professorial lectures on higher mathematics similar to those given in the German universities.

The recently introduced changes in the 'Mathematical Tripos have afforded a striking object-lesson as to the prevailing state of feeling at Cambridge. In the fly-sheets that were issued nothing was said about the failure of Part II. It was not the Cambridge mathematicians who carried the day, but the teachers of physics, chemistry, engineering, and other applied sciences. The object of the reform was not to remodel the teaching of mathematics on lines that would bring Cambridge into line with German and American universities, but to reduce Part I. to the level of a B.Sc. examination suitable for applied science students. The opposition made no attempt to defend the present Part II., but based their arguments to a considerable extent on statistics of the number of senior wranglers who had attained eminence at the Bar or in other professions—not in the mathematical world. The University professors and lecturers and nearly all the mathematicians were on the side of reform, but no weight was, so the opposition said, to be attached to their views because the professors took no part in the teaching of the students. Fancy the leading University of Great Britain boasting that it awards the highest honours in mathematics to candidates who have never attended a single professorial lecture!

The far-reaching improvements that have taken place in recent years in the teaching of geometry in schools have their own story to tell. It will, I think, be generally admitted that it was Prof. John Perry who first set the ball rolling. Now Perry spoke as an engineer, not as a mathematician. Had he posed as a mathematician it is very doubtful if the movement would have succeeded. The work of reform was carried out mainly by the Mathematical Association, which thus ably fulfilled the objects for which it had been originally founded under the title of Association for the Improvement of Geometrical Teaching.' But it was Euclid, and not algebra, against which Perry's attacks were mainly directed. The result has been the evolution of a fairly consistent scheme of teaching geometry, and it is interesting from an historical point of view to notice that this scheme was largely anticipated by the late Rev. H. W. Watson in a book published years previously in the 'Text Books of Science' series. But in algebra a perfect chaos now reigns. The old useless drudgery still lingers both in text books and in certain examinations. The headmistress of a large girls' school said to me a few weeks ago: 'We should only be too glad to teach mathematics on

more rational and practical lines, but the examiners for the "Locals” will go on setting the old questions, and what are we to do?'

But it is to our university colleges and provincial universities that we must turn for the strongest proofs of England's neglect of mathematics. When I visit a foreign university for the purpose of calling on my colleagues in accordance with the universal custom, I sometimes ask for the Professor of Mathematics, and the invariable answer is 'Which do you wish to see?' Pisa has six professors, Padua has five, all teaching special subjects, such as higher analysis, geometry, rational mechanics, or geodesy, and mostly provided with competent assistants. In an English university college, the whole responsibility of the mathematical teaching, both pure and applied, usually falls on the shoulders of a single professor, often with only one assistant, while for the teaching of languages seven professors is not an unusual number, and in the technical departments separate professors are appointed for such specialised subjects as metallurgy, mining, mining engineering, civil engineering, mechanical engineering, book-keeping, and dyeing. A considerable portion of the time of the English professor is taken up in teaching the rudiments of trigonometry, and the substance of the sixth, if not of the first four books of Euclid to classes of beginners who do not want to learn mathematics and who will never get beyond this elementary stage. In the German universities a professor may be called on to hold a junior course in the Differential Calculus, but he will not go any lower, and it is generally being recognised that the subject ought to be taught in the schools. Passing next to America, I find in the calendar of a typical American university a collection of mathematical notes far above the heads of students of the English B.Sc. standard. On turning to the enumeration of classes, I find that the subjects taught in many departments correspond fairly well with those of one of our university colleges. The standard may be rather higher, but the difference is one of degree rather than of kind. In mathematics, on the other hand, the absence of elementary teaching, the high standard of the advanced classes, and the small number of hours allotted to each member of the teaching staff form a marked contrast to the conditions that exist in our universities.

The result of this difference is that while English mathematicians are a small and powerless body, who are far too overworked to be able properly to defend their own interests, a strong and powerful organisation has developed in the last twenty years in the

American Mathematical Society, which sends representatives to attend meetings and conferences in Europe, and to report on the latest investigations and discussions, and whose Bulletin is a chronicle of every event taking place in the mathematical world.

The fourth International Congress of Mathematicians which met this year at Rome from April 6th to 12th, was attended by about 530 mathematicians from all parts of the world. Of this number only about four per cent. came from Great Britain. The French Government, on the other hand, was represented by eight official delegates, while several other governments, of course not including ours, sent representatives. A number of foreign insurance companies were also represented officially, and a special sub-section met for the purpose of discussing actuarial problems. No English companies availed themselves of this advantage.

A small but amusing illustration of England's neglect of mathematics exists at my college. The Surveyors' Institute annually awards a scholarship to one of our students, and it might be naturally supposed that mathematics was the one subject of which every surveyor ought to know something. But the examination syllabus includes chemistry, physics, geology and botany, while mathematics is excluded.

If England is to hold her own against foreign competition by the endowment of research, adequate provision must be made for mathematical research. The mere building of laboratories fitted with costly apparatus, may be a necessary but it is not a sufficient condition of success. The mathematician carries his laboratory in his own brain. This laboratory requires much more careful maintenance and handling than one built of bricks and mortar. It is much more liable to injury if its resources are unduly taxed, and instances of breakdown are common. There is a considerable difference between mathematical research and research in the applied sciences. I have known a student put on to do research in chemistry who could never learn his Euclid. Such research may be purely unskilled labour. All that the student has to do is to put up certain apparatus and perform certain experiments under the direction of his professor. If the experiments are new their performance is described as a research. The performance of student research has been likened to the weighing of a brickbat fifteen times, obtaining a different result each time. There are so many investigations in physics and chemistry that are really worthy of the title research' that the above remarks must be

taken as illustrating the abuse rather than the use of this term. But the mere accumulation of statistical data can never produce any lasting addition to our knowledge unless the facts can be made the subject of a connected theory, and for this the services of the mathematician are indispensable.

Mathematical research consists essentially in the performance of original work. It involves continuous and concentrated use of the brain. There is no opportunity for the investigator to rest his thoughts while attending to the minutiae of a laboratory experiment. When he is not at work he must obtain all the relaxation he can, and he should therefore be freed from all the petty troubles and anxieties which are not directly associated with his work.

What, then, are the best means of maintaining at its proper efficiency the research laboratory which exists in the brain of the mathematician? The offering of prizes is a very inefficient method. If ten people compete for the same prize, they will all be engaged on the same research instead of on different subjects, and only one of them will secure any return for the hours of original work which he has devoted to the investigation. The loss of efficiency will therefore be 90 per cent. Even if only two compete the loss will be 50 per cent., and this loss cannot be safely spared in the present age of international competition. A further loss of efficiency also arises from the fact that the competitors have to carry out the investigations not on the subjects on which they have previously formed their own opinions, but on the themes proposed for the competition. Nevertheless prizes are not to be altogether condemned. The competition for the Smith's prizes is responsible for a large proportion of the original work in mathematics now done in this country and these prizes only involve an expenditure of about 50%. per annum. The most promising field for reform appears to lie in the better endowment and multiplication of the mathematical chairs of our University colleges, the provision of more adequate assistance for relieving professors of the elementary teaching necessitated by the present backward state of English mathematics, and the foundation of studentships and prizes awarded on the results of work actually published without restriction as to subject or age of candidate, not for the purpose of enabling the holder to work at some prescribed centre on investigations which may or may not be successful. At the same time the problem is a most difficult one and many points have been left untouched in this article for want of space.

VOL. XXV.-NO. 146, N.S.

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Let us in conclusion turn to a comparison which has been made in certain recent articles between the cost and national importance of endowing universities as centres of higher study and research, and the cost and national importance of building battleships. The nation which provides for research in every department except mathematics may be likened to a nation which builds a battleship armed with the most modern machine guns, but which fails to make adequate provision for that part of the ship under the control of the chief engineer. Perhaps the engines may be of an antiquated type, quite unadapted to modern requirements, and the chief engineer may not be furnished with an adequate staff of assistant engineers. The mathematical teaching of B.Sc. standard required by the students of applied science may be likened to the steam supplied from the boilers to the donkey engines required for moving the guns. This ship will make a very good show in the first stages of the battle, but when the race starts for the conquest of fresh territory she will be left hopelessly in the lurch by her foreign competitors. The first thing that the officers will do will be to blame the chief engineer, but they will soon find that they have no one to take his place, and that their gunners can only manipulate guns, not engines. By the time new engines have been put in and a more adequate staff of engineers has been provided it will be found that the rival ships of other countries have still further increased their vantage, have further improved the efficiency of their engines, and have again increased the staff of assistants working under the direction of their chief engineers.

It is for these reasons that I apprehend a serious danger lest England's neglect of mathematics should culminate in England's ultimate disappearance from the sphere of international enterprise and activity in which England once occupied the premier position.

G. H. BRYAN.