to keep up her physics, an auxiliary subject in applied mathematics.

In conclusion, it may be observed that the suggestions and plans described above are the result of experience and experiment, and that the views put forward, it may be somewhat dogmatically, as to the value of science training and knowledge for girls, are not those of a science specialist, but of one whose personal interests are humanistic and literary. Even on the transcendental side, physical science, like abstract mathematics, has its element of imagination, poetry, beauty and reverence. To know, like the wisest of kings, all the trees “from the cedar that is in Lebanon even unto the hyssop that springeth out of the wall," to discern with the Roman philosopher "the courses of the stars in heaven and the tumid surging of the seas," to catch some whisper of the mighty harmonies force and matter weave and interweave through the universe of phenomena, is not without a message to the soul within us, nay, is to some more eloquent of all that is truest and best in the life of reason, than even the glories of literature, or the vocal and storied record of cities and empires and the deeds of man.

For Girls Specialising in Science.

Form V-Botany.-Types of cryptogams, cells and cell structure done microscopically. About twelve Natural Orders. Natural History.-Same as Class IV. Also some invertebrates.

Form IV., Upper.-Heat. Experiments on expansion of solids, liquids and gases. Thermometers; kinds; how to make and test them. Freezing points and melting points. Specific heat of water and other things determined and compared. Latent heat of water and steam. Transmission of heat-radiation, conduction, convection.

For Girls Specialising in Science.

Form V.-Physics.-General properties of matter. Heat and its effects. Specific heat and latent heat. Light-reflection from plane surfaces, refraction, shadows, prisms and decomposi tion of white light.

Form V.-Chemistry.—Study of air and water and their constituents. The chief non-metals. Simple theory.

Form VI.-Metals in general. Alloys, &c. The chief metals in detail. Equivalents determined. All done practically.

RECENT DEVELOPMENTS IN MATHEMATICAL EXAMINATIONS.

TH

By C. ALMERIC RUMSEY, M.A. Dulwich College.

HERE is an old complaint that governments are slow in their movements: that reforms necessary to the well-being of communities are not infrequently initiated by unofficial action from below before the powerful machinery which alone can make them effective is set in motion by the force of public opinion.

At first sight it would appear, from an inspection of certain data, that in the process of the reformation through which mathematical examinations have passed during the last two decades a remarkable series of exceptions to the usual course of events has been exhibited, in that time after time changes of the most radical character have been made under the direct auspices of state departments, while other bodies have lain dormant in the grip of conservatism. It might perhaps be inferred that high authorities have recently become imbued with a loftier view of their responsibilities than heretofore, and that an application of the same zealous

usher in the millennium.

But this inference is not altogether legitimate; for in its application to the present instance the word "government" must be held to denote not the powers that dominate the British Empire, but those which wield the paramount control over things mathematical. It would, therefore, be well, before deducing from these special considerations any general theorem as to an improved morality in rulers, to make an investigation into the conduct, not of the state departments whose attention has been accidentally called to the matter, but of the universities. Such an inspection, though it reveals much backwardness in the past, yet yields some hope for the future. Committees are now actively at work, and new regulations have been published for Responsions, and for the Oxford and

Cambridge Locals, which are perhaps the most important elementary examinations which these universities conduct. But the improvements which have just been made here are in the Woolwich and Sandhurst papers already a matter of history; and some of the papers on elementary subjects which are set to undergraduates are still in need of radical reform. But though, ideally no doubt, everything should be conducted on the best possible lines, the universities are perhaps not greatly to blame for the lack of reforming spirit which has hitherto existed in their dealings with the examinations for poll degrees and Little Go. The dons know very well that these examinations, however well appointed, will never be taken seriously by the candidates. We must, however, hope to see them brought, in the near future, into line with others of the same order of difficulty; if this is not done, a very awkward situation will be created in some of our largest public schools.

