but keep together from beginning to end. Now, these two lines BC and EF are sup- posed to be straight lines, and you have shown that they begin together and end together. Say now, do they separate like the pens, or keep together like the pen-holders ? 59 Tliey all answer promptly and rightly this time. The Master now tells the boy whom he has called up, to draw his finger along the space filled by EF; he does so ; And again to draw his finger along the space filled by BC ; he draws his finger along the same space. Eight, and yon have discovered (he adds) that the two lines fill the same space, i. e., so much length, you need take no account of their breadth because lines have not any. Now, let us look back and see what we have done. We have shown that if any two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the included angles equal, — and that if one of the triangles be applied to the other as a triangle of any form ivhatever might he applied to any other, — THE two BASES WILL OF NECESSITY FILL THE SAME IDENTICAL SPACE. Now, turn to your eighth axiom and read it. " Magnitudes which coincide, that is, which exactly ^/Z the same space, are equal." What, then, can you now say of BC and EF ? That they are equal. Quite right, well said. After a minute or two's rest for breathing, the lesson is resumed. Now, tell me, where does AB lie ? On DE. How do you know ? He hesitates. The Master passes on the question. At last, one, a little clearer or bolder than the rest, ventures to say, " Because we put it there." Quite right. A very 60 good and sufficient reason, we put it there. Shake hands for that. And where does BC He ? On EF. How do you know ? If he is very bright he will say, '' We have just proved that it must come there," and the Master may well say, "Capital." If he does not see the right answer, he must be led on to it. When he sees this, the Master says. Now put your finge]' into the space filled by the angle ABC, now into the space filled by DEF. He puts his finger both times into the same space. So you have discovered that the angles ABC and DEF both fill the same space. And as they fill the same space, what does the eighth axiom tell you ? That they are equal. Eight again. Now, once more. Where does AC lie ? On DF. " How do you know ? Don't say because we put it there." No, but we put AB on DE, and we proved that, in that case, AC must come on DF. A triumph ! you deserve exemption for that. Of course, you know where BC lies, you have already told me. Now put your finger into the space filled by the angle ACB, and now into that filled by DFE. Twice again he puts his finger into the same space. So you have discovered for the third time that the two things which you wish to prove equal (that is, the second pair of remaining angles) fill the same space. How do you finish ? Therefore by the eighth axiom The pupil finishes the sentence by say- 61 ing, "They are equal." Right agam. "You are gomg ahead — one 'bout more, and you're safe in port ! " Once again, then, where was it you put AB ? On DE. And where did you prove AC would come ? On DF. And where did you show that BO must lie ? On EF. Now, then, put the flat of your hand into the space filled by the triangle ABC, now into the space filled by the triangle DEF ; twice he puts the flat of his hand on the same space. So you have for the fourth time made the discovery that the two things you want to prove equal (now the two triangles) fill the same space, and, therefore, by the eighth axiom in tone of sympathetic triumph, the pupil takes up the word, and adds, the triangles are equal. You have thus proved, in accordance with EucVuVs eighth axiom, that if two triangles have two sides of the one equal to two sides of the other, and have the included angles also equal, thcg are equal in every respect. This is the grand fourth proposition. It is, as I said, the key to Euclid. I congratulate you on what you have done. Go now to your place, read over the proposition, as it is written out in Euclid, once or tv/ice, carefully, then close the book and write it out, while I take another boy in hand. I have here imagined a very fair (a far too fair) 62 specimen of an average pupil. A Master must not be disappointed if he has to lead a boy on and encourage him many a time, before he wins the victory. Many and many a time has a boy said to me, *'AB lying along DE, AC will lie along DF, because AC is equal to DF." And I have had to answer, " My dear fellow, there is ex- actly as much sense in what you have now said to me, as if you were to tell me it is twenty-two miles to London, because butter is eighteen-pence a pound." Boys are most apt to say words without attach- ing a meaning to them. This is often found to be the case where it would not be expected. They must be shaken out of the habit, and this is exactly one of those things that Euclid when properly learnt does. I remember once hearing a boy saying the pro- position thus : — Because AB is equal to DE, and the angle BAC is equal to EDF, therefore the line AC lies on DF. No, no, said I. What ought he to say ? The next boy answered. Because AB lies on DE. A little fellow sitting by the side of the speaker (he is a post Captain in the Navy now) looked up eagerly into my face and said. Will you tell me why it is not as good to say it one way as the other ? — " Do you wish to know why ? I'll tell you willingly." Look at those two triangles on the black board, 63 ABC and DEF. Remember AB and AC are respectively equal to DE and DF, and the included angles