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As civilisation advances, an increasing toll is levied on the resources of the world, and as these resources are limited it becomes daily more and more important to utilise them to the greatest advantage. Thus greater use is made of the methods of exact science in the problems of everyday life, and as problems of increasing complexity arise the investigations are bound to assume more and more of a mathematical character. No better illustration could be afforded of the changes that are taking place in this direction than the modern applications of mathematics to statistics, biology, and eugenics.

It might be supposed that, as a consequence, the services of highly trained mathematical specialists were in considerable demand at the present time, and that a man who devoted the best years of his life to the study of mathematics would have plenty of openings offered him for a successful career.

This is not the case. Indeed, it is not improbable that the position of the mathematical specialist is worse now than it was thirty years ago.

It is very important that English mathematicians should face these questions, and should not shut their eyes to the conditions under which they labour. Otherwise they may incur serious responsibility in advising their pupils as to the choice of a future career.

The son or daughter of a clergyman, railway guard, or pork butcher shows a taste for mathematics at school. In time he (or she) gains a scholarship at one of the older Universities, obtains a degree with first-class honours, and becomes qualified to embark on original study and research. But the family purse then becomes exhausted, and the boy or girl has to seek some means for earning a livelihood. What chance have they of doing so ? They have had the whole of their time fully occupied during their college days in learning mathematics, and their distinction in this subject is the one qualification they can produce. What is the earning power of this qualification ? And does the well-trained English mathematician obtain the remuneration which his services deserve, and which he

would probably have received, if he had undergone an equally thorough training for a different career ?

The International Congress of Mathematicians at Cambridge afforded a unique opportunity for calling attention to the position of the English mathematical specialist. It is probable that reports of the proceedings as published in the Press might have brought the question before the unmathematical section of the British public in a way that will now be impossible for many years to come. But instead of arranging a discussion on this subject, the late Sir William White was appointed to deliver a lecture on The Place of Mathematics in Engineering Practice, and thus the main result of the Congress was to advertise the claims of English engineers, and, indeed, to place mathematicians in a very subordinate position.

Now English engineers are not backward in calling attention to their successes, they have plenty of opportunities of doing so, and their achievements are so well known to everybody that no harm would have been done to them by keeping them out of the Congress altogether. It is different with mathematicians, who, as a rule, possess little or no public influence. It is even not improbable that, if an English mathematician had ventured to address an engineering congress in the same spirit that Sir William White addressed the Cambridge mathematicians, the whole of the engineers would be up in arms against him.

So far as I can make out, the Cambridge Congress has done nothing whatever to improve the position of the English mathematical specialist, and thus a chance has been thrown away that may never occur again.

However, a useful purpose will be served by inquiring what are the present prospects of the student who has carried his mathematical studies at the University up to the highest standard, and who is capable of doing research.

Had a similar situation arisen in America, I have not the slightest doubt that discussions of the situation would have appeared in the 'Bulletin of the American Mathematical Society,' a journal that (unlike anything in England) gives considerable prominence to questions affecting mathematicians, and does not merely deal with mathematics itself. I hope the present paper may induce the Mathematical Association (before which it was read) to give a little more attention to the interests of mathematicians and little less exclusive attention to 'pretty' theorems (so-called). Personally I feel that it is difficult for one mathematical specialist



to advocate the claims of others without producing the impression that he is advocating his own claims, especially if nobody else has the strength of mind to back him up in his efforts. May I therefore say once and for all that the world has treated me very well, and has amply recognised any work in which I have been engaged ; therefore, when I hear of the difficulties of other mathematicians and know the merits of their work, it seems to fall upon me to call attention to the subject.

The recent discussion in the Press on ‘Business Careers for University Men’ will, it is hoped, deter parents from committing their children to a University education without carefully considering their subsequent prospects. It has also done much to call public attention to the desirability of providing openings for University graduates outside the teaching profession, and it will undoubtedly encourage graduates to take up any job that is offered them instead of at first thinking (as so many mathematicians have done) that no job is good enough for them, and then finding that no one will employ them.

Now in subjects such as physics, chemistry, and electrical engineering a demand has arisen for men with University degrees for work of a commercial character, and appointments sometimes fall to candidates of quite moderate attainments, or even go begging (there are recent instances).

There are no similar openings at present in existence for the mathematical graduate, consequently he has no alternative but to regard the whole of his University education as a course of training for the teaching profession.