The changes which have recently taken place may, in regard to the causes which have produced them, be classified under two main heads. The first, those which are ordained by regulations, issued at the instance of controlling councils, are by far the most important. They are the result alike of careful consideration of a responsible and constituted body, and of an explicit statement which it is not easy to revoke: they alone can cause radical alteration in the teaching of subjects, and form a determining factor in educational progress. But if of materials furnished by the past a basis for conjecture as to the future is to be formed, it is frequently necessary to look behind these indications of syllabi, and to draw conclusions from the changes made by individual examiners. It is often found that a certain paper progresses in difficulty from year to year, or in some other way alters its character, and though no new regulations may have been published, the circumstances suggest that this will be the case in the near future. The geometry papers in the military Entrance examination form an interesting case in point. These, during the late 'nineties, passed through a period of evident unrest. The examiners were, to judge from the questions which they set, dissatisfied with the syllabus, and did their best, without overstepping its limits, to adapt it to meet the modern improvements in educational method. There was a frequent admixture of drawing and mensuration questions with those in formal geometry; and finally, in 1901, a regulation dispensing with Euclid's order of propositions was issued by the board. This supersession of the old text book in order to make way for more modern methods is a change in comparison with which all others sink into insignificance. The pioneers in the movement were not the military authorities, but the heads of the Science and Art Department, who many years ago decided not to make a knowledge of Euclid's Elements a sine quâ non for securing a pass in geometry.

This example has now been followed in the naval Entrance examinations, London Matriculation, lower Civil Service, Oxford Responsions,

and many others. The Oxford Local regulations for 1903 contain the following important notice:

Questions will be set so as to bring out as far as possible a knowledge of the principles of geometry, a smaller proportion than heretofore consisting of propositions as enunciated in Euclid. Any solution which shows an accurate method of geometrical reasoning will be accepted. Geometrical proofs of theorems in Book II. will not be insisted upon.

The new syllabus issued on behalf of the "Cambridge Locals" gives a very complete account of the type of questions that will be set in 1903, the whole being entirely on British Association lines. Specimen papers in geometry (Preliminary and Junior) are to be published with the book of papers for December, 1902. In the meantime, we are told that Euclid's order of propositions is to be dispensed with, the papers are to consist of two parts, one a practical section, for which compasses, protractor, set squares, and an inch and centimetre ruler will be required, the other theoretical, in which proofs of propositions will be demanded. Hypothetical constructions are admitted,―ad lib. apparently, there being no statement to the contrary. This is an omission which cannot but lead to difficulties, but such must undoubtedly occurand in many forms-during a period of transition. There are some, however, which can be avoided by forethought, and it would be well if the example set by the Science and Art Department were followed in a certain particular, with a view to preventing imposture: at the head of its geometry papers occurred the following notice to candidates:

Unless you expressly state the contrary, it will be assumed that you have read GEOMETRY in Euclid, and you will be expected to follow Euclid's sequence, otherwise you must state what text-books you have used in geometry.

It is scarcely possible to find words which will sufficiently animadvert against the folly of those examining bodies which have made the announcement that they will not insist on Euclid's sequence, without accompanying it by this precautionary clause. One or two instances illustrate the class of difficulty that must inevitably arise if this procedure is not adopted: Euclid I. 18 is set: a demonstration similar to that of I. 19, mutatis mutandis, is sent up, the result of . 19 being assumed; or III. 26 is proved by means of III. 27. Now, how is the examiner to know that the candidate has not been taught on a system in which I. 19 and III. 27 are proved independently of their converses, the latter being subsequently deduced from them? He has no choice but to give full marks, though in all probability both answers are what schoolboys expressively call a "fudge." A similar predicament is liable to occur in the case of any two consecutive converse propositions the second of which is deduced from the first. There is, as a rule, no intentional dishonesty on the part of the candidate; he has simply forgotten. Such instances are of frequent occurrence. Less frequently, but sufficiently often to make the case worthy of consideration, are first-book propositions

made to depend upon the theory of proportionalsand all these proofs might conceivably be placed on a logical basis.

Again, the examiner will frequently find himself on the horns of more subtle and philosophical dilemmas than the above. Consider, for instance, the following typical question and a possible

answer:

Give reasons to show that similar polygons are proportional to the squares of corresponding sides.

Let ABCDE, abcde, be two similar polygons.

