to-day is of no more value than was that which it has displaced, and in as far as more time is being given to it is a loss rather than a gain. Knowledge per se is of little value unless it vivifies those who have it, and as a test of this let me take geometry, partly because in it has been the greatest change, partly because during the past five years I have been brought in contact with so much of it in many schools scattered all over the country; nor is it an unfair test for other reasons, success in it is incompatible with failure in other elementary mathematical subjects, it requires perseverance and resource, imagination, and constructive effort, a sense of style and a logical sense which present even in a small degree-and that is all that can be looked forwill be a sure indication that other work is not bad. So if I state facts noted in examinations in Geometry and come to the conclusion that far too many boys and girls are spending their time at mathematics which is utterly repugnant to their natures, I shall not feel that my deduction is unfair or unjustifiable.

During the past few years I have examined in the papers set by the Joint Board of Oxford and Cambridge, papers set for the pupils of average standard; those above this are scarcely catered for at all-indeed, there is a danger of boys and girls of real mathematical ability being discouraged by being overlooked. In each of the papers to which I refer there have been three drawing questions of an elementary character involving only the intelligent use of a ruler and a pair of compasses—one of the main features in the new teaching-six straightforward propositions, five at least being among the simplest in every text-book, and six riders, if such they can be called, for they are so very nearly only special cases of the propositions, and certainly nothing like those in the Todhunter of old, and here are the results as indicated by marks.

No candidate in any year obtained full marks; 2 per cent. obtained three-quarters of full marks; 76 per cent. obtained less than a quarter of full marks (the results year after year are so much alike that there is no need to do more than state the results of 1926); and let it be noted that more than a quarter of full marks could have been obtained by success in the drawing questions alone; Vol. 249.-No. 493.

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only about 25 per cent. wrote out correctly one of the three familiar propositions dealing with the congruence of triangles, and only 5 per cent. made a numerical calculation which depended upon the theorem of Pythagoras.

Bad as these facts are, and they are very bad, they do not reveal the worst, marks do not measure impressions; very few papers showed any idea of what a proof really means- -a statement such as this was extremely common, 'The thing must be so because if it were not a different result would follow': and the fact to be proved was constantly and expressly assumed as part of the data; and this after twenty years of the replacement of Euclid by geometry, with all that this means, the introduction of practical measurement and of experimental work! Euclid was said to be too abstract, too severely deductive, too theoretical, he was said to fail when put to the test of practical application, but the new geometry has led to no better result, it has added nothing to intellectual development or equipment and it adds no interest and no thrill, I doubt even if it furnishes the mind with more facts; but even if it did fulfil this humble purpose it would not justify the many hours spent over it; for education, school education especially, is not intended to treat the young, delicate, growing mind as though it were a granary to be crammed quite full with facts. In looking over the work of school after school I could come to no other conclusion than that geometry does not bring out principles or conduce to clearness of mind, it is not educative, but is as a dead wall surrounding a dead heap of things called propositions to more than 75 per cent. of those who are doomed to do it. Latin and Greek verse-making in the past was no worse waste of time than is this difficult subject.

And the bright visions which characterised the Revolution fade away in the realm of fact; as is often the way with revolutions, they break against the hard realities of life, the imperfections and limitations of human nature. A feeling for geometry and a geometrical sense has not been given to most boys and girls-why assume that it has? Many are deficient in other senses, they fail to appreciate music or colour or art, it is not given to all to play any instrument or to sing or to paint, why assume that they can enter into the spirit of

geometry, with its power, its beauty, its truth, and above all its need for the exercise of imagination?

So I would plead for a reconsideration of the school syllabus, as a step towards the drastic reduction of subjects so imperatively needed and as a recognition of the conditions of human minds. I would try to teach to all boys and girls arithmetic, some geometrical drawing, the use of a ruler, a pair of compasses, a protractor with which to measure angles, a little algebra up to the solution of simple equations, to establish familiarity with generalised numbers, and also a little numerical trigonometry of a simple practical kind-the suggested limitation in algebra would be quite enough for it. Beyond these limits I would not go except in the case of the specialists, and they would do better, find more interest and stimulus in their work, because they had not to travel along the same road and at the same pace as their slower and weaker schoolfellows, who stumble at every difficulty and look for help at every fall.

The carrying out of this suggestion would free many hours for other work, our own splendid literature, art, workshop, music-so that boys and girls might thus leave school with some interest which they had made their own, and would take pleasure in pursuing when school days were over. As it is they are all, or nearly all, kept at mathematical drill long after it has served any educational purpose.

But is it really necessary to cater for so many hours at all? Are not many boys and girls, perhaps most, being kept at school longer than is desirable? Are they kept there solely to keep them out of mischiefas so many letters in the Times' about the length of holidays would lead one to suppose? Are they not only having pumped into them smatterings of knowledge forgotten as soon as they have been poured into the sink of an examination?

Confessedly examinations are the bane of all educational ideals, but democracy will never return to the system of nomination in vogue some seventy and eighty and more years ago, its various classes are too suspicious of each other to make this possible, though probably it fitted men to their posts in the public service better than does a series of written answers to many examination

questions. It has never been proved that because at a certain period of time a boy or a girl, a young man or a young woman, can translate a piece of Latin or Greek or solve a difficult problem more satisfactorily than his or her competitors, he or she will make a better public servant, a better leader in any position, or even a better worker in the trivial round and daily task; examinations do not and cannot take into account so many of the factors necessary to the carrying on the organisations of our complex society; the system has many drawbacks, though on the whole it seems to work without as many disasters as might a priori be expected, and will continue because the frailty of man cannot at present evolve anything better, but that is no reason why it should dominate the work of schools, or necessitate the teaching of subjects which are of no educational value to the great majority of pupils; and so much mathematics, even of the new mathematics, as a matter of course to every boy and girl is a gross waste of time, a refusal to recognise the facts of life, and a great injustice to very many individuals.

C. H. P. MAYO.

Art. 6.-THE PROBLEM OF DISARMAMENT.

DISARMAMENT has long been regarded as the first postulate for the maintenance of peace, and most of the schemes for averting war have been based on some form of reduction of armaments. Of the attempts made before the World War it is unnecessary to speak, as the fact that the war occurred proves them to have been failures. It was not, indeed, until after the war that any systematic plan was evolved to achieve some practical measure of disarmament. This was one of the main objects which the League of Nations was expected to attain, although plans for reducing armaments were also discussed independently of the League.

The steady increase of armaments during the years preceding the World War was, and is still, believed in many quarters to be one of the main causes of the conflagration, and large sections of public opinion, comprising not only professed pacificists, but also numbers of men who had done their duty in the war and some who had played a distinguished part in it, were strongly in favour of such international agreements as would effectively reduce armaments, and consequently, they believed, make war more difficult if not impossible in the future. This feeling found expression in Art. 8 of the Covenant, which provides for the reduction of armaments, and is perhaps the most definite and imperative clause in the whole instrument, proclaiming as it does that disarmament is an essential condition of peace. Members of the League,' it states, recognise that the maintenance of peace requires the reduction of national armaments to the lowest point consistent with national safety and the enforcement by common action of international obligations.'

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The arguments in favour of a reduction of armaments are, of course, fairly obvious. In the first place, the maintenance of large armed forces places a heavy strain on the budgets even of the wealthiest nations; and if this burden was serious before the war it is felt far more acutely to-day when every effort is required to make good the losses suffered and when, in view of the experience acquired during the late war and of the general rise in prices, armaments would necessarily be