increasing needs, we have hurried too far from an ideal which transcends time to the immediate requirements of our own period.

It is not easy for those actively engaged in teaching to be the best judges of their aims; they are so prescribed in their limits, so busy in their routine, so intent upon their actual task, and withal so narrowed by that impending examination-the very antithesis of all real education-that they cannot see the forest for the trees. For many years it was my function to try to guide the mathematics at Harrow, and I realise how difficult it was to grasp what (beyond getting boys up to such a standard of superficiality as would enable them to get marks in certain examinations) was the educational value of the work we were doing with much patience and expenditure of energy; wider opportunities have since made me wonder whether school mathematics to the extent to which it is now being carried as a general subject is not just as tyrannical as was the Classical domination of old. Has not the subject really suffered from a swing of the pendulum which has gone too far? A reaction which swept away the one-sided curriculum of the mid and later Victorian age has assumed that where Classics failed, Science and Mathematics must succeed: and the assumption is not proven.

Schools formerly had a much easier time-table than they have to-day. Latin and Greek were the main subjects of intellectual supply: from a very tender age we learnt the Latin Grammar, with its syntax from Kennedy or the Public School Primer; we knew and could repeat with unconscious effort:

'Many nouns in is we find

To the mascula assigned,'

and could apply these rhythmical rules more or less accurately; we stumbled through Farrar's unintelligible Greek Syntax with difficulty. We hitched Latin verses into shape with the aid of a Gradus as an intellectual jigsaw puzzle not devoid of pleasure when the words went together. We read Latin and Greek authors; those who reached the VIth Form-for all boys were treated alike, there were no specialists-were familiar with Cicero and Tacitus, Thucydides and Plato; Ovid,

and Virgil, and Horace, Euripides, Eschylus, and Sophocles-and that, too, in the little Oxford Press editions without notes, without divisions into Acts and Scenes, and with the enigmatic and dubious utterances of the Chorus to unravel or not as best we could. The work was not made as easy as it is to-day with the annotated editions, the expunged choruses, the marginal explanations, and the many subdivisions, all bypaths to the royal road to learning. Then there was French, still in the hands of the old-fashioned French pedagogue, for no school was properly staffed without a Frenchman whatever might be the measure of his inefficiency; German, if taught at all, was in a more parlous state; Science was an extra and so was Drawing; and then there was Mathematics, which came a long way behind Classics in pride of place, but it was the only other study seriously regarded. It was Classics first, Mathematics a very poor second, only indeed placed at all because it was in a better position than the other work, which was nowhere.

And for the greater part of the school-indeed, for all but for those who could be counted on the fingers of one hand-the only mathematical subjects were arithmetic, algebra, and Euclid; of these arithmetic was the staple work. We worked at sums, applied certain rules and got answers the main thing was to get a right answer, method, style, continuity were little seen. 'I will read out the answers to the sums; 5 marks each for those right, 0 for those wrong,' was a common dictum heard in many a room from a master who faced his class with an air of boredom. Times must then have been easy for the masters, for we were left alone to do long exercises to a pattern type; it was all drill, little thinking. Much the same prevailed in algebra, an explanation followed by long, wearisome exercises; and then Euclid, how familiar was that little brown-backed Todhunter, how thumbed, how dreaded! there was a tradition of impossibility associated with the pons asinorum, that was the real test; the first four propositions, except, perhaps, the fourth, were fairly easy, though the first with its formality, its apparent unnecessary verbiage, its seeming attempt to obscure the obvious, gave one a strange feeling of dabbling with the unreal. Many years

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