Describe squares on AB and ab.

Then the whole figures thus drawn are similar, and hence corresponding parts of them are proportional: therefore the polygons are as the squares on AB and ab.

Now this argument has probably no philosophical basis in the mind of the candidate. Yet it is absolutely convincing to anyone who has a sense of proportion, not only as a proof of VI. 20, but as a substitute for both VI. 19 and VI. 20. It is, therefore, worth some, if not full, marks. Moreover, it is conceivable that in the text-books used by the candidate the following sequence of propositions occurred: (1) Similar triangles are proportional to the squares of corresponding sides. (2) If similar rectilinear figures be divided by the joins of corresponding points, their corresponding parts are in proportion. (3) Similar polygons are proportional to the squares on corresponding sides.

This is a reasonable arrangement; it differs from Euclid's only by the substitution of ratio of squares for duplicate ratio, and the division of VI. 20 (with the first part slightly altered in form) into two propositions. On this supposition the answer deserves full marks.

It would be easy to multiply instances to show that the present generation of English mathematicians have by no means discharged their duty to posterity by abolishing the use of Euclid. Á new set of definitions and axioms, and a new order of propositions, must be established, backed by sufficient authority to ensure recognition throughout the country. When it is remembered that the whole science of geometry is based upon experience, that to some minds the existence of a plane, as defined by Euclid, is a matter of doubt, while to others the above inclusive proof of VI., 19 and 20, would appear perfectly rigid, the folly of leaving each teacher to propound his own axioms must become too palpable to be tolerated. A scheme of propositions for a revised text-book on geometry has been included in the pamphlet on Teaching of Elementary Mathematics" by the committee of the Mathematical Association. This committee was composed of masters from nearly all the great public schools, and representatives from other prominent educational bodies. recommendations consist mainly of omissions of useless propositions and of alterations in the order of others; but Euclid's "logical order" has been retained i.e., no change has been made which would render any of his proofs invalid. Also certain hypothetical constructions are recommended, such,

"The

for instance, as the bisection of a line or angle, where the possibility is obvious.

The changes which have taken place in ALGEBRA papers are far less noteworthy than those in geometry; in fact, the only innovations which are of great importance are really geometrical in character, and arise from the feeling that the two subjects ought to be interwoven with each other at a much earlier stage than has been usual heretofore. The feeling originally vented itself in the creation of "mensuration," which has formed a section in a large number of examinations; but questions which were at first classified under this head are now frequently set in the Euclid and algebra papers at most Government examinations and in many others. The plotting of curves for statistical purposes or for the solution of equations forms a prominent feature in training colleges, and has recently found a place in the naval and military Entrance papers. There are minor alterations which, though not so easy to place upon formal record as the above, indicate a trend of opinion among examiners, and should, therefore, be not altogether overlooked. Questions involving long analysis are less in evidence than formerly, a larger proportion being of the kind that require an understanding of principles. It is, of course, not to be expected that young boys will be able to discuss the ultimate bases of the laws of algebra. But verifications by substitutions in formulæ and illustrations are frequently a means of bringing home to the learner the issues involved, and, if not formal proofs, supply at any rate strong circumstantial evidence. In this connection that hitherto unprofitable servant, the second book of Euclid, is much in evidence. An Army Cadet paper for July, 1902, contains the question: "Draw figures to show that (1) (a±b)3=a3±2ab+b2, (2) a2—b2 = (a+b) (a—b)." The Cambridge Local examination for December, 1903, will demand "illustration or explanation by means of rectangular figures of the identity" k (a+b+c . . . ) = ka+kb+kc . in addition to those just mentioned.

There is naturally little to record on the side of ARITHMETIC. In the Naval, Military, and Lower Civil Service examinations it has become customary to set two papers, one designed to test accuracy, the other containing questions of mathematical difficulty to test resource. A prominent feature is the requirement of approximate calculations by which answers can be obtained to a given degree of accuracy, recurring decimals having been placed in the background as of less practical importance.

In the region of higher mathematics, such as the Cambridge Tripos, there is continual progress, as might naturally be expected, since it is in this quarter that the attention of prominent mathematicians most naturally concentrates itself. But there has been an extraordinary conservatism shown in the matter of Entrance Scholarship examinations at all colleges. Excepting for the addition of differential calculus, no change in syllabus has been made since time immemorial, although there is abundant evidence, both internal,

furnished by the papers themselves, and external, furnishable by teachers outside the universities, that such is eminently needed.

The internal evidence consists in the ever increasing difficulty of these papers: modern improvements in teaching have rendered it impossible to separate the candidates who present themselves by means of questions demanding only a working knowledge of the subjects below the integral calculus; and examiners have in self-defence had recourse to many of their less important ramifications. This will always supply a solution of the difficulty, but one which is by no means satisfactory. Any tolerable mathematician can, by piling up successive wedges, create with a stroke of the pen a dynamical system the accelerations of whose parts no boy-or man-could discover within the space of three hours, or with Hobson's "Trigonometry" in front of him devise a dozen questions which might serve to differentiate a candidature composed of Senior Wranglers: but the question whether a schoolboy's time is well employed in attacking problems of this character is now being discussed on all hands: nor is there much doubt but that the discussion will shortly bear fruit.

As to the external evidence, it is well known that many of the competitors, especially those who come from university colleges, have actually read subjects above the differential calculus. Moreover, a strong feeling is growing up that a school course should be such as to give a wide grasp of mathematical principles rather than great skill in solving fanciful problems of a highly specialised character. An able boy would have no difficulty in acquiring by the age of 19 a working knowledge of integral calculus, particle and rigid dynamics, and threedimensional analysis, in addition to the subjects now required of him. Such a course of work would lend an intensified interest to school mathematics, and obviate the tendency to "staleness," which cannot but be engendered by the continual plodding over the same ground which is necessary to success under the present system. Moreover, it would form a preliminary not only to the Mathematical but also to the Science Tripos. If men are to become first-class physicists they must acquire some knowledge of mathematics; and this should mainly be done at school in conjunction with elementary practical work, the higher experiments being in most cases postponed: because, though most schools are able to supply good mathematical masters, few have at their disposal sufficient funds to furnish laboratories suited to advanced work.

A word as to the supersession of Euclid's Elements. This movement, which is a natural consequence of the evolution of geometrical thought, must not be confused with another, the reasons for which are purely didactic, namely, the separation for teaching purposes of the subject into practical and theoretical courses.

The intense difficulty experienced in learning Euclid under the old system has arisen from the fact that the pupil has been required to call

simultaneously into great activity two totally uncorrelated faculties, the geometrical and linguistic. This to an ordinary boy is almost impossible. The two faculties must be trained separately before they are used in combination. Some familiarity with lines and circles must be gained before an attempt is made to argue in concise language as to their properties. If this is not done, the same kind of difficulty, though no doubt in less degree, will always be felt in the teaching of formal geometry, however excellent the system and arrangement of propositions.

That a new system will shortly be adopted may now be taken for granted. But if we are to consign Euclid's Elements to the silence of the upper shelf, we must do so in no contemptuous spirit but with feelings of the deepest reverence and respect. As a text-book it possesses a unique history. A manual of science composed three hundred years before the birth of Christianity, it is to-day, after centuries of scientific discovery, a volume of recognised utility and a model of logical precision. It forms a colossal monument to the intellect of a remote age, demonstrating that our superiority to the Greeks is due only to accumulated knowledge and in no way to an accession of mental acuteness.

The setting aside of this extraordinary work in favour of more modern methods is but a part of a revolution which is taking place in the education of the country, and but one result of the great truth which is being forced upon her schools. These schools have set the noble ideal of Athenian thought and culture before many generations of Englishmen. If future generations would emulate this ideal they must do so by discovering new sciences and creating new systems; nor must they think, as men have thought in the past, that by gloating over the words of Plato they become the successors of the Greek philosophers.

THE HEADMASTERS' CONFERENCE.

A

T the first meeting of the Headmasters' Conference, held at Uppingham in 1869, Thring, the founder of this important educational association, said, "Our schools depend absolutely and entirely on the vitality of progressive work "; and it was this belief which inspired him to set about the arduous work of securing a hearty co-operation between the headmasters of the public schools of England. Of the difficulty of Thring's task there can be no doubt. As Mr. G. R. Parkin says, in "The Life and Letters of Edward Thring" (Macmillan), the Conference "has broken down a deadening isolation, induced a healthy interchange of ideas between public schools, given them a united voice in time of need, exercised a powerful influence on educational questions"; and to accomplish a task of this sort

is never easy.

The formation of the Conference is described in

one of the most interesting chapters in Mr. Parkin's book. The headmaster of Canterbury School, Mr. Mitchinson, afterwards Bishop of Barbados, invited, in 1869, a number of headmasters to meet in London to discuss the Endowed Schools Bill then before Parliament, and eventually persuaded Thring to attend. At the close of the meetings Thring rose and proposed that such a gathering should become an annual institution, and then and there invited the first Conference to Uppingham the following December. The meeting in London took place on March 1st, 1869, and on October 23rd of the same year Thring sent out to the headmasters of the public schools the letter of invitation to attend the first Conference to be held at the beginning of the next Christmas

But the conservatism of the great schools was soon overcome. After the second meeting, which was held at Sherborne, Thring writes in his diary, "The seven school delusion broken up." The Headmasters of Winchester and Shrewsbury had attended the second meeting, and the Headmaster of Eton had joined the Conference soon after. From this time the Conference steadily gained the confidence of public-school headmasters, and increased in public importance.

The annual meetings have since taken place regularly, being held in succession at Highgate, Birmingham, Winchester, Dulwich, Clifton, Rugby, Marlborough, Harrow, Eton, Wellington, University College School, Charterhouse, Oxford, Merchant Taylors' School, Shrewsbury, and Bradfield College. Three meetings have been held at the College of Preceptors, and two meetings each at Eton, Winchester, Rugby, and Sherborne. The meetings of 1901 took place in the Senate House at Cambridge, and those of 1902 at Tonbridge.

The executive of the Conference is its committee of nine members, three of whom retire each year, and can only be re-elected after the expiration of a year. The committee for 1902 was as follows:

From a photograph by Messrs. Elliot and Fry.]

THE HON. AND REV. CANON E. LYTTELTON, M.A. Master of Haileybury College; Chairman of the Executive Committee of the Headmasters' Conference.

holidays. The following sentences from this letter indicate clearly what Thring thought such meetings could accomplish :

"Government is dealing with school bye-laws recently passed, other measures are contemplated, and future Governments will most assuredly take up the question.

"Nothing has been more remarkable than the absence of any decided voice from the great body whose work is being handled by external power.

"Yet a profession involving experience and practice of the most varied and intricate kind ought not to be without a common voice under such circumstances."

Between sixty and seventy invitations were sent out, and twelve headmasters attended at the first Conference. The numerous refusals showed clearly that there were prejudices to be broken down.

In addition to this there are several standing sub-committees charged with special duties. These are as follows:

Parliamentary: Revs. the Hon. E. Lyttelton (chairman), G. C. Bell, Dr. Fry, W. H. Keeling, R. D. Swallow.

Universities: Revs. Dr. Gray (chairman), H. M. Burge, A. H. Cooke, Dr. Field, Dr. Rendall.

Public Examinations: Revs. Dr. Gow (chairman), M. G. Glazebrook, and S. R. James, and Messrs. J. E. King and A. T. Pollard.

Professional Questions: Revs. G. C. Bell (chairman), Dr. Flecker, H. W. Moss, Mr. J. S. Phillpotts, and Rev. Dr. Tancock.

With reference to the chief matters which have engaged the attention of the Conference and its committee, we cannot do better than quote from an article by the Master of Marlborough in the current issue of the "Public Schools' Year Book" (Swan Sonnenschein): these have been :

The examination of schools by the Universities; the higher and lower certificate examinations conducted by the joint board of Oxford and Cambridge.

The conditions and arrangements for awarding entrance scholarships at Oxford and Cambridge. The training and registration of teachers.