In support of this statement it will be useful to examine some of the alternative courses which naturally suggest themselves.

I. The Indian Civil Service absorbs a few good mathematicians, owing to the nature of the competitive examinations. Although high marks are given for mathematics, the advanced questions are very hard, and I am not quite sure whether the

classical' candidate does not stand a better chance of success than the mathematician. In any case the successful candidate does not use his mathematical ability directly to a very large extent (perhaps not at all) in the duties of his future career, and he treats mathematics mainly as an examination subject, instead of proceeding to undertake research in it.

II. Observatory Work. Perhaps this offers the most suitable opening to mathematiciang outside the teaching profession. But

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there are only about twenty-four observatories in the British Empire, of which many are private or are connected with Universities, so that probably only about a third of the number would afford openings for a man (other than a mechanic) wishing to earn his income exclusively from observatory work. It is clear that these openings could only absorb a very small percentage of our mathematical graduates. A candidate who set out with the object of qualifying for such a post might have to wait a long time before a vacancy occurred, and then might have to be contented with a subordinate appointment at a very low salary. Moreover, a good deal of modern observatory work falls into the province of the physicist rather than the mathematician, as is indeed abundantly shown by the recent changes connected with the Plumian Chair at Cambridge, rendered vacant by the death of Sir George Darwin.

III. Actuarial Work. Much of this work is highly mathematical in character, and can be taken up by University graduates. It does not open up any very brilliant prospects, however. The mathematical expert in an insurance office does not often get promoted to the most important appointments, for which business capacity is usually a main qualification. An old mathematical friend of mine (we were College Fellows together) worked for many years in a Life Office and has now obtained an appointment under the Insurance Act, so that there is one thing at least for which the present Government must be thanked !

IV. Electrical Engineering. Many problems in electrical engineering involve the highest use of mathematics, and indeed such problems have indirectly done much to advance modern developments of higher analysis. But the mathematician who embarks exclusively on such studies will have to regard them altogether in the light of pure research, and he can never expect to utilise them as a source of income. On the other hand, there are at the present time numerous vacancies in the electrical engineering profession for students who have undergone a technical training and possess some knowledge of physics, and it is a remarkable fact that these appointments often fall into the hands of students who are anything but brilliant at mathematics (often very much weaker than ' Arts' students). My own opinion is that & mathematician who is out of work and has no prospect of employment in his own subject might with advantage take the necessary course of technical training, especially if he is good at physics as well as mathematics. I am




sure he would find the whole of his mathematical knowledge very helpful to him in his subsequent career. Whether he would be able to earn an increased income at the outset as a result of his higher mathematical knowledge is doubtful. So far as I can judge, the posts now vacant might require him to start on an income of 1001. a year, with prospects of an increase. At any rate these prospects, though not grand, are at least not quite so unpromising as they might be, and the mathematician would probably have a better chance of ultimately falling into a lucrative berth than he would in many alternative careers. It would be more healthy than unemployment.

V. Fellowships at Oxford and Cambridge are only tenable for six years, and cannot be regarded as a permanent source of income unless they are associated with teaching duties, in which case the holder becomes a member of the teaching profession and is no exception to the general rule. Dublin Fellowships are different tenure, but the competition for them is so strenuous that a brilliant mathematician may waste the best years of his life in preparing for the examinations, and may sacrifice his prospects of a successful alternative career in consequence of his efforts.

VI. Engineering other than Electrical. The average mathematician is mortally afraid of the engineer and will do anything to conciliate him. When an engineer abuses mathematicians they are too weak-kneed to stand up for themselves, and instead of this they adopt an attitude of grovelling servility and allow themselves to be led by the nose in the most humiliating way. The origin of most of the changes that have taken place in the modern teaching of mathematics can be traced to the demands of the engineering students, and no better illustration can be afforded than the way in which graphs were recently ridden to death, in season and out of season. Thus mathematicians have failed to realise that the class of pupil for whom reformed methods were most needed was not the engineering student at all, but the student who is learning mathematics as part of a general education.

Now I think I have some claims to speak with authority on the relation between mathematics and engineering, because I have always found engineering problems interesting as affording a source for original work in applied mathematics. I believe there are a great many obscure points which might be cleared up by a mathematician who possessed no practical training whatever. Indeed I go

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