later I felt a thrill of sympathetic understanding with a boy, who in answer to my question, 'How much Euclid have you done?' said that he had been as far as the first proposition many times.

Only a few regarded Euclid as anything but an unintelligible world, it had to be done, but why we did not know, it was part of the worry of school like mumps and measles, which every one in his teens-not in her teens in those days-had to endure. And as for the riders at the end of Todhunter, they for most boys were altogether an unknown quantity. I doubt if more than half a dozen boys in any school ever knew what those exercises were for, or wandered into the pages where they were to be found. We learnt the propositions more or less by heart, said them by rote if. we could, and without any real grasp of their meaning and with no thought of ever applying them.

Beyond these three main subjects a few boys-perhaps two or three-read trigonometry and other branches of mathematics: Todhunter's Trigonometry,' a book for the specialists dealing with difficult general propositions at the very beginning, and developed in a manner likely to appeal only to those with a mathematical bent, without easy and interesting practical and numerical applications. All mathematical work in those days was alike and made the same appeal to idealisms. It confirmed the conviction in boys' minds, if they thought at all, that things in a book were totally different from things out of a book, and mathematics was either a wearisome round of tricks to bring out answers, or an appeal to abstractions which did not and never could exist.

Gradually, but quietly and surely, there grew up a conviction, which found fuller and more continuous expression as time went on, that there was something wrong in school work, that the Mathematics were unreal and the Classics unintelligible, that both, the latter more especially, occupied boys for many years and provided no stimulus to intellectual effort, and left them unable to read a single page of either Latin or Greek. There was a revolt, which showed itself in many ways. No longer were only Classical masters thought to be eligible for Headmasterships, and for the guidance of studies in

general; mathematics came into greater prominence, and a very different type of mathematics from that which had been taught in the 'eighties and 'nineties of the last century. And yet again another very important change. In my own school days of which I write mathematical teaching was for the most part not in the hands of qualified mathematicians, but in the hands of classical masters who knew very little about the subject. They could do a few sums, but probably embarked upon them with anxious misgiving, and were only too glad to be reassured by the answer. One such I knew who when asked how he scaled his marks, replied, 'I multiply them by 5 and then by 2, and cut off the last figure, sometimes it comes the same, then I do it again.' How was it possible for such a man, and he was typical of many others, to do geometrical riders which required both knowledge and imagination and constructive capacity? Such masters heard and valued Euclid in proportion to the exactitude of the reproduction of the words of the text-book; very much as the old drill sergeant in the days before the Boer War expected the Army Drill book to be learnt and repeated word for word. One of my own masters was wont to hear the propositions in a unique way. We all must have our books open on our knees before us, but with a piece of paper held over the print in which a hole had been cut sufficiently (and sufficiently was naturally a term of elastic meaning) large to allow the figure to be seen, and from that we repeated what we knew, not excluding the final Q.E.D. or Q.E.F. No notice was taken of the kind of paper; tracing paper or very thin transparent tissue paper was not unknown.

All this, amusing if ineffective, was fundamentally changed with the Revolution, when mathematics entered into the realm of the serious. Schools were staffed quite differently and by men who knew what they had to teach, believed in their work, its educational value and possibilities; with their advent came other changes. Euclid was at last given up to make way for Geometry, whether the change has been beneficial I will consider a little later; let it only be said that the geometry taught in schools to-day is not merely Euclid writ large, it is different; arithmetic and algebra have undergone

changes, too, all tending to simplification, to replacing the unreal by the real, to training and expanding the mind by interesting it and stimulating it, so that there may be no need to commit to memory many formulæ and unsatisfying rules, but that boys shall know how to do their work because they find interest in it and think about it and can understand it and reduce it if possible to something within their own experience. They are trained to develop their intelligence and not merely to function a machine, and thinking grows by thinking just as apathy is fostered by the routine of a mechanical drill which makes the mind insensitive to all delicate work whether of brain or of hand. Further, to add to the brightness and thus to the attractiveness of their work, and also to bring home to them its practical utility, trigonometry of a non-specialised type has been introduced quite low down in very many schools, and where fifty years ago one boy was doing this subject, there must now be at least twenty or thirty; but it is different trigonometry and of everyday application, devoid of all those general propositions of an abstract character which were a stumbling-block in the past; it is intended to help young minds to bring the lessons learnt in their class-rooms to the survey of their country, to estimate the height of hills, the distance of ships out at sea, of objects on the horizon, of landmarks far away. It is as different from the trigonometry of Todhunter as if it were another subject altogether. And the same applies to other branches of school mathematics, Mechanics and Calculus. No longer are they upon a lofty pedestal looked at only by the few; they have been simplified beyond all recognition by those who are only familiar with the text-books of the past.

All this is excellent and full of promise that schools are turning out boys and girls, not more learned, for learning does not instruct the mind, as the old Greek Heracleitus wrote, but with intelligence stronger, more flexible, more adaptable, more willing to learn: butand there is a big but which looms very large and holds up an arresting hand to those who have seen the work which is the result of all this modern simplification and effort—most reluctantly the conclusion has been forced upon me that much of the mathematical work done

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 I 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 for→ will 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 experi mental 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 paint, why assume that they can enter into the spirit of

or to