also. Yes. Now, if I were to take a saw and cut down through the board between the two triangles, and were to send one-half of it, that on which the triangle ABC is drawn, to Oxford, and the other, that on which DEF is drawn, to Cambridge, the lines and angles would still be equal, would they not ? Of course. But if one of the triangles was in Oxford and the other in Cambridge, could the line AC lie oil DF ? Of course not. But if the triangles are neither at Oxford nor at Cambridge, but there on the board before us, and I conceive the triangle ABC to be laid on DEF, laying A on D, and AB along DE, then indeed AC ti-ill lie along DF, because the included angles are equal. A rich, deep blush forthwith suffused the boy's ingenuous countenance. "Thank you, sir," said he ; " I see it perfectly now, — I never saw it before." When the whole class have been thus set on the right tack for appreciating the demonstration, and by their own efforts have mastered it, and shown themselves able to produce what they know by writing it out correctly, then follows another exer- cise most interesting and instructive. The aim of it is to show the application of the fourth proposition. But it is high time the class should have a 64 little rest, and while they are resting, I should like to have a little talk with my readers about all this. Will one of those friends who have been reading these last pages feel ready to exclaim, "What labour, what misdirected effort have we here ! all this work to prove (in subjection to an artificial test of equality) an axiom ! for the fourth pro- position is nothing else ! " " My friend," I answer, " let me speak as a Schoolmaster. Euclid is the book which the traditions and the general con- sent of the country put into our hands, and my business as a Schoolmaster is to try and make it do what it is put into our hands to effect. Now, what do I find, by experience, is the result of this mental ordeal I have been describing ? I reply, confidently, the boys come out of it intellectually changed. They are lifted into a higher mental level from that in which they stood before, never to descend again into it as long as they live." After this badgering (it might be called, if there were not a perfectly good understanding on both sides) on the fourth proposition, there is comparatively no more difficulty in Euclid. Henceforth they take kindly to it ; they icill know the meaning of what they are doing. They never think of learn- ing a proposition by rote. It is too much trouble ; there is no pleasure in it ; but there is very great pleasure in understanding it. The sort of mental activity that is henceforth 65 evoked by Euclid, and the satisfaction of the EucHd lesson, I can best illustrate by referring to the actual facts of those weekly evening Euclid lessons of last school-time with the class of St. Mark's boys that I have alluded to. I can speak more freely of them than I can of others. When we had come to the third proposition of Book III., I told them of a mistake which had been made by some great men* in a mathe- matical examination, who tried to prove the second part of the proposition by the B.I. 4., and I asked them to explain where their mistake lay. They did so. One of them, however, observed, '' But it might be proved by the fourth." " How so?" "Because two of the angles of each of the triangles are equal, so by the thirty-second proposition the third must be, and that is the included angle." Another of the lads here interposed : "It might be proved by the forty- seventh proposition." * I am afraid tliese " great mea " here spoken of were none other than the scholars of Trinity in their fellowship examination. When I was trying for a fellowship, (which I did not get,) one of the examiners told me that all the candidates but two had made this mistake. Now, the story is so valuable for making boys give an intellectual spring, for giving them what the French would call elan, that I am afraid it often does duty in my class-room. So I must here humbly apologize to that most select and respected body of men the scholars of Trinity, and ask them, " for a great good," to let me " do them a little wron"; ? " 6G There is not much in this ; but it shows mental activity, and an appreciation of Euclid. The study cannot be repulsive to boys who have this kind of Imowledge of it, and can link on readily one proposition to another ; and I can assure my readers that as I led them on to discover for themselves the different unexpected properties of the circle, which Euclid brings out in admirable continuity in the latter half of the third book, so entirely interesting was the exercise, that they could hardly refrain from a cheer of satisfaction at the winding up of each proposition. Come here, my friend, and stand by my side. Look into those boys' faces. They are merry- hearted boys ; they laugh like others, and sing (that they do, and sing very well too !) Look at them now. Mark the fixed eye — the rivetted attention — the animated countenance — the smile of satisfaction — they are not conscious of all this, but you and I looking into their faces can see it. What, I ask you, is all this the sign of ? The interest and internal j^leasure felt in the exercise of the mental faculties in the following out intel- lectual truth. Give me then your sjmapathy, my friend, and tell me if you do not think that Euclid is doing them a great good. I could speak of others besides St. Mark's boys, and say what pride I have in finding myself sur- rounded by boys in whose minds has grown that 67 grand capacity which EucHd emphatically imparts, *' the capacity to know when they know a thing, and to hnow when they don't hioiv it.'' I rejoice to see them reject an explanation if they do not see their way to link on all that is said, from sentence to sentence, in a manner satisfactory to themselves. I rejoice to see their quickness at putting their finger on the weak or obscure point of an expla- nation. I rejoice to he pinned down by them by a searching question. " Why is this ? You have not explained that." But we must leave this. It is time to go back to the Euclid. We left the pupils resting ; but they were only half through the battle. To appreciate the fourth proposition the learner must know, not only its demonstration, but its application. Of course, the fifth and sixth pro- positions are exercises on the fourth ; but it so happens that they are particularly hard exercises. There are a number of far easier ones scat- tered about Euclid, if only it were permitted to take a few things for granted, which they have no difiiculty in doing; for instance, that an angle may be bisected, and that a perpendicular may be drawn from a point in a line. (If they have done any practical geometry, they have learned how to do such problems,) then very easy exer- cises on the fourth proposition may be found Figs.5,Q. in Euclid {e. g., the tenth prop.) — or framed. E 2 68 Agaiu, if we assume that on opeiiiug a pair of scissors the angle between the blades is equal Fiij. 7. to the angle between the handles, then the six- teenth proposition furnishes an exercise. If we Fig. 8. write a figure like a Z on the black board, and assume that the angles in the knees of the Z are equal, then the thirty-third is an exercise. If we Fi(i. 9. assume that a square may be described, the forty- seventh furnishes a capital exercise. And if they are told that all these things, which they are asked to take for granted now, will be regularly proved hereafter, no harm is done. The two triangles to be proved equal being set before the pupil, he examines them and finds out for himself the two sides of the one that are equal to the two sides of the other, observing that the equal angles in each triangle must be those that are contained by the respectively equal sides ; then he writes formally out the exercise in five lines or paragraphs : — 1. He names one side in each triangle that is equal, giving the reason why. 2. He names a second side in each that is equal, giving the reason why. 3. He recapitulates, — wherefore the two sides named of the one triangle are respectively equal to the two sides named of the other ? 4. He names the included angles, and gives the reason why they are equal. 5. He concludes, therefore, hij the fourth pro- 69 position, the triangles are equal, in every respect, naming the respects if necessary. It is hitting a nail on the head that drives it home, and that is just what these exercises do. After doing two or three of them, they get quite clear and hrisk. They enjoy the exercise. They examine the triangles ; pick out the sides ; write out the five formal paragraphs ; come up with it. It is examined. Right. Off they are to do another. One hears occasionally, in a tone of triumph, " Therefore, hy the fourth proposition," &c. The last exercise is always that crucial test, the second part of the third proposition of Fiy. lo the third book, and the scholars of Trinity do duty again and again. Now, among these exercises, we introduce the following in an off-hand way, without letting the class into the secret of what we are about. Draw, says the Master, a couple of circles, with Fiy. 1 1. the same centre one inside the other ; and draw now a couple of radii (two spokes of the wheel) to the rim of the outer circle ; call them AD, AE ; put B and C where these lines cut the inner circles, join BE and CD. Now take as an exercise to prove that the triangles AEB and ADC are in every respect equal, and name the respects. They set to work ; in two or three minutes they bring up the exercise, the five paragraphs written correctly out. . Then the Master says, Now join BC, and. 70 remembering what you have proved already, try and prove that the triangles DBC and BCE are also equal in eveiy respect, observing to them that the bases of both triangles are the same line. Down they go to work, and this exercise is brought uj) in a few minutes. Then the Master again quite casually says. Let me see, we proved these big angles ACD and ABE to be equal ? — Yes, say they, in the first of the two exercises. And we have just now proved that BCD is equalto CBE ? — Yes. Suppose we take these little poky angles from the big ones. Then ABC and ACB remain ; And if equals are taken from equals, what do we know about the remainders ? — They are equal. Exactly, then what angles are equal ? ABC is equal to ACB. The tutor's face, which has been solemn and somewhat constrained all this time, now relaxes into a serene smile of triumph and congratulation. ** Gentlemen," says he, "I have the pleasure to an- nounce to you that you have done the fifth propo- sition. The ' Bridge ' is passed." They are taken aback. " Quite true. It's all over ! The tooth is out ! It's all right ! " So, then, we are aU very happy together. In doing these exercises with the boys, I have had reason to admire Euclid's judgment in speak- ing of the two sides and the base of a triangle, instead of talking, as we very commonly do, of the three sides of a triangle. To observe which dcoJ./^