THE enquiry of your correspondent calls my attention to the fact that I did not specifically mention the publishers of the maps I recommended for use. I really thought them matters of common knowledge among teachers. The sheets of the Ordnance maps of all scales can most easily be obtained in the country by ordering through a head post-office: a charge is made for postage in addition to the price of the maps. I have tried this method and have been very satisfied with the ease and promptness with which I obtained the sheets. In London, any map can be obtained immediately from Mr. Stanford, Long Acre, London, W.C., who also publishes a very useful Résumé of the Ordnance Survey publications. It is well to note that the maps on larger scales than six inches to the mile have no contour lines.

Bartholomew's cycling maps (2 miles to an inch) may be had from the Edinburgh Geographical Institute. As far as the limited stock allows, the copies without black printing should be asked for, with an explanation that they are wanted for teaching purposes. The "Diagram" hand-maps, which are all very small scale, may be ordered through Messrs. Philip and Son, 32, Fleet Street, London.

I take this opportunity of repairing another omission from my article. It is necessary to treat large paper-maps with great care in class-work. As far as possible, they should be always kept flat, not rolled or folded. Rolled sheets, when unrolled, are very liable to tear. Folded sheets wear badly at the creases they can, of course, be cut into rectangles, and pasted on holland or linenette with a slight margin between the pieces to allow of folding, but this method, though excellent for outdoor use, is bad for practical indoor work, as it prevents the drawing of straight lines of section across the map. After trying several other methods, I have found the following answer very well for large maps on thin paper, such as [Bartholomew's cycling maps. Obtain a sheet of millboard rather larger than the map itself, and fasten the map to it by means of the gummed tape which is sold in reels for mending music. Great care must be taken to lay this tape straight, or the map will cockle up. It is best first to fix down one of the longer edges of the map, then the other long edge, and then the shorter edges. On no account work round the edges, finishing at the startingpoint, for cockling is inevitable that way.

A. MORLEY DAVIES. The Geometrical Treatment of Angles and Parallels.

THE suggested proof of Euclid I., 32, given in Mr. Woodall's paper under the above title in the May number of THE SCHOOL WORLD, is open to the gravest possible objection, and that is, that it contains the assumption of a principle not contained in the definition of angle on which the proof professes to rely, and that assumption is involved in such a way that it would be absolutely impossible for a beginner to detect it.

If now the angle B of the triangle BAC be transferred to the position IAK, BC along AI, BA along AK,

BC has turned through zero angle from BI. ... BA rigidly attached to BC has turned through zero angle from BI.

But, unless we assume Euclid's parallel axiom (or its equivalent), we do not know whether AK coincides with AE (AB produced) or not. We must therefore assume that EAK is an angle, a which may not be zero.

Therefore, in order to bring BA along its old direction AE, we shall have to turn it through an additional angle + a from AI, i.e., from BI making zero angle with AI.

That is, as the line BA slides along itself to AE it turns through an angle + a from BC.1

Now let us apply to the triangle BAC the process of Mr. Woodall's paper. I have only modified it so as to bring B back to its original position, and to pivot only about that point in the line moved. This serves to simplify the issue.

(i.) Pivot at B, turn BC into position BA, angle turned through from BC = + B. (i.) Slide BC along BA to AE, angle turned through from BC KAE = + a. Total angle turned through from BC + B+ a.

=

=

=

(iii) Pivot at A, turn BC through angle A into position AF. Total angle turned through from BC + A+B+ a. (iv.) Slide BC along AC to CD, as in (i.) angle turned through from BC = LCH 7,2 where LCI CAI. Total angle turned through from BC + A+B+a-y. (v.) Pivot at C, turn BC through C to lie along BC, and slide along CB to BC'. This last slide adds no angle. ... Total angle turned through from BC + A+B+aY+ C.

=

=

Thus, A+B+C+ a Y = two right angles. And without assuming the parallel axiom, or an equivalent, we cannot show that a- y is zero, and therefore we do not know that A + B + C is two right angles.

1 EAI is EBI under the conditions chosen. 2 HCI is > HAI under the conditions chosen.

WITH regard to Mr. Budden's criticisms, which I have read with some interest, I should like to make three remarks :

(1) Mr. Budden's objections seem to depend on the introduction of unusual sliding motions, which, being unnecessary, were not mentioned in my article. He also seems to have overlooked the fact that, for the benefit of our "beginner," I use a straightedge, and do not suppose the sides of the triangle to be moved at all. The lines which form the sides of my triangle do not necessarily terminate at the corners, and if the portions of them which form the sides are, say, 8 inches, 9 inches, and 10 inches long, then I use a straight-edge a yard long and no sliding is required. It is an advantage to let some portion of the straightedge lie across the base while turning through the angle pivoted at the opposite corner. At the same time sliding would introduce no difficulty, for the straight edge, however short it may be, will slide along the side on which it lies (obviously without turning) until the corner towards which it began to slide lies somewhere between its two ends, and then it can be turned through the angle pivoted at that corner. I prefer not to have an end of the straight-edge at the corner in question. These details of actual demonstration were not necessary in the statement of the general principle, but I think they will help Mr. Budden to see that the straight-edge may, without sliding, turn in succession through the three angles of the triangle, and that in so doing it turns neither more nor less than two right angles.

(2) The "corresponding proof that the exterior angles of a polygon are four right angles" was given by Playfair and Hamilton (his quaternion proof), who, moreover, used it to prove that the angles of a triangle are equal to two right angles, and I fail to see that Mr. Budden has proved it to be vicious. Of modern books giving it I may mention Casey's "Elements of Euclid,” and Minchin's "Geometry for Beginners." The latter book makes very full use of the "turning " definition of angle.

(3) In conclusion, I venture to think that not only would it be "absolutely impossible for a beginner to detect" in my proof the assumption of the principle omitted from my definition of angle, but that it would be equally impossible for that same beginner to understand Mr. Budden's explanation of the deficiency. At all events, I will suggest that he should try the experiment of explaining to some beginner, or class of beginners, my proof as given by me, then explaining his objection to it, and letting the beginner say which he finds to be the more convincing.

“Book IV.” of the "Britannia History Readers," published recently by Edward Arnold (price 1s. 6d.) will supply your correspondent's needs. That is a very good book.

For the teacher E. W. Kemp's "History for Graded and District Schools" (Ginn, 4s. 6d.) may be useful as the record of attempts made (allegedly successful) in the U.S.A. But it is quite adapted to schools for that country only. The Britannia Reader is only a European history. Kemp's book treats of Hebrews, Egyptians, &c. A. J. E.

[AN article by Mr. C. S. Fearenside in THE SCHOOL WORLD for October, 1901, contains a list of pupils' books of European history.-EDITORS.]

The Education of Pupil Teachers.

THE appointment by the Board of Education of a small Committee from the Inspectorate to enquire into and make suggestions regarding the training of young teachers is an earnest of the Board's intention to remedy some of the defects of the pupil. teacher system as at present worked.

The recommendations of the Committee of 1898 (Report, vol. i.) have to some extent become operative, but not to any great extent, and by no means generally through the country. This Report is still a valuable mine of suggestions, although its recom mendations, largely based on the assumption that an improve ment in the material would be brought about, lose weight from the very reason that things are, as regards the sources of supply. very much as they were four years ago.

The pupil-teacher system being the only available source of supply at all ample and regular, of primary teachers its improvement, rather than its abolition in favour of some other plan, must be looked for. I propose to examine a few of its defects, and to suggest some form of solution.

The establishment of central classes is now practically un versal, and considerable improvement in respect of staffing, appliances, and curriculum is noticeable. The best of them, having regard to the unpromising class of candidates admitted, and the limited time at their disposal, are doing thoroughly good work. It is perhaps unsound to pay much attention to examination results; but if these are worth anything, the record is satisfactory. Central classes pass annually about 350 boys and girls at London matriculation and higher examina tions, immediately, and mediately through the training colleges a number about as large. At Wales, Victoria and Birmingham the numbers are correspondingly ample. The pupil teachers entered at Cambridge and Oxford are sound and trustworthy students, and as a rule take an honours degree.

Now for the chief defects. There is often no entrance examination conducted by the local authority, and the Govern ment test for candidates is often a mere farce. An entrance examination is absolutely necessary, unless the entrant has spent at least a year at a secondary school and can produce evidence of fair ability and industry, such as a Junior Local Certificate. With a rational entrance qualification, pupil teachers should attend at least half time. The standard they are expected to reach is as high for the average student as in the full-time secondary schools; and it is obviously unjust to expect good work from young people fagged out with a day's work before a class. Again, it is wrong to compel girls of pupil-teacher age to be in the streets of large towns late in the evening, for the heart of a large town is anything but a savoury place after nightfall. This question demands an immediate solution, and possi bly furnishes a reason for careful parents declining to allow their daughters to undertake the work of primary teaching.

The amount of recreation possible under present conditions must be but small. In addition to Saturday afternoon, a weekly half-holiday should be general; and every pupil-teachers' centre should have its sports clubs, and in addition, chess and debating societies for both sexes.

To the practice of apprenticing pupil teachers to particular schools the limited outlook of the elementary teacher is largely due. I would have the pupil teacher articled to the Education. Authority, and during the middle years of his apprenticeship he should visit all classes of schools in the area, and so broaden his knowledge of teaching by observation of the less stereotyped methods of secondary teachers.

When we come to the problem in rural districts we find that a complete overhauling of the system is necessary. Though there are still many teachers in rural schools whose scholarship is adequate, the conditions of their work are such that their energies are fully exercised in the conduct of their schools; and

the majority of these teachers would gladly be relieved from instructing pupil teachers. The following plan has already been submitted to some of those best fitted to judge, and has been pronounced perfectly workable. In each of the lesser towns, a small centre, staffed by two or three well-qualified instructors, should be establshed. Each "year" of pupil teachers would attend two whole consecutive days in each week and part or the whole of Saturday. A half-holiday in the week would bring their school work down to that of half-time teachers. Mainteance scholarships would be provided by the county councils to cover the cost of bed and board while away, and by means of approved-lodging houses the difficulty of young people spending a night in each week away from home would be met. Voluntary supervision and occasional hospitality would be freely given by clergymen, teachers and others with a view to safeguarding the morals and extending the outlook of the village girl or boy. In very few cases would the pupil teacher have to travel more than six miles, and if no bicycle were at hand the farmer's or carrier's cart would be available. The rural pupil-teacher is a source of supply well worth considering, especially as the number of boys in towns willing to take up primary-school work is fast diminishing.

If I were to summarise the wishes of those teachers engaged in the instruction of pupil teachers, the list would run somewhat as follows:

(1) A proper entrance examination for all candidates for pupilteachership, followed by small scholarships to enable those selected to spend a year in a good secondary school.

(2) Increased Government grants to pupil-teachers' centres. (3) The abolition of all evening classes for pupil teachers. (4) Oral collective instruction for all pupil teachers, particularly those in the rural districts.

(5) A matriculation examination common to all universites. (6) More thorough technical training in the schools, and more direct responsibility attached to head teachers for this part of the pupil-teacher's preparation for his profession.

(7) Greater accuracy and resourcefulness in the work of candidates from primary schools, and more vigour in the work of many of those from secondary schools.

(8) Sufficient training-college accommodation, unhampered by religious tests, for all who are fit to profit by a normal

THE general recognition of the value of graphs as a powerful factor in arousing interest in the early stages of a boy's mathematical work ought to be sufficient to compel its inclusion in a school curriculum. By the decision, however, of the Universities of Cambridge and London to include questions in graphs in the algebra papers of the Preliminary, Junior and Senior Locals and the London Matriculation, no choice is let to the teacher whose work lies in preparing pupils for these examinations but to adopt the subject at once. The following notes which I have made after a year's work with three forms may be of assistance to those who have not yet mapped out a course for their own classes.

First, as regards the mathematical attainments of the forms in question. Form A consisted of boys who had only begun algebra some little time previously, and therefore were not able to do much more than the simple rules and the solution of simple equations. In form B the boys were able to solve simultaneous equations and resolve expressions into factors. The boys in form C had covered the ground up to progressions and indices.

Time-Table.-It was found impossible to obtain a separate period in an already crowded time table, and I was, therefore, compelled to take one-third of the time devoted to a lesson in algebra or Euclid. The results more than justified themselves. Note Books consisting of alternate leaves of squared and ordinary ruled paper will be found most satisfactory.

As

Scheme of Work.-In form A we began by discussing positive and negative quantities illustrated by numerous examples of the type, "Prove on squared paper that 35+1 + 2 = I." Then followed the co-ordinates of a point with plenty of oral work at the black-board. The areas of various geometrical figures formed by the straight lines joining points whose co-ordinates were given were next determined. The next step was the drawing of simple graphs, and here the purely mathematical part of the subject ended so far as form A was concerned. soon as they had had sufficient practice in drawing graphs, they began to represent graphically various data in which they were personally interested; e.g., the scores made at cricketimaginary in many cases, I am afraid-the rise and fall of the barometer and thermometer, the number of marbles, marks or chestnuts gained per diem, and so on. Gradually they were able to tackle questions in which they had not only to plot curves but to deduce the answers to questions arising from the curve they had plotted. Questions of the following type were worked with great interest: "Given the lighting-up time for cyclists for various dates, find whether a cyclist could be summoned for not having his lamp lit at such a time on such a date." The use of graphs to find the number of inches in a given number of centimetres and vice-versa, and other practical questions of the type afforded a considerable amount of interest. ing and useful practise.

In form B the same ground was covered more quickly, and the solution of simultaneous equations by graphs and the verification of the results by algebra was the next step. More difficult questions in plotting curves were given, and the limit of the work was reached by their ability to find the equation of a given straight line.

The boys in form C required a considerably less amount of time to reach the position attained by form B. They were then able to proceed with the solution of quadratic equations by means of graphs, and this part of the subject occupied some time, but the work proved interesting and suggestive. For example, in solving two equations such as 6x + 9 = 0 and x2-6x+8= o, a boy is apt to give the answer to the first equation as x = 3 and to the second as x = 4 or 2 without stopping to ask himself the question as to why one equation has apparently only one root when as a quadratic equation it ought to have two. A comparison of the graphs of the equations, however, at once enables him to see the reason, and the liability to future mistakes of this type is thus reduced to a minimum. The solution of simultaneous quadratic equations by means of graphs, and the introduction of the geometric figures of the circle, ellipse, parabola and hyperbola and their respective equations, afford ample practice and practically mark the limit of the purely mathematical work in the form. More advanced questions arising from the plotting of curves should alternate with the purely mathematical work, and in this respect Whitaker's "Almanac" will be found a veritable El Dorado of suggestive statistics.

Newark Grammar School.

R. B. MORGAN,

Viva-Voce Examinations in French.

I MUST thank Mr. Conacher for his kind expression of approval of my short article on this subject. I do not know whether I understand aright his remarks on nasals, but I think Prof. Passy will convince him that mg and ng are as different

THE members of the party which I propose to take to Switzerland this summer cannot be described as "schoolgirls," under which title you referred to them in the notice which you were kind enough to insert in your April number. On the contrary, most of them will be engaged in some professional work.

We shall leave London on August 4th, and I have arranged with M. Dessoulavy that those who travel out with me and wish to do so can take a course in French at Neuchatel (fee £1) for about three weeks from August 6th. They can then spend several days in the Oberland, and return vid Lucerne and Paris. As this is not a commercial venture, I shall feel obliged if enquirers will send a stamped addressed envelope.

38, Woodberry Grove,

Finsbury Park, London, N.

L. EDNA WALTER.

Physical Geography at the Cambridge Locals. THOUGH rather late, I feel bound to draw attention to the criticism on the Junior candidates of the examiner in physical geography in the Cambridge Locals. He says: "The practical part of the subject as defined by the schedule issued by the Syndicate had evidently been studied in a practical manner in very few cases. For instance, in the majority of the papers in which a question referring to a rainbow was attempted the colours of the rainbow were given in the order exactly opposite to the correct one, and had obviously been learned by rote." This criticism seems unfortunate. (i.) The time set apart for teaching geography in schools is necessarily short, and, however much excursions may be indulged in and lectures given on the spot, it is not likely that a rainbow will present itself for examination at the proper time, and however much observation apart from the teacher be insisted on, it is scarcely the fault of the "instruction" if the children do not notice the particular order of the rainbow's colours. (ii.) Even if the fact is observed, it is detached from everything else in the subject of physical geography and therefore useless educationally. (iii.) The "observation of rainbows" is stated in the printed schedule, referred to by the examiner, to be "for seniors only;" and with the examiner I hold that, if the question is to be set at all, it must be set as a question on observation and not as cram-work, and consequently it ought not to have been set in an examination from which the observation is specially excluded.

Perhaps as an isolated question it may not do much harm, but there is a distinct tendency to set questions on physical geography which are not geography at all, much less physical geography, and the tendency should be checked.

who named the six books correctly.
The second prize is taken by
C. Newdigate,

Stonyhurst, Blackburn,
who named five of the winning books.
Edith C. Stent's list was the next in order of merit.

No. 19.-Most Popular School Class-Books of
General Geography.

WHICH six books of general geography are most widely used in schools at the present time? Answers to this question are required in the competition for this month. Each competitor must send a list of the titles, &c., of six school-books of general geography that he considers are the most popular ones now in use in schools.

For the purpose of this competition, those books will be judged the most popular which are most frequently named in the lists received.

We offer two prizes of bookɛ, one of the published value of a guinea, the other of half-a-guinea, to be selected from the cata logue of Messrs. Macmillan and Co., Limited. The prizes will be given for the two lists which most resemble that drawn up as a result of the voting of the competitors.

In naming a book, its title, author, publisher and price should be given. Each list of books sent in must be accompanied by a coupon printed on page v., though a reader may send in more than one list provided each has a coupon attached. Replies must reach the Editors of THE SCHOOL WORLD, St. Martin's Street, London, W.C., on or before Thursday, June 11th, 1903. The decision of the Editors in this, as in all competi tions, is final.

The result will be published in the July number, when the successful list will be published.

ST.

The School World.

A Monthly Magazine of Educational Work and
Progress.

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THE SCHOOL WORLD is published a few days before the beginning of each month. The price of a single copy is sixpence. Annnal subscription, including postage, eight shillings.

The Editors will be glad to consider suitable articles, which, if not accepted, will be returned when the postage is prepaid. All contributions must be accompanied by the name and address of the author, though not necessarily for